Abstract

The focus of this project is not found solely in locking the output of a diode laser, but also concentrates on the methods leading up to this end.  At the outset, the laser is collimated, then "flashed" in order to ensure that the laser is in fact receiving external feedback.  This feedback significantly narrows the natural bandwidth of the free running laser and thereby makes it possible to probe the hyperfine energy states of atomic transitions.  The external cavity is a Fabry-Perot resonator whose reflecting facets include the rear facet of the diode chip and a diffraction grating.  By rotating and longitudinally moving the grating, selected frequencies may be sent back into the lasing cavity of the diode, making it possible to both tune and stabilize the laser at a particular frequency.  The laser frequency is accordingly scanned by modulating the external cavity length through the use of a pulsating PZT attached to the rear of the grating.  Course tuning is accomplished by varying the temperature of, and the injection current through the diode laser.   By patiently and systematically adjusting each one of these tuning parameters, the laser is tuned to the ⁸⁵Rb and ⁸⁷Rb D₂ resonance.  Frequency scanning then produces a Doppler broadened absorption spectrum over this resonance due to the Maxwell-Boltzman distribution of atomic velocities in the vapor cell probed with the laser beam.  A technique called saturation spectroscopy subsequently enables one to produce Doppler free absorption peaks that resolve a level of precision equal to the atomic hyperfine transition width of Rb, allowing a suitable reference for certain optical experiments.  However, at this level of precision the laser is not adequately stabilized by simply adjusting the parameters mentioned and holding them constant.  After further manipulation of these parameters, one may hone in on the very summit of a single Doppler-free peak for reference.  By feeding this signal into a lock-in amplifier as feedback, an error signal is established that repeatedly forces the laser back onto the peak of the hyperfine transition as the laser output naturally drifts in one direction or the other, away from the peak.

Introduction

            The emission line width of a free running diode laser (DL) is relatively wide, it is also both quite divergent and unstable in mean frequency and power.  The divergence is easily fixed by an aspheric collimating lens, however the instability in the output is precisely what makes the laser tuneable.    That is, if one can manage to control the otherwise unregulated parameters that govern the behavior of the lasers output, one may transform instability into tunability.  These parameters typically include the temperature of the laser and its injection current.  In our case we have artificially inserted an additional parameter that has a considerable effect on the constitution of lasers emission.  Namely, our laser apparatus includes an extended cavity that provides frequency selective optical feedback to the lasing medium which forces the stimulated emission of light at particular wavelengths and leads to the further narrowing and stability of the lasers output.  It is due to this increased tunability, stability, and bandwidth narrowing that makes an extended cavity DL appropriate for probing atomic energy states.

Throughout the remainder of this discussion a basic understanding of laser operation will be assumed. 

Frequency Selective Optical Feedback

 Longitudinal Modes

            Some characteristics of a lasers output that are relevant here include the lasers modality, the bandwidth of a single mode and the free spectral range.    Modality is a general term that can be refined into the following two categories:  transverse modes and longitudinal modes.  Longitudinal modes refer to the simultaneous lasing at several discrete frequencies within the lasing cavity, all of which compete for gain dominance within the lasing medium.  Each mode can be thought of as an intensity (Lorentzian) peak with a given frequency bandwidth characterized by the full width at half maximum (FWHM) of the peak.   The free spectral range (FSR) is characterized by the frequency spacing between these neighboring peaks.  It is the job of the extended cavity feedback to contribute to the reduction of the modality of the laser to single mode operation, to narrow the bandwidth of a single mode, and all the while to keep the mode spacing (FSR) far enough apart, for purposes that will become clear shortly.  The external cavity accomplishes this feet by helping to dictate what wavelengths of light have dominance within the lasing cavity. 

            As is the case with the lasing cavity, the external cavity is essentially a Fabry-Perot resonator composed of two confronting and parallel, reflecting surfaces separated by a given length.  Light contained within this cavity is subject to the condition that the magnitude of the electromagnetic field must be zero at the cavity boundaries.  As such, for a given length of the extended cavity (L), only photons with wavelengths (l_o) an integer number (m) of half wavelengths (l/2) as long as the cavity may resonate and thus be sustained within this cavity and provide feedback; i.e. ml_o/2 = nL, where n = refractive index of the cavity medium and nL = optical path length.  Since n~1.003 at STP, nL~ L, and ml/2 ~ L for the external cavity only.  Like standing waves, these modes are established by being able to reproduce themselves after one round trip in the cavity, where the phase difference is some multiple of 2p.  By augmenting the optical path length of the lasing cavity through the addition of an external cavity, the mode structure changes in a dramatic and advantageous way, allowing for a narrow and stable output as described below.  The fact that the lasing medium is a part of the extended cavity allows the output to be coupled directly into the external cavity and also provides a direct return path for feedback from the extended cavity into the lasing medium.

Transverse Modes

           As mentioned, the modes that have been discussed so far are referred to as longitudinal modes and correspond to the longitudinal direction of the resonating cavity.  Notwithstanding, there also exist transverse modes that correspond to the spatial variation in the beam intensity cross section in the transverse direction of the lasing medium.  These spatial variations are Gaussian in shape and are the result of the spatial variation of the injection current density and the refractive index across of the lasing material.   Typically, modes of different orders with multiple peaks overlap, resulting in an output intensity profile with a given number of spatially separated and overlapping peaks.  Thankfully, the DL’s available to atomic physics experiments are index-guided AlGaAs lasers that typically operate in a single spatial, or transverse (TE₀₀) mode, resulting in a uniform and symmetric intensity cross section with a profile that is highly intense at the center and falls off towards the edges of the beam, greatly simplifying the spectroscopic analysis.

One final point should be made about the polarization of the TE spatial mode.  This will become important once the output beam is processed by external optical elements that are sensitive to the beam polarization.  At the threshold injection current, there is a sharp rise in the degree of linear polarization along the major axis of the elliptical profile of the beam.  One may take advantage of this linear polarization for laser alignment purposes, to attenuate the intensity of the beam and to isolate optical feedback, described later.  

Extended Cavity Feedback

            Although the free running DL has a good chance of operating in a single longitudinal mode, this output may be too unstable and its single mode bandwidth may be too broad for probing atomic transitions.  It is precisely the role of the external cavity to both tune the laser frequency and to narrow the tuned bandwidth of this single mode.  The output stability will be discussed later.  DL’s used in experiments that require a high degree of tuning and narrowing require a high degree of feedback with a lot of influence over the lasing cavity.  Normally, any feedback into the lasing medium sets up a competition for stimulating the emission of inverted atoms between the lasing photons of the lasing cavity and the feedback photons.   As mentioned, feedback photons not only satisfy the ml/2 = L_ext condition for the external cavity length, but must satisfy the same condition for the lasing cavity length as well: ml/2 = L_int , since they were created and sustained within the lasing cavity to begin with.  So, the stimulated emission of photons from external feedback will be amplified within the lasing medium just the same as normally lasing photons. The degree to which the emission photons are stimulated by external feedback photons is directly proportional to the fraction of feedback to lasing photons present within the lasing cavity.  Accordingly, feedback control increases with the increased decoupling of light out of the lasing cavity and into the external cavity.  Although feedback photons are always resonant within the lasing cavity, lasing photons are not necessarily resonant within the external cavity, which is at the heart of the operating concept of external feedback.  Like an optical filter, the extended cavity continually sorts through incoming photons and returns a selected set.

Cavity Q-Factor

Feedback control is enhanced by putting an anti-reflection (AR) coating on the exit facet of the lasing cavity.  The rear surface of the lasing cavity remains highly reflective.  The result is that more light will be coupled out of the internal lasing cavity and into the external cavity.  Hence, the external cavity receives a greater fraction of photons and the internal cavity maintains a smaller fraction for stimulating the emission of photons.  In other words, the AR coating decreases the Q factor of the lasing cavity.  The Q factor is an index proportional to the ratio of the energy stored inside a standing wave and the wasted energy from the wave during its trip between the cavity facets.  The “wasted” energy in this case is not completely wasted though.  The energy gained by the external cavity is structured into narrowed modes that are then fed back into the lasing medium in order to stimulate the emission of similar photons.  Therefore the AR coating favors the feedback photons in the struggle between the internal cavity photons and the external cavity photons, finally allowing the external feedback to dominate over the internal feedback.  Furthermore, by decreasing the Q factor of the lasing medium, the external cavity gains influence over the lasing processes which allows one to manipulate the output of the laser by having access to and manipulating the external cavity itself.  There is one drawback to the feedback scenario however, and it lies in the fact that the more the Q factor is decreased within the lasing medium, the weaker the output intensity will be.  Consequently, there must be a trade-off between control and power output, where 20% power decoupled into the external cavity seems to be a good compromise. 

Central Wavelength, Modal Line width, Free Spectral Range

             In a broad sense, the sustainable wavelengths within the external cavity are a subset of the input wavelengths from the lasing cavity.  And ultimately, it is out of this subset that the lasers output arises, resulting in the overall purification of the lasers emission.  As mentioned, the wavelengths permitted to exist within the external cavity conform to the simple geometric relation, ml₀/2 = L.  Now, l₀ only refers to the central wavelength of a single mode, and gives no insight into the mode structure within the cavity, including the mode line width and the FSR of the cavity.    Utilizing ml/2 = nL, and c = nl, mc/2n = nL, where c is the speed of light in vacuum.  Thus, the following discrete frequencies are sustainable within the external cavity: n = mc/2L.  Each discrete frequency corresponds to a given mode, and therefore the FSR can be calculated as follows: Dn_FSR = n(m=2) – n(m=1) = 2(c/2nL) – (c/2nL) = (c/2nL), which = (c/2L) for n ~1.  Evidently, the frequency separation between all neighboring modes is constant, despite the fact that the FSR in terms of wavelength is not.  It is also apparent that as L is increased, the FSR is decreased. 

The Fabry-Perot modal line width is given by:  Dn_FWHM =  Dn_FSR/F, where F is the finesse of the Fabry-Perot cavity.   F = p(R₁²R₂²)^1/2/(1- R₁²R₂²), where R is the reflection coefficient of the reflecting facets of the F-P cavity.    Although the FSR is simply a function of L, the modal line width is a function of both L and R.  From the expression for F, it is apparent that F increases with R, thus larger values of R result in smaller values of Dn_FWHM .   The value of R depends on the surface smoothness of the reflecting facets.  Variations in surface smoothness and quality result in variations of L in different portions of the reflecting surface that enables wavelengths other than l₀ to satisfy the F-P condition, which accordingly leads to the broadening of a single mode.  In the end, high values of the cavity finesse result in narrow mode peaks.  Up to this point, we have also concluded that increasing values of L result in decreasing values of  Dn_FWHM  and Dn_FSR.  Now, it will become apparent later that from a tuning standpoint, this poses the problem of choosing a value of L such that Dn_FSR is sufficiently large and Dn_FWHM  is sufficiently narrow. 

Also, as mentioned above, as the quality factor, Q of the lasing cavity is decreased, more lasing control is surrendered to the external cavity.  Q = n₀/Dn_FWHM, where n₀  = c/l₀, and like the finesse (F), Q is also a measure of the sharpness of the cavity modes, and as Q increases Dn_FWHM decreases.  Thus, it is favorable for the external cavity Q to be as large as possible.  This way, the more energy the external cavity is able to retain, the narrower one of its modes will be.  It follows, therefore that as the lasing cavity Q is decreased for the sake of greater external control, the lasing cavity modes will be broadened.  It will soon be apparent however that this broadening is unexpectedly advantageous from a tuning standpoint, meanwhile the enhanced feedback mode structure from the external cavity is vital to the performance of the laser.

