Download Report

OPTI 511R, Spring 2015
Problem Set 10
Prof. E. M. Wright
Due Thursday April 23, 2015
1. 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,
ˆ When the atom returns to its ground state
the atom gets a kick of momentum p~p = h
¯~k = h
¯ |k|k.
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 kˆ 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
o
A21
I/Isat
F~ = d~
p/dt = h
¯~kγscat = kˆ · ¯h|k|
·
o + 4(∆/Γ)2 .
2 1 + I/Isat
Doppler cooling utilizes the scattering force to remove kinetic energy from a gas of atoms via momentum 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 +kˆ 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 < ω0 . 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 |k0 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: ∆ ∼ |k0 |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 +kˆ 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 temperatures that can be achieved by Doppler cooling: TD = h
¯ A21 /(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 undergoes 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 A21 ≈ 2π × 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 tempera2 = (1/2)k T , with m
−25 kg.
ture T = TD ? Use (1/2)mRb vD
B D
Rb = 1.5 × 10
1
(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 2k0 u
¯/A21 , 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 atpT = TD . In
which of these cases is the transition Doppler broadened? Use the mean speed u
¯ = 2kB T /mRb .
Also, use k0 = 2π/λ0 , with λ0 = 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 ∼ ¯hω0 γscat . If a single atom
scatters X photons, it gains on average an amount X ·[p2p /(2m)] of kinetic energy, where pp = h
¯ k 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 )?
Lasers and Optical Amplification
2. The overall gain G(ω) (as opposed to gain coefficient g) 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 (I0 ), so that
Iout = G(ω) · I0 .
We have seen that if saturation effects can be neglected, Iout = eg0 (ω)L · I0 for an amplifier of length
L with a small-signal gain coefficient of g0 (ω). Thus eg0 (ω)L is the small-signal gain, which we can
label as G0 (ω). If saturation can not be neglected, G(ω) 6= G0 (ω).
(a) A commercially available ruby laser amplifier using a 15-cm-long ruby rod has an on-resonance
small-signal gain G0 = 12. What is G0 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 G0 = 10
at λ0 = 1.06 µm. Using the maximum on-resonance value σ(ωo ) ≈ 3 × 10−20 cm2 for this laser amplifier, determine the population difference (N2 − N1 ) required to achieve this gain. (This will be some
number of Nd3+ ions per cm3 .)
2
3.
Recall that in a homogeneously broadened CW laser operating under ideal conditions, lasing
will occur for a single axial mode (also referred to as longitudinal modes). 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 (gef f = g − 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?
x
y
Active medium
z
Laser output
4. 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 P03 .
(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 Γ21 .
(iv) the decay from level 1 to level 0 occurs at the rate Γ10 .
Now consider that there is an additional pumping process that excites atoms from level 1 and promotes them to level 3 at the rate P13 . 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 appropriately.
(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 N0 , N1 , and N2 , solve them for the
steady-state ratios N1 /N0 and N2 /N1 .
(d) Under what conditions can a steady-state population inversion be established simultaneously between 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.)
3
5. In class, we determined the maximum light intensity that can be obtained within a long amplifying medium, given some loss coefficient β. In this problem, you will work out the maximum
light intensity that can be extracted from a laser resonator (as was also discussed in class), and what
the output coupler’s transmittance should be in order to obtain this intensity. For all expressions in
this problem, it is assumed that the laser light is at a frequency equal to the atomic resonance frequency, so all explicit references to frequency and detuning are neglected (∆ = 0). It is also assumed
that there is no spatial dependence to the intensity. Of course, real beams have a transverse profile,
plus convergence and divergence, so this assumption is just an approximation to make the math easier.
