•Optical instruments
–Prisms
ECE 5616 OE System Design
Prisms
What are they good for?
• Fold
– “Erect” or “rotate” images
– Change direction of propagation
• Fold system for compactness
• Retroreflect
• Disperse (vs. )
• Control beam parameters
– Anamorphic telescopes
– Vary angle, position, path length
• Amplitude or pol. division
– Beam splitters
Robert McLeod
139
•Other important optical components
–Prisms
ECE 5616 OE System Design
Folding prisms
Bouncing pencils to analyze image orientation
Right angle prism
Retroreflector
Penta prism
Compare orientation to plane mirror
Even # reflections = constant deviation
Such prisms are often used with telescopes to
“erect” the image (flip upright).
objective
e.p.
object
Roof (Amici)
Same function as RAP but add roof to get extra reflection

Robert McLeod
virtual
image
image
Dove prism inverts image. Rotation of
prism rotates image at twice the angle.
Dispersion at first interface correct at output.

140
•Other important optical components
–Prisms
ECE 5616 OE System Design
Tunnel diagrams
Tool to simplify ray-tracing
2A
Porro erecting prisms – very common
in binoculars
3D tunnel diagram
A
A
Ray trace trough glass slabs
2A
2A
Replace glass with air and effective
thickness (paraxial only!).
Insert into paraxial, unfolded ray-trace.
2A/n
Robert McLeod
2A/n
141
ECE 5616 OE System Design
Anamorphic prisms
Often better than cylindrical telescope
h

’
M
cos  
cos 
Mh
Non-deviating prism telescope
h
M2 h
Problems:
• Compression largest near TIR – tolerances and polarization dependence
• Angular bandwidth quite low (works best for collimated beams)
Advantages
• Lower aberrations than cylinders
• Cheap
Robert McLeod
142
•Passive components
–Prisms
ECE 5616 OE System Design
Thin prism tricks
Risley prisms

n
Beam is deviated by angle (n-1)If prism
is rotated about its axis, the beam is
deflected in a circle. Two cascaded prisms
give arbitrary x,y deviation. For small ,
control of deviation can be quite fine.
Sliding wedge
As above, if  is small, control of
displacement can be quite fine.
Focus adjust
Robert McLeod
Yoder,
Design and Mounting of Prisms and Small Mirrors, SPIE, 1998
Variable path length.
143
•Passive components
–GRIN lenses
ECE 5616 OE System Design
GRIN lenses
Common in fiber/telecom applications
“Full pitch” = imaging lens
“Half pitch” = collimating lens
Fractional pitch (<.5) as typically used
Robert McLeod
144
•Passive components
–GRIN lenses
ECE 5616 OE System Design
GRIN lenses via eikonal
Good example and important lens technology
m

 
n   n0 1    
 a  

Power-law radial index distribution

d 
dr 
n      n    
ds 
ds 

d 
dr  d 
 d

 n    n  ̂  zˆ  
dz 
dz  dz 
 dz

d
d 
d 
n     n   
d
dz 
dz 
Plug into eikonal
Paraxial approx.
Simplify with known dependencies
m
1 d
1 d  
d2
    

n  
n0 1     
dz 2
n  d
n  d  
 a   
m 1
m 1
1 
m    
m   

  
 
 n0
n  
a  a  
a a
d2
2 



  2 
2
d
a a
  z    0 cos z  
Robert McLeod
 0
sin  z 

Plug in n()
Special case m=2
Solution for ray trajectory
145
•Passive components
–GRIN lenses
ECE 5616 OE System Design
GRIN lenses
Derived properties
For a lens of pitch P where P = ½ is a half pitch = Fourier
transform or collimation lens.
 L  P or
LP
n0



Pa
a 

2n
2
OK, how do we pick the lens radius, a? The lens NA or largest
acceptance angle is
NA  sin  ext  n0 sin  int  n0  0
The most extreme ray had better not hit the edge of the lens, so
 0
a

