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