MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.097 (UG) Fundamentals of Photonics
6.974 (G) Quantum Electronics
Spring 2006
Final Exam
Time: May 23, 2006, 1:30-4:30pm
Problems marked with (Grad) are for graduate students only.
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This is a closed book exam, but three 81/2”x11” sheets (both sides) are allowed.
At the end of the booklet there is a collection of equations you might find helpful for the exam.
Everything on the notes must be in your original handwriting (i.e. material cannot be Xeroxed).
You have 3 hours for this exam.
There are 8 problems for undergraduate and 9 problems for graduate students on the exam with
the number of points for each part and the total points for each problem as indicated. Note, that
the problems do not all have the same total number of points.
Some of the problems have parts for graduate students only. Undergraduate students solving
these problems can make these additional points and compensate eventually for points lost on
other problems.
Make sure that you have seen all 29 numbered sides of this answer booklet.
The problems are not in order of difficulty. We recommend that you read through all the
problems, then do the problems in whatever order suits you best.
We tried to provide ample space for you to write in. However, the space provided is not an
indication of the length of the explanation required. Short, to the point, explanations are
preferred to long ones that show no understanding.
Please be neat-we cannot grade what we cannot decipher.
All work and answers must be in the space provided on the exam booklet. You are welcome to use
scratch pages that we provide but when you hand in the exam we will not accept any pages other
than the exam booklet.
Exam Grading
In grading of the exams we will be focusing on your level of understanding of the material
associated with each problem. When we grade each part of a problem we will do our best to assess, from
your work, your level of understanding. On each part of an exam question we will also indicate the
percentage of the total exam grade represented by that part, and your numerical score on the exam will
then be calculated accordingly.
Our assessment of your level of understanding will be based upon what is given in your solution. A
correct answer with no explanation will not receive full credit, and may not receive much-if any. An
incorrect final answer having a solution and explanation that shows excellent understanding quite likely
will receive full (or close to full) credit.
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No work on this page will be evaluated.
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.097 (UG) Fundamentals of Photonics
6.974 (G) Quantum Electronics
Spring 2006
Final Exam
Time: May 23, 2006, 1:30-4:30pm
Full Name: ______________________________________________
Are you taking 6.974 ____ or 6.097 ____ ?
Your Points
1
2
3
4
5
6
7
8
9
Total
Fabry-Perot Filter
ABCD Matrices
Gaussian Beams/Resonators
Polarization of Light
Harmonic Oscillator
Operators/Commutators
Two-Level Atom
Laser
Three-Level System
Max points
(undergrad.)
20
12
5
18
20
10
15
25
125
Max points
(grad.)
20
12
5
18
20
10
15
25
25
150
3
Problem 1: Fabry-Perot Filter
(20 points)
In most fiber optics transmission systems a single fiber is used for transmission of multiple channels,
each of which has a different carrier frequency. Suppose that you have a system with 40 channels; each
channel has 40 GHz bandwidth and these channels are spaced by 100 GHz. A part of this spectrum is
illustrated below.
At the output of the fiber, each of these channels must be separated from the others before information
carried by this channel can be used. Suppose we want to use a Fabry-Perot resonator as a filter to select
the single channel at 194 THz. The two mirrors of the Fabry-Perot are identical and are separated by
material with a refractive index of n = 1.5 .
(a) (5 points) What is the requirement for the free spectral range of the Fabry-Perot filter to be used for
selecting 1 of these 40 channels?
4
(b) (15 point) Choose the length L of the Fabry-Perot cavity and the reflectivity of the mirrors, R , that
can be used for extracting the channel at 194 THz. We have the constraint that R cannot
exceed 99%. If you think that the answer to this problem is not unique, give a combination of L and
R that will work.
5
(b) continued
6
Problem 2: ABCD Matrices
(12 points)
In each figure below, we have an input plane and an output plane separated by some unknown optical
elements. The arrows in these figures denote optical rays. What can you say about the matrix elements
of each optical system’s ABCD matrix?