Length can be expressed
NA n0
2 n n0
aa
NA  2 n n0
L  P
n0 a
NA
The NA of a slab waveguide (a rectangular GRIN, if you will) is
NA 
n0  n 2  n02
 2 n n0  n 2  2 n n0
This is because the index profile can be cast as a potential well
for the photon and thus the transverse momentum is limited by
the depth of the well.
Robert McLeod
146
•Design of ideal imaging systems with geometrical optics
–Ray and eikonal equations
ECE 5616 OE System Design
Numerical solution of eikonal
For arbitrary index distributions
3D gradient index distribution
30
20
20
10
10
0
0
-10
-10
-20
-30
-20
XY slice of n
-30
-20
-10
0
10
20
30
XZ slice of n
-20
0
20
40
60
Ray-trace in XZ
Robert McLeod
147
•Diffractive optics
–Introduction
ECE 5616 OE System Design
Diffractive optics
Introduction/terminology
• Classes
• Diffractive optical element: Modification of the optical
wavefront via subdivision and individual modification of the
phase and/or amplitude of the segments.
• Grating: linear segments = uniform diffraction angle
• Computer generated hologram: A DOE in which the structure
has been calculated numerically
•Holographic optical element: DOE in which the structure is
generated by the interference of optical wavefronts.
• Discretization
•Binary optic: phase or amplitude structure with two levels.
Typically created via a single etch step.
• Dammann grating: Binary optics with repetitive pattern,
generates N beams (fan out)
• Multilevel optic: Same as binary but with M etch steps to
achieve N=2M levels.
• Kinoform: Phase DOE with smoothly varying profile (limit of
N→∞)
• Blazed: Grating with linear (sawtooth) segments
• Fabrication
• Direct machining: aka “ruling” or diamond turning, fab via
mechanical machining. Often used for masters.
• Lithography
• Direct write: scan laser or e-beam over photoresist
• Interference (holography) inc near field
• Masks: grey-scale, multiple exposure
• Replication
Robert McLeod
O’Shea, Diffractive Optics: Design, Fabrication and Test
148
•Diffractive optics
–Diffraction gratings
ECE 5616 OE System Design
Diffraction gratings
Basics
Real-space
Fourier-space
kz
k+1
km=0
k+2
k-1
x
n
d
k+3
L
k x in  K G
 x   jk n d sin  K G x 
Eout  x   Ein  x  rect e 0 2
FT
L

x
 Ein  x  rect   J m (k0 n d2 ) e  jmKG x
 L m  
kx
k x in  2 K G
k x in  K G
k x in  3K G
k x in
 L
Eout k x   Ain k x  k x in * sinc k x  *
 2

J
m  
 Ain
m
(k0 n d2 )  k x  mK G 

J
m  
m
L

(k0 n d2 ) sinc k x  k x in  mK G  
2

What are angles of diffracted waves?
k xm 
2

sin  m
Conservation of transverse k
 k x in  mK G

2

sin  in  m
2
G
sin  m  sin  in  m
Robert McLeod

G
Grating equation
149
•Diffractive optics
–Diffraction gratings
ECE 5616 OE System Design
Resolving power
aka number of spots
Real-space
Fourier-space
kz
k+1
km=0
 blue peak
 red  peak
kin-B
KG
kx
2
L
kin-R
 red  null   blue peak
sin 1
Rayleigh resolvability criterion
mK G  2 L
mK G
 sin 1
k red
k blue
k red
m 2  G  2 L

k blue
m 2  G
1
…algebra…


 1 G
0
mL
R
0
 mN

Robert McLeod
where N =
L
= number of grating lines
G
illuminated
150
•Diffractive optics
–Diffraction gratings
ECE 5616 OE System Design
Estimation of grating R
Why gratings are interesting
Holographic gratings of 1800 lp/mm are typical in the visible.
A 10 mm beam and first-order diffraction would yield
R  10  1800  18,000
or a minimum resolvable wavelength shift of .03 nm in the
visible.
For a prism at the minimum deviation condition (symmetrical
incident and exit angles) the resolving power can be shown to
be
R
0 dn
n  nB
b
b R
b  V 1 nY  1

 d
R  B
170 nm
In the visible a b = 25 mm prism would give resolving power
R
1100 Crown
3400 Flint
or ~ 0.5 to 0.17 nm,
roughly an order of
magnitude lower
resolution than a grating.
Robert McLeod
S2
S1
 2 w2
w1 1
b
S1  S 2  nb
151
•Diffractive optics
–Diffraction gratings
ECE 5616 OE System Design
Bandwidth
aka Free spectral range
kz
k0
k+1
k0
k+2
k+1
KG
2 KG
kx
When will diffractions be confused with the neighboring order?
 red  m   blue m1
sin 1
m  1K G
mK G
 sin 1
k red
kblue
kblue blue  

 1 m
k red
blue
  mblue
Thus first-order grating spectrometer could operate from 400 to 800 nm.
Robert McLeod
152
•Diffractive optics
–Diffraction gratings
ECE 5616 OE System Design
Efficiency
Overview by type
1
Sinusoidal phase
0.6
h
Thin phase grating:
Typically many orders,
can’t reach 100% in any
(e.g. sinusoidal).
/2


k x in
0.8
k x in  2 K G
k x in  1K G
0.4
k x in  1K G
0.2
-6
-4
Exception: blazed phase grating:
Can be 100% in single order
-2
0
kx
2
4
6
n
n

Thin amplitude grating:
Lossy (by definition),
typically many orders.
Exception: Sinusoidal amplitude grating:
DC and +/- orders
Thick phase grating: Bragg selectivity can give
single order and theoretically 100% DE. BUT,
very sensitive to incident wave (unlike thin).
Robert McLeod
153
•Diffractive optics
–Diffraction gratings
ECE 5616 OE System Design
Multilevel DOEs
Why you pay for them
Phase (not physical) profiles:
N=2


