# Research Journal

### Sunday, 07 April 2013

Casey and I have started our project. We are in the process of writing an abstract, and setting up the equipment. We will be using tools such as an interferometer, spectrum analyzer, and oscilloscope to analyze temporal (longitudinal) and spatial (transverse) modes of HeNe lasers. Our project's emphasis will be on temporal modes, and is geared towards a further understanding of how lasers work. This research log will go over the aforementioned tools, science, and what seems to be our plan for the project.

As Dr. Noe has stated, transverse mode patterns of a laser with cylindrical symmetry (which seem to be what we have in the lab, since they are cylinders) can be described with a “Gaussian beam profile and a Laguerre polynomial”. A Gaussian profile is a bell curve, and the Laguerre polynomial is a solution of Laguerre's differential equation. According to this article on Wikipedia, “transverse modes occur because of boundary conditions imposed on the wave by the waveguide.” A waveguide is a structure that guides waves (commonly a hollow conductive metal pipe). The waves travel in a “zigzag” motion through the waveguide, bouncing off of the sides as it proceeds forward. At the ends of the waveguide, the electric field must be zero so that the waves reflect inside in a constant pattern. The pattern of the electric field is limited to those that fit between the walls. The allowed modes supported by the waveguide are quantized, so they can be found by solving Maxwell's equations for the boundary conditions of a given waveguide.

There seem to be a few different ways to classify the modes of transverse waves. The Transverse Electric and Magnetic (TEM), Transverse Electric (TE), or the Transverse Magnetic (TM). A TEM mode is a mode that has no electric field nor magnetic field in the direction of propagation. A TE mode is a mode that has no magnetic field in the direction of propagation. A TM mode is a mode that has no magnetic field in the direction of propagation. In our study, we are using optical resonators, and since these exhibit linearly polarized TEM modes, the following will be a further research into the TEM mode.

Tangent From Topic:
Upon further research, it seems like our the lasers we will be analyzing are not Laguerre-Gaussian lasers, because these types produce a transverse mode with a circular appearance. Instead, they are Hermite-Gaussian due to the rectangular shapes of their mode patterns. By the aforementioned “cylindrical symmetry”, they seem to actually mean that the cavity design is radially symmetrical, so the modes have rotational symmetry along the axis of propagation, which explains the circular transverse mode patterns. In the Hermite-Gaussian mode patterns, there is a rectangular symmetry along the axis of propagation due to the design of the resonator. A Hermite-Gaussian resonator seems to consist of a parabolic mirror, and a planar (flat) mirror. With these facts in mind, I can conclude that the lasers we work with are actually Hermite-Gaussian, and not Laguerre-Gaussian. Furthermore, it seems safe to say that I can conclude that I now know the structures of the resonator cavities. What makes the pattern rectangular is unclear to me, as I have not found any explanation online.

Return to Main Topic:

Now onto the “further research of the TEM modes” as promised... All I can find is that it is a mode that has its electric and magnetic field vectors perpendicular to the longitudinal axis. Light waves has this property as well, which further explains why optical resonators are TEM.

A longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves in the cavity. The amount of half-wavelengths in the cavity is the representation of what mode the wave is in. When a wave in a cavity reflects off of an end surface, its phase changes by a half wavelength. At each half-wavelength interval along the standing wave, there is what we call a node. A node is a point of destructive interference that gives the standing wave the appearance of “bouncing up and down”, or “standing”. At the points of “bouncing up and down”, there are things called antinodes. This bouncing pattern is due to the fact that the wave reflects off of the ends of the cavity with an “antiphase” (half wavelength in difference from the incident wave). The mode order of the wave in a cavity is equal to twice the length of the cavity divided by the wavelength of the wave.

The modes of HeNe lasers, according to Laser Sam's website, operate at what seems to be a nominal wavelength of 632.8 nm. However, the output will not always operate under this single peak, due to the fact that it must satisfy the equation for the possible frequencies of oscillations (located at http://www.repairfaq.org/sam/laserhen.htm#hentoo1). He mentions that in order for a linear (or Fabry-Perot) cavity to resonate strongly, a standing wave pattern must exist. This will only occur when an integral number of half-wavelengths fit between the two mirrors, so this restricts the laser modes to the equation. The laser will operate with a series of peaks around 632.9 nm spaced at a frequency difference of the speed of light divided by twice the length of the resonating cavity. A short cavity will have larger frequency differences, which will result in the peaks “dropping off” at a higher rate (relative to a longer cavity) as the modes move away from the peak. It seems like he is referring to “mode sweeping” when he says “dropping off”.

