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PrismsWhat 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
•Optical instruments–Prisms
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Folding prismsBouncing pencils to analyze image orientation
•Other important optical components–Prisms
Right angle prism RetroreflectorCompare orientation to plane mirror
Penta prismEven # reflections = constant deviation
Roof (Amici)Same function as RAP but add roof to get extra reflection
object virtualimage
image
e.p.
objective
Such prisms are often used with telescopes to “erect” the image (flip upright).
Dove prism inverts image. Rotation of prism rotates image at twice the angle.
Dispersion at first interface correct at output.
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Tunnel diagramsTool to simplify ray-tracing
Porro erecting prisms – very common in binoculars2A
A
A
3D tunnel diagram
2A 2A
2A/n 2A/n
Ray trace trough glass slabs
Replace glass with air and effective thickness (paraxial only!).
Insert into paraxial, unfolded ray-trace.
•Other important optical components–Prisms
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Anamorphic prismsOften better than cylindrical telescope
h
M h
’
cos
cos M
h
M2 h
Non-deviating prism telescope
Problems:• Compression largest near TIR – tolerances and polarization dependence• Angular bandwidth quite low (works best for collimated beams)
Advantages• Lower aberrations than cylinders• Cheap
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Thin prism tricks
•Passive components–Prisms
Yoder, Design and Mounting of Prisms and Small Mirrors, SPIE, 1998
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 wedgeAs above, if is small, control of
displacement can be quite fine.
Focus adjustVariable path length.
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GRIN lenses Common in fiber/telecom applications
“Full pitch” = imaging lens
“Half pitch” = collimating lens
Fractional pitch (<.5) as typically used
•Passive components–GRIN lenses
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GRIN lenses via eikonalGood example and important lens technology
m
ann
10
zdz
dn
dz
d
dz
rdn
dz
d
ds
rdn
ds
dn
ˆ̂
dz
dn
dz
dn
d
d
11
0
02
2
1
111
mm
m
aa
m
aa
mn
n
an
d
d
nn
d
d
ndz
d
22
2 2
aad
d
zzz sin cos 00
Power-law radial index distribution
Plug into eikonal
Paraxial approx.
Simplify with known dependencies
Plug in n()
Special case m=2
Solution for ray trajectory
•Passive components–GRIN lenses
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•Passive components–GRIN lenses
GRIN lenses Derived properties
For a lens of pitch P where P = ½ is a half pitch = Fourier transform or collimation lens.
Pan
naPLPL
22or 0
OK, how do we pick the lens radius, a? The lens NA or largest acceptance angle is
00int0 sinsin nnNA ext
The most extreme ray had better not hit the edge of the lens, so
2
2
0
0
0
0
nnNA
aann
nNA
a
The NA of a slab waveguide (a rectangular GRIN, if you will) is
22 02
020
20 nnnnnnnnNA
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.
NA
anPL 0
Length can be expressed
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Numerical solution of eikonalFor arbitrary index distributions
-20 0 20 40 60
-20
-10
0
10
20
-30 -20 -10 0 10 20 30
-30
-20
-10
0
10
20
30
3D gradient index distribution
XY slice of n XZ slice of n
Ray-trace in XZ
•Design of ideal imaging systems with geometrical optics–Ray and eikonal equations
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Diffractive opticsIntroduction/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
• ReplicationO’Shea, Diffractive Optics: Design, Fabrication and Test
•Diffractive optics–Introduction
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Diffraction gratingsBasics
kx
kz
km=0k+1
k+2
k-1
k+3
m
xjmKdmin
xKnjkinout
G
Gd
enkJL
xxE
eL
xxExE
) (rect
rect
20
sin 20
x
dn
L
mGinxx
dmin
mG
dm
xinxinxout
LmKkknkJA
mKnkJ
LkkAkE
2sinc ) (
k ) (
*2
sinc*k
20
x20
x
inxk
Ginx Kk
Ginx Kk 2
Ginx Kk 3Ginx Kk
Real-space Fourier-space
Gin
Ginx
mmx
m
mKk
k
2sin
2
sin2
What are angles of diffracted waves?
Ginm m
sinsinGrating equation
Conservation of transverse k
FT
•Diffractive optics–Diffraction gratings
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Resolving poweraka number of spots
kx
kz
km=0
k+1
GK
Real-space Fourier-space
kin-B
kin-R
L
2
peakred
peakblue
Lm
m
Lm
k
k
k
mK
k
LmK
G
G
G
blue
red
blue
G
red
G
peakbluenullred
11
2
22
sin2
sin
0
11
Rayleigh resolvability criterion
…algebra…
where N = = number of grating lines illuminated
mNR 0
G
L
•Diffractive optics–Diffraction gratings
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Estimation of grating RWhy gratings are interesting
Holographic gratings of 1800 lp/mm are typical in the visible. A 10 mm beam and first-order diffraction would yield
000,18180010 R
or a minimum resolvable wavelength shift of .03 nm in the visible.
