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OPTI 511R, Spring 2009 Problem Set 9: Part I Prof. R.J. Jones
Absorption and Saturation Due: in class, Thurs, April 30
The absorption coefficient α is related to many other concepts in optical physics, such as laser cooling, the optical dipole force, and optical lattices. In this assignment, many new terms and definitions are introduced. Throughout, the word intensity is used to represent a quantity with units of power per unit area, consistent with most optical physics textbooks.
- Atom-photon collision cross section. The absorption coefficient α can be used to derive many quantities important for describing a realistic interaction of a beam of light with an atomic gas. In this problem, we first consider the steady-state absorption coefficient for a two-level atom interacting with a monochromatic polarized beam of laser light, and we assume that the gas is not Doppler-broadened, that state ψg is the ground state, and state ψe is the excited state. The absorption coefficient is
α = (N (^) g^0 − N (^) e^0 )
2 ω℘^2 c≤ 0 ¯hΓ
1 + 4(∆/Γ)^2 + 2(Ω 0 /Γ)^2
Notice that the steady-state absorption depends on the population density difference that would ex- ist without the light turned on, usually a thermal distribution where N (^) e^0 /N (^) g^0 = e−¯hω^0 /kB^ T^ , so that N (^) g^0 >> N (^) e^0. Thus N (^) g^0 − N (^) e^0 ≈ N (^) g^0 , which is approximately the total population density N of the gas, with units of number of atoms per unit volume. With that approximation, α can be written as α = N σ. Since α has units of inverse length, the new variable σ must have units of area per atom. Specifically, σ is interpreted as the cross-sectional area for an atom-photon interaction. If we think loosely (and incorrectly) of a photon as a particle propagating through space, then the bigger σ is, the “bigger” the photon and atom look to each other. For a given photon flux and atomic density, a larger σ will mean that an absorption event is more likely to occur.
In the questions that follow, you will need to note the following important points: (i) For a strictly two-level system, ℘^2 is the same as |℘~|^2 , which is used if light is randomly polarized. (Remember, | ℘~|^2 is the quantity that appears in the Einstein A coefficient). Previously, we discussed a factor of 3 difference between these two quantities; however, as it turns out, this is only appropriate when different ml states are available in a transition (such as between the hydrogen 1S and 2P levels). (ii) The quantity Γ that appears in the expression for α is the full-width at half-max (FWHM) of the non-power-broadened Lorentzian absorption lineshape, which is the same as the Einstein A coefficient: Γ = A = ω^30 ℘^2 /(3π≤ 0 ¯hc^3 ). You will use this expression in the following problems. In the remainder of this problem set, we’ll use Γ as the natural linewidth of the transition, and τn = 1/Γ as the natural lifetime. (iii) The average intensity of a laser beam with electric field amplitude E 0 is given by I = c≤ 0 |E 0 |^2 /2, so I is proportional to |Ω 0 |^2.
(a) By manipulating
σ = α/N =
2 ω℘^2 c≤ 0 ¯hΓ
1 + 4(∆/Γ)^2 + 2(Ω 0 /Γ)^2
show that σ can be written in the following form:
σ = σ 0 ·
1 + I/Isat + 4(∆/Γ)^2
where σ 0 and Isat are collections of constants. You will need to make the approximation that ω = ω 0.
(b) Express σ 0 and Isat in terms of other variables and constants. To verify that your expressions are correct, show that σ 0 can be reduced to the form σ 0 = 3λ^20 /(2π), where λ 0 is the wavelength of the atomic resonance (λ 0 = 2πc/ω 0 ). Also show that Isat can be written in the form Isat = ¯hω 0 /(2τnσ 0 ), which can be immediately seen as an intensity: an energy per time (which is power) per area. The meaning of Isat will be discussed in the next problem. σ 0 is the atom-photon interaction cross-section for on-resonant light and low light levels.