             If one were to compute the FSR (Dn_FSR) and line width of a single longitudinal mode (Dn_FWHM) of the external and lasing cavities as outlined above, one would get for the external cavity: Dn_FSR = 7.5 GHz and Dn_FWHM  = 4.6 GHz (where R₁=.9, R₂ =.2, L =20 mm, and n =1) and for the lasing cavity: Dn_FSR = 171 GHz and Dn_FWHM  = 45 GHz (where R₁=.9, and R₂=.5, L=250 mm, and n = 3.5).  Although the values of the FSR seem reasonable enough, it is immediately apparent that the linewidths as described here are simply unacceptable for doing any sort of atomic spectroscopy, since atomic spectral lines have linewidths on the order of tens of MHz, which in turn require a probe much finer than that to scan them.   Thankfully, the values for the emission linewidths obtained in practice are orders of magnitude narrower than the modal linewidths calculated here, and are about 30 MHz for a free running laser and around 100 kHz for a laser with external feedback.  On the other hand, the values of FSR determined here agree relatively well with those values obtained experimentally. 

Gain Narrowing, Gain Selection

               The basis for this alarming discrepancy between the mode and emission linewidths comes from the fact that a real laser cannot simply be reduced to a Fabry-Perot resonator, but is composed of a gain medium that adds a high degree of complexity to the expression of the emission line width.  The mechanism largely responsible for the narrowing is the exponential growth factor of the gain medium, where the emission line width is the result of the interaction between the gain curve of the amplifying medium and the Fabry-Perot resonance.  The developing light intensity inside the lasing cavity after the electromagnetic field has traversed a distance, d within the lasing-gain medium is given by :I(d, n) = I₀ exp[g(n)dp], where I₀  is the initial intensity before amplification, g(n) is the wavelength dependent gain coefficient stemming form the Lorentzian atomic line width of the gain medium, and p is an integer representing the number of times the electromagnetic wave traverses the gain medium due to reflection off of the cavity facets before it exits the lasing cavity.  The gain of the laser is thus  G = I(d, n)/I₀ = exp[g(n)dp].  It is thus apparent that even small differences  in n which result in small differences in g(n) will result in substantial differences in intensity.  For a typical GaAs DL,  g(n₀) = 100,000/m at the peak of the gain profile, L=.000250 m, and p=1.5, thus the initial intensity at the frequency n₀, will grow according to I₀G = exp(37.5) ~ I₀ (1E16) ! 

             Consequently, the frequency at the highest value of gain will develop and reach its saturation intensity first.  It will continue to develop at an exponential rate, all the while removing atoms from the upper lasing level, thereby reducing the number of excited atoms available to amplify other frequencies.  Stimulated emission at this particular frequency, therefore has a considerable advantage over even closely neighboring frequencies.  This effect is much more drastic for mode competition, where a mode near the peak of the gain profile has an enormous advantage over an adjacent mode which is relatively much further from the peak, making it very difficult for other modes to lase with any significant gain.  Ultimately, most of the amplifying energy of the laser is diverted into one single mode, where its intensity is roughly 100 times greater than any other spectral feature.  Now, as mentioned above, the DL used here operates in a single mode, but with the caveat that the gain is high enough such that it will select a single mode as described.  Accordingly, for low values of injection current around the threshold value, the laser operates in multi-longitudinal modes.  Only at values of injection current high enough above threshold will the laser operate in a single longitudinal mode.  In addition to this mechanism of mode selection by the gain profile, optical feedback of a single mode contributes heavily to the single mode operation of the laser as well, essentially by the same mechanism of gain selection.  This is illustrative of the fact that optical feedback simply provides a means of control over steering the gain of the laser. 

Emission Line width

              Unfortunately, the situation is more complex than stated, for the lasing medium of the DL is a non-linear quantum object whose output is the result of quantum processes not adequately described by the model outlined above.   Notwithstanding, although the description above is only an approximate qualitative model of laser emission linewidths, it is quite useful for describing the tuning characteristics of our laser apparatus.  A more precise quantitative model is provided by the modified Shawlow-Townes formula:  Dn_FWHM  = (hn/8pP₀) (c/nL)² [aL + ln(1/R)]  ln(1/R) n_sp (1+a²), (P₀ is the output power, n_sp is the spontaneous emission factor, and  (1+a²) is a tacked on factor that takes into account the dependence of the refractive index on the carrier density).  This last factor, (1+a²) (where a ~ 5) represents the strong coupling between amplitude and phase fluctuations and increases the line width by roughly an order of magnitude.  As with the Fabry-Perot mode width, the emission line width as predicted by this formula decreases with increasing L.  An interesting and significant feature of this formula is that there is an inverse relationship between power and line width.  Therefore, since the injection current scales linearly with power, it is preferable to operate the laser at higher currents.  Incidentally, this formula is also proportional to 1/Q² , which agrees with the statement above, that larger values of Q result in narrower widths.  Solitary laser emission linewidths of tens of MHz are obtained using the modified Shawlow-Townes formula, in good agreement with experiment.  Not to mention, with the introduction of the extended cavity, there is over a magnitude of additional improvement over the solitary laser in the experimentally established value of the emission line-width.

External & Lasing Cavity Mode Structure

            It is already established that one facet of the external cavity is the highly reflecting facet of the lasing medium.  The other facet of the external cavity can be any dispersing element, but as one will see, a holographic diffraction grating has various favorable qualities.  To begin with, the dispersive resolution of the grating allows the selection of a single lasing cavity mode at a time to be reflected back into the external cavity, which is absolutely critical to the performance of the laser, as described above. Starting from the grating equation:  ml = d(sina + sinb), (where a and b are respectively the angle between the incident beam and the normal to the grating surface and the angle between the diffracted beam and the normal, d is the groove spacing and m is an integer representing the order of diffraction), it is apparent that  a and b have to be equal in order for light to get back into, not only the external cavity, but into the lasing cavity as well for feedback.  Such a grating arrangement is known as the Littrow configuration, where the grating equation now looks like this: ml = 2d sina_L, where a_L is referred to as the Littrow angle.  Consequently, the wavelength diffracted into the laser is changed by simply changing a_L as the grating is rotated, such that the dispersion plane remains perpendicular to the surface of the grating.  Although the grating is capable of dispersing the incident light into a various number of diffraction orders, it is readily apparent that the gain profile of the laser is narrow enough such that there is no second order light produced by the laser, since second order wavelengths would have values of l₁/2 = 780nm/2 = 390 nm.  The same is true of course for subsequent orders, where l_m = l₁/m.  Thus, the grating equation can be further simplified to:  l = 2d sina_L, where m = 1 only, and the first order diffracted wavelengths are sent back into the laser exclusively.  For l₁ = 780.00 nm, and d= 0.833 mm, a₁ = 27.9^o.  Accordingly, as the wavelength selected feedback is varied over the full range of first order diffracted wavelengths, the full scope of lasing cavity modes under the gain profile will be encompassed, all the while the Littrow condition is satisfied, allowing a path for feedback.  A typical tuning rate for a holographic grating with 1200 rulings/mm is about da_L/dl = .08^o/nm.

          Notwithstanding, the diffraction grating not only serves as a dispersive medium, but also as a simple mirror.  The first, second, …, etc. orders of reflected light are dispersed by the grating, with an angle-wavelength relationship described by the grating equation, however the zeroth order, where m=0, is a special case. In this case a_L and b must be equal and l can take on an infinite number of values.  In actuality, l will take on all the values from the set of wavelengths incident on the grating.  In summary, the grating will reflect some percentage of all the incident wavelengths of light into the undispersed zero order and in this case acts simply as a mirror, where the angle of reflection equals the angle of incidence. This undispersed light is the output of the extended cavity diode laser apparatus with the corresponding line width described above.  Thus, some of the light is used for feedback and the rest is the tuned and narrowed output, that is cyclically a product of the feedback.  The grating efficiency is typically such that about 21% of the light is diffracted back into the first order, 62% is sent into the zeroth order and decoupled from the external cavity, leaving 17% left that is probably absorbed by the material composition of the holographic grating. 

Blazed Holographic Grating

             The level of grating efficiency depends on how much the laser in use needs to be dragged from its solitary operation in a particular mode to a mode that is desired for the experiment at hand.  In other words, the further a desirable mode is from the peak of the lasers gain profile, the more feedback is required to drag the laser output to the desired point away from its intrinsic point of operation.  As mentioned, in order to increase the level of external feedback, the higher the Q factor of the external cavity needs to be, which directly depends on the degree of reflectivity of the cavity facets.  Aside from losses due to absorption, increasing the efficiency of the grating involves redistributing and concentrating most of the diffracted energy into a particular frequency and/or diffraction order, which effectively enhances the reflectivity of the cavity for that particular wavelength.  This is accomplished by “blazing” the grating, which basically alters the ruling profile of the grating such that the groove shape is in the form of a right triangle with a characteristic angle known as the blazing angle.  The efficiency of the grating at a particular wavelength is optimized by setting the grating in the Littrow configuration, then altering the blaze angle until the concentration of diffracted energy falls into a particular Littrow angle with a corresponding wavelength.  As such, gratings come blazed for different wavelengths.  Consequently, by incorporating such a suitable blazed holographic grating into the external cavity in the Littrow configuration, with a blazing angle corresponding to the desired wavelength, the Q at that particular wavelength goes up and feedback is enhanced.   The greater the need for efficiency and therefore feedback, the closer the blazing wavelength should be to the desired wavelength. 

If a blazed grating is to be used, the orientation of the grating is very important when it comes time to install it.  Although, it is obvious that the grating rulings should be vertical, this requirement can result in two different ways of mounting the grating.  If a blazed grating is used, one should note that the blaze arrow, which points from the grating surface normal to the blaze facet normal, also points towards the zero order output.  Therefore, the blaze angle should be oriented in the direction in which the zero order output ought to go.      

External Cavity Modes, Mode Hops

              As expected from the superposition of the lasing cavity mode structure on top of the external cavity mode structure, a tuning configuration emerges that immediately illuminates the tuning limits of the external cavity apparatus.  That is, as the laser is continuously tuned by changing the external cavity length, one is presented with the unappealing fact that the lasers output abruptly and discontinuously shifts or “hops” from one external mode to another.  Although an external mode may be swept through the span of the lasing mode (~45 GHz), the range of continuous tunability is limited to slightly less than the FSR of the external modes only (<10 GHz).  As the external cavity length is continuously altered in one direction, a particular external mode, in turn shifts in one direction underneath the lasing cavity mode, with a resulting gain incurred by the lasing mode profile.  All the while, one should note, the lasing cavity mode remains perfectly still since none of its governing parameters are touched.  As such, while the external mode continues to moves underneath, its amplitude is continuously modulated and eventually starts to lose gain as it approaches the tail end of the lasing profile.  At the same time, an adjacent external mode is shifted towards the peak of the lasing profile and its gain is strongly enhanced, and the balance of power is again transferred from one mode to another.  The magnitude of this hop between modes is naturally equal to the FSR of the external modes as well (10 GHz!), which is quite a large and unpleasant discontinuity.

Expansion of Continuous Tuning Range

           Although there is little one can do about the magnitude of this mode hop, a novel technique has been developed to increase the range of continuous tunability of the external cavity length.  Though by in large, the function of the diffraction grating is to select a particular lasing cavity mode, in this case the grating can also be used to expand the tuning range.  Essentially, the effectiveness of this technique hinges on the placement of the axis of rotation of the grating.   When the axis of rotation rests on the surface of the grating itself, any rotation about this axis results in a pure rotation only, without translating the grating.  However, if the grating is mounted such that it can be synchronously translated and rotated, then the external cavity modes will be shifted at the same time the grating profile is.  This is accomplished by installing an actuator (e.g. a piezoelectric transducer or PZT disc) on one side of a rotating dynamic optical mount, between the push screw and the moveable base plate which holds the grating.  This configuration ensures that the axis of rotation is set at a given distance away from the surface of the grating, which forces the grating to rotate somewhat as it is translated. 