Mode sweeping occurs when the laser cavity is turned on, and heats up over time. This heating up of the cavity causes the length of the cavity to change slightly, which results in a continuous changing of which modes are underneath the gain peak. There is a point at which the laser heats up to its maximum temperature, and in turn, mode sweeping stops because the cavity is no longer expanding.

An interferometer is an optical device that superimposes waves in order to extract information about the waves. In our project, we will be using a Michelson interferometer to analyze the temporal modes in HeNe lasers. By adjusting the distance (away from the beam splitter) of one the mirrors, we can change the patterns we see on the detector (usually a white piece of paper). This process changes the relative phase of the two modes, which creates a different superimposed image. And contrary to my previous understanding, these modes are actually transverse, and not longitudinal.

A spectrum analyzer measures the magnitude of an input signal versus the frequency within the full frequency range of the given instrument. We will presumably be using it to analyze the spectral compositions of HeNe lasers. According to this article, we can do this with the added application of a transducer (converts one form of energy to another).

An oscilloscope is actually a tricky device to describe. According to many online sources, it is a device that measures voltage versus frequency. An optical signal (fed in by a photo detector) is converted to a voltage and displayed on the CRT. I believe we are able to see the beats of modes in a laser with this device. I will have to ask Dr. Noe to explain this to me.

### Tuesday, 05 March 2013

Today in lab Casey and I practiced adjusting the “adjustable cavity” laser (I think that's what it's called). We also placed hairs in front of the beam so we could see wave modes.

Adjusting the mirror to get the laser to lase was a challenge. At the base of the mirror, there is this screw that when tightened, moved the mirror forward about two millimeters, throwing off our adjustments. You might ask “why didn't you just screw the mirror in place and then adjust with the knobs?” Perhaps that would have been easier now that I think about it, but sweep scanning horizontally was much easier with it unscrewed. To compensate for the screw moving the mirror, I applied a force with my hand to move the mirror/rod device as forward as possible in its stand. I think I will draw this in my notebook to illustrate what I mean. After scanning with this “force” adjustment that I applied, I was able to screw the mirror in, but this time with minimal consequence from using the screw to support it. Since I am now very close to the lasing spot, we can use the mirrors to sweep until I find the right spot. After all this is done, we now have laser feedback.

To create modes, we put hairs in front of the laser beam. Adjusting the hairs was also a process that required a steady hand. According to Dr. Noe, nobody really knows why the hairs do this.

### Tuesday, 26 Feb 2013

Today in the lab I learned a lot about positioning mirrors and other devices to allow for light to pass through a fiber-optic cable. This log will cover that, and research on a SHG crystal, Sam's green laser, and a bit about waves.

There is a process called scanning that is used to align laser light in a way that will achieve the desired output. For example, we could scan to get the laser light in just the right spot that allows it to pass through a fiber-optic cable, or we could scan to achieve lasing between mirrors. The scanning processes that I was involved in had similar procedures to them. In each case, we had the light producer, a fixed mirror, and a moveable (scanning) mirror. In the case that we are trying to angle light into a fiber-optic cable, we set up a lasing laser, a stationary mirror, and then a moveable mirror to precisely direct the light into the cable. The scanning mirror has two knobs on it: One for adjusting the vertical angle (of the mirror), and another for adjusting the horizontal angle (of the mirror). These knobs produce very small changes in the mirror's angle, allowing for a higher precision adjustment. The process of scanning goes a bit like this: In general, we choose a knob (either vertical or horizontal) to be one that we move in small increments. Next, we use the knob that was not chosen to make larger sweeps in that knob's respective direction in which it operates. For example, if we choose the vertical knob to be the increment knob, and the horizontal knob to be the sweeping knob, we would proceed in this manner: We start by angling the light to a position in the general area of where we would like to aim the laser. Next, we start sweeping with the horizontal knob. By sweeping, I mean moving (in this case) to the left and right a large distance relative to the incremental distance. If sweeping at this vertical angle does not achieve our desired outcome, we move the vertical knob slightly upwards or downwards (towards the desired location) and repeat the sweeping process. This incremental sweep process proceeds until the desired outcome is reached. In the specific examples of making a laser, or angling laser light into a fiber-optic cable, we would scan to find the lasing angle for the laser, and the internal reflection angle for the fiber-optic cable.