- Saturation. In thermal equilibrium, with only blackbody radiation present, at any instant in time the vast majority of atoms in a gas will reside in the electronic ground state ψg. In steady-state equilibrium after near-resonant light has been turned on, each atom will then have a higher probability of being found in the excited state, due to interaction with the light. Consider the probability that a two-level atom is in its excited state in steady-state; this probability is denoted by pe. With all light off, pe ≈ 0. With light on, pe will depend on the light intensity and frequency (detuning): the higher the intensity and smaller the detuning (so that ω ∼ ω 0 ), the more likely it is that an atom will absorb energy from the beam of light. Note that N (^) ess = N pe, where N (^) ess is the steady-state atomic density of atoms that are in their excited state, so pe is also the approximate fraction of atoms in state ψe at any instant. Also in steady-state, the population densities N (^) ess and N (^) gss do not change with time (the definition of steady-state).
With high intensities of near-resonant light, an atom can spend at most half of the time in its excited state (remember that in a gas of two-level atoms, Ne can never exceed Ng in steady state). So at most, pe ≈ 1 /2, and the total time that an atom spends in an absorption-spontaneous emission process (a “scattering event”) is τscat ≈ 2 τn. This says that the atoms are spending as much time absorbing light as they are spontaneously emitting light, so we can estimate the scattering time. (In photon scattering, we’re interested in processes that remove a photon from a beam. Since absorption followed by stimulated emission puts the photon right back into the beam right after absorption, such a process is not considered here to be a photon scattering event). When pe ≈ 1 /2, further increases in the light intensity have very little effect - the atoms are already absorbing as much light as they can. We say that the transition is saturated.
Actually, saturation of the absorption process starts to apply at much lower light levels than would be needed to reach pe ≈ 1 /2, and saturation is better described as being when increasing intensity levels start to have markedly less efficiency in stimulating a transition. The strict definition of that saturated absorption regime is when pe ≥ 1 /4. Note that it is not just intensity that determines whether a transition has reached saturation; the detuning of the light from the transition frequency is also critically important.
It is useful to define a saturation parameter s:
s = 2
|Ω 0 |^2 /(Γ^2 )
1 + 4(∆/Γ)^2
I/Isat 1 + 4(∆/Γ)^2
where Isat is the the constant discussed in problem 1. In the above expression, it is easy to see that 0 < s < ∞, depending on I and ∆. This new parameter s can be used in writing an expression for pe:
pe =
s 1 + s
From the above expression (stated without derivation), it is easy to see that 0 < pe < 1 /2, and that saturation (pe ≥ 1 /4) is reached when s ≥ 1. It is now easy to see the meaning of the quantity Isat: it
- Scattering force. Consider a beam of light of frequency ω and wavevector ~k. The beam is incident on a gas of two-level atoms. As a ground-state atom absorbs a single photon from the beam, the atom gets a kick of momentum ~pp = ¯h~k = ¯h|k|ˆk. When the atom returns to its ground state by spontaneous emission, it also receives a momentum kick, this time in the direction opposite to the direction of the emitted photon, which is assumed to be in any random direction as is usually appropriate for spontaneous emission. If an atom scatters photons at a rate γscat, all of the absorption events produce a net force in the ˆk direction. The emission events, which occur in random directions, average to a net force of zero. Thus, the total scattering force of light on atoms is given by
F^ ~ = d~p/dt = ¯h~kγscat = ˆk · ¯h|k| Γ 2
I/Isat 1 + I/Isat + 4(∆/Γ)^2
Doppler cooling utilizes the scattering force to remove kinetic energy from a gas of atoms via mo- mentum removal from the atoms, thus cooling atoms. Consider an atom moving in the −kˆ direction, opposite to the propagation direction of a laser beam. The scattering force will push the atom in the +ˆk direction, and will thus slow the atom down. However, the Doppler effect must be considered when atomic motion is relevant. To effectively slow down and cool an atom, we should have a beam of light tuned below the atomic resonance: ω < ω 0. This is called red detuning (the light is closer to the low-frequency, or red, end of the spectrum). If the atom moves toward the beam with a velocity −|vk|, the atom will see the beam’s frequency as being Doppler shifted closer to resonance by an amount approximately given by |k 0 vk|. Thus, the atom will scatter light most effectively from this beam when it moves into the beam, and when the amount of detuning is close to the Doppler shift: ∆ ∼ |k 0 |vk.