As the grating is translated, it changes the length (L) of the external cavity and in turn alters its mode structure.  Specifically, the modes shift as the cavity length changes.  The trick here lies in installing the grating on the dynamic optical mount such that while the grating is rotated to, for example, smaller angles of incidence (a_L), the external cavity length is shortened at the same time.    Accordingly, both the mode structure and grating feedback condition move towards shorter wavelengths simultaneously.  The only constraint here is that the grating must sweep across the wavelength axis at a relatively slow rate such that it doesn’t hop onto another lasing cavity mode in the process of being rotated.  In other words, the rotation shouldn’t shift the grating profile by more than the FSR of the external cavity modes.  As the grating is rotated towards shorter wavelengths, the grating bandpass profile sweeps across the lasing cavity mode towards shorter wavelengths as well.   And thus, the gain towards the shorter wavelength end of the lasing mode is enhanced.  All the while, the external cavity mode is also moving towards the shorter wavelength end of the lasing mode.  As a result, the gain of the external cavity mode diminishes at a slower rate, and therefore the endurance of the mode at shorter wavelengths is favored.   Specifically, the external mode persists over a longer range before it runs out of sufficient gain and a neighboring mode is significantly amplified. 

             It should be noted that the grating configuration described above isn’t employed in the following discussion.  Thus it is necessary to point out that in this case, and others for that matter, where the PZT is simply mounted between the back of the grating and the moveable base of the dynamic mount, the axis of rotation is also offset from the surface of the grating.  However, although this means that the length is altered as the grating is rotated, the grating may be translated without subjecting the grating to rotation.   

           The following method attempts to align the grating with respect to the laser output such that the Littrow condition stated above is satisfied.  Namely, by ensuring that the angle of incidence equals the angle of the first order diffraction, the frequency selective optical feedback is able to enter the lasing cavity. Due to the fact that the aperture of the lasing chip is on the order of .1 mm by .3mm, physically positioning the grating based on some simple geometry is not practical.  By taking advantage of the peculiar gain characteristics of the lasing medium, the following procedure enables one to confirm that light is indeed getting back into the lasing medium by simply witnessing a sudden increase in the light intensity of the output light for a given grating angle(s).  Initially, the injection current must be set a few mA below threshold.  The sudden “flash” that follows is caused by the fact that once light is fed back into the lasing medium, the power Vs current curve is shifted down by about 15 mA, which in turn drops the threshold current by about 15 mA, resulting in an increased intensity at the given values of injection current.  Just below threshold, spontaneous emission followed by absorption is dominant, resulting in an energy density not high enough to cause significant lasing to occur.  Since the external cavity is essentially an energy storage device, once this energy is sufficiently accumulated, the external feedback becomes intense enough so as to initiate significant lasing (through stimulated emission) in the lasing medium. 

In order to find this particular angle(s) at which the grating sends light directly into the lasing medium, a Euclidean grid should be devised and set up about 30 cm in front of the grating where the output light intercepts it.  While the current is set to just below threshold, the entire grid should be scanned with the output light by varying the grating angle in both the polar vertical and horizontal angular directions (f,q respectively) until the flash is noticed.  As the angles of the grating change, the angle of the output light changes, causing the output to change its position on the grid.  A neat trick for expediting this method is to look for the small reflection of light off of the surface of the collimating aspheric lens.  The reflected light is easier to notice if the injection current is brought about 20 mA above threshold.  Once this reflected light is observed, the grating should be moved such that this beam and the output beam overlap. At this point, the injection current should be brought back to the value just below threshold.     Once this is done, the angle of the grating is in the neighborhood of where it should be in order to send light into the aperture of the lasing cavity.  Due to the fact that the grating dispersion plane is oriented horizontally, the flash will occur for only one value of f.  However, due to the same reason and others described later, more than one value of q will result in a flash and a small flashing region may be mapped out on the grid horizontally, recording the points at which to move the output light to ensure flashing in the future.  Of course the grid’s position on the table should be marked and the grid should be set up in exactly the same orientation each time one checks to see whether the laser is flashed or not.

Flashing Region

Bandgap of lasing chip

             The tuning parameters of the solitary, free-running laser are, as mentioned the DL temperature and injection current.  But, the dominant parameter in producing the output of the laser is the bandgap of the chip’s p-n-p junction that is set according to the level of Al doping during manufacturing.   Although the DL temperature impinges on a mixture of attributes of the DL, the bandgap also changes with temperature through the expansion of the semiconductor lattice constant and by altering the lattice vibrations.  For a given bandgap, electrons fall from the conduction band to the lower energy states of the valence band, recombine with holes and emit photons of specific wavelengths in the process of crossing the p-n-p junction.  It is these processes that are largely responsible for the broad gain profile of the lasing medium, alluded to earlier.  The DL temperature, in turn shifts the entire gain profile by modifying the bandgap.  The bandgap depends on the DL temperature according to the following approximate expression: E_g(T) = E_g(0) – aT²/(T + q), where  E_g(0) is the extrapolated bandgap at absolute zero, and a and q are characteristic constants for a given semiconductor.  Given E = hn, l₀ = hc/E_g (where h is Planck’s constant and l₀ is the intrinsic wavelength of the laser at a particular temperature at the peak of the gain profile), describes how the bandgap modifies the wavelength, with a corresponding broad bandwidth, ~ +/- 10 nm from l₀.    According to the manufacturers specifications, typical DL temperatures may be increased and lowered below room temperature by about 30 K, which shifts the central wavelength, l₀ of the gain profile by 21 nm.  In this way, the bandgap effectively provides the laser apparatus with a broad input range of wavelengths which are later refined and narrowed by the rest of the laser apparatus.  Operating the laser at the extreme temperatures however, severely lowers the lifetime of the laser and is not recommended.

Response of Optical Path Length to Temperature

Lasing Cavity Mode Hops

             Unfortunately the tuning curve as a function of temperature is not a continuous one, but is shaped in a staircase-like pattern with sloping steps as the laser output hops from one longitudinal mode to another as the laser is heated or cooled.  Typically mode hops occur in intervals of 1.2 ^oC, corresponding to the 170 MHz mode spacing, mentioned above.   Specifically, the “hopping” occurs due to the fact that the optical path length changes by about 0.06 nm/K, while the gain curve changes by about .25 nm/K.  That is, as the DL temperature is changed, the gain curve and the lasing cavity modes are shifted simultaneously at the different rates above.   Since the gain profile favors the lasing mode directly under or closest to the gain peak and moves relative to it at a faster rate, the gain profile continuously hops onto or selects successive lasing modes as the temperature is altered.  Put another way, each step represents the slow tuning rate of the optical path length over a single cavity mode, while the hop represents the quicker tuning rate of the gain curve hopping from one longitudinal mode to another.   The range of continuous tuning corresponds to somewhat more than the line width of the lasing cavity mode (~45 GHz), whereas, if one will recall, the continuous tuning range for the external cavity modes corresponds to somewhat less than the FSR of the external cavity modes (~10 GHz).  Although the former tuning range is greater, there are other drawbacks with scanning the laser frequency by shifting the lasing cavity mode, discussed below. 

            Although the following statement currently has no cogent theoretical basis, it has been verified empirically that the emission line width of a solitary laser has an inverse relationship with respect to the DL temperature, so if in doubt, one should choose a higher operating temperature over a lower one when using a solitary laser. 

           The injection current alters the temp. of the laser through joule heating: P_DL = IR²  ~ mcDT, where R is the effective resistance of the lasing chip, I is the injection current, m and c are the effective mass and specific heat capacity of the DL.   Altering the injection current, therefore is simply a way of indirectly changing the temperature.   The injection current also increases the carrier density within the lasing medium, which affects the refractive index.  However, once the injection current reaches threshold, the carrier density is clamped and the injection current has command of the laser through temperature only.   Unlike direct temperature control, which is usually accomplished externally, the injection current raises the laser temperature internally and thus elicits a much faster response from the laser.   A typical frequency tuning rate for the injection current is about 4 GHz/mA.  The injection current is a tuning parameter that shifts both the lasing cavity modes and the gain profile simultaneously, and at different rates, producing the staircase, mode hopping tuning curve described earlier. 

Within one of the continuous tuning steps of the lasing cavity mode structure, the lasing cavity mode is shifted continuously.  For a free-running laser without the addition of the external cavity, the output frequency would ultimately follow the lasing cavity mode (after the gain medium subjects the lasing to some processing).  However, with the addition of the external cavity, the external cavity imposes a dramatically new frequency mode structure, with amplitude modulation incurred by the encompassing lasing mode profile.  So, if the current is changed within the limits of a continuous lasing mode step, an external mode hop may also be incurred by the shifting lasing mode, if the shift is on the order of the FSR of the external modes.  By the same token however, the injection current greatly enhances the range of possible emission frequencies by shifting the lasing mode into frequency domains that were inaccessible before.  Until the lasing mode is shifted, particular, perhaps desirable frequencies are inaccessible due to the minimal gain gap between successive lasing modes.  The magnitude of this gap is considerable given by the difference between the FSR and line width of the lasing modes:  ~170 GHz – 45 GHz = 125 GHz, illustrating just how advantageous this ability to DC offset the frequency is. 

            As alluded to above, the power output of the laser depends linearly on the injection current beyond the threshold current.  The power is given by: P₀ = n_ex(hn/e)(I-I_th),  where n_ex is the differential external quantum efficiency equal to the flux per unit change of current above threshold, and I_th  is the threshold current value.  In this respect, the injection current doesn’t simply change the DL temperature, but concurrently changes the power output as well.  The power Vs. current curve is an important characteristic of the laser and should be among the primary measurements made.                                                                                                                       

             It should also be noted that the entire power Vs. current curve is shifted a constant amount as the temperature is altered, such that :  I_th(T) is proportional to exp(T/T₀), where T₀ is the characteristic temperature of the DL material.  The curve shifts by about  .4 mA/^oC, for typical DL’s. 

Tuning to an atomic Transition Frequency

            Ultimately, the final or master tuning curve is a convolution of all the tuning curves discussed so far, where the laser output corresponds to the point where all these individual tuning curves overlap.  By altering the individual parameters that steer these curves one way or another, one can tune the laser.  Although, the laser may also be tuned by changing the ambient pressure and magnetic field of the laser as well, these tuning parameters will not be (deliberately) changed in order to tune the laser here.  To summarize, the individual curves employed here include the gain profile of the laser medium, the lasing and external cavity mode structure, the grating profile, and the power Vs. current curve.  As a result, one may regard the master tuning curve as a multi-dimensional surface or map, where each tuning parameter may be viewed as a variable.  Although the shape of this tuning map is largely unknown, by following a method that effectively scans the surface of this tuning map, one may eventually hit upon an atomic transition frequency of interest.  It should be noted however that depending on the specific DL, the surface of the map is not perfectly continuous, but may have holes and discontinuities in it resulting from gaps and mode hops in individual tuning curves.  These gaps are unavoidable, short of getting a new laser.  This structure of this map is further complicated by the fact that the preceding variables are not necessarily independent from one another.

            Going from course to fine tuning, scanning across the surface of the map starts out with setting the temperature since it impinges on everything but the grating and the external cavity modes. The temperature controls the optical path length and the bandgap (T à n, L_int, E_g).  Though modifying the optical path length results in fine tuning, it is the modification of the bandgap that results in the course tuning.  In addition, altering the temperature of the DL externally can take up to half an hour at a time, by the time the temperature stabilizes, and is impractical for scanning the entire surface of the tuning map.  After the temperature is set, the next parameter to change is the position of the grating profile over a particular lasing cavity mode (Grating q à l₀, L_ext).  By doing so, the range of wavelengths emitted from the bandgap may be scanned.  Usually, after rotating the grating enough and selecting different lasing cavity modes, a particular frequency mode is selected that coincides with at least a portion of the atomic transition width.

The Rb “transition” may be quickly confirmed and monitored through the use of a CCD camera focused onto the atomic vapor cell probed with laser beam.  Once transition is reached, the gas will absorb the tuned frequency of light and re-emit it in all directions in a process called fluorescence.  The infrared fluorescent light is then detected by the CCD and displayed on a TV monitor.  

