•Optical instruments Prisms - University of Colorado...

ECE 5616 OE System Design

Robert McLeod 139

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


Robert McLeod 140

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.


Robert McLeod 141

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


Robert McLeod 142

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


Robert McLeod 143

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.


Robert McLeod 144

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


Robert McLeod 145

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



Robert McLeod 146


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


Robert McLeod 147

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


Robert McLeod 148

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


Robert McLeod 149

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


Robert McLeod 150

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



Robert McLeod 151

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.


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.


Robert McLeod 152

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.



Robert McLeod 153

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


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


Robert McLeod 154

Multilevel DOEsWhy you pay for them


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:


Robert McLeod 155

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).


Robert McLeod 156

Diffractive lenses dependence of angles


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


Robert McLeod 157

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


Which gives us the diffracted electric field vs. angle for a uniform Einc


Robert McLeod 158

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.


1

1

n

n


Robert McLeod 159

Hybrid refractive/DOEs


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.

•Optical instruments Prisms - University of Colorado...

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