If a second red-detuned beam, with wavevector −~k is also incident on this atom, the atom will see this new light as being even further away from resonance (since the atom is moving in the same direction as this second beam), and the atom will scatter light from this beam much less effectively than from the first beam of light propagating in the +ˆk direction. Thus, no matter what component of velocity the atom has along the kˆ direction in this two-beam field, the atom will slow down due to unbalanced scattering forces that always oppose the direction of atomic motion. This concept is called Doppler cooling (one type of laser cooling). When 3 pairs of red-detuned laser beams are configured such that there is always a scattering force in any direction (3 orthogonal pairs of beams), atoms with a velocity vector in any direction can be cooled. There is however a lower limit on the temper- atures that can be achieved by Doppler cooling: TD = ¯hΓ/(2kB ), where kB is Boltzmann’s constant. (This limit originates in the fact that when the atom has reached low enough temperatures and thus very small velocities and Doppler shifts, the scattering forces are roughly balanced, and the atom un- dergoes a random walk in momentum space and scatters light from all beams at nearly the same rate.)
(a) What is the Doppler-limit temperature TD for an atom with a linewidth of Γ ≈ 2 π × 6 MHz? This number is appropriate for the primary excited states of lithium, sodium, potassium, rubidium, and cesium.
(b) What is the natural lifetime of these atomic excited states?
(c) What is the one-dimensional velocity vD of a rubidium-87 atom if it is in a gas of tempera- ture T = TD? Use (1/2)mRbv^2 D = (1/2)kB TD, with mRb = 1. 5 × 10 −^25 kg.
(d) What is the velocity of a rubidium atom at room temperature (T = 300 K)?
(e) What is the deBroglie wavelength of a rubidium atom for each of the two velocities found in (c) and (d)?
(f) Calculate 2k 0 ¯u/Γ, the ratio of the Doppler-broadened linewidth to the natural linewidth, for a sample of rubidium atoms at temperature T = 300 K, and for a sample of atoms at T = TD. In which of these cases is the transition Doppler broadened? Use the mean speed ¯u =
√ 2 kB T /mRb. Also, use k 0 = 2π/λ 0 , with λ 0 = 780 nm, the main resonance wavelength for 87 Rb.
(g) Suppose an isolated sample of atoms is very cold, say T << TD. If a single beam of light at frequency ω interacts with the gas, it will heat each atom at a rate of ∼ ¯hω 0 γscat. If a single atom scatters X photons, it gains on average an amount X ·[p^2 p/(2m)] of kinetic energy, where pp = ¯hk is the momentum of a photon. Thus the kinetic energy gain per photon scattering event is the photon recoil energy, Erec = (¯hk)^2 /(2m), where k is the wave number of the applied light (not necessarily at atomic resonance). Calculate the photon recoil energy for a rubidium atom scattering resonant light. Also, express the answer in temperature units: Trec = 2Erec/kB (which comes from (1/2)kB Trec = Erec). Each scattered photon thus raises an atom’s temperature an average of approximately Trec.
(h) About how many photons will each atom of a sample need to scatter in order to take a gas of atoms at T ∼ Trec up to the Doppler temperature TD (i.e., what is the ratio TD/Trec)?
- The optical dipole force. The scattering force considered so far is dissipative: it removes energy from an atomic sample via conversion of kinetic energy (of the atom) to photon energy (of the spontaneously emitted photons). In that sense, Doppler cooling is similar to friction, which opposes the motion of an object and whereby kinetic energy is converted to heat.