            If fluorescence is not observed at all, the temperature should be changed and the process iterated until some fluorescence is detected.  On the other hand, once fluorescence is achieved, it can be maximized by altering the next parameter in the sequence of tuning, namely the injection current (I_DL à T, n(below I_th), P₀).  That is, if a portion of the transition width is excited by a particular mode, the laser current may be changed to quickly tune the output by shifting the lasing mode by a small enough increment until the mode engulfs the transition width.  As discussed, the lasing mode, without the help of the external cavity results in an emission line width of ~30 MHz to 45 MHz which is broadband relative to a transition width (~15 MHz) and is therefore unsuitable to probe the atomic transition.  However, with the addition of the external cavity, the current shifts the lasing mode, which in turn envelopes successive narrower external modes as it moves.  Once the laser current is altered and consequently selects a specific external mode that causes fluorescence in the vapor cell, the frequency of the external mode may still not be centered on the atomic transition width.  Centering is accomplished by shifting the finest tuning parameter, specifically the optical pathlength which is modified by changing the external cavity length.  Finally, in order to maximize the gain of the shifted external mode, the laser current may be used again to shift the lasing mode such that the external mode lies beneath its peak, thereby maximizing the external mode amplitude.  Since the cavity length needs to be changed on the order of a few to fractions of a micron, a PZT installed on the rear of the grating longitudinally translates it and affords this level of precision in the cavity length (PZT à L_ext).   Since, the external mode results in an emission line width of ~500 – 100 kHz, the external mode may be swept through the span of the transition width, giving the laser the ability to probe the transition width, as described below.         

Structure of Rb

The D₂ Transition

            The structural underpinnings of Rb responsible for the transition of an electron from one state to another is as follows:  the ground state electron configuration of Rb is [1s², 2s², 2p⁶, 3s², 3p⁶, 3d¹⁰, 4s², 4p⁶], 5s¹.  Only the valence shell (5s) is unfilled with one electron.  As a result, the structure of energy levels is similar to that of hydrogen.  As for the core Rb+ ion, it is spherically symmetric, resulting in a total angular momentum (L), spin angular momentum (S), and thus spin-orbit L-S coupled angular momentum (J = |J| = |L + S|) of zero.   Consequently, with all the core ion quantum numbers equal to zero, the observed energy transitions with the associated changes in L-S coupled quantum numbers comes from the valence electron only.   Stated in L-S coupling notation: (nl)^2S+1L_j, (where n and l are the principle and angular momentum quantum numbers respectively), the ground electronic state is (5s)² S_1/2 , and the first electronic excited state is found in the (5p)² orbital.    Here S = ½ and L = p = l = 1, leaving two possible values for J (i.e., |L–S|, …, J, …, |L+S|; in integer steps) equal to 1/2 and 3/2 .  As a result, two possible energy levels exist for these given n and l values.  The first, less energetic excited state is referred to as the D₁ line and has the following quantum numbers:  (5p)² P_1/2 .  The second, more energetic excited state is referred to as the D₂ line and has these quantum numbers:  (5p)² P_3/2.     This splitting of the S=1/2 and L=1 state into two finer (J = ½, and 3/2) states is known as the magnetic fine structure states of the atom, where the former state corresponds to a transition from the ground state with a wavelength of 794.8 nm, whereas the latter transition corresponds to a wavelength of 780.0 nm.   Our DL can only be tuned in and around 780.0 nm, and therefore we concentrate on the D₂ line.     

Controlling the Tuning Parameters

Laser Driver

             Each of the four parameters mentioned has a method by which they are controlled.  The injection current is set by a commercial DL current source that is quite stable up to +/- .05 mA.  Commercial laser drivers also provide a level of surge protection.  The DC current ranges from 0 mA to about 150 mA, with typical operation at about 65 to 95 mA.  When installing the DL to the laser controller, one must ensure that 1) grounding straps are used to avoid electrostatic shock and 2) the DL is installed with the correct, manufacturer specified polarity, i.e., either cathode to ground (CG), or anode to ground (AG).   

Spatial Orientation of Beam

            The grating and the laser collimating tube are both mounted onto a standard, but modified dynamic mirror mount in the Littrow configuration as described in the paper by Arnold et.al., entitled: “A simple extended cavity diode laser”.  According to this paper, the grating is mounted on a custom-made arm machined out of aluminum.  The arm is shaped like an “L” (~1.5 x 2.5 cm) with the horizontal portion cleaved at an angle of 45^o.  The arm is rigidly attached to the moveable base of the dynamic mount and the PZT is sandwiched between the grating and the surface of the cleaved section.  In this arrangement, the grating angle is easily adjusted using the adjustable screws on the dynamic mount.  By securing this arm to the dynamic mount and effectively making the facets of the external cavity into a single mechanical unit, vibrational differences between the grating and the laser (i.e. the two facets of the external cavity) may also be minimized.  As a result, an absorbed vibration will cause the entire external cavity apparatus to vibrate together as a whole and the relative movement between cavity facets, which leads to broadening is reduced.  Another plus of this design is that the external cavity is in thermal contact with the laser and reaps the benefits of temperature stabilization as well.  If the temperature of the external cavity fluctuates the lengths does also, which also leads to broadening.  

Once the grating is installed, the following guidelines should be followed:  The grating can be mounted using five minute epoxy such that its rulings are perpendicular to the optical table surface, and the beam is centered on the grating surface.  The collimating tube should be rotated such that the minor axis of the elliptical cross section is perpendicular to the optical table as well. 

            One disadvantage of any external cavity geometry is that as the grating is rotated, the beam placement also changes and the beam with respect to the optics is continually repositioned while the laser is being tuned.  As a result, the optics must frustratingly be realigned after each rotation, although for rotations within the flashing region, the transverse deflection is small enough such that the repositioned beam isn’t detrimental to the alignment.  There exist mounting configurations however, suggested by, for example, R.E. Scholten et.al. in a paper entitled “Littrow configuration tunable external cavity diode laser with fixed direction output beam” that significantly circumvent this problem.

            The most course tuning parameter comes from directly changing the temperature of the laser.  However, the temperature must not only be changed, it must also be maintained with a high degree of stability.  Since the laser tunes with changing temperature roughly at a rate of .06 nm/^oC to about .008 nm/^oC in a continuous tuning region, a temperature excursion of only a millidegree will result in a laser frequency shift of ~ 30 MHz (max.).  Taking into consideration that a temperature unstabilized laser will drift by about 1^oC/hour, temperature stabilization is absolutely necessary if high resolution spectroscopy is an issue.  Therefore, a feedback mechanism is put into use that both monitors and changes the temperature as the temperature drifts away from a given setpoint temperature.  For the successful locking of the laser, the temperature stability should achieve < +/- .001 K in drift magnitude. 

In our apparatus, the laser is actively cooled and heated with a thermoelectric cooling element (TEC) that utilizes the Peltier effect as its working concept.  The TEC is sandwiched between the laser mount and a large metal block with good thermal conductivity (1.5’’x1.5’’x4’’), again by using five minute epoxy.  The block acts as a large heat sink with large thermal inertia, helping to ensure temperature stability.  The metal block is also mounted firmly to the optical table.  The quality of thermal contact between these parts is checked by ensuring maximal surface contact between adjacent surfaces.  Some thermal grease may be used to reduce thermal resistance.  An AD590 thermocouple monitors the temperature, and a plexiglass box covers the entire laser apparatus and blocks any air currents that may otherwise destabilize the temperature.  Air currents are particularly detrimental to locking the laser. 

A side effect of bringing the temperature of the laser to a specific value should also be brought to light.  Aside from the mechanical stress that the laser apparatus will endure due to fast temperature fluctuations, there can even be dire consequences if the temperature is varied slowly.   That is, if the temperature is brought below the dew point, water vapor will condense on the cooled surfaces which can be detrimental to all the electronics, including the laser itself.  Therefore it is advantageous to operate the laser at temperatures above the dew point.  If the tuning map is such that transition can only be reached at lower temperatures, the laser must be encased in a moisture tight container with some desiccating crystals placed inside.

              The basic idea behind the feedback mechanism responsible for maintaining a stable temperature is as follows:  given a specific setpoint temperature, the feedback circuit subtracts this value from the actual temperature and this signal then is used, most likely after some more sophisticated signal processing, as an error signal that tells the Peltier element to either heat or cool the laser depending on the sign or polarity of the difference.  Once the setpoint equals the actual temperature, the difference between the two is zero and the resultant error signal is zero, telling the TEC element to do nothing. 

              The preceding method is known as proportional feedback temperature control and is superior to a “ON/OFF” method of feedback control which simply turns a heater and/or a cooler on (at some constant intensity) or off depending on whether the setpoint is above or below the actual temperature.   Proportional feedback control can be illustrated by the following formula: F = P (T_a – T_s); where F is the feedback signal to the TEC, P is a proportional gain controller, T_a is the actual temperature, and T_s is the set point temperature.  Proportional feedback thus applies just enough signal in proportion to what is needed and thereby significantly reduces the magnitude by which the actual temperature overshoots the set point value. The ON/OFF method results in a tremendous amount of overshooting, and often leads to a critically unstable system whereby the actual temperature never stabilizes to the set point value.  Nevertheless, often enough proportional control may still be inadequate and other methods that process the difference between the set point and actual temperature in yet a more complex fashion are employed. This additional signal processing is vital in many cases and serves to eliminate runaway oscillations altogether.  These other standard forms of control simply involve modifications to the proportional control.  Such forms include derivative, and integral control. By taking the time derivative of (T_a – T_s), the system takes on some damping characteristics that allay over and undershooting the set point, whereas by taking the time integral of the difference helps to eliminate the steady state error often caused by derivative control.  The resulting feedback signal has the form:  F = P{(T_a – T_s) + D d/dt(T_a – T_s) + I  (T_a – T_s)dt}, where D is the damping constant, and I is the integral gain parameter. 

Unwanted Optical Feedback

Producing an Absorption Spectrum

Photodiodes

           Once transition has been achieved, an absorption spectrum may be produced through the use of photodiodes that are sensitive to the laser light (IR) intensity and are capable of producing and transmitting signals faster than they are received.  Assuming the transition occurs in Rb vapor at an adequately low pressure, in the natural ratios of ⁸⁵Rb [72%] and ⁸⁷Rb [28%], at the D₂ transition, the four peaks  produced span a frequency range of about 6.5 GHz.  In order to produce the spectrum of peaks, the lasers output frequency needs to be rapidly changed or swept over the range of these peaks.  A photodetector  known as a photodiode (PD) is then employed to monitor the output of the laser once it has traversed the length of the vapor cell. 

           A photodiode consists of a p-i-n-p junction with a depletion layer that absorbs incident photons and converts them into an electron-hole pair.  Only photons with energies above the bandgap energy will be absorbed.  The production of electron-hole pairs creates a current proportional to the incident flux in the presence of a biased voltage, which comprises the signal from the PD.  When the laser is tuned to transition, the lasers output will be absorbed while traversing the vapor cell and subsequently re-emitted in all directions, thereby reducing the number of photons that continue along the path of the beam to the PD.   This diminished intensity at the PD is exhibited as absorption resulting directly from the fact that the lasers output is tuned to the transition frequency of the Rb vapor.  Although there is a reduction in light intensity due to absorption, the remaining intensity that passes to the PD may still be large enough to saturate the PD.  That is, the number of photons coming in, in a given time may be higher than the rate of photon to electron conversion within the cladding layers of the PD,  leaving many photons unconverted, and therefore undetected.  As a result, one must make sure the PD is not operating at its maximum photon to electron conversion rate and attenuate the beam illuminating the PD as needed.  A typical PD circuit is such that the following relation holds:  V_o = P R(l) R_L, where V_o  is the generated signal, P is the power of the input beam, R(l) is the spectral responsivity of the PD (which is typically ~.56 A/W at 780.0 nm), and R_L is the load resistance. 

           If the tuning parameters of the laser were to be cyclically changed or modulated fast enough, the lasers output may be tuned over a broad range of frequencies (~7 GHz max. => ~0.014 nm without external mode hop) and will reveal a spectrum around 780.0nm .  Specifically, by monitoring the output of the PD with an oscilloscope, as the lasers output is cyclically tuned over a certain range of frequencies, the oscilloscope will reveal an absorption spectrum in time.  The relative amplitudes of the peaks is a direct result of the abundance ratio of each isotope, scaled by the signal gain of the PD. 

For a free running laser, a common way to frequency modulate (FM) the lasers output is by modulating the lasers injection current.  Cyclically changing the temperature is both too course of a change and too slow, and therefore impractical.  However, modulating the injection current also has the drawback that there is a high degree of amplitude modulation (AM) coupled to frequency modulation (FM).  This is illustrated by the dual dependence of the power output and the single lasing mode frequency on the injection current. 