(a) (3 points)
Input plane
Output plane
Input plane
Output plane
(b) (3 points)
7
(c) (3 points)
Input plane
Output plane
Input plane
Output plane
(d) (3 points)
8
Problem 3: Gaussian Beams and Resonators
(5 points)
Suppose you have a resonator with identical mirrors as shown below. If we increase the radius of
curvature of both mirrors but don’t modify the resonator length, will the waist radius of the resonator
mode increase or decrease? Why?
9
Problem 4: Polarization of Light
(8 points)
A plane electromagnetic wave propagates in free space along the positive z-axis. The electric field
vector of the wave is given as
r
r
r
E ( z, t ) = E0 x cos(ω t − k z )ex + E0 y cos(ω t − k z + ϕ )e y (1)
with
ϕ = π / 2 , E0 y = 3E0 x
(a) (3 point) What is the polarization of the wave i.e. is the light linearly polarized, circularly polarized,
or elliptically polarized?
(b) (15 points) We would like to transform this wave into a circularly polarized wave using one or
several half-wave and/or quarter-wave plates. Give a sequence of half-wave and quarter-wave plates
and their rotation angles with respect to the x-axis such that the output is circularly polarized. If the
answer is not unique, please give at least one combination that works.
10
(b) continued
11
Problem 5: Harmonic Oscillator
(20 points)
In class, we determined the wave functions, ψ n ( x ) , of a one dimensional harmonic oscillator with
V ( x) =
1
mω 2 x 2 . Below are the solutions for the first 4 modes: (note n indicates the state number)
2
Now consider a new potential defined as:
⎧ ∞ for x < 0 ⎫
⎪
⎪
V ( x) = ⎨ 1
⎬
2 2
⎪⎩ 2 mω x for x > 0 ⎪⎭
This is a model potential for a spring which can only be extended but not compressed.
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(a) (5 points) Sketch the eigenfunctions of the ground and first excited state of this new potential. These
are the first two states of this new oscillator. We will call these normalized eigenfunctions φ1 ( x) and
φ2 ( x) .
Hints: It is not necessary to solve the Schrödinger Equation. Consider the value of φn at x = 0.
(b) (5 points) Suppose at t = 0 the state of this system is
φ ( x, t = 0) =
1
(φ1 ( x) + φ2 ( x) ) .
2
What is the time dependant wave function, φ ( x, t ) , for t > 0.
13
(c) (5 points) If we measure the energy at a time t > 0, what is the expected value of the energy?
Express your answer in terms of hω .
(d) (5 points) What is the uncertainty associated with measuring energy?
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Problem 6: Operators and Commutators
(10 points)
We are given an operator b such that
b = μ a + υ a†
where a is the creation operator, a † is the annihilation operator, and μ and υ are real constants related by
μ 2 −υ 2 = 1 .
(a) (5 points) Show that ⎡⎣b, b† ⎤⎦ = ⎡⎣ a, a † ⎤⎦ .
(b) (5 points) If X =
1 †
j †
b + b ) and P =
(
( b − b ) , are X and P Hermitian? Justify your answer.
2
2
15
Problem 7: Two-Level Atom
(15 points)
(a) (6 points) You are given a 2-level atom in a resonant interaction with a monochromatic, classical,
electromagnetic wave. The levels of this system are separated by an energy (Ee-Eg) and initially the
atom is in the excited state. Sketch the evolution of the occupation probabilities of the ground and
excited states. Label your sketches appropriately and physically interpret this time evolution.
(b) (9 points) We found for the average dipole moment and inversion of an ensemble of two level atoms,
the following equations of motion:
•
⎛
1⎞
d = ⎜ jωeg − ⎟ d
T2 ⎠
⎝
(i)
•
and w = −
w − w0
.
T1
What do the time constants T1 and T2 represent? In real systems, which value is most often
smaller and why?
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(ii)
What does the term w0 represent? At thermal equilibrium, is w0 positive or negative? Justify
your answer.