N=4


N=∞

First order diffraction efficiency vs. number of levels
0
h dB
- 0.5
-1
1  sinc 2  / N 
- 1.5
-2
- 2.5
2
Robert McLeod
4
6
8
10
N
12
14
16
154
•Diffractive optics
–DOEs as lenses
ECE 5616 OE System Design
Diffractive lens design
Multilevel on-axis Fresnel
Fermat (refractive) = “all rays have = OPL”
Diffractive = “all rays have = OPL modulo m”
Thus refractive is limit of diffractive w/ m=0.
r2  f 2
r
f
Spherical converging wavefront
What is the radial location of the pth zone for a mth order DOE
fabricated with N layser?
2
OPL of each zone differs
rp2  f 2   f  p m0 N 
by m 
2
2
rp  2 f p m 0 N   p m 0 N 
For 2 f >> p 
 2 f p m 0 N
What is the radius size of the pth zone?
rp2  2 f p m 0 N
Radius of pth and p+1th zone
rp21  2 f  p  1m 0 N
rp21  rp2  2 f m0 N


 rp 1  rp rp 1  rp
 r 2rp 

r  f m0 rN   2m0 F #  N 
Take difference
Expand
Local grating period
Note for minimum feature size, N reduces F/# linearly (ouch).
Robert McLeod
155
•Diffractive optics
–DOEs as lenses
ECE 5616 OE System Design
Diffractive lenses
 dependence of angles
Reading a diffractive optic at ’ and order m’ that was designed
for  and order m.
kz
k’0
’
k0
k’m’
f
Local grating
period G
f’
km

m KG

m 2 
m
2 

m 2 

sin   
 m
2  

m K G
h
sin  
sin   neff sin  
m


 neff m
neff 
m 
m

Change in angle is perfectly analogous to
refracting into a slab of index neff. Note
that this index can be < 1.

f  h sin 
f   h sin  
f f
m
sin 
 f neff  f
m 
sin  
Robert McLeod
DOE
kx
Definition of focal length.
1. Diffracts to ∞ set of focii.
2. For neff≠1, each suffers
spherical aberration.
156
•Diffractive optics
–DOEs as lenses
ECE 5616 OE System Design
Diffractive lenses
 dependence of efficiency (1/2)
For a kinoform (N= ∞)
h x  
h(x)
x
t

t
x

S x  
x
m
constant

 n h x   t  h x   t  n  1h x 
x
 n  1 t

tm
OPL by design is m at .
Calculate from profile

n 1

x
n  1
S  x   m
 n 1
 x  
2
2
 n  1
x
S  x  
m

 n 1 
2     n 1 

x   j  m  n 1     x 


x   rect  e
TGrating
     x  l 

 

 l
Substitute h(x)
Solve for step height.
OPL at shifted ’
Phase at ’
Transmission mask




k x    sinck x  2 m   n  1    1    k x  m 2 
TGrating
 
  n  1     2   m 

Which gives us the diffracted electric field vs. angle for a uniform Einc
Robert McLeod
157
•Diffractive optics
–DOEs as lenses
ECE 5616 OE System Design
Diffractive lenses
 dependence of efficiency (2/2)
Efficiency of a blazed grating designed for wavelength and order m
with index n read at wavelength ’ and order m’ with index n’
 
   n  1  

 



n
1

 

  sinc 2  m  m
m
 
   n  1 

  1 a blazed grating
If 

 n  1   
has 100% theoretical DE in the
design order and (conveniently) 0%
in all other orders.
-2
-1

1
m 1
0

1
1
   n  1 
  1 / 2 a blazed grating

If 
 n  1    
has 40.5% theoretical DE in the
design order and an equal amount in
the next lowest order. An infinite #
of orders are present.
-2
Robert McLeod
-1
0
2
m
   n  1 



 n  1    
m 1
1
2
m
158
•Diffractive optics
–DOEs as lenses
ECE 5616 OE System Design
Hybrid refractive/DOEs
m
m 

f f
   
From page 182
  

 
B  R  B R Y
Y
 Y VDOE
Y
VDOE 
B  R
If used at same order (m=m’)
Find change in power over 
From page 170
Solve for V.
589.6
486.1  656.3
 3.46

This is a) the same for all DOEs, b) negative and c) very strong.
Let’s design an achromatic f=25.4 mm BK7 singlet:
BK 7
64.2
  DOE

DOE
 0 ,  BK 7  DOE  1 / 25.4
 3.46
 1 496.695 mm, BK 7  1 26.769 mm
Achromatic
conditions
Note the refractive power is nearly unchanged and the DOE is quite
weak.
Robert McLeod
159