Research Journal

where e is the elementary charge and m_e is the electron mass. If the atom is in the prescence of a magnetic field which is not aligned with the electron's magnetic moment, it will exert a torque on it, causing it to precess at a frequency that is proportional to the strength of the magnetic field.

                    

Above images taken from cronodon.com and hyperphysics.phy-astr.gsu.edu

This means that there are more possible energy levels that the electron can exist in within a particular subshell (s, p, d, f etc) depending on the orbital angular momentum. For a particular subshell with an azimuthal quantum number ℓ, "there are 2 ℓ + ℓ integral magnetic quantum numbers m ranging from -ℓ to ℓ, which restrict the fraction of the total angular momentum along the quantization axis [parallel to the magnetic field] so that they are limited to the values m"(Wikipedia, Magnetic Quantum Number). The relationship between the quantum numbers is demonstrated in the table below:

Above Image taken from Wikipedia

The additional available states in the subshells result in the Zeeman splitting of the atomic energy levels. The new possible states will be equally spaced and displaced from the zero field (the state within that subshell for which m = 0) by

ΔE = m_ℓ eℏ/2m_e B = m_ℓ μ_B B

where μ_B is the Bohr magneton, the magnitude of magnetic dipole moment of an orbiting electron with an orbital angular momentum of one ℏ, which Niels Bohr called the ground state.

Above image taken from ucdavis.edu

The image above shows the Zeeman effect on atomic energy levels. It can be seen that there are not only more possible energy states for the electron, but more possible energy transitions as well. It is evident that this process obeys the selection rules, that the magnetic quantum number cannot change by more than one unit in any single energy transition.

When Pieter Zeeman discovered the effect bearing his name the electron spin had not yet been discovered. It is now known that the spin of an orbiting electron will affect its total angular momentum and thus its magnetic moment. If the magnetic moment is affected, based on the equations above, the energy difference of the Zeeman splitting will be affected as well. The Anomalous Zeeman effect takes this into account.

                   

Above images taken from nitt.edu and chemwiki.ucdavis.edu

The images above demonstrates the angular momentum contribution from the spin of the electron. A rotating charged body, like an electron, will create a magnetic dipole moment with a north and south end. A magnetic field will exert a torque on this magnetic moment just as it did with the dipole created by the orbital angular momentum of the electron, which will contribute to the energy of the Zeeman splitting.

When accounting for the spin angular momentum of the electron, the magnetic dipole moment is calculated as follows:

μ = -μ_B g J/ℏ

= -μ_B ( g_ℓ L + g_s S )/ℏ

J is the total angular momentum of the electron, while L and S are the orbital and spin components. The g-factor is related to the gyromagnetic ratio between the magnetic dipole moment and the angular momentum of the electron. The two factors, g_ℓ and g_s have been measured experimentally to be 1 and approximately 2.00231... respectively, the latter being one of the most accuratelly measured constants in physics to about twelve decimal places.

As stated before, due to the torque that the magnetic field exerts on the magnetic dipole moment, the rotational axis of the electron will precess. In the case of a weak magnetic field, the orbital angular momentum and spin angular momentum are no conserved separately, but the total angular momentum is conserved. The angular and spin momentum vectors precess about the constant the total angular momentum vector. The average of these vectors over the time domain are simply the projections of those vectors onto the total angular momentum vector:

S_avg = ( S · J )J / ^2

L_avg = ( L · J ) J / ^2

Substituting these average values into the equation for the magnetic moment of the electron, then plugging that into the equation for the magnetic potential energy, we obtain:

μ = -μ_B ( g_ℓ ( L · J )J / ^2 + g_s ( S · J )J / ^2 ) / ℏ

< V_M > = μ_B J(( L · J )J / ^2 + g_s ( S · J )J / ^2 ) · B / ℏ

The conclusion to this derivation was taken from Wikipedia

The term m_j is the z component of the total angular momentum, and g_j is the Lande g-factor:

g_j = 1 ± g_s - 1 / 2ℓ + 1

It can be seen that taking into acount the spin angular momentm of the electron has a significant effect on the values of the split energy levels. Without considering this, one's experimental data would not match their predicitions with these formula.