Background. If we neglect collisional broadening in a 3-level laser system and assume level 2 of a
−1
gain medium only decays radiatively to level 1, then ∆ω = A21 = Γ21 , and τ2 = A−1
21 = Γ21 . For the
case of a homogeneously broadened gain medium, the gain coefficient can then be written (in terms
of the angular frequency ω) as:
g(ω) =
g0 (ω)
g0 (ω0 )
,
=
o
1 + I/Isat
1 + I/Isat + 4(∆/Γ21 )2
where here ∆ = ω − ω0 , the small signal gain coefficient is given by g0 (ω) = ∆N 0 × σ(ω), and the
saturation intensity for a 3-level system can be written Isat = ¯hω0 /[2τ2 σ(ω)] (recall the factor of 2 is
not there for a 4-level laser). In this problem, we will also just consider the on-resonance case (i.e.
o =h
∆ = 0), such that Isat = Isat
¯ ω0 Γ21 /(2σ0 ).
In steady-state, the total intracavity light intensity in a single mode of a laser resonator is constant
(by definition), with some value I. The steady-state on-resonance intensity-dependent gain coefficient
is given by the expression
g0
gss =
o .
1 + I/Isat
Solving this expression for the intracavity intensity that saturates the gain down to this steady-state
value gives
o
Iss = Isat
(g0 /gss − 1).
If the high reflector has a reflectance of 1 (and hence a transmittance of 0), and the output coupler
has a transmittance of T , the output intensity of the exiting laser beam is
o
Iout = (1/2)T · I = (1/2)T · Isat
· (g0 /gss − 1).
The factor of 1/2 comes into this equation since we can think of half of the intracavity intensity I
coming from light moving towards the output coupler, and the other half of I coming from light moving away from the output coupler. (We’ll assume that these are two equal quantities, which is fair as
long as T is very small.) However, only the half that corresponds to light moving towards the output
coupler has a chance of leaving the cavity, hence the factor of 1/2 above.
A real output coupler can be characterized by its reflectance R and transmittance T , and also by
its absorption or scattering losses, which we’ll characterize by S such that
1 = R + T + S.
The loss coefficient for the laser (as opposed to the amplifier) is the same thing as the required threshold
1
gain coefficient discussed in class, which can be written as gt = β = 2L
(1 − R) (given that the high
reflector has reflectance of 1, and in the limit that R is close to 1). We can now write this as:
β=
1
(T + S).
2L
4
o .
(a) Given that gss = β, solve for Iout in terms of g0 , T, S, L, and Isat
(b) Determine Topt , the optimum value of T that will maximize Iout , assuming that all other parameters are held fixed. Your final answer should be a function of L, g0 , and S.
opt
(c) Determine Iout
, the output light intensity when an output coupler of T = Topt is used.
(d) For lasing to begin, g0 must exceed the threshold gain coefficient gt (which is equal to β and
to gss ). Write an expression for gtopt , the value of the threshold gain coefficient for the case T = Topt .
Write your answer as a function of L, g0 , and S.
(e) Given your answer to part (d), what is the minimum value of g0 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 g0 >> S/(2L). From your answer to part (c), write an expression
opt
o , L, and g .
for Iout
in this limit. Your answer should depend only on Isat
0
(g) An Ar+ laser has a gain medium (a gas of argon ions) characterized by atom/photon cross section
σ = 3 × 10−12 cm2 , and a natural lifetime of τn = 10 ns. Assuming a population density inversion of
∆N o = Ne − Ng = 1012 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 I0 of the beam is related to the total beam power P
and beam radius w by the relation I0 = 2P/(πw2 ). 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 I0 is equal to the value
calculated in (g)?
6.
Suppose that a gaussian laser beam is propagating along the z direction. At some position
z0 , it has an electric field amplitude with a profile proportional to exp (−r2 /w2 ), where r is the distance 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) = I0 exp (−2r2 /w2 ). (Both I0 and w are functions of z, but that won’t matter for
this problem.) Derive the gaussian power-intensity relationship I0 = 2P/(πw2 ) used in the previous
problem by integrating I(r) = I0 exp (−2r2 /w2 ) over all values of r. You can do this with a cylindrical
coordinate system, or transform to Cartesian coordinates using r2 = x2 + y 2 .
5