Light can also exert a conservative force on an atom: you can think of this force as resulting from matched absorption and stimulated emission events. Each such event leaves the atom in the same state (and velocity) it had before the absorption, but the scattering force (or the spontaneous emission rate) must be negligible for this conservative force to dominate. In this case, during interaction with the light, the state of the atom is perturbed, and the energy eigenstates that are used to describe an atom in the absence of the light must be modified in order to properly describe the atomic state when the light is on. It turns out that in the presence of light, the energies of the atomic states ψg and ψe are different than if the light were not present. A good way to picture the force is by first realizing that the light will induce an electric dipole moment in the atom. Since the light field oscillates at an optical frequency ω, the induced dipole moment will oscillate at that same driving frequency, but with a frequency-dependent phase shift. Depending on what that phase shift is, the atom will see a time-averaged lower energy or higher energy when in the presence of the light than if the light were not present. In the limit ∆ >> Γ, if the light is red-detuned, (i.e., if ∆ = ω − ω 0 < 0), the atomic dipole is attracted to the highest intensity regions of the optical field. If ∆ > 0 (blue-detuning), the atom is repelled from the highest intensity parts of the field. In a potential well, an atom will try to minimize its internal energy; this is the origin of the optical dipole force, which depends on the dipole interaction energy, and which either pushes atoms towards or away from (depending on detuning) regions of high intensity. Because a force is the gradient of a potential, a light intensity gradient can act as a potential well for atoms. This may all sound confusing, but it comes down to this: the optical dipole potential for a beam of light with intensity I and detuning ∆ is given by
Vdip(~r) ∼= ¯hΓ 8
I(~r)/Isat ∆/Γ
where it is explicitly indicated that the field intensity has a spatial dependence (such as being due to the spatially varying profile of a focused laser beam).
Problem Set 9: Part II
Lasers and Optical Amplification
- The overall gain G(ω) (as opposed to gain coefficient γ(ω)) of a laser amplifier is defined as the ratio of the intensity at the output of the amplifier (Iout) to the intensity at the input (I 0 ), so that
Iout = G(ω) · I 0.
We have seen that if saturation effects can be neglected, Iout = eγ^0 (ω)L^ · I 0 for an amplifier of length L with a small-signal gain coefficient of γ 0 (ω). Thus eγ^0 (ω)L^ is the small-signal gain, which we can label as G 0 (ω). If saturation can not be neglected, G(ω) 6 = G 0 (ω).
(a) A commercially available ruby laser amplifier using a 15-cm-long ruby rod has an on-resonance small-signal gain G 0 = 12. What is G 0 for a 20-cm-long rod?
(b) A 15-cm-long rod of Nd3+:glass used as a laser amplifier has a small-signal gain of G 0 = 10 at λ 0 = 1. 06 μm. Use the data in Table 13.2-1 at the top of page L-42 of the class notes (taken from page 480 of Saleh and Teich, Fundamentals of Photonics) to determine the population difference (N 2 − N 1 ) required to achieve this gain. (This will be some number of Nd3+^ ions per cm^3 .)
- (a) Write the rate equations for a 3-level atomic system, assuming the usual conditions and notation: (i) the lasing transition is between levels 1 and 2, where level 1 is the ground level, (ii) pumping is from level 1 to 3, the highest-energy level, and (iii) instantaneous decay occurs from level 3 to 2.
(b) Solve the rate equations for steady state population densities N 1 and N 2. However, do not assume that field intensities are small. Keep the absorption and stimulated emission rate R(I) in your equations. (In class, we had neglected these terms when finding the steady state population densities in order to find answers valid in the low-intensity limit.)
(c) Examine your answers, and notice that if R ∼ 0, N 1 and N 2 match the expressions stated in class. What happens to the ratio N 2 /N 1 as R(I) becomes large? You should find this a mathemati- cally simple method to see that saturation occurs in the gain medium.
- (a) Suppose that the small-signal gain coefficient γ 0 (ω) of a Doppler-broadened gas laser has a peak γ 0 (ω 0 ) (at at the atomic resonance) that is twice the threshold gain coefficient γT. If γ 0 (ω) has a FWHM of 3 GHz, about how many TEM 0 , 0 axial modes could simultaneously lase if the cavity has a length of 1 m?
(b) What would the cavity length have to be (approximately) in order to make this laser operate on only a single axial mode? Does this seem like a realistic length for a gas laser?
(c) A CO 2 laser has an unusually narrow Doppler-broadened transition linewidth of ∆ω ∼ 2 π × ( MHz) at FWHM. Approximately what cavity length would allow a CO 2 laser to operate in a single longitudinal mode?