Monday, June 24th, 2013

Fabry-Perot interferometers are very usefull for studying spectral lines that are very closely spaced. They make it easier to study extremely narrow line widths and, apparently, allow virtually no loss in intensity from the incident to transmitted light. This last fact seemed peculiar to me at first, because, surely some light must be transmitted every time a light wave reflects back and forth inside the cavity and some light is absorbed or scattered, resulting in losses of intensity. At a single instance in time it doesn't make any sense, but the key is to observe the system over a period of time. In order to analyzer the spectra, the laser will be left on for at least a few seconds or minutes, which is more than enough time for the light to make an uncountable number of trips back and forth inside the cavity. As the laser is left running, light with a particular phase lag will be amplified by the continous light that follows the same path. This is simply contructive interference. The light collected by the output lens will approach the intensity of the input light, especially in cases with very high-reflecting mirrors of about 99%, it will virtually match the input intensity. The loses due to light being absorbed, scattered or transmitted through the input coupler mirror are still present, but insignificant when compared to the total build up of the output intensity.

Wednesday, June 19th, 2013

I began reading a book titled, "Fabry-Perot Interferometers", by G. Hernandez, in order to gain a better understanding of how the device works, and how I might use it in an experiment.

The diagram below represents the typical Fabry-Perot Interferometer:

According to G. Hernandez, there will be a phase lag between the waves due to the difference in the lengths of the paths that they take through the interferometer. The lag between waves A and B can be calculated as follows:

Φ = 4 π ν μ d c^-1 cos(θ)

where ν is the frequency of the wave, μ is the refractive index of the material between the two reflective surfaces, d is the distance between the reflective surfaces and θ is the angle of the incident light with respect to the normal of the transmitting surface.

If μ d c^-1 = t/2 , where t is the time for the light ray to transmit through the first layer, reflect off of the second, then transmit through the first again, then it follows that:

Φ = 2 π ν t cos(θ)

If θ = π/2, then there is no phase lag(the light did not transmit or reflect, in fact it simply ran along the surface. Interestingly, the phase lag increases as θ increases, until θ = 0. At that point, waves A and B will be out of phase by a multiple of the period, so they will be technically in phase once again.

To calculate the amplitude of each wave, one must know the reflection and transmission coefficients of the surfaces.
Ex: Amplitudes of sucessive waves reflected from plates (a, b, etc.):

r₁ , t₁² r₂ e^iΦ , t₁² r₂³ e^2iΦ

I was confused as to where the exponential terms came from, so I decided to learn more about superposition of wave functions.

The amplitude of any sinusoid function can be modelled by the following functions:

x = r cos(θ) , y = r sin(θ)

The following Argand diagram in the complex plane reveals how they can be incorporated into the complex amplitude of the wave, z:

Above diagram obtained from daviddarling.info

So z can be expressed as:

z = x + i y = cos(θ) + i sin(θ) = e^iθ

In this case, we are considering the phase lag between waves, so:

z = e^iΦ

When the reflected light is focused by the first lens, the total amplitude is given by:

A_r(Φ) = r₁ + t₁² r₂ e^iΦ + ...

= [ ( 1 - e^iΦ ( R + τ) ) R^1/2 ] ( 1 - R e^iΦ)^-1

and for the transmitted light:

A_t(Φ) = t₁ t₂ + t₁ t₂ r₂ e^iΦ + ... = τ ( 1 - R e^iΦ )^-1

Note that:

t₁ t₂ = τ

r₁ = - r₂

r₁² = r₂² = R

δ = Φ + 2χ

χ represents the phase changes upon reflection on each surface (when a wave reflects off of a fixed surface, the phase flips):

These amplitudes are complex and cannot be observed. By multiplying them by their complex conjugate, we obtain the intensity of the wave.