Generally speaking, the laser frequency can be modulated only by varying the finest tuning parameter of the apparatus, if the course and fine tuning parameters are independent of one another - the way the lasing and external cavity modes are.  Thus, with the presence of the external cavity, external feedback is responsible for modulating the laser frequency as the external cavity length is modulated.  If the laser current is modulated in the presence of external feedback from the external cavity, conflicting wavelengths will assuredly develop.  That is, if a particular lasing mode is shifted over a few immobile external modes such that only one of these external modes is gain selected, the lasing mode would simply serve to modulate the amplitude of the single external mode until the lasing mode shifts so much that it mode hops onto or gain selects an adjacent external mode.  Further, as the lasing mode shifts such that its peak doesn’t coincide with the peak of the external mode, it sends those peak wavelengths into the external cavity, which subsequently do not resonate there and are decoupled from the laser and lost.  The likely outcome in modulating the current in the presence of the external cavity then, is that there will be amplitude modulation of a single external mode without changing the frequency much, except for occasional mode hopping if the lasing mode is shifted enough.  Despite its drawbacks however, the injection current may still be used to optimize the relative positions of the lasing and external cavity modes. 

 Nevertheless, there is still some minimal AM to FM coupling when the external cavity length is modulated.  As the external mode is cyclically shifting underneath the stationary lasing cavity profile, as before, the relative motion of the lasing mode over the external mode results in amplitude modulation of the external mode.  Modulating both the injection current and cavity length at the same time, judiciously, however may serve as an option to avoid this unwanted AM coupling, not to mention mode hopping, and can dramatically increase the continuous tuning range well beyond the FSR of the external cavity modes. 

Here we’re able to modulate the external cavity length by attaching an actuator (in our case a PZT disc) to the back of the grating and drive it with a saw tooth signal that cyclically expands and contracts the PZT, moving the grating back and forth.  This mechanism cyclically and continuously sweeps through different wavelengths for feedback and consequently modulates the lasers output as described above.  An additional DC “offset” signal applied to the PZT may also be added for moving the range of scanning higher or lower in frequency underneath the lasing mode profile.  One should recall that the magnitude for which the external modes may be offset is equal to less than the FSR of the external modes.  Also, by decreasing the amplitude of the saw tooth signal, it is possible to zoom in on a particular scanning range as the laser probes the gas sample.  In other words, the resulting spectrum displayed on the scope by the PD will be dedicated to a smaller frequency range and will therefore yield greater detail of that segment of the spectrum being scanned.  Together with the DC offset, a frequency window is established that can zoom in on and be repositioned along different segments of a given broad spectrum. 

Through the following simple relation one may calculate just how much the output is modulated given a change in the external cavity length:  Given: n= c/l, then Dl = c Dn/n² = (l/n) Dn (where n is the central frequency of a given mode, c is the speed of light in vacuum and l is the corresponding wavelength of a resonant wave within the cavity).  Utilizing the Fabry-Perot relation: ml/2 = L, DL = mDl/2.  Taking the ratio of the two equalities reveals:  DL/L = Dl/l  (L is the extended cavity length, and m is an integer).

The linear expansion of the PZT goes as follows:  Dt = V_PZT d₃₃ (Dt is the change in PZT thickness as a result of the applied voltage, V_PZT; and d₃₃ is the expansion coefficient in the direction of the PZT thickness).  Assuming the PZT is mounted squarely behind the diffraction grating, Dt = DL.  Given the following input values: d₃₃ = 3.8E-10 m/V (depending on the PZT material), L ~ 20mm, l = 780.00nm, and V_PZT = 1kV, Dt = DL = 3.8E-6 m, which corresponds to a scanning range of about 7 GHz.

            Since the frequency scan is accomplished by exclusively altering the external cavity length, it is useful to explore the sequence of cause and effect, from the point the external cavity length is altered to the point a spectrum is observed on the oscilloscope.  Generally, the external cavity length, L modifies the frequency output, n(L) which in turn interacts with Rb atoms and results in a particular output intensity, I(n) at the PD.

            Starting with the external cavity length, it may be reduced to the following components: L_ext = L₀ + DL_DC + DL_AC , where L₀ is the relaxation length of the external cavity before any voltage is applied to the PZT that either expands or contracts in thickness, DL_DC is the change in the external cavity length resulting from a constant DC voltage applied to the PZT, and DL_AC is the modulated change in the cavity length resulting from an AC voltage applied to the PZT.   

             The PZT expands or contracts depending on the polarity of the applied voltage, which implies that the PZT material is polarized.  From now on it will be assumed, for the sake of clarity, that the voltage leads are attached to the PZT electrodes such that an applied positive voltage results in contraction, while a negative voltage results in expansion of the PZT thickness.  A contraction in the PZT thickness then corresponds to a positive increase in the external cavity length (where DL = -Dt).  The salient point that is relevant to the rest of the discussion however is that a positive applied voltage brings about a positive change in cavity length.

             The two components under our control are of course DL_DC and DL_AC, which depend on the voltage we choose to apply to the PZT.  As discussed, in the present case, frequency scanning is accomplished through a “saw-tooth” or ramping signal plus a DC offset.  The AC saw-tooth voltage signal has the form:  s k t {s = +1, (nT t (n+1/2)T); s = -1, (n+1/2)T t (n+1)T), for n = 1, 2, … }, where T is the period of oscillation and k is a constant value of slope; whereas the DC signal is a changeable constant V₀.  The voltage applied to the PZT, thus has the form:  V(t)=skt + V₀ and the resulting change in the cavity length is DL_ext(t) =  DL_DC + DL_AC = d₃₃(V_DC + V_AC) = d₃₃(skt + V₀), and the total length at a given point in time is: L_ext(t) = L₀ + DL_ext(t) = L₀ + d₃₃(skt + V₀). 

            The central wavelength of an external cavity mode depends on the cavity length as such:  ml₀/2 = L, where m is the mode number and is equal to the number of half wavelengths that will fit within the cavity facets.  Assuming m is a constant referring to a given mode, we have:  l₀(t) = l₀ + Dl₀  = (2/m)(L₀ + d₃₃(skt + V₀)). 

            It is necessary at this point to explore the inverse relationship between the output frequency and wavelength and determine how this fact comes into play once scanning is initiated.  Although it would be preferable to directly modulate the laser frequency and wavelength, this option isn’t available.  Instead, the laser is tuned indirectly by means of adjusting the external cavity length.  For the case of wavelength modulation, the cavity length is linearly related to the wavelength as described:  l = (2/m) L.  The change in l with respect to the change in L is constant:  Dl/DL = l/L = 2/m.  It is apparent that the given slope remains constant so long as m remains constant over the interval of the scanning, which effectively limits L to changing over a continuous scanning range of a single external cavity mode, before a mode hop is encountered.

            For scanning ranges of about 7 GHz, changes in the wavelength are barely noticeable, and therefore it is more appropriate to express the scanning variable in terms of frequency instead of wavelength.  Unfortunately, this forces one to deal with the problem that the external cavity length is inversely proportional to the frequency: n = (mc/2) (1/L).  As a result, if the cavity length is altered linearly, the frequency spacing of the absorption spectrum becomes non-linear.  The problem isn’t as dire as implied however.  If one calculates the change in output frequency with respect to cavity length, the following expression results:  Dn/DL = - (mc/2) (1/L²).  Clearly n is a non-linear function of L, however if the magnitude of the scanning interval is considered (~7GHz), L changes by the following over the range of the scan:  DL = (2L²/mc) Dn = (lL/c) Dn = (7 GHz * 780.00nm * .2m/c) = 3.6E-6 m, which is a mere .0018% of the relaxation length.  That is, over a range of 7 GHz, Dn/DL is virtually constant, which means that the frequency, to a very good approximation, scales linearly with the change in cavity length as well.  As such, if the change in L remains small enough, it may be assumed that the value of L in the following expression is constant:  Dn = -DL (mc/2L²).  Given the modulated cavity length above, Dn = -(mc/2L²) d₃₃(skt + V₀), and n(t) = n_o + Dn = (mc/2L²)( L₀ - d₃₃(skt + V₀). 

            The resulting frequency output has the following form, expressed in terms of angular frequency (2pn=w): w(t) = w₀ + Dw_DC +  Dw_AC(t), where each frequency component corresponds to the three respective components of the cavity length. To be more precise, w₀ is the central wavelength of one of many external cavity modes and is selected according to the point where the gain curve, the grating profile and a single lasing cavity mode overlap.  Mode selection is a form of course tuning and is itself governed by the injection current, the laser temperature, and the grating angle, and ultimately serves to select the mode number m.  Fine adjustment of the value of w is accomplished by shifting the external cavity mode over relatively small increments and is achieved by changing the external cavity length, which is the point at which the other two terms, Dw_DC and Dw_AC emerge.  

            Once the laser light enters the Rb vapor cell, it interacts with atomic energy states and finally arrives at the PD.  However, the PD isn’t sensitive to the wavelength of the incident light.  The PD is only capable of producing a signal proportional to the incident light intensity which has the form: I(w(t)) = I₀(w₀, i) +  I_{AM to FM}(w) – I_GG(w), where i is the injection current, and I_G is the amplitude of the Gaussian profile, dependent upon both the incident beam intensity and the density of Rb atoms in the vapor cell.  The first term represents the laser intensity stemming from unabsorbed photons, but more importantly from the power Vs. injection current curve and the gain incurred from the respective tuning curves.  The second term represents the AM to FM coupling discussed earlier.  Since the external cavity mode shifts underneath the lasing cavity mode profile (linewidth = 45MHz) for a range of ~ 6 GHz, which corresponds to only ~ 1/8 of the lasing cavity width, the external mode amplitude is modulated roughly linearly.  Finally, the last term represents the absorption of light incurred through the interaction of the laser light with the Rb atoms, which as we already know, takes the form of a Gaussian in the case of the Doppler broadened peaks.   

            Thus the frequency output to the laser is conveyed to the PD and finally produces a frequency dependent spectrum through the vehicle of absorption.  That is, the relative degree of absorption corresponding to a location on the absorption profile discloses the frequency at that location.  In this way, the Rb vapor cell serves as a medium to convert frequency into a detectable intensity profile.

Calibrating the Frequency Axis

            Scanning the frequency for a maximum range of 7 to 9 GHz by modulating the cavity length is a large feet and may not be possible with most PZT’s.   The PZT contracts and expands increasingly with greater voltages applied to it, however the PZT material has a voltage limit that should not be exceeded in order to keep the PZT crystal in tact.  Since the voltage limit is stated in terms of applied voltage per change in unit thickness, the magnitude of expansion may be enhanced by simply employing a PZT with larger thickness.  Another option would be to utilize a PZT with a larger expansion coefficient.

A function generator signal fed through a high gain AC amplifier drives the PZT.  The modulation speed or the function generator output frequency is not very pertinent at this point and is typically set to about 200 to 300 Hz.  Although if one would like to see four quick flashes on the CCD corresponding to the four Doppler peaks as a way of “eyeballing” the range of the frequency scan, the function generator should be set to well below 1 Hz.   Typically the PZT tuning rate ranges from about 10 to 60 MHz/V depending on the thickness and material of the PZT used.  Usually, two to three peaks worth (~ 4 to 6 GHz) of scanning is possible, with a maximum peak-to-peak voltage amplitude applied to the PZT of a few hundred volts to 1 kV.  The PZT also exhibits some hysterisis, and only expands nearly linearly with applied voltage, although these effects are minimal over small changes in thickness.  Displaying the input voltage to the PZT (saw-tooth curve) on the same oscilloscope that the absorption spectrum is displayed on provides a PZT signal monitor and frequency scale that reveals the range of frequency that is being scanned.  Most scopes have a max. input voltage of ~25 V, so the input voltage should first pass through a voltage divider.