- Recall that in a homogeneously broadened CW laser operating under ideal conditions, lasing will occur for a single axial mode. However, also recall that for each axial mode number q, there are two independent axial modes having orthogonal electric field polarizations. For laser output that consists of a single frequency, as in a homogeneously broadened laser, the output will have only one mode with a well-defined polarization: the one with the higher effective gain coefficient (g = γ − β). If the active medium within a laser resonator is terminated by Brewster windows as shown in the following diagram (the plane of the windows is normal to a vector in the x − z plane), will the laser output propagating along z be linearly polarized in the x direction, y direction, or some other direction? Why?
z
x Active medium y (^) Laser output
- Consider the 4-level system discussed in class, with the following usual specifications: (i) the pump rate from level 0 to level 3 occurs at a rate P 03. (ii) the decay rate from level 3 to level 2 is instantaneous. (iii) the decay from level 2 to level 1 is by spontaneous emission, at the rate Γ 21. (iv) the decay from level 1 to level 0 occurs at the rate Γ 10.
Now consider that there is an additional pumping process that excites atoms from level 1 and pro- motes them to level 3 at the rate P 13. Also assume that the decay from level 1 to level 0 is by optical spontaneous emission, rather than some other process. Finally, assume that there are no allowed transitions between levels 0 and 2.
(a) Sketch the energy level diagram for this system, and indicate the possible transitions, labeling each with an arrow between levels and a rate. Include the stimulated processes of absorption and stimulated emission between levels (where applicable), and label the rates for these processes appro- priately.
(b) Write down the rate equations for this system.
(c) Now assume that we are interested in steady-state population densities in the low-intensity limit, and thus absorption and stimulated emission rates can be neglected from the problem. Rather than solving the rate equations for steady-state population densities N 0 , N 1 , and N 2 , solve them for the steady-state ratios N 1 /N 0 and N 2 /N 1.
(d) Under what conditions can a steady-state population inversion be established simultaneously be- tween levels 1 and 0, and between levels 2 and 1?
(e) Are the population inversions achieved under the conditions you specified in part (d) most like a 3-level system, a 4-level system, or both at the same time? Would it seem like a system of this type could lase at two different wavelengths at the same time? (Note that we would really need to include stimulated radiative transitions between levels 0 and 1, and levels 1 and 2, in order to determine the steady-state population densities for a system operating near saturation.)
(c) Determine Ioutopt, the output light intensity when an output coupler of T = Topt is used.
(d) For lasing to begin, γ 0 must exceed the threshold gain coefficient γt, which is equal to β and to γss. Write an expression for γtopt , the value of the threshold gain coefficient for the case T = Topt. Write your answer as a function of L, γ 0 , and S.
(e) Given your answer to part (d), what is the minimum value of γ 0 in terms of S and L only that must be achieved for lasing to occur?
(f) Suppose that for high enough pump rates or small enough mirror absorption and scattering losses, it is possible to achieve the limit that γ 0 >> S/(2L). From your answer to part (c), write an expression for Ioutopt in this limit. Your answer should depend only on Isat, L, and γ 0.
(g) From the table at the top of page L-42 of the class notes, you can see that an Ar+^ laser has a gain medium (a gas of argon ions) characterized by atom/photon cross section σ 0 = 3 × 10 −^12 cm^2 , and a natural lifetime of τn = 10 ns. Assuming a population density inversion of δN = Ne − Ng = 10^12 cm−^3 and a gain medium of length L = 100 cm, determine an optimum value for the output laser light intensity if scattering losses at the mirror can be neglected.
(h) For a gaussian beam, the maximum intensity I 0 of the beam is related to the total beam power P and beam radius w by the relation I 0 = 2P/(πw^2 ). Given the answer to (g), what is the laser beam’s power if the beam has a radius w = 0.2 mm and the maximum beam intensity I 0 is equal to the value calculated in (g)?
- Suppose that a gaussian laser beam is propagating along the z direction. At some position z 0 , it has an electric field amplitude with a profile proportional to exp (−r^2 /w^2 ), where r is the dis- tance away from the z axis in a direction normal to the z axis. The intensity profile of the beam can be written as I(r) = I 0 exp (− 2 r^2 /w^2 ). (Both I 0 and w are functions of z, but that won’t matter for this problem.) Derive the gaussian power-intensity relationship I 0 = 2P/(πw^2 ) used in the previous problem by integrating I(r) = I 0 exp (− 2 r^2 /w^2 ) over all values of r. You can do this with a cylindrical coordinate system, or transform to Cartesian coordinates using r^2 = x^2 + y^2.