Y_r(δ) = A_r(δ) A_r*(δ)
= R [ 1 - 2 ( R + τ ) cos(δ) + ( R + τ )² ] [ 1 + R² - 2 R cosδ ]^-1

Y_t(δ) = A_t(δ) A_t*(δ)
= τ² [ 1 + R² - 2 R cos(δ) ]^-1

Expressed in terms of the absorption scattering coefficient (because, realistically, not all light is either reflected or transmitted) A, where

R + τ + A = 1

It follows that:

Y_r(δ) = R [ 1 + ( 1 - A )² - 2 ( 1 - A ) cos(δ) ] [ 1 + R² - 2 R cos(δ) ]^-1

Y_t(δ) = [ 1 - A (1 - R )^-1 ]² ( 1 - R )² ( 1 + R² - 2 R cos(δ) )^-1
= [ 1 - A ( 1 - R )^-1 ]² ( 1 - R ) ( 1 + R )^-1 ( 1 + 2 Σ^{m=1 → m=∞} R^m cos(mδ)
= [ 1 - A ( 1 - R )^-1 ]² [ 1 + 4 R ( 1 - R )^-2 sin²(δ / 2) ]^-1
= [ 1 - A ( 1 - R )^-1 ]² ( 1 - R ) ( 1 + R )^-1 2 ln(R^-1) × Σ^{n=-∞ → n=∞} { [ ln ( R )^-1 ]² + ( δ - 2 π n )² }^-1

The plots below demonstrate the "transmitted and reflected radiation of a Fabry-Perot Etalon as a function of the phase retardation of the beams for various reflectivities. The values of the latter are: 0.98, 0.81, 0.65 and 0.45 for the narrowest to the widest profile":

Above images taken from "Fabry-Perot Interferometers"

It is important to take away from the diagrams above that a Fabry-Perot interferometer with higher reflectivity on it's mirros is ideal. This is demonstrated by the fact that at higher reflectivies, it is easier to differentiate between lines that are close together--there is an increase in spectral resolution. It is also interesting to note that, if A=0, meaning that the reflective material of the mirrors does not absorb nor scatter light, that

Y_r(δ) + Y_t(δ) = R ( 2 - 2 cos(δ) ) ( 1 + R² - 2 R cos(δ) )^-1 + ( 1 - R )² ( 1 + R² - 2 R cos(δ) )^-1

= [ R ( 2- 2cos(δ) ) + ( 1 - R )² ] ( 1 + R² - 2 R cos(δ) )^-1

= ( 1 + R² - 2 R cos(δ)) ( 1 + R² - 2 R cos(δ)^-1

= 1

So, for all reflectivities R, in the absence of absorption and scattering, the intensities of the transmitted and reflected light add to unity. This simply means that the energy of the waves if conserved. This is the result that I would always expect, but it is nice to check just to make sure that I understand the physics system and how it conserves energy.

What kind of interference pattern should one expect to see from such an interferometer? Due to the circular symmetry of the Fabry-Perot, one will see an interference pattern of bright concentric rings, like the one illustrated in the image below:

Above image taken from Johnwalton on Wikimedia Commons

G. Hernandez explains that, for the case of transmission, there will be maxima at which constructive interference occurs at specific angles related to the phase lag between succesive transmitted waves:

n = δ (2π)^-1 = 2 μ d σ cosθ

where σ is the inverse wavelength, or wavenumber, given by:

σ = ν c^-1 = λ^-1

It can be seen that a single wavenumber corresponds to multiple interference maxima at specific angles givn by the equation above. The difference in wavenumber between these successive maxima is known as the free spectral range, and is given by the following:

Δσ = ( 2 μ d )^-1 = σ / n₀

where n₀ is the central order of interference for transmission ( the bright circle at the center of the interference pattern ).

It is interesting to note that,

Δσ^-1 = Δλ = c / Δf = 2 μ d

Which follows that,

Δf = c / 2 μ d

So, the formula for the free spectral range of a Fabry-Perot interferometer is the same as that for the beat frequency between adjacent laser modes of a laser. It makes sense why this is the case, because the two systems are very similar. In both, the frequencies of light that are sustainable inside the cavity that get amplified can fit an integer number of half-wavelengths between the two mirrors of said cavity. It should be noted that this discovery disproves an assumption I made in a much earlier journal entry a few months ago. Upon my derivation of the beat frequency of a laser, I assumed that it would be the same, regardless of the medium is was in, because I ended up with Δf = c / 2 l, which has no factor related to the medium of the cavity. The reason for this, is because I chose a particular medium without realizing it: vaccum. I simply didn't incorporate the index of refraction into the speed of light in a particular material, which was incorrect. The speed of light in a given material with index of refraction n is given by:

v = c / n

Friday, June 14th, 2013

Transverse modes were a subject of one of the sub-topics of my report on my laser modes project. I discovered that some of Dr. Noe's past students did work on creating these kind of modes as well.