Doppler Broadened Peaks

Ground States

Maxwell-Boltzman Velocity Distribution

            This Doppler broadening of the D₂ transition peaks refers to the fact that within the Rb vapor cell, the atoms are moving with a Boltzman velocity distribution corresponding to room temperature.  The Boltzman velocity probability distribution is as follows:  P(V_x)exp(-MV_x²/2k_BT), and the Doppler width is given by:  Dn_FWHM = (8k_B ln2)^1/2 (T/M)^1/2, where x refers to the direction of the beam, V_x is the x component of the velocity of the Rb atoms in the vapor cell, M is the atomic mass, T is the temperature, and k_B is the Boltzman constant. Using the laser beams direction of propagation as a reference for the positive direction, entering the cell from one end and exiting at the other, the atoms move with a velocity component either parallel to (x-direction) or anti-parallel to the beam.  To use a second convention, the atoms speed is gauged with reference to the lab frame, and depending on the velocity of the atoms relative to the direction of the beam, in the frame of the atom, the frequency of light appears to be either red shifted or blue shifted, resulting in the absorption of light slightly above and below the transition frequency in the lab frame.   Thus, the Doppler profile is a direct verification of the fact that atoms in the vapor cell will absorb light that is detuned off of the lab frame transition frequency.  The zero velocity group or class of atoms is the only group that does not contribute to the Doppler broadening and corresponds to the summit of the Doppler peak and the lab frame transition frequency.  Atoms with positive velocities with respect to the beam see the incident beam red shifted to the transition frequency and correspond to the higher frequency side of the Doppler peak, whereas atoms with negative velocities see the incident beam blue shifted to the transition frequency and correspond to the lower frequency side of the Doppler profile.  This is the very source of the Doppler broadening as seen in the lab frame.   That is, each portion of the Doppler profile corresponds to a different velocity class of atoms which see different degrees of Doppler shifting of the incident beam frequency to the lab frame transition frequency. 

            Since the magnitude of the detected absorption is proportional to the number of atoms there are to do the “absorbing”, the Doppler profile thus has the shape of the Maxwell-Boltzman distribution, i.e., a Gaussian, which gives the relative number of atoms in each velocity class.  The relation between the velocity of the Rb atoms and the Doppler shifted frequency is:  n’ = n_o  (1+ V_x/c), where n_o is the lab frame transition frequency of the Rb, and n’ is the incident beam frequency.  Since n_o  is constant, the preceding equation nicely illustrates the one-to-one correspondence between the absorbed (incident) frequency n’, and the atomic velocity V_x.  Doppler broadening leads to a FWHM of about 505 MHz around room temperature, although other broadening mechanisms contribute to the width as well, described below. 

Saturation Spectroscopy

Transition Bandwidth Reduced

            For experiments requiring greater precision, i.e. a smaller FWHM, a technique known as saturation spectroscopy has been developed that significantly reduces the FWHM of the Doppler broadened peaks.  By overlapping the probe beam that produced the familiar Doppler broadened peaks with an intense, counter-propagating saturating (or pump) beam (~an order of magnitude more intense and derived from the same laser) within the vapor cell, the spectrum from the zero velocity group of atoms is able to be selected out. The technique of sat. spec. operates by taking advantage of a peculiar interaction between atomic velocity classes and the overlapping beams mentioned, in a way such that when the laser is tuned to precisely the lab frame transition frequency, the fine structure, initially obscured by Doppler broadening is exposed.  These Doppler free peaks with a FWHM of about 10 to 40 MHz depict the hyperfine structure of the atomic transitions. 

Beam Layout

             In this technique of laser saturated absorption spectroscopy, three beams are utilized, all split off from the same laser and thus have the same frequency.  All three beams are directed into the Rb vapor cell.  There is one saturating, intense beam called the pump beam and two other probe beams.  It is these probe beams whose light is ultimately detected by the PD in producing the Doppler free spectrum.  One probe beam is overlapped with the pump beam such that the two beams are counter-propagating and the greatest angle between them doesn’t exceed 1 or 2 degrees.  The purpose of the saturating beam is to saturate the transition of those atoms in its path such that the overlapping probe beam cannot interact with them.   The reference probe, meanwhile is displaced far enough from the pump beam within the vapor cell such that the saturating beam cannot impinge upon it. This is for the purpose of providing a reference frequency sweep.  

            Now, once the counter-propagating pump beam overlaps one of the probe beams, the situation remains relatively unchanged with respect to the absorption profile, since the two beams interact with different velocity classes of atoms.  That is, since the two beams are of the same frequency they each interact with atoms moving with similar speeds and thus similar Doppler transition frequencies. But since the beams are counter-propagating they interact with atoms with opposite directions of movement.  At one particular value of frequency in the sweep, each of the two beams interacts with atoms on opposite sides of the Doppler profile.  The absorbed light in both cases is either blueshifted or redshifted.  Assuming it is blueshifted, once the sweep goes up in frequency, the two beams switch sides of the Doppler profile, and the respective atoms now see the light redshifted.   This is just another way of saying that the production of the Doppler peak is independent of the direction of the laser beam in the lab.  And since the two beams interact with a different set of atoms altogether, they have no effect on one another.  As a result, the probe beam reports a signal to the PD that is unperturbed by the presence of the pump beam. 

However, when the laser is nearly tuned to the lab frame or zero velocity transition frequency during the scanning, the two beams have no choice but to interact with the same set of zero velocity atoms.   The result is that the absorption of the probe beam is reduced at one spot along the Doppler broadened profile. Namely, during the course of a sweep across an absorption peak, the PD reports the familiar Doppler broadened profile, but the absorption is suddenly reduced and the intensity goes up at one location once the sweep reaches the transition frequency of the zero velocity atoms.  Once the laser is tuned to the lab frame transition frequency, both beams interact with the same atoms, but the pump beam, being much more intense, has a much larger probability of interaction.  Once an atom is excited it cannot be re-excited until it decays.  In other words, once the pump beam excites a certain atom, it is removed from the pool of excitable atoms available to the probe beam.  And once the two beams are overlapped, the absorption of the probe beam is reduced by a factor governed by the number of zero velocity atoms that are excited by the pump beam.  As a result, the signal reported by the overlapping probe beam tuned to the lab frame transition reveals Doppler broadened peaks with small dips (called Lamb dips) corresponding to the absorption frequencies of the zero velocity class of atoms only.  Since the un-broadened absorption profile is a direct result of the natural linewidth of the atomic transitions, which have a Lorentzian lineshape, these Lamp dips inherit the very same Lorentzian lineshape.

             Now, by subtracting off the fully Doppler broadened signal provided by the remaining reference probe beam (whose signal gain is set to the same level as that of the overlapping probe beam), we may finally reveal the Doppler free spectrum.  The subtraction is accomplished through the use of two separate PD’s wired such that their signals are subtracted from one another.  One PD detects the signal from the overlap beam, the other from the reference beam.  Usually a low gain AC signal amplifier is used to amplify the small signal current coming from the PD’s.

Some Tricks

1)     A quick way to verify whether an observed peak is truly a Doppler free peak is to simply block the saturating beam, at which point the peak should disappear if it is indeed a Doppler free peak and not some artifact of the apparatus.

2)     While in pursuit of finding a Doppler free peak it is good to start out by maintaining a high ramping amplitude (corresponding to a few GHz scanning range) and verifying whether the peaks that are observed are Doppler free or not by the method just described.  It is often easier in the long run if the reference probe beam is blocked in the meantime, in which case the Lamb dips will appear on a Doppler background.  After a satisfactory peak is found then the ramping amplitude is turned down as the DC offset is shifted in an effort to keep the scanning window on the peak of interest during the process of zooming in on it.  After the zoom is such that the majority of the oscilloscope window is occupied by a given peak, then the reference beam may be introduced and its gain adjusted such that the two probe beams subtract to zero once the saturating beam is blocked.  The easiest way to adjust the gain of the reference is simply to slightly misalign the beam on its respective PD aperture, thereby reducing the incident intensity by a constant amount.

3)     The amplitude of the Lamb dips may be maximized and the width decreased by simply adjusting the mirror that steers the saturating beam into the vapor cell so as to increase the amount of overlap between the probe beam and pump beam.  In order to get a reasonable amplitude, the beams should overlap by no more than 1 to 2 degrees, as mentioned.

4)     Since there is usually a lot of beam steering going on due to the fact that the beam position moves while the grating angle is tuned, and the overlap conditions for the probe and pump beam must be maintained, not to mention the fact that the entrance window of the vapor cell must be kept at an angle in order to reduce unwanted optical feedback, etc. it is a good idea to mount the vapor cell in a way that affords the greatest flexibility in position and its angular orientation.  A commercial platform (~3” x 3”) mounted on a standard optical adjustable post has proved to be quite flexible.   Once the post is mounted in a suitable position on the optical table and the height is adjusted, some double-sided tape may be applied over the surface of the platform, and the vapor cell may simply be placed and effortlessly replaced in the optimized position on the platform as needed.  The  angular orientation may be fixed by simply adding more tape to side than the other.  This angle usually requires no further alteration after the initial adjustment, but the vapor cell will have to be translated in one direction or another from time to time, which makes this setup tailored to the needed flexibility and easy to implement.   Other mounting strategies, where the cell is fastened into a rotating commercial joint that can be height adjusted, provides a lot of angular flexibility, but virtually no lateral repositioning once the mount is installed in one position onto the table.  The advantage of the setup above is that it gives just enough angular adjustment, but also provides the large range of translation that is really needed in the apparatus.

5)     Given the setup just described, an easy way to verify whether a given peak on the  oscilloscope is a Doppler broadened peak and not simply an artifact of the apparatus is simply to remove the vapor cell and see whether the peak disappears or not.  If it disappears then the vapor cell is indeed responsible for the peak, and one may be confident that the observed peak is a Doppler broadened Rb absorption peak.  Instrumental artifacts generally appear often and it isn’t enough to simply verify by eye that a certain dip resembles an absorption peak.  As a general cautionary note, it is a necessary, but insufficient condition that the absorption peak has a Gaussian profile.                           

            The final spectrum reveals not only the Doppler free peaks that have been discussed, but additional spurious peaks, appearing between each real absorption peak.  These spurious peaks are known as cross-over peaks and can arise when there is more than one hyperfine transition under the same Doppler profile, allowing the laser beam the possibility of interacting with two different velocity classes of atoms at the same time.   At this point it is useful to recall that the Doppler profile is nothing more than a composite effect of the different velocity classes of atoms that see a range of frequencies Doppler shifted to the lab frame transition frequency of Rb.  As such, given two different hyperfine transition frequencies under the same Doppler profile: n₁ and n₂, corresponding to two different nonzero velocities, V₁  and V₂  respectively, there are potentially certain laser frequencies,  n_L that will be concurrently Doppler shifted to both n₁ and n₂ and absorbed by the two different sets of atoms at the same time.  In particular, it turns out that the frequency spacing between the hyperfine transitions is such that a single beam must interact with atoms moving in opposite directions for cross-over peaks to arise.  Thus, for a given output frequency, blue shifted transitions in atoms moving in one direction and red shifted transitions in atoms moving in the opposite direction are both excited to their respective transitions by the same beam at the same time.   It is important to emphasize that the atomic interaction in each of these two cases corresponds to two completely different hyperfine transitions, for it is impossible for a single frequency to be simultaneously red shifted and blue shifted to the same transition frequency.  Now, if both the probe beam and saturating beam are present, the following scenario is established such that each beam interacts with two groups of atoms moving in opposite directions.  Since the probe beam and saturating beam are of the same frequency, they each interact with two distinct atomic velocity classes having the same speed, but with opposite directions of movement. 