•Diffractive optics–Diffraction gratings
For a prism at the minimum deviation condition (symmetrical incident and exit angles) the resolving power can be shown to be
w2
b
2S
2
1S
nbSS 21
w1 1
nm 170
110 bnVb
nnb
d
dnR Y
BR
BR
In the visible a b = 25 mm prism would give resolving power
Flint
Crown
3400
1100R
or ~ 0.5 to 0.17 nm, roughly an order of magnitude lower resolution than a grating.
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Bandwidthaka Free spectral range
kx
kz
k0
k+1
GK
k+2
GK2
k+1
k0
mk
k
k
Km
k
mK
blue
blue
red
blue
blue
G
red
G
mbluemred
1
1sinsin 11
1
bluem
When will diffractions be confused with the neighboring order?
Thus first-order grating spectrometer could operate from 400 to 800 nm.
•Diffractive optics–Diffraction gratings
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EfficiencyOverview by type
-6 -4 -2 0 2 4 6kx
0.2
0.4
0.6
0.8
1
h
Ginx Kk 1Ginx Kk 1
Ginx Kk 2
inxk /2
•Diffractive optics–Diffraction gratings
Thin phase grating:Typically many orders, can’t reach 100% in any (e.g. sinusoidal).
Sinusoidal phase
Exception: blazed phase grating:Can be 100% in single order
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).
n
n
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Multilevel DOEsWhy you pay for them
•Diffractive optics–Diffraction gratings
2 4 6 8 10 12 14 16N
-2.5
-2
-1.5
-1
-0.5
0
hBd
N/sinc21
First order diffraction efficiency vs. number of levels
N=2
N=4
N=∞
Phase (not physical) profiles:
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Diffractive lens designMultilevel on-axis Fresnel
•Diffractive optics–DOEs as lenses
f
r
22 fr
Spherical converging wavefront
What is the radial location of the pth zone for a mth order DOE fabricated with N layser?
Nmpf
NmpNmpfr
Nmpffr
p
p
0
200
2
20
22
2
2
OPL of each zone differs by m
For 2 f >> p
What is the radius size of the pth zone?
NFmrNmfr
rr
rrrr
Nmfrr
Nmpfr
Nmpfr
p
pppp
pp
p
p
#2
2
2
12
2
00
11
022
1
02
1
02
Radius of pth and p+1th zone
Take difference
Expand
Local grating period
Fermat (refractive) = “all rays have = OPL”Diffractive = “all rays have = OPL modulo m”Thus refractive is limit of diffractive w/ m=0.
Note for minimum feature size, N reduces F/# linearly (ouch).
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Diffractive lenses dependence of angles
•Diffractive optics–DOEs as lenses
Reading a diffractive optic at ’ and order m’ that was designed for and order m.
kz
m
mfnfff
hf
hf
effsin
sin
sin
sin
kx
k0
k’m’
GKm
km
k’0
GKm
’
m
mn
mnm
n
mm
mm
eff
eff
eff
sinsin
2
2sin
2
2sin h
f
f’
Change in angle is perfectly analogous to refracting into a slab of index neff. Note
that this index can be < 1.
Definition of focal length.
1. Diffracts to ∞ set of focii.2. For neff≠1, each suffers
spherical aberration.
DOE
Local grating period G
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Diffractive lenses dependence of efficiency (1/2)
For a kinoform (N= ∞)
h(x)
x
t
tx
xh
tx
n
xhntxhtxhn
mx
xS
1
1
OPL by design is m at .
Calculate from profile
Substitute h(x)
1
nmt
Solve for step height.
11
nn
mx
xS
OPL at shifted ’
xn
nmxSx
1
1
22Phase at ’
l
xn
nmj
Grating lxex
xT 1
1
2
rect Transmission mask
mxxxGrating mk
n
nmkkT
21
2
1
1
2sinc
constant
•Diffractive optics–DOEs as lenses
Which gives us the diffracted electric field vs. angle for a uniform Einc
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Diffractive lenses dependence of efficiency (2/2)
m
n
nmm
1
1sinc2
Efficiency of a blazed grating designed for wavelength and order mwith index n read at wavelength ’ and order m’ with index n’
11
1
1
n
nIf
m
has 100% theoretical DE in the design order and (conveniently) 0% in all other orders.
2/11
1
n
nIf
0 1-1-2 2
a blazed grating
1m
1
m0 1-1-2 2
1m
a blazed grating
has 40.5% theoretical DE in the design order and an equal amount in the next lowest order. An infinite # of orders are present.
•Diffractive optics–DOEs as lenses
1
1
n
n
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Hybrid refractive/DOEs
•Diffractive optics–DOEs as lenses
46.33.6561.486
6.589
RB
YDOE
DOEY
YY
RBRB
V
V
m
mff
From page 182
If used at same order (m=m’)
Find change in power over
From page 170
Solve for V.
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:
mm 769.261 mm, 695.4961
4.25/1 , 046.32.64
7
77
BKDOE
DOEBKDOEBK
Achromatic
conditions
Note the refractive power is nearly unchanged and the DOE is quite weak.