17
Problem 8: Laser
(25 points)
We will consider the ring laser shown in the figure below. It has the following parameters:
stimulated emission cross section: σ = 3 ⋅10−19 cm 2 , upper state lifetime τ L = 3μ s , effective mode cross
sectional area Aeff = 10−4 cm 2 , cavity length L = 1m , output coupling mirror transmission 1%, V is the
volume of the laser mode V = Aeff · L, and τp the photon lifetime.
The reduced rate equations for the population in the upper laser level, N 2 , in the four level laser
material and the photon number, N L , in the laser mode are
dN 2
1
σc
= − N2 −
N 2 ⋅ N L + RP
τL
dt
V
σc
1
dN L
= − NL +
N 2 ⋅ ( N L + 1)
τp
dt
V
(1)
Where RP is the pump rate of the laser and c = 3 ⋅108 m / s is the speed of light which shall be for
simplicity equal to the group velocity in the laser material.
(a) (5 points) Give a physical interpretation of each of the terms in equation (1).
−
−
1
τL
σc
V
N2 :
N2 ⋅ N L :
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RP :
−
+
1
τp
σc
V
NL :
N2 :
(b) (5 points) Determine the minimum pump rate, RP , necessary to reach the lasing threshold. How
much pump power does this pump rate correspond to, if optical pumping with photons with 532nm
wavelength would be used?
19
(c) (5 points) Let’s assume now, that the laser has in addition to the losses due to output coupling also
internal losses of 0.5%. Derive an expression for the output power for the laser as a function of the laser
saturation power, the small signal gain per roundtrip, the internal losses and the output coupling mirror
transmission, T.
(d) (10 points) For a given small-signal gain per round-trip, what is the optimum output coupling mirror
transmission T to achieve maximum output power.
20
(d) continued
21
Problem 9: Three-Level System
(Graduate problem, 25 points)
Consider an atom with three energy levels as shown in the figure.
E − E1
The pump at frequency f p = 3
causes stimulated tranh
sitions between the 1st and the 3rd levels; the cross-section of this
process is σ 31 . The 3rd level exhibits a spontaneous decay rate to
the 2nd level of γ 32 . The second level is long-lived with a spontaneous decay rate of γ 21 < γ 32 . With sufficient pumping, population
inversion can be achieved between levels 2 and 1 so that the light
E − E1
at frequency f L = 2
will be amplified. The cross-section of
h
stimulated transitions between levels 2 and 1 is σ 21 .
(a) (8 points) In some systems of practical interest the relaxation from the 3rd to the 2nd level is not fast
enough to ignore the population of the 3rd level. Write the rate equations for the number of atoms in
levels 1, 2, and 3, N 1 , N 2 , and N 3 . The pump intensity is I p . Assume that the intensity at
frequency f L is very small so that it does not change the population of the energy levels. The total
number of atoms is N .
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(b) (8 points) Find the steady-state population of the second level N 2 .
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(c) (2 points) What is the population of the 2nd level for very large pump intensity?
For the remaining two questions, consider the case of very fast relaxation from the 3rd level, γ 32 >> γ 21 .
(d) (2 points) What is the population of the 2nd level for very large pump intensity? Explain the physical
difference between this answer and that of part (c).
24
(e) (5 points) What is the lowest pump intensity needed to achieve amplification?
25
Equation Sheets
r
r
∂B
∇×E = −
∂t
r
∇⋅D = ρ
r
r ∂D r
∇×H =
+J
∂t
r
∇⋅B = 0
r
r r
r
D = ε0E + P = ε E
r
r
r
r
B = μ0 H + μ0 M = μ H
r
r
P = ε0χeE
r
r
M = χ mH
Index of Refraction
n2 = 1 + χ ,
for χ << 1: n ≈ 1 + χ / 2
Poynting Vector
r r r
S = E×H
Energy density
we =
r 1 r r*
T = E×H
2
1 r
wm = μ H 2
2
Snell’s Law
n1 sin θ1 = n2 sin θ 2
Brewster’s Angle
tan θ B =
Reflectivity
r TE =
Maxwell’s Equations
Material Equations
1 r2
εE
2
TE
TE
Power Refl. Coef.
