ece 6006 numerical methods in photonics ray and gaussian...

ECE 6006 Numerical Methods in Photonics

Robert R. McLeod, University of Colorado 78

Ray and Gaussian beamlet tracing

•  Goals/outline –  Understand the basis of ray tracing

(the most common optical modeling tool by far)

–  Use finite differences to propagate paraxial rays in gradient index media

–  Apply the tools derived in ray tracing to Gaussian beams

–  Break arbitrary fields into sums of Gaussian beams

–  Put it all together to do wave optics simulations in slowly varying media and paraxial optical systems

•  Not a goal: The use of ray tracing to model discrete optical systems (e.g. lenses) is the topic of another course – Optical System Design.

• Ray tracing



The ray equation Approx. solution of ME when n(r) is slow

( ) ( ) ( )rSjkerrE!!!!

0−= EAssume slowly varying

amplitude E and phase S

( ) ( )rnrS !! 22 =∇Substitute into isotropic wave equation and retain lowest terms…

( ) zkykxkrSk zyx ++=!0 E.g. plane wave

( ) 22200 zyxkrSk ++=! E.g. spherical wave

Contours of S(r) at multiples of 2π

S∇n(r)

S(r) Optical path length [m]

Ray equation – approximation of Maxwell’s equations.

“Ray” = curve orthogonal to S(r)

( )dsrnB

A∫≡ !

• Ray tracing – Derivation



The eikonal equation Equation for the evolution of ray trajectory

( ) ( ) ( ) ( )dssrdrnsrnrS!

!!! ==∇ ˆ

s Parametric distance along ray [m] Ray trajectory vs distance along ray, s [m] ( )sr!

( )[ ] ( ) ⎥⎦⎤

⎢⎣⎡=∇

dsrdrn

dsdrS

dsd !

!!then take derivative in s,

Rewrite

( )[ ] ( )[ ] ( ) ( ) ( )[ ]rSrSrn

rSdsrdrS

dsd !!

!!

!! ∇∇⋅∇=∇∇⋅=∇ 1

finally apply the chain rule identity for

( ) ( ) ( )[ ] ( ) ( )[ ] ( )rnrnrn

rSrSrn

!!!

!!! ∇=∇=∇⋅∇∇= 2

21

21

( ) ( )rndsrdrn

dsd !

!! ∇=⎥⎦

⎤⎢⎣⎡

Take square-root of ray equation,

r!

rdr !! +

sdssd

dsrd ˆ==

!!

Ray

Saleh & Teich 1.3

• Ray tracing – Derivation

rd!

r!

from above

∇⋅=∇⋅= dsrdsdsd !ˆ

( )BA!!

⋅∇



Finite difference solution to paraxial eikonal

nrdsdn

dsd ∇=⎟

⎠⎞⎜

⎝⎛ !

• Ray tracing – Finite difference solution

Non-paraxial eikonal

nrdzdn

dzd ∇=⎟

⎠⎞⎜

⎝⎛ !

Paraxial eikonal

( )( ) ( )( )zkyx

kkkk

kkkk

kknzxxzkyxn

zxxzkyxn

Δ

−−−

+++ ∇=

Δ−Δ−−

Δ−Δ+

,2

1

21

2

1

21 ,,,, 2

121

21

21

Apply centered differences to paraxial, write for x component

( )( ) ( )( ) ⎥⎦

⎤⎢⎣

⎡∇+

Δ−Δ−

Δ+Δ+=

Δ

−−−

+++

zkyx

kkkk

kkkk

kknzxxzkyxn

zkyxnzxx

,,2

1

21

21

21 ,,

,,21

21

21

21

And solve for xk+1

Evaluation of n at xk+1/2, yk+1/2 is problematic. Can approximate as n(xk, yk)or estimate based on linear extrapolation from ray direction. For additional accuracy, iterate several times until stable.



FD of paraxial eikonal Typical results

0 100 200 300 400Z@mmD

-‐ 20

-‐ 10

0

10

20

30

Y@mmD

• Ray tracing – Finite difference solution

Fractional pitch gradient index lens surrounded by free space.

Exhibits spherical aberration, just like it should.