Thus, in order to produce cross-over peaks at least four groups of atoms are involved, moving with different velocities: +/-V₁, and+/-V₂ corresponding to n₁ and n₂ respectively, where the probe interacts with, for example, the +V₁ and –V₂ velocity class and the pump beam with the –V₁ and +V₂ velocity classes.  (Here the direction of the probe beam is taken as positive.)  So far, given the present situation, the pump beam and probe beam do not interact with same atoms.  The probe beam then, although interacting with two atomic velocity classes at the same time, is simply contributing to the Doppler profile as before, but in a two-fold manner.  In fact, it is safe to say the reference beam interacts in exactly the same way the probe beam does here and any effect like the sharing of beam intensity between two transitions (resulting in a sudden increase in absorption) can be canceled out.  Now it is conceivable that the laser frequency can be swept to a given frequency, n_LC such that the -V₁ velocity class (red shifted to n₁) of the pump beam is the very same velocity class as -V₂ (i.e., -V₁ = -V₂) corresponding to the probe beam (blue shifted to n₂).  The same would of course be true for the +V₁ and –V₂ velocity classes due to the symmetry of the velocities described above.  A circumstance has thus evolved where the probe and pump beam interact with the same nonzero velocity class of atoms.  As a result, like the case for the zero velocity class of atoms discussed earlier, there is a reduction in the absorption of the probe beam and consequently a detected absorption peak.  Unlike the case for the lab frame transitions we are after, these blue shifted and red shifted transitions correspond to nonzero velocity classes of atoms and are therefore dubbed spurious. 

           In summary, a crossover occurs when the laser frequency n_LC is such that atoms with velocity -V see the pump beam Doppler red shifted to the transition frequency at a lab frame value of n₁ and at the same time, see the probe beam blue shifted to the lab frame value of n₂ and vice-versa for atoms moving with velocity +V, where n₁ = n_L(1 - v/c), and n₂ = n_L(1 + v/c).  Since the pump beam is so much more intense, the interaction with the pump beam is dramatically favored.  Solving for n_L gives: n_L =(n₂ + n₁)/2, which places each crossover halfway between successive Doppler free transitions.

Broadening Mechanisms

Doppler, Natural and Pressure Broadening

             The final type of broadening is pervasive in laser spectroscopy and is known as power broadening.  In this type of broadening, the light output from the laser saturates the transitions of the atoms in the vapor cell.  Normally, once an atom is excited it remains so for a given natural lifetime before the energy state decays to the ground state and becomes available again for excitation.  While excited, the atom is removed from the pool of available atoms ready to be excited.  Now if there is more than enough light to excite all the atoms in the path of the beam, the remaining photons that are not absorbed go through the vapor cell undeterred and lead to a reduction in the absorption signal as detected by the PD.  Up to this point, it seems as though it is the amplitude of the absorption peak relative to the beam intensity that is lowered only, but surprisingly the peak width is increased as well.

If one considers a single absorption peak with a Gaussian profile at the point of saturation, the peak summit of this profile will not exhibit further absorption due to an increased intensity of the transition tuned light. The light will then continue through the vapor cell, not interacting with any atoms, and will be detected by the PD.  However, at the wings of the Gaussian profile the absorption frequencies are not saturated yet and the absorption continues to increase with increasing beam intensity.  The final result is the broadening of the profile as the wings are heightened in absorption with respect to the peak.  This effect is readily apparent in the Doppler-free profiles since the broadening can be up to tens of MHz. 

The natural width of a transition is dominated by spontaneous emission in the de-excitation process.   Power broadening, on the other hand, is the direct result of the shortening of the natural lifetimes of the excited atoms due to the stimulated emission of these atoms as a greater number of photons pass by.  Thus, if the beam intensity is high enough such that there are more photons available for stimulating the emission of excited atoms than photons produced through spontaneous emission, then the lifetime will be reduced beyond its natural point, leading to a broadening of the transition energy according to the uncertainty principle, given above.  Hence, as the beam intensity increases, so will the rate of spontaneous emission, leading to further broadening. 

A quantity known as the saturation parameter (s) is defined as the ratio of the induced to the spontaneous transition rates while the laser output is tuned to transition, and is a measure of the number of atoms whose lifetimes have been reduced beyond the natural limit.  A corresponding quantity known as the saturating intensity (I_s) is the intensity of light needed to achieve the ratio just stated.  To avoid saturating the atomic transitions of Rb, the laser output should not exceed I_s ~ 2.0 mW/cm².   It should be kept in mind that although the probe beams of the sat. spec. apparatus should be attenuated well below the limit just stated, the saturating pump should be kept just below the saturating intensity of ~2 mW/cm² = .08mW/4mm² (assuming the area of the beam cross section is about 4mm²).  Although the probe beam of the sat. spec. setup is the one being detected, the pump beam interacts with the very same atoms.  Namely, as a specific zero velocity class atom interacts with the radiation field composed of the two overlapping beams, the process of stimulated emission becomes a cumulative effect of the two beams, which again leads to broadening.  The broadening effect is then conveyed to the PD through the loss of probe beam photons to absorption corresponding to the broadened transition.  Since the pump beam should be about an order of magnitude more intense than the probe beam, the pump beam intensity is placed at a value just below I_s , allowing the probe beam to be as intense as possible without resulting in power broadening.

            It should be pointed out that in the preceding discussion it was assumed that the radiation field of the laser beam has no perturbitive affect on the atomic energy states.  Although this assumption may be an adequate approximation in most cases, a real quantum electronic model would take into account the affect of the radiation field on the atomic states, which in this case depends disfavorably upon the intensity of the laser light.  Once the incident laser light becomes too intense, the energy density of the radiation field increases to the point where the radiation field perturbs the atomic energy states, and new atomic “dressed” states are manifested.  These dressed states are the result of the coupling of the ground and excited states, which in turn modify the Hamiltonian of the system.  The details of these processes are beyond the intention of the present tract, but the salient point to be made is that these new states alter the absorption spectrum in such a way that leads to the overall broadening of the absorption profile.    

Frequency Locking with a Lock-In Amplifier

Frequency Drift

At this point we introduce two new sinusoidal signals generated by the same oscillator.  One is the reference signal of the lock-in and the other is referred to as the dither signal.  The oscillator from which these two signals are derived may either be external to or an internal oscillator of the lock-in amp.  In any case, while the dither is sent to, and drives the PZT, the reference is used exclusively by the lock-in, and will be clarified later.  As its name implies, the dither is a small jittering signal that affords a level of precision capable of scanning over very small frequency intervals.  Ultimately the dither replaces the saw-tooth signal as the input signal driving the laser.  To be more precise, the dither drives the external cavity length as usual, which in turn elicits a frequency response from the laser at a rate specified by the dither frequency.  The signal produced by the PD is then proportional to the incident intensity, “dithered” at a controlled rate.  The dither and the lock-in reference signals are essentially the same, except for a possible phase difference due to the fact that the dithered signal must travel though various lengths of waveguides and through some other peripheral electronics.  Other contributors to the phase differential include the capacitance and inductance of the transmission cables, and the effect of the low and high pass filters of the lock-in.  It will become evident below why this phase difference is deleterious to locking the laser.  Fortunately however, the relative phase between the reference signal and the dithered signal can be compensated for on the lock-in itself. 

To expound on what was said, this dither is essentially a sinusoidal carrier signal that replaces the saw-toothed ramp signal in driving the scanning of the very summit of the absorption peak under scrutiny.   Until the very summit of the peak is centered and zoomed in on (~10 MHz sweep) by continuously decreasing the amplitude of the saw-tooth (and shifting the DC offset), the dither may be turned on and the saw-tooth eliminated altogether.  As a helpful visual cue, the saw-tooth amplitude should be decreased to the point where the summit curvature is just barely noticeable.  At this point, the dither replaces the saw-tooth.  But, the dither amplitude should start out high enough though, such that the very summit of the peak remain visible, and then may be ramped down slowly during the locking procedure described below.   The resulting absorption signal produced at the PD then conveys information about the shape of the region of the peak being scanned, and should subsequently be channeled into the input of the lock-in amp.  As mentioned, the dither signal itself is also picked up by the PD and returns to the lock-in, not in the form of an alternating voltage, but by being encoded in the rate or frequency at which the detected absorption profile is scanned.

            To summarize, the dither signal starts out as AC voltage applied to the PZT, the PZT then expands and modulates the external cavity length.  The external cavity then elicits a frequency output (with minimal AM) in the laser which interacts with Rb atomic energy states.  The PD finally detects the resulting power output after being convoluted by the absorption features of Rb and converts it into a proportional voltage signal.   This “dithered” signal is in turn fed into the noninverting input of the lock-in.  All the while, it is critical to “T” off a connection from the PD output to both the input of the lock-in and the oscilloscope in order to maintain a visual monitor of the absorption peak.

The conveyed dithered signal, now convoluted with the shape of an absorption peak and some AM coupling, enters the lock-in amp where it undergoes filtering through a low pass and high pass filter in order to eliminate unwanted noise.  The signal then goes through the phase sensitive detector of the lock-in, where it is multiplied by the reference signal of the same frequency.  The phase sensitive detector then continues to take the integral of the product, and because the two signals are of the same frequency a term proportional to the derivative of the region of the peak over which the dither was scanning and a sinusoidal term are produced.  If the scanning interval, and therefore the dither amplitude is kept small enough, the derivative term is a DC constant.  Now, by initially ensuring that the dither and reference signal are of high frequency (e.g. 10 KHz), the lock-in can filter out the remaining and unneeded sinusoidal portion of the sum with a final low pass filter.  The enduring DC portion is the dispersive output of the lock-in: the error signal, and the fourth and final signal that is needed to lock the laser.

It should be pointed out that there are limiting factors governing just how high the dither frequency can be.  For example, if the frequency of the dithered signal, transmitted through waveguides, is too high, the ensuing capacitance and inductance of the cable will obscure the signal beyond the point of recognition, by the time the dithered signal enters the lock-in.  The result is that the lock-in simply will not “lock” onto the input signal.  

The lock-in is “locked” onto the input signal only when the homodyne mechanism of the lock-in determines that the input has the same frequency as the reference signal.  This is why it is imperative for the dither to replace the saw-tooth signal in driving the PZT.  Because it has the same frequency as the reference, the returning dither frequency is identified by the lock-in as the driving signal of the experiment.  It should be pointed out that the input gain of the lock-in, (controlled by the sensitivity scale) is largely responsible for whether the lock-in will lock or not and should be adjusted such that the highest gain is obtained while maintaining the lock. 

 What is pertinent about the error signal is its sign and the fact that it absolutely must be DC.  The reason for this urgent necessity is two-fold:  First off, if the error is AC it will get filtered out by the final low pass filter of the lock-in, leaving behind some DC average that doesn’t necessarily conform to the derivative of the scanned region.  And, assuming the frequency bandpass of the filter is such that low frequency components of the signal could get through the filter, the AC error signal consequently wouldn’t allow the laser output to lock to the reference peak, since an AC signal cannot equal a constant value of zero.  The entire driving force behind locking the laser frequency depends on the fact that the error signal has the peculiar propensity towards assuming a constant value of zero.   Because of the un-halting action of AC signals however, such an AC error signal simply would have the harmful effect of contributing to the instability of the frequency output.  As explained above, the error signal is the derivative of scanned portions of the reference peak, where on one side of the summit the sign is positive, on the other it is negative and at the very top, it is zero.  The “DC” constraint can only be satisfied if the dither scans over a small linear region of the absorption profile where the derivative is sure to be constant.  As a result, the dither amplitude strictly determines the AC/DC behavior of the error and must be kept small enough to satisfy the preceding requirements. 

As stated, as the output of the laser drifts to one side of the peak, the calculated derivative becomes non-zero, and once the consequent error signal is applied to the PZT it will contract or expand it (depending on which side of the peak the laser has drifted to) in an effort to bring the lasers output back on top of the peak.  The polarity of the error may be corrected on the lock-in by simply changing the relative phase increment between the dither and reference signals by 180 deg., (once the phase differential is offset to zero) so that the error works in the right direction.  The “right” direction, as mentioned is such that the error signal tends towards eliminating itself.  To be exact, as the error signal works to bring the lasers output back on top of the reference peak, its magnitude gets less and less until it becomes zero. 