The simplest transverse mode has a Gaussian intensity profile. Transverse modes can be created by putting a thin obstacle in the lasing path (such as a hair), or by slightly changing the orientation of the output coupler mirror. These patterns can actually be modeled mathematically if one multiplies the Gaussian function that represents the 0-0 mode by a Hermite Polynomial.

Similar work was done by a former student of Dr. Noe named Justin Tian. In his report, "Creating Higher-order Mode Laser Beams with an Open Cavity Laser", he produced some quality transverse modes:

Interestingly enough, he used a different method than we did in order to produce these modes. He used a scratched microscope slide instead of a hair to obstruct the beam. In order to minimize reflections he tilted the slide at the Brewster angle. This angle can be calculated utilizing the Fresnel equation, if one knows the index of refraction of air (1.00) and of the glass slide (about 1.518):

θ_B = arctan(n₂/n₁)

Thursday, June 13th, 2013

I'm very excited to be back at Stony Brook for the summer. Being here for research gives me a completely different feeling from being here during the semester, mostly because I don't have all the stress when worrying about homework and exams. I can focus my time on learning what I'm interested in instead of focusing on a curriculum preset for me. Every day I have the opportunity to learn from the many scientists and grad students here.

We have already had the opportunity to attend two seminars given by Professor Eden Figueroa's former colleagues back in Germany about their experiments, after which he showed us around his lab. His experiment greatly intrigued me. Eden is the leader of the Quantum Information Technology group here at Stony Brook. They are aiming to develop the fundamentals of Quantum Information Technology (as the tech is still in its infancy), such as developing quantum memory technologies as well as quantum transistors and gates with the use of cavity electromagnetically induced transparency(EIT).

The main goal of Quantum Information Technology it to be able to create systems in which information can be stored in the form of quantum states. The unit of of information is the quibit which is a two-level system, but can also be in a superposition of the two states (this is the main difference between classical and quantum information systems).

This quantum information is stored in atoms which act as quantum nodes, and is transmitted between nodes by photons. In order for the photons to carry the information about the atoms quantum state the photon and atoms must become entangled. That is one of Eden's goals, but first he must be able to isolate the single entangled photon from the laser fields. This is a very difficult task because the laser produces millions of photons over a short period of time. He is able to do this through the use of etalons, which are solid cavities consisting of highly reflective surfaces. If the etalons have the proper finesse (q-factor, which is related to losses of the system and bandwidth) and free-spectral range, all cavity modes except for one can be suppressed. This technique can be used to isolate the single photon.

In order to create a quantum memory, so that the information carried in the quantum state can be stored for a later release on command. Eden uses the technique of EIT in order to slow down the light so he can store it within a rubidium cell; normally, the cell would not be able to contain the light, as it would be opaque. This proces involves tuning two lasers to interact with three quantum states of the rubidium. This creates a spectral window of transparency in the absorption frequency.

It can be seen below how the finesse of an etalon cavity affects the allowed modes. Higher finesse (less losses) results in a narrower line-width of wavelength, which results in a smaller frequency range and thus less allowed modes. The diagram shows that the "allowed" modes have high transmission and low reflection. It would be ideal to make the peaks as narrow as possible in order to allow virtually one mode, especially in the case of Quantum Information experiments such as Eden's.

Monday, April 22nd, 2013

There haven't been recent updates to my journal because the focus of our project has changed quite a few times. Measuring the coherence length of a laser is a very difficult and delicate experiment that will run into numerous problems. It will certainly not be a clean experiment to do, indeed the data may be impossible to take properly because it is very difficult to tell where the two modes lose their coherence because it is not easy to see when the fringe patterns completely overlap or not. The geometry of the beam splitter actually leads to some fringe like effects of the screen which can be confusing while trying to take these measurements.