μ1/ 2
1
ε 1/ 2 cos θ1/ 2
t TE =
2Z 2
TE
TE
Z1 + Z 2
t TM =
t TE =
2n1 cos(θ1 )
n1 cos(θ1 ) + n2 cos(θ 2 )
t TM
r TM
Power Transm. Coef.
T
=t
TM
2
TE 2
TM
μ1 / 2
cos θ1 / 2
ε 1/ 2
n2
n
− 1
cos θ 2 cos θ1
=
n2
n
+ 1
cos θ 2 cos θ1
n1 cos θ1 − n2 cos θ 2
n1 cos θ1 + n2 cos θ 2
R TE = r TE
Z1 − Z 2
TM
TM
Z1 + Z 2
Z1TM
/2 =
r TE =
TE
w = we + wm
TM
r TM =
TE
Transmitivity
μ = μ0 (1 + χ m )
n2
n1
Z 2 − Z1
TE
TE
Z1 + Z 2
Z1TE/ 2 =
ε = ε 0 (1 + χ e )
2 Z1
TM
TM
Z1 + Z 2
n2
2
cos(θ 2 )
=
n2
n1
+
cos(θ 2 ) cos(θ1 )
R TM = r TM
TE
TE
Z1TE
4 Z1 Z 2
=
TE
TE
TE
Z2
Z1 + Z 2
2
T
TM
=t
TM 2
2
TM
TM
Z2TM
4Z1 Z2
=
TM
TM
TM 2
Z1
Z1 + Z2
26
Pulse Dispersion
k ′′ ∂ 2 A( z, t ′)
∂A( z, t ′)
= j
2
∂t ′2
∂z
Gaussian Pulse
⎛ k ′′L ⎞
τ ( L) = τ 1 + ⎜ 2 ⎟
⎝τ ⎠
Fabry Perot
S 21 =
(1 − R ) 2
(1 − R ) 2 + 4 R sin 2 (φ / 2)
where φ = 2kL , k =
fm = m
c0
2nL
FSR =
2
F=
E (r , z ) =
F=
Resonator Stability
Harmonic Oscillator
⎞
⎟⎟
⎠
2
⎞
⎟⎟
⎠
1− R
≈
π
1− R
⎡ ⎛ zR ⎞2 ⎤
R ( z ) = z ⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ z ⎠ ⎥⎦
I (r , z ) = I 0
k w02 π w02
=
2
λ
θ=
⎡ 2r2 ⎤
w02
exp
⎢− 2 ⎥
w( z ) 2
⎣ w ( z) ⎦
λ
π w0
1
1
λ
=
−j
q( z ) R( z )
π w2 ( z)
q ( z ) = z + jz R
⎡
r2 ⎤
1
exp ⎢− jk 0
⎥
q( z)
2q ( z ) ⎦
⎣
0 ≤ g1 g 2 ≤ 1 g1 = 1 −
Schrodinger’s Equation j h
π R
⎡
⎤
r2
r2
exp ⎢− 2
− jk 0
+ jζ ( z ) ⎥
2 R( z )
π w( z )
⎣ w ( z)
⎦
⎛ z
⎝ zR
E (r , z ) =
c0
2nL
2P
ζ ( z ) = arctan⎜⎜
q-parameter
2π f
n
c0
jt ⎞
2
2
⎟ , with r + t = 1
r⎠
⎛ z
w( z ) = w0 1 + ⎜⎜
⎝ zR
zR =
τ FWHM = 2 ln 2 τ
FSR
Δf FWHM
⎛r
Beam Splitter S-matrix S = ⎜
⎝ jt
Gaussian Beams
2
L
R1
g2 = 1 −
L
R2
d Ψ (r , t )
h2
=−
ΔΨ (r , t ) + V (r )Ψ (r , t )
dt
2m
1⎞
⎛
E = hω ⎜ n + ⎟
2⎠
⎝
27
h
2
Heisenberg’s Uncertainty Principle
ΔxΔp ≥
de Broglie’s Formula
p = hk
Einstein’s Energy/Frequency Relation