             Before attempting to lock the laser, it is recommended to check whether the lock-in is producing an error signal with the correct shape and polarity.  By turning up the amplitude of the dither such that it scans roughly over 30% of the peak including the cusp, the lock-in will take the “AC” derivative of the scan and hopefully reveal the expected shape of the error signal, i.e. the derivative of a Lorentzian, which may be confirmed by monitoring the error signal on a scope.  In doing so however, the high amplitude of the dithered signal may overload the lock-in, not to mention convolute the derivative with a sinusoidal function.  In order to avoid this, the saw-tooth signal may be employed concurrently while the dither is active.  The ramp signal will provide the large scanning amplitude necessary, over the cusp of the peak, while the dither (with small amplitude) will simply take the derivative in small increments, corresponding to contiguous segments of the cusp.   The error signal will thus appear to be segmented, but will outline the profile of the derivative of the peak just as well, and all the while will avoid overloading the lock-in.   Just make sure this AC error signal isn’t received by the PZT in the process.       

Clearly, the uncertainty in the locked frequency value ultimately depends upon the uncertainty in the reference frequency itself.  Theoretically, the uncertainty in the value of the reference frequency at the very summit of the peak is infinitesimally small.  In the present feedback circuit, this value of frequency is unambiguous and very well defined as that unique and single value for which the derivative is zero.  Realistically however, the dither and the lock-in have a limit to how well they can, together resolve this unique frequency value with zero derivative.  That is, as the dither seemingly scans continuously across the summit of the peak, the PD relays a voltage signal back to the lock-in in the form of a quantized string of voltages, where there is a finite limit to how well the PD and the lock-in can discriminate between two adjacent values of incident flux density and voltage respectively.  This leads to a finite frequency interval over the cusp of the peak where the derivative is calculated as zero, and corresponds to an uncertainty in the reference frequency.  When the technique of sat. spec. lowers the reference frequency linewidth by an order of magnitude, what it really does is increase the intensity and voltage drop for a given change in frequency and thus allows the PD and  lock-in to discriminate between the same values of frequency more readily.

            One final note should be made that lowering the dither amplitude such that the scanning range is smaller than the magnitude of the reference uncertainty interval is not only fruitless but will lead to a shorter integration time, which naturally leads to further uncertainty.  Thus, the dither amplitude should be kept small enough such that the calculated derivative over the region that is scanned remains constant.  But at the same time, the dither amplitude shouldn’t be reduced beyond the limits where the lock-in can discriminate one voltage from another. 

Formulaic Summary

                The following attempts to put the preceding discussion into a more rigorous formulaic form: 

It has already been established that:

1)     The PZT expands and contracts virtually linearly with the applied voltage, thereby changing the external cavity length in suit.

2)      The changing cavity length alters the output frequency of the laser linearly to a very good approximation within the limits of our scan, with little amplitude modulation.

3)     The ramp and DC offset voltage signals are both initially sent to the PZT which linearly modulate the external cavity length, and in turn modulate the frequency output of the laser which has the resulting form:  w(t) = w₀ + Dw_DC +  Dw_AC(t) . 

              Now, as the ramp amplitude is gradually brought to zero, it is replaced by the dither signal which has the form:  Acos(Wt), where A is the dither amplitude, and W is the dither frequency.  The resulting dither-modulated laser frequency has the form: w(t) = (pmc/L²)( L₀ - d₃₃(Acos(Wt)+ V₀), after the PZT modulates the external cavity length.  This result follows directly from the similar result derived earlier for the ramp signal.  In this case the ramp is simply replaced with the dither.  (However, unlike the earlier case, while the frequency remains a linear function of the cavity length, the cavity length here is a non-linear function of time and the time axis of the scope is consequently not linear in frequency.  At this point we aren’t interested in calibrating the time axis, so this fact is immaterial here.)

              Nevertheless, the resulting detected intensity has the same form described earlier when the ramp was doing the frequency scanning: I(w(t)) = I₀(w₀, i) + I_{AM to FM}(w) – I_GG(w).  The only difference, since the substitution of the ramp for the dither is that the frequency of scanning has changed.  Additionally, after the reference probe beam signal from PD(1) is subtracted from the overlapping probe beam signal from PD(2), the Doppler background (encompassed in G(w))  is canceled out, including I₀(w₀, i) and I_{AM to FM}(w):  I_PD(1) – I_PD(2) = I_LL(w).  L represents the frequency dependent, Doppler-free  Lorentzian profile brought about by the hyperfine transition width of atomic Rb, and I_L is the Lorentzian amplitude dependent upon the degree by which the overlapping pump beam saturates the hyperfine transitions of intercepted Rb atoms. 

             Since w₀ and Dw_DC are both time independent adjustable constants, they may be combined into one single term and define the central frequency around which the frequency is modulated, where the dither-modulated frequency now has the form w_m(t) = w₀ + Dw_AC(t) = w₀ + Mcos(Wt).  The ensuing signal reported to the lock-in is proportional to the intensity at the PD:  I_m(t) = I_LL(w_m).  Although the dither signal from the oscillator isn’t relayed to the lock-in as a fluctuation in power detected by the PD, it is itself the rate of laser frequency change, and is encoded in the frequency at which the absorption profile is produced.  That is, the rate of frequency modulation (procured by the dither) translates directly into the rate of intensity modulation at the PD.  Hence, the resulting light intensity detected by the PD is a convolution of both the modulated frequency output of the laser and the absorption profile of the transition that is scanned. 

For small dither amplitudes (M<< L_L, G [FWHM of L(w_m)]), the Taylor expansion of  L(w_m) yields, L(w_m) ~  L(w₀) + L’(w) (w_m - w₀) + … . 

           Once the PD signal from the modulated light intensity I_m(t), reaches the lock-in, the noisy component are first filtered out by a high pass and a low pass filter.  The signal is then multiplied by the reference signal (with the same frequency as the dither (W)), and integrated in time:  I_m(t) Bcos(Wt + q) dt , where q is the phase difference between the dither and the reference, and B is the reference signal amplitude.  

          Now, by plugging in the truncated Taylor expansion for L(w_m) in the expression above we have:                    

 I_L{L(w) + L’(w) Mcos(Wt)}B cos(Wt + q)} dt

= I_L L(w) B cos(Wt + q) dt + I_L L’(w) Mcos(Wt) B cos(Wt + q) dt

 Utilizing the following trigonometric identity in the second integrand … 

Acos (Wt) Bcos(Wt+q) =  ABcos(Wt){cos(Wt) cos(q) – sin(Wt) sin(q)}   

=AB{cos(2Wt) cos(q) – cos(Wt) sin (Wt) sin (q)}                  

=AB{(1/2+1/2cos(2Wt))cos(q) – 1/2sin(2Wt) sin (q)}

=1/2AB{(1+cos(2Wt))cos(q) – sin(2Wt) sin(q)}

=1/2AB{cos(q) + cos(2Wt) cos(q) – sin(2Wt) sin (q)}

=1/2ABcos(q) + 1/2AB{cos(2Wt) cos(q) – sin(2Wt) sin(q)}

=1/2ABcos(q) + 1/2ABcos(2Wt+ q)

 As a result, the second integral is equal to

1/2 I_L L’(w) MBcos(q) dt + 1/2 I_L L’(w) MBcos(2Wt+ q) dt

=1/2 I_L L’(w) MBcos(q) + 1/2 I_L L’(w) MBcos(2Wt+ q) dt .

                Now, if the dither frequency (W) is initially set to a high enough value, the lock-in can filter out the first and last terms in the sum above with a low pass filter.  Once these two terms are filtered out, a constant term is left proportional to L’(w), the derivative of the Lorentzian absorption profile.   The phase difference between the dither and the reference is offset on the lock-in in order to maximize the error signal, resulting in a value of q = 0 and a constant value for cos(q) = 1.     

It should be noted that all the signals to the PZT are summed up together in a variable gain summing amplifier.  Considerable knob turning is usually required in order to sum each of the respective signals in just the right amounts in order to lock the laser.  As mentioned, a neat trick that allows an additional degree of freedom in adjusting the relative gain of each PD signal consists of slightly misaligning each of the probe beams on their respective PD apertures.  However, too much misalignment may have some non-linear effect and is not recommended.  After the error signal is turned on and the dither amplitude minimized, the DC offset may be adjusted in order to position the dither sweep across the peak summit.  Meanwhile, the error signal amplitude may be adjusted as needed in order to maintain a good lock.   The error signal should also be “T’d - off” and monitored on an oscilloscope while attempting to lock because the success of the lock may be confirmed by the fact that the error signal stops moving and finally settles to a value of zero. 

Although it has already been suggested that the error signal should be maximized, this was meant to be done in an effort to secure a reserve of signal gain at the lock-in that the summing amplifier may draw upon in order to establish a lock.  Like in any feedback circuit, if the error signal is too high in amplitude, overshooting the reference will occur, thereby destabilizing the system.  Thus, the magnitude of the error should be ramped up to the point where it forces the dither on top of the peak quickly, but keeps it there without overshooting back and forth.  The gain of the error is also controlled by the output sensitivity control on the lock-in.  At higher sensitivities, the error experiences higher gain.  However, if the sensitivity is too high the limits of the lock-in are breached and some data in the output is lost.  As a result, the sensitivity is set to a value, one or two notches below that at which the lock-in overloads.

References

¹A.S. Arnold, J.S. Wilson;  “A simple extended cavity diode laser”  Rev. Sci. Inst.  69 (3), Mar. 1998.

²H.A. Bethe, R. Jackiw; “Intermediate Quantum Mechanics” , 3^rd edition. Benjamin/Cummings, Reading, MA 1986.

³J.C. Camparo;  “The diode laser in atomic physics”  Contemp. Phys. 26, (443, 447), 1985.

⁴C.J. Hawthorn, K.P. Weber, R.E. Scholten; “Littrow configuration tunable external cavity diode laser with fixed direction output beam”  Review of Scientific Instruments,  72 (12), Dec. 2001.

⁵Charles H. Henry; “Theory of the linewidth of semiconductor laser”  IEEE Journal of Quantum Electronics, QE-18, (2), Feb. 2, 1982.

⁶P. Horowitz, W. Hill, “The Art of Electronics” 2^nd edition  Cambridge University Press, Cambridge, MA  1998.

⁷J.M.W. Kruger; :A novel technique for frequency stabilizing diode lasers”  Thesis, univ. of Otago, Oct. 1998.

⁸K.B. MacAdam, A. Steinbach, C. Wieman;  “A narrow-band tunable laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb”  Am. J. Phys. 60 (12), Dec. 1992.

⁹Patrick McNicholl, Harold J. Metcalf; “Synchronous cavity mode and feedback wavelength scanning in dye laser oscillators with gratings” Applied Optics.  24 (17), Sept 1, 1985.

¹⁰Harold J. Metcalf, Peter van der Straten;  “Laser Cooling and Trapping”  Springer, New York, NY  1999.

¹¹Frank L. Pedrotti, Leno S. Pedrotti;  “Introduction to Optics”  2^nd edition  Prentice Hall, Upper Saddle Rive, NJ  1993.

¹²Stephan Petra;  “Development of frequency stabilized diode lasers for building a magneto-optical trap”  Doctoral Thesis, Univ. of Amsterdam, Aug. 1998.

¹³A. Siegman;  “Lasers”  University Science Books, NY  May 1986.

¹⁴William T. Silfvast; “Laser Fundamentals”  Cambridge University Press, Cambridge, MA  1996.

¹⁵G.M. Tino;  “Atomic Spectroscopy with Diode Lasers”  Physics Scripta T51 (58), 1994.

¹⁶Carl E. Wieman, Leo Hollberg; “Using diode lasers for atomic physics”  Rev. Sci. Instrum. 62 (1), Jan. 1991. 

¹⁷Diffraction Grating Handbook

¹⁸Sharp Diode Laser manual/handbook

¹⁹Lock-in amp Model 126 Manual

Acknowledgements

This project wouldn’t have been possible without the invaluable advice, endless tolerance, and dedicated interest of my thesis advisor, prof. Harold Metcalf, SUNY Stony Brook; Dr. John Noe of the Laser Teaching Center, SUNY Stony Brook; Dr. Matt Cashen, Stony Brook; and the following graduate students from the department of Atomic, Molecular, and Optical physics, SUNY Stony Brook:  Benjamin Dietchek, Matt Partlow, and Seth Aubin.  Finally, we thank Inna Shvartsman for her hard effort in helping to construct this web-page.