We decided to return to the roots of the project and look into the physics of modes inside the laser's resonant cavity. We observed the frequency differences (we like to call them "mode beats") on an oscilloscope. The actually frequency of the modes cannot be seen on the oscilloscope because they are too high. One can observe the differences between these differences, the "beats of beats", by inscreasing the sweep speed. The number of these beats of beats oscillates with time. This effect can be clearly seen on both the oscilloscope and the RA spectrum analyzer, which plots the gain in dB (voltage) against the frequency instead of time like the oscilloscope.

This phenomena is due to the changing environment inside the laser's optical cavity. It can be described as the result of "mode pulling", which is similar to mode competition. This mode pulling is the non-linear interactions between the modes and the Neon gas (the gain medium in a Helium Neon laser). Modes are electromangetic radiation, which is a changing electric field. If the electric field which affects the atoms of the gain medium is changing, then their rate of stimulated emission will be affected as well which in turn instigates the development of the other modes. In this way, the modes indirectly affect eachother through their direct interactions with the gain medium.

One can see this effect ont the spectrum analyzer when one particular beat frequency is being analyzed. With time, it will change from being a single smooth peak, to become greatly chaotic with several, seemingly random small peaks, only to return to the smooth peak again. This process occurs over the course of a few minutes. The smooth peak situation represents the state in which the interactions between the modes and the gain medium cause the beat frequencies to have no difference (no beat of beats), which the state in which the peak is broken up into a rough, jagged, mountain-like peak represents the situation in which these "beats of beats" exist because of the differences between the beat frequencies.

Interestingly enough, no one, not even Laser Sam (Mr. Goldwasser) is sure why, in the "jagged peak" state of the beat frequency, why there appear to be too many beats of beats, when one considers how many actual beats there are available to interfere with one another. Perhaps this is the topic for a future project.

Although the jagged peak may seem random, it is not. If one watches several oscillations, it can be seen that the same patterns arise again and again with very little, perhaps no differences. It would be interesting to observe these patterns with more or less combinations of beats of beats, by trying longer or shorter lasers than the 44 cm HeNe which will support more or less modes.

The rate at which these oscillations occur is not constant with time. Dr. Noe left the laser on over night, and returned to find that the oscillations had become much slower. We believe that this is due to the "mode sweeping" phenomena, that occurs as a result of the increasing temperature of the laser. When the laser is first turned on, it will heat up, which will cause the cavity length to change. If the cavity lenght changes, then the supported modes will change as well, and the modes that fit under the gain curve will sweep back and forth, which will affect the mode pulling phenomena, and thus the change in the beats of beats as well. The rate of temperature change will decrease over time (it's a typical thermodynamic system) as it approaches equilibrium, which means that the oscillations in the changes of the beats of beats will become less frequent. It would be interesting to see if the patterns in the jagged peak representing the beat of beats stays constant over long periods of time, or changes due to this phenomena.

Thursday, March 28th, 2013

One misunderstanding that we had was whether the waves completely destructively interfere after one coherence length. The key to understanding this is that modes outside of the cavity are travelling, NOT standing waves. The waves do destructively interfere copmletely after they travel one coherence length, but we cannot see this because this point is traveling at the speed of light. The time it takes for this to occur is

t = L_c / c = Δf / 2

Where L_c is the coherence length of the laser, c is the speed of light, and Δf is of course the mode separation.

This means that the modes cancel out completely at a frequency of

f_d = 2Δf

This is not dependent on c, so we don't have to know the speed of light accurately in the gain medium in which the light is moving in order to calculate this rate at which the modes go dark.

Tuesday, March 26th, 2013

My misconception about the origin of the fringe patterns was not entirely undone as I had previously thought; I didn't really understand why the modes couldn't interfere with eachother. Surely two different waves can still interfere with one another even if they have different frequencies and thus different wavelengths. As it turns out, the two modes (in the case of the short 10 cm Helium Neon laser) are oppositely polarized; they have an orientation difference of about 90 degrees. This is the reason that they are unable to interfere, construcvtively or destructively with one another. Dr. Noe explained that this was due to "mode competition" inside the laser. The fringe patterns do result from the modes interfering with themselves, after the beam splitter splits each in to two plane waves, they reflect off of the mirros and can come back at a slightly different angle depending on the arm length difference (the difference in the distances that the components of the separated beam travels). Fringes form where the plan waves intersect one another.

