PHYS 408
Applied Optics
(Lecture 11)
JA N - A PRIL 2 0 1 6 E DI T ION
JE F F YOUN G
AMPEL RM 113
Quiz #5
1) You can use M and S matricies either from input to output, or from output to input T/F
2) The input to output direction allows one to evaluate the internal field distribution in a fairly
straight forward fashion T/F
3) When two high reflectivity dielectric stacks face each other, light between them is trapped
forever at any wavelength T/F
4) There is a midterm exam in this course this coming Friday T/F
Quick review of key points from last
lecture
M matricies can be used to propagate either left to right or right to left (input to output or output
to input).
The output to input formulation is both a bit more numerically friendly, and much more useful if
you want to explore the field distribution inside the dielectric stack.
High reflectivity mirrors surrounding a “defect” region (uniform region of arbitrary thickness) can
“trap” light of certain wavelengths, which corresponds to light bouncing back and forth many
times between the mirrors before “leaking out”.
This is perhaps the simplest form of an optical cavity.
Cavities and Gaussian Beams
In this module, we are going to consider non-planar waves and how
they interact with optical elements.
An important class of non-planar waves that find many applications
are Gaussian Beams.
Recall
n1
n2
d1
d2
…
nlayers-1
n1
n3
n1
n2
n1
n2
d1
d3
d1
d2
d1
d2
…
nlayers-1
n1=2; n2=sqrt(12); n3=4
d1=300 nm; d2=173 nm; d3=10*d2
10 periods SYMMETERIZED
Cavity Modes
1
0.9
0.8
Transmission
0.7
0.6
0.5
0.4
0.3
0.2
0.1
3500
100
200
90
180
80
160
70
140
60
120
4000
4500
Wavenumber 1/
5000
35
20
2
50
25
|E|
2
|E|
|E|
2
30
100
40
80
30
60
20
40
10
20
15
10
5
0
-7
-6
-5
-4
-3
Z (cm)
-2
-1
0
1
x 10
-4
0
-7
-6
-5
-4
-3
Z (cm)
-2
-1
0
1
x 10
-4
0
-7
-6
-5
-4
-3
Z (cm)
-2
-1
0
1
x 10
-4
n1=2; n2=sqrt(12); n3=4
d1=300 nm; d2=173 nm; d3=20*d2
10 periods SYMMETERIZED
Cavity Modes
1
0.9
0.8
Transmission
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
3400
3600
3800
4000
4200
4400
Wavenumber 1/
4600
4800
5000
What is a “cavity mode”?
First ask what is a “photonic mode”?
- Stationary solution of Maxwell Equations in a lossless medium for nonzero w
What is a “stationary solution”?
-
One that is harmonic in time…i.e. proportional to exp-iwt, where w is real
Have we already encountered examples?
Cavity defined by two perfectly reflecting
planar mirrors
vacuum
Boundary conditions?
Form of allowed fields inside?
Allowed solutions?
0
z
d
Planar cavity modes: standing waves


ikz

 ikz 
E ( z )  ( E exp  E exp ) y
Equations to satisfy boundary conditions?


E ( z  0)  E ( z  d )  0

E ( z  0)  0  E   E   0

E ( z  d )  0  (expikd  exp ikd )  0
 kd  m  k  m / d with m  1,2,....
Observations?
Amplitude unrestricted…make sense?
Consistent with results shown earlier?
Generalize/application
3 dimensional cavities
1
0.9
0.8
Transmission
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
3400
3600
3800
4400
4200
4000
Wavenumber 1/
4600
4800
5000
Stability/sensitivity issues
If you can’t ensure both mirrors are perfectly parallel, modes mess up very
quickly (eg. HeNe laser lab)
Lateral size of the mode not so easy to control using macroscopic mirrors/gain
media.
Does this remind you of anything we did
earlier in the term?
q
Gaussian Beams
Can you guess why these Gaussian beams may be relevant to the modes that
can be supported by two curved mirrors?
Next lectures: Geometric ABCD matrix
and Gaussian Beam propagation
There is something quite profound/pathological about Gaussian beams, so
they deserve some serious consideration.
Magic!
It turns out that the propagation properties of Gaussian beams that satisfy the
“paraxial approximation” (essentially the slowly varying envelope
approximation), through collections of optical elements (thin films, prisms,
lenses, mirrors etc.), can be derived using simple equations that describe
classical ray optics (which totally ignore the wavelike properties of the light).
The diffraction properties are “magically” taken account of.
… from Steck
It seems amazing that a solution to the wave equation can be propagated using classical rules. But really
what this is saying is that, in some sense, wave phenomena are absent in the paraxial approximation.
As we will see, this transformation rule applies in more general situations as long as the paraxial
approximation holds. But let痴 focus on the Gaussian beam for concreteness. The Gaussian beam stays
Gaussian through any optical system as long as the paraxial approximation is valid. As soon as the paraxial
approximation breaks down (i.e., nonlinear terms become important), the beam will become non-Gaussian.
…
… just ray optics; Gaussian beams do not show any manifestly wave-like behavior, at least
beam optics is really
within the paraxial approximation. All the diffraction-type effects can be mimicked by an appropriate ray
ensemble.
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