Paraxial ray tracing Derivation of refraction & transfer equations

kkkk tuyy ′′+=+1 Transfer equation

yk Height of ray at surface k [m] fk Power of kth surface [diopters]

Paraxial ray angle incident on surface k [radians] Paraxial ray angle exiting surface k [radians]

• Paraxial ray tracing of optical systems – Basics

fk

yk

k

kk

k

kk u

ytuyt

′−=′=− ,

k

kk

k

k

yu

yu −=′

− φ

kkkk yuu φ−=′

Paraxial tangents

Substitute into thin-lens equation

Refraction equation

ku′−ku

yk+1 yk ku′

kuku′

kt− kt′

kt′



ABCD matrices Matrix formulation of paraxial ray-tracing

⎥⎦

⎤⎢⎣

⎡′

≡⎥⎦

⎤⎢⎣

⎡′⎥

⎦

⎤⎢⎣

⎡ ′=⎥

⎦

⎤⎢⎣

⎡

+

+

k

kk

k

kk

k

k

uy

uyt

uy

T10

1

1

1

⎥⎦

⎤⎢⎣

⎡≡⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡′ k

kk

k

k

kk

k

uy

uy

uy

R101

φ

Transfer equation

Refraction equation

N is “conjugate matrix” when planes zero and K+1 are conjugates

f1 fk

-fK y0 0u′ y1

1u′−-yK+1

1u

1+Ku

M

N

Kt′

0t′

• Paraxial ray tracing of optical systems – Matrix formalism

Typical use to propagate rays through a system:

⎥⎦

⎤⎢⎣

⎡′

≡⎥⎦

⎤⎢⎣

⎡′

=⎥⎦

⎤⎢⎣

⎡−−

+

+

0

0

0

001111

1

1

uy

uy

uy

KKKKK

K NTRTRTRT …



Gaussian beams

Saleh & Teich chapter 3

( ) ( )( ) ( ) ( )zj

zRjkjkz

zwezwwArE

ζρρ +−−−= 20

0

2

2

2

!

( )2

00 1 ⎟⎟⎠

⎞⎜⎜⎝

⎛+=zzwzw ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞⎜

⎝⎛+=

201zzzzR ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= −

0

1tanzzzζ

r

z

Solution of scalar paraxial wave equation (Helmholtz equation) is a Gaussian beam, given by:

where

Real part of E vs. radius and z

• Paraxial ray tracing of optical systems – Gaussian beams

This is not a solution of the eikonal and thus shouldn’t be something we can model with ray tracing…



Gaussian parameters Relationships between parameters

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

2

0

22 1zzww o

I(wo)=I(0) /e2

θ0

wwo

z = 0 0zz =

( ) 00 2wzw =I(z0) = I(0)/2

000

1 θπ

λπλ == zw

0

020

200 θλ

πθπλ wwz === −

000

11zw π

λπλθ ==

At z=z0 Line shows

Everything in terms of everything else




Gaussian beam parameter q(z)

( ) ( )( ) ( ) ( )zj

zRjkjkz

zw

D

ezww

wArE

ζρρ

π

+−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛= 20

2/

00

2

2

2

1!

( ) ( ) ( )zwj

zRzq 2 11

πλ−=

( ) ( )( ) jkzzq

jkD

ezqz

wAjrE

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛= 20

2/

00

2

1 ρ

π!

( ) 0

1

20

20

2

00

20

1

1

1

11 zjz

zzzjz

zzz

j

zzz

zq +=⎥⎦

⎤⎢⎣

⎡+

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞⎜

⎝⎛+

=−

The complete Gaussian beam expression normalized to intensity

Define the complex radius of curvature

What is q(z)?

Can now write Gaussian above as

arg[ j/q(z)] = z


( )zzq 01tanarg −= ( )

0

2

00

1wzw

zz

zq

=⎟⎠⎞⎜⎝

⎛+=

D is number of transverse dimensions

=1,2



How does q change with transfer and refraction?

Free space:

( )121022

011

zzqzjzqzjzq

−+=+=+=

Thin lens

( ) ( ) ( )zwj

zRzq 2 11

πλ−=

fRR111 −=

′

fqq111 −=

′

1+−=′

fqqq

Start with expression for q(z)

so q2 = q1 + Δz

Start with expression for 1/q(z)


Thin lens equation expressed as change in curvature of wave

Apply to 1/q

Solve for q



ABCD approach to q

⎥⎦

⎤⎢⎣

⎡ ′=

101 k

k

tT⎥

⎦

⎤⎢⎣

⎡−

=101

kk φ

R

DCqBAqq

++=′

kk tq

qtqq ′+=+′+=′10

1Check for free space:

1101

+−=

+−+=′

fqq

qqqkφ

Check for thin lens:

Remember the ABCD matrices for thin lens refraction and free-space transfer

and define the evolution equation for q

Which says, rather remarkably, that we can model the propagation of a Gaussian beam through a paraxial optical system using ray matrices. But there’s a better way to do so (at least I like it better).