I am still confused by this mode competition phenomena, however, so the confusion has not been fully cleared up for me. According to the RP Photonics Encyclopedia, this phenomenon can be described as follows: "In a situation with strong mode competition (in the sense of strong overlap of mode intensity distributions), which may occur, e.g., in a unidirectional ring laser, a single mode may be excited in the steady state (? single-mode operation, single-frequency lasers): the mode with the highest net gain will saturate the gain so that it exactly balances its losses, and any other mode will then experience a negative net gain, which causes its power to fade away." This means that the modes can limit eachother through spatial overlap and through their interactions with the gain medium. I'm still not entirely sure why this leads to the modes being oppositel polarized however... perhaps the modes can limit eachother so that only certain polarizations are allowed?

Thursday, March 14th, 2013

Our current project Idea involves measuring the coherence length of a laser. Our interest clearly lies in understanding the concepts of lasers as resonant systems and how the frequencies of light that can be amplified by a laser exist as modes within the laser's optical cavity.

This idea can be easily understood through observation of a laser's gain curve. This gain curve is the plot of gain in power vs. frequency of the light inside the laser's cavity. The shape of this curve and what frequencies it covers depend on the material that comprises the gain medium. The modes that are stable are the ones that lie under the gain curve. As it turns out, if the lenght of the laser is increased it can support more modes, and they will be closer together, so the beat frequency between two modes will be smaller.

A short laser will generally have a more narrow gain curve and will thus support less modes at any given time. Suppose we have a short laser of length 10cm or so that will support either 1 or 2 modes at any given time (more on the number of modes being amplified at any given time later). If the modes wavelenths differ by λ/2, then the difference between the supported modes underneath the gain curve is Δf = c/2L, where c is the speed of light and L is the length of the optical cavity.

In the case of two stable modes, after the modes leave the laser cavity, their coherence will deacrease, and the fringe pattern seen on the screen will become blurred. Two waves are said to be coherent if the realtion between their two phases is constant.

The length of the cavity is equal to the coherence length, in the case of a short laser that can support a maximum of two modes, (which can be verified with the equation, Δf = c/2L, if one knows the width of the gain curve of that laser).

In order to adjust the length between the laser and the screen, a Michelson Interferometer can be used, which is comprised of a beam splitter that separates the laser beam into two, one of which is reflected back by a moveable mirror, the other of which is reflected by a fixed mirror back through the beam splitter onto a viewing screen on which interference patterns can be observed. A misconception that I had was that these fringe patterns were due to the different modes interfering with eachother constructively and destructively, but it is actually due to the modes interferring with themeselves after they are split and meet back at the beamsplitter. The arm length can be adjusted to change the toal distance between the laser and the viewing screen.

Tuesday, March 12th, 2013

Since we still haven't decided on the exact topic of our project, I figured that it would be a good idea to review the basic concepts of the general subject of lasers. I picked up a book from the Math, Physics and Astronomy Library here at Stony Brook Univeristy called Principles of Lasers. I was unable to take out the book titled LASERS by Peter Milloni because it cannot be taken out of the library. It is only up for two hour loans. I did some digging and found a book titled Principles of Lasers, by Orazio Svelto from the Polytechnic Institute of Milan, which is probably the next best thing. The first chapter, titled Introductory Concepts, focuses on the general theory behind "the laser idea".

The first section discusses spontaneous emission in general. In nature, Atoms will tend to decay to lower energy levels. When the electrons orbiting the atom go through this decay to the lower energy level, the energy must go somewhere because energy must always be conserved. This energy will be radiated away in the form of an electromagnetic wave. If the two energy levels are E₂ (higher) and E₁ (lower), then the frequency of this electromagnetic wave, or how many times the electric and magnetic field components that comprise this wave oscillate per second can be calculated as follows:

f = ( E₂ - E₁ )/ h

Where h is of course Planck's constant. It must be noted, however that the energy can also be released kinetically to surrounding atoms and molecules. It must be noted that there is no definite phase relation between the separate atoms so the wave can be emitted in any direction. This means that light emitted spontaneously will be very incoherent. The rate of decay to due spontaneous emission can be modeled as a differential equation:

(dN₂/dt)_sp =-A_2,1 N₂

Where N₂ is the number of atoms per unit volume in level two and A is the spontaneous emission probability, or the Einstein A coefficient which depends on the particular element and energy levels involved in the transition. The time that the electron will stay in the upper energy level, or the spontaneous emission lifetime, can be calculated by integrating the rate of decay formula above:

t_sp =1/A_2,1

The second section focuses on stimulated emission, which is at the heart ot the laser's functionality. Incident electromagnetic radiation can cause an atom to decay in energy, if and only if its frequency is equal to the atom's frequency. This is very strange to me and I do not understand it. I can imagine it as "knocking the electron down" because it has the same energy as it, but then why would it not work if the incident photon has more energy? I need to learn more about this, but I have the feeling that it requires knowledge of quantum mechanics. Anyway, when the electron decays to the lower energy level equal to the energy of the incident photon, it will then emit a photon of not only that same energy, but also with the same phase as the incident radiaion. This must then be why lasers are nearly monochromatic and very coherent.

The formula for the rate of decay due to stimulated emission is slightly different than the rate of decay formula for spontaneous emission: (dN₂/dt)_st = -W_2,1 N₂

Where W_2,1 is the stimulated transition probability which depends on the particular transition and the intensity of the incident electromagnetic wave:

W_2,1 = σF

Where σ is the stimulated emission cross-section, and F is the photon flux of the incident wave. This makes sense: if the area of atoms and/or the amount of incident photons is larger, one would expect the transition to be more likely, and the rate of decay should increase.

The third section focuses on absorption. This works oppositly to stimulated emission. If an incident photon has more energy than an atom, it will increase the atoms energy by an amount equal to its own, if there energies would sum to a stable level. However, the probabilites of either stimulated emission and absorption are equal. The formula for the rate of absorption is the same as that for the rate of stimulated emission, except N₂ is replaced with the number of atoms per unit volume in level one, N₁. The stimulated emission probability is also the same and the transition cross section in this formula, _1,2 is equal to σ_2,1.

(dN₁/dt)_st =-W_1,2 N₁

The fourth section of this chapter final introduces the idea of a laser, that a material with energy levels E₁ and E₂ and population densities of those energy levels N₁ and N₂ will cause a change in flux in a plane wave that passes through it due to stimulated emission and absorption, given by:

dF=σF(N₂ -N₁)dz

This is very interesting. The material will behave as an amplifier if N₂ > N₁ because (dF/dz)>0 and if N₂ < N₁ then the material behaves as an absorber because (dF/dz)<0. If the material is in thermal equilibrium, which means no heat is entering or leaving the material, then the ratio of the level populations can be calculated as follows:

(N₂/N₁) = e^{(-E₂ - E₁)/(kT)}

Where k is Boltzmann's constant and T is the absolute temperature of the material. It should be noted that T will represent the kinetic temperature if and only if the material is also in thermodynamic equilibrium, meaning that the total emission is equal to the total absorption. In thermal equilibrium, N₂ < N₁ so the material acts as an absorber. If nonequilibrium is acheived for which N₂ > N₁ then the material will act as an amplifier, in which case there is a population inversion, and the material is considered to be "active".

LASER stands for Light Amplification by Stimulated Emission of Radaiation. The typical laser consists of a resonant cavity with resonance at the frequency of the transition of stimulated emission. The cavity is enclosed by two higlhy reflective mirros so the plane electromagnetic wave will reflect back and forth, getting amplified (increase in flux) each time it passes through the active material. One mirro is made slightly more transparent so an output can be achieved. The "lasing threshold" is reached when the gain of the active material fully compensates for the losses due to output coupling. The gain per pass in the active material is given by:

Φ_output / Φ_input =e^{(σN₂ - N₁)l}

where l is the length of the active material. The lasing threshold will be reached when:

R₁ R₂ e^{2σ(N₂ - N₁ ) l = 1}

Where R₁ and R₂ are the power reflectivities of the mirros (if losses are only due to transmission through the mirros because of their imperfect reflectivity and not due to any other flaws of the cavity). The critical inversion for the lasing threshold will be reached when:

(N₂ - N₁)_c = -ln(R₁ R₂)/(2σL)

When this critical inversion and lasing threshold is reached the oscillations will build up from the spontaneous emission. These spontaneously emitted photons along the cavity axis will initiate the amplification process.