SHG stands for second harmonic generation. A simple definition of this process is that photons interacting with a nonlinear material are effectively 'combined' to form new photons with twice the energy, and therefore twice the frequency and half the wavelength of the original photons. A nonlinear material is one in which the dielectric polarization P responds nonlinearly to the electric field E of the light. Unfortunately I am in a very distracting environment at the moment, so I cannot begin to go into detail on this (Been trying for at least 30 minutes. My roommate it playing Guitar Hero; one of the loudest games of all time). I will have to ask Dr. Noe what this stuff means. Furthermore, I cannot find Sam's green laser online, and I can't find anything about green lasers. However, I think I can figure out how they can work. If we took infrared radiation with wavelength of 1000nm and applied it to SHG medium, according to the definition of a SHG medium, the radiation's frequency would double, and its wavelength would be cut in half, essentially creating a green light wave with a wavelength of 500nm. According to what information I could find, this can only happen with high intensity incident light since it acts nonlinearly and the superposition principle no longer holds. I don't really know though, since this is all speculation.

To conclude, I will do a segment on waves. A wave is an oscillation that transfers energy from one point to another. There are mechanical waves and electromagnetic waves, but I want to focus more on electromagnetic, since that is what I work with in the lab. Looking at from what I found on the web, I will need to break this topic into several sections because there is a lot of information to cover. I think I will dedicate this log to the general definition of a wave, and then go into specifics in later logs.

Electromagnetic waves do not require a medium in order to oscillate. They use the periodic oscillations of electrical and magnetic fields generated by charged particles. This property allows them to travel through a vacuum. Electromagnetic waves are transverse waves, and so their oscillations are perpendicular to the propagation, also known as the direction of energy transfer.

### Tuesday, 19 Feb 2013

After discovering that I want to focus my research on resonance, I have decided to dedicate this log to "modes." First, I will try to define what a "mode" is. Through research online, I cannot find a definition for "mode" specifically, but I will use what I can find to try and figure it out. Next, I will analyze a few different things that I come across during my search.

It seems like a mode is part of a description of an oscillating system. For example, if a system were oscillating in its "normal mode", it's pattern of motion is sinusoidal, with a single frequency and fixed phase relation. According to online sources, the most general motion of a system is a superposition of its normal modes.

Superposition
Superposition is the process of adding waveforms together. Superposition applies to linear systems only. This principle has many applications in optics because the laser beam can be modeled as a linear system. Well, that's about all I can really grasp on that, so I'm going to move on.

So, a superposition of a system's normal modes is the most general motion of said system. This is pretty confusing. Can't there only be one normal mode? If there can only be one, how can I add all of them? The rest of them (excluding one) do not even exist. Hold on, I think I know what this means. Multiple factors are causing the object to oscillate and the summation of all of these factors is called the superposition. That seems correct since each one of those things has one normal mode, they can contribute to the oscillation of an object, and if there are multiple, they can be added.

Anyway, it seems like a mode is just the oscillation pattern of an oscillating object/wave.

Cool beans. I think my next log will be dedicated to waves in general.

### Tuesday, 12 Feb 2013

Today in the lab we talked about many strange (some difficult to understand) optical topics. We spoke about the maser, MRI machine, laser cavities, resonance, Lorentzian oscillators, cavity ring down spectroscopy, some aspects of laser engineering, polarization, and Brewster's LTC.

The maser was the first device to produce coherent electromagnetic waves through amplification by stimulated emission. Only slightly different in title from laser, MASER stands for Microwave Amplification by Stimulated Emission of Radiation. The first maser used ammonia molecules to amplify microwaves, and was built by Charles H. Townes, James P. Gordon, and H. J. Zeiger.

The physical definition of resonance is "the tendency of a system to oscillate with greater amplitude at some frequencies than at others." A "resonant (or resonance) frequency" is a frequency that causes the response amplitude to become a relative maximum. A good description of this phenomena is Dr. Noe's swing set example: "If you push the swing at just the right moment, even with a small amount of force, the amplitude of the oscillating swing will increase. " This is because the system stores vibrational energy. In a previous entry I described the resonance effect in a laser cavity.

MRI, or Magnetic Resonance Imaging, uses resonant frequencies as part of its main function. A radio frequency current is turned on for a short duration, producing a varying electromagnetic field. This field has a resonance frequency that is just right so it is absorbed by and to flip the spin of protons in the material being analyzed. The field is then turned off, and the spins of the protons return to thermodynamic equilibrium and the bulk magnetization becomes realigned with the static magnetic field. During this time, a radio frequency signal is generated by the protons, which can be measured with receiver coils.