E = hω
Wien’s Law
w( f ) =
Wien Displacement Law
λmax =
Rayleigh-Jeans Law
w( f ) =
Planck’s Law
w( f ) =
Einstein’s A and B Coefficients
A21 =
Waveguide Coupling
2
⎛ 2
⎞
⎛ Δβ ⎞
⎜
P1 ( z ) = P1 (0) cos γ z + ⎜⎜
⎟⎟ sin 2 γ z ⎟
⎜
⎟
⎝ γ ⎠
⎝
⎠
P2 ( z ) = P1 (0)
| κ 21 |2
γ
2
sin 2 (γ z )
if Γ = π :
Rabi Frequency
hc
4.965kT
8π 2
f kT
c3
8π f 2
hf
3
c exp hf − 1
kT
8π f 213
B12
c3
γ = Δβ 2 + | κ 12 | 2
B21 = B12
Δβ =
β1 − β 2
2
β0 =
β1 + β 2
2
⎛ − jΓ / 2
⎞
⎛Γ⎞
⎜e
− j sin ⎜ ⎟ sin(2ψ ) ⎟
cos2 ψ + e j Γ / 2 sin2 ψ
⎝ 2⎠
⎟
W =⎜
⎜
⎟
⎛Γ⎞
e − j Γ / 2 sin2 ψ + e j Γ / 2 cos2 ψ ⎟
⎜ − j sin ⎜ ⎟ sin(2ψ )
⎝2⎠
⎝
⎠
Polarization, Retardation Plate
if Γ =
8π hf 3 − hf kT
e
c3
π
2
:
⎛ cos(2ψ ) sin(2ψ ) ⎞
⎟⎟
W = − j ⎜⎜
⎝ sin(2ψ ) − cos(2ψ ) ⎠
⎛ 1
[1 − j cos( 2ψ )]
⎜
2
⎜
W =
⎜ − j 1 sin( 2ψ )
⎜
2
⎝
uur uu*r
M e p
Ωr =
E0
2h
1
⎞
sin( 2ψ ) ⎟
2
⎟
1
[1 + j cos( 2ψ )] ⎟⎟
2
⎠
− j
28
Annihilation and Creation Operators
Bloch Equations
⎡⎣ a, a † ⎤⎦ = 1
•
⎛1
⎞
d = − ⎜ − jωeg ⎟ d + jΩ*r e jωtω
⎝ T2
⎠
•
ω=−
Trigonometric Identities
ω − ω0
T1
+ 2 jΩ r e− jωt d − 2 jΩ*r e jωt d *
sin 2α = 2 sin α cosα
cos 2α = cos 2 α − sin 2 α
1
1
sin α sin β = cos(α − β ) − cos(α + β )
2
2
1
1
cosα cos β = cos(α − β ) + cos(α + β )
2
2
1
1
sin α cos β = sin(α + β ) + sin(α − β )
2
2
1
1
cosα sin β = sin(α + β ) − sin(α − β )
2
2
Helpful Integrals
∫
∞
2
e −a x dx =
−∞
∫
∞
−∞
2
π
∫
−∞
a
x 2 e −a x dx =
∞
2
xe −a x dx = 0
1 π
2 a3
∫ α sin α dα = sin α − α cos α + C
∫ α cos α dα = cos α − α sin α + C
Constants
εo = 8.85 × 10-12 C2/Nm2
μο = 4π × 10-7 N/A2
m = 9.11 × 10-31 kg
e = 1.60 × 10-19 C
k = 1.380650 × 10-23 J/K
c0 = 2.997925 × 108 m/s
permittivity of free space
permeability of free space
mass of an electron
charge of an electron
Boltzmann's constant
speed of light in free space
29