Tuesday, February 26th, 2013

Most photodetectors aren't fast enough to measure the duration of ultrashort pulses. One way to get around this difficulty is by using an interferometric autocorrelation setup. This setup involves focusing a pulse from a mode-locked laser with a collimating lens onto a retroreflector, so the light would be reflected back in the same direction parallel but slightly displaced, so it could then be put through a beam splitter. This beam splitter is a partially reflecting mirror that splits the pulse into two separate pulses. The arm length of the interferometer is chosen to be small so that the split pulses overlap and constructively interfere. This arm length can be adjusted to vary the time delay between the pulses, which will ultimately be used to estimate the duration of the original. After this, the pulses propagate collinearly(in parallel) into a second harmonic generation crystal. When pulses pass through these crystals they are non-linearly polarized in such a way that the outgoing pulse has half the wavelength of the ingoing pulse, and thus twice the frequency. This is a form of sum frequency generation, a non-linear optical process, in which the pulses polarity is changed in a non-linear fashion due to the electric field, which typically occurs at high intensities. With a much higher frequency the amplitude of the pulse will be much higher than the background noise, so its duration can be more accurately estimated. The pulses will have the same polarity when they exit the crystal. It is important to note that sum frequency generation conserves energy, so the angular frequency of the outgoing pulse is simply the sum of the angular frequencies of the combined photons, all multiplied by Max Plancks constant divided by 2 (ℏ).

h/23=h/2(2+1)

The autocorrelation signal, or the average power, which is recorded by a photodetector, can then be related to the time delay through the following equation:

IM()=-+(E(t)+E(t-))22dt

The duration of the pulse can be estimated from the plot of the autocorrelation signal by measuring its width. Characterizing ultrashort pulses has many uses, such as determining the wavelength of the output light of a mode-locked laser.

Tuesday, February 12th, 2013

Today Dr. Noé explained that a laser is in fact a resonant system. When making a laser one cannot just have any size cavity. In order to create a resonant system with the laser, the incident frequency of the light must match the cavity mode, so a standing wave can form inside the cavity. The ends of the cavity must be constructed of mirrors: one that is 95% reflective, so the laser light can eventually escape, and one at the other end that is 100% reflective. Because of this, what is called circulating power develops: when light makes its first go around in the cavity, it reflects off the 95% reflective mirror, but some of it is lost. Whats left reflects off the other mirror back to the 95% mirror, and some is lost again, but even less than before, and then the process continues until the incoming energy equals the outgoing energy(this is known as cavity buildup, which occurs in all resonant systems). Dr. Noe explained that because of this, there is always more power on the inside of the cavity than the output, but that power is unattainable because one would have to interrupt the system to harness it.

Dr. Noé also taught me something about the polarization of light. Unpolarized light, which is often encountered in daily life, is oscillating in various directions. Light can be polarized with a polarizing filter, which only lets light oscillating parallel to a certain plane in three-dimensional space pass through. This is known as linear polarization. There are other kinds, along with [such as?] circular polarization, which is particularly interesting. By rotating a filter in front of linearly polarized light, Dr. Noe showed me that it was possible to completely dim it. However, attempting to do the same with circularly polarized light proved unsuccessful. In order to polarize circularly polarized light, one must pass it through not just any ordinary filter, but a quarter-wave plate. The construction of one requires a birefringent material, which refract light differently depending on the orientation of the light passing through. Wave plates shift the phases between two orthogonal components of the light wave. The birefringent crystal is cut so that an ordinary and extraordinary axis form. Both axes have a different index of refraction, so light components that travel along each of them will have different speeds in the crystal, which results in a phase difference when they exit the medium. In quarter-wave plates, the birefringence and thickness of the crystal result in a phase shift of π/2, so that the exiting light is elliptically polarized. If the axis of polarization is made so that it makes a 45 degree angle with the ordinary and extraordinary axes, the light will be circularly polarized. It is important to note that the crystal must be cut into the plate so that the optical axis(the axis along which light experiences no birefringence) is parallel to the plates surface, in order to form the ordinary and extraordinary axes.