About Bulk Magnetization and Static Magnetic Field: Nuclear Magnetic Resonance = NMR The nuclear spin has an interaction with an applied magnetic field, which gives rise to an NMR spectrum, and energy levels. It seems like this spectrum is unique to the nanoscopic structure (such as a molecule) or atom being analyzed, and that NMR spectroscopy is the technique used to analyze these spectrums. NMR spectroscopy is used to discover the properties of an atom or nanoscopic structure. In many ways, it is permissible to think of the nucleus as behaving like a a small bar magnet, or to be more precise, a magnetic moment. According to quantum mechanics, the magnetic moment can be aligned in any direction. In an NMR experiment, a large number of nuclei are analyzed (for example, 10^20). In this experiment, what needs to be analyzed is the net effect of all these nuclei. According to observation, when the magnetic moments are in equilibrium, their magnetic field contributions are added up to create a net magnetic field along the direction of the applied field; this is called the bulk magnetization of the sample.
The static magnetic field is a strong magnetic field at the center of the MRI scanner whose strength does not change over time.

It seems like these resonance frequencies are described mathematically with Lorentzian functions. A Lorentzian oscillator is any oscillating physical system that has resonant frequencies that can be analyzed with a Lorentzian function.

During one part of the lab period, we experimented with polarizing surfaces a bit. A liner polarizer has slits in it that only allow light in that is at a certain oscillation pattern. This pattern allows it to pass through the slit. Imagine two vertical lines. Now picture a light wave whose crests and troughs are angled so that they point vertically along with the lines. Now I will describe what happens to the light in terms of a piece of unfolded letter paper and a slot mailbox. If the slot of the mailbox is horizontal, and you are trying to put the paper while it is pointed in a vertical direction, the paper will not enter the slot because it's edges will bang against the material surrounding the slot. But if we turn the paper so that it is horizontal with the slot, we can now slide it in without any contact. If we replace the slot with the slits, and the paper with the light wave, we can picture the exact same thing happening, except now we are describing how a linear polarizer functions, and not a mailbox. If we have two polarizers and lined them up so their slits are parallel to each other, and we shine a light through them, we will see that polarized light exiting from the polarizers. But if we turn one of the polarizers so that the respective slits on both are perpendicular to each other, we will see no light passing through. This is because one of the polarizers has prevented the light from entering the slit.

Circular polarization is a bit hard to explain in my opinion, so I will do the best that I can. According to diagrams and websites, linear polarized light is converted to circularly polarized light by slowing one component of the field. When a linear polarized light wave passes through the circular polarizer at an angle of 45 degrees, the vertical component of the field is slowed down so that it is out of phase with the horizontal component. When it is a quarter wavelength slower, the wave turns into a helix. The electric field of this light will now appear to rotate instead of just oscillating up and down.

There is a laser in the lab that has a piece of glass at the feedback end of a laser cavity. This glass is angled so that most of the light exits the cavity instead of being reflected back in by a vertical surface of glass. The angled glass prevents reflection of the light. The physics behind this is unbeknownst to me. I tried to draw a diagram but it still looks like the light would reflect even though the glass is at an angle. Perhaps that only applies to surfaces that are not see through. In that case, incident light perpendicular to the surface of the glass would reflect, whereas angled light would pass through. If all of that is correct, then that would mean that each material has its own characteristics when it comes to how they interact with light.

I thought Dr. Noe mentioned something about Brewster's LTC, but it seems like Brewster is a special angle, and not a place (according to Google searches, anyway). Brewster's angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection (that sure does seem like a good explanation for why the glass is angled). This angle is named after physicist Sir David Brewster.

Upon further research, I have discovered that Brewster's angle is indeed why the glass is angled in gas lasers. The Brewster's angle reflects some s-polarized light, but no p-polarized light. The laser's output is p-polarized, and allows lasing with no loss due to the window. P-polarized is light whose electric field is polarized in the same plane as the incident ray and surface normal. S-polarized light is light that reflects from a surface at a Brewster's angle which is entirely perpendicular to the incident plane. With this, a glass plate placed a Brewster's angle in a light beam can be used as a polarizer.

After this second day of research, I think I want to study resonance effects. I think it would be really cool.