Representation of Gaussian beams by complex rays (1)

J. Arnaud, Applied Optics, V24, N4, p. 538, 15 Feb 1985 A. W. Greynolds, SPIE V 560, p. 33, 1985


Define the following three rays. Note their suggestive names and relationship to the Gaussian beam.

Define the complex ray trajectory

( ) ( ) ( )zjzz Ω+Δ=ΓYou can then show that this ray contains q(z)

( )

( )zqzjzjwzujuyjy

dzdz

=+=+=

++=

ΓΓ

ΩΔ

ΩΔ

00

00

θθ

w0 θ0

Ω= Waist ray

Chief ray

f Pa

raxi

al im

age

plan

e

Wai

st lo

catio

n

Wai

st lo

catio

n

⎥⎦

⎤⎢⎣

⎡=Δ

⎥⎦

⎤⎢⎣

⎡=Ω

00

00

0

0

θ

w

( ) 0wz =Ω

Paraxial ray trajectory form

ABCD vector form

( ) 0θzz =Δ

This is Greynolds’ definition and yields the proper form of q. Arnaud’s definition yields q*.

E.g. at z=0

Ray heights over ray slopes




R. Herloski, S. Marshall, R. Antos, Applied Optics, V22, N8, p 1168, 15 Apr 1983


By brute for tracing of the rays, we can find the following Gaussian parameters based on the two rays at that point:

( ) ( ) ( )zzzw 22 Ω+Δ=

( )πλ0=− ΩΔΔΩ uyuyn Which is the Lagrange invariant

First we note:

We could use these two and the expressions for the Gaussian beam parameters to generate the complete Gaussian, but this would be a bit tedious. A more elegant way is to use the complex ray formalism:

( ) ( )dzdzzq

ΓΓ=

Which, apart from the on-axis phase k0S gives the full Gaussian beam at this plane z. Keeping track of this plane-wave phase requires us to trace the “chief ray” and calculate its OPD at every position.

As shown on the previous page

( )22

⎟⎠⎞⎜

⎝⎛∂Ω∂+⎟

⎠⎞⎜

⎝⎛∂Δ∂=

zzzθ

Gaussian divergence is quadrature sum of ray angles

Gaussian radius is quadrature sum of ray heights



Gaussian beam superposition Tracking phase

• Gaussian beam superposition – Tools

( ) ( )( )zjjkz

D

ezww

wAzE ζ

πρ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛== 0

2/

00

1,0

( )jkz

D

ezqz

wAj −

⎟⎟⎠

⎞⎜⎜⎝

⎛= 0

2/

00

1π

Consider the field on axis:

0 500 1000 1500 2000z@mmD-‐ 30

-‐ 20

-‐ 10

0

10

20

30

y@mmD

Chief ray

So our complex ray (which gives q(z)) clearly does not keep track of the plane-wave portion of the phase. So we must explicitly calculate

Including the OPL of thin phase structures (e.g. the lens) – next

( )∫=Chief

dsrnSk !

00

2λπ





On-axis examples: 1) λ = 1 µm, w0 = 10 λ, f = 500 µm, 1 f – 1 f system.

2) λ = 1 µm, w0 = 10 λ, f = 500 µm, 2 f – 2 f system.

Notes •  In (1), second waist is at Fourier plane, as expected. •  In (2), second waist occurs before image plane, as expected. •  In (3), as distance to lens increases, waist moves to paraxial image plane •  To conserve energy, must use complete Gaussian expression on pg 76 •  No restrictions on size or sampling of space (e.g. BPM).

3) λ = 1 µm, w0 = 10 λ, f = 500 µm, 3 f – 3/2 f system.




Via standard ray-tracing we can trace the ray trajectory through 3-space. The definitions of the waist and divergence rays are in a coordinate system normal to the chief ray. Thus to transform the 3D ray coordintes to the 2D (x,y) system normal to the chief ray (z).

⎥⎦

⎤⎢⎣

⎡DCBA0Δ

0Ω

0C

1+ΔK

1+ΩK

1+KC

( ) ( )( ) svhzC

zzuyzC

CC

kCC!!!

+=

−+=

z( )kzyxP ,,=

!

kz

( ) ( )( ) svhz

zzuyz k

ΔΔ

ΔΔ

+=Δ

−+=Δ!!!

To find q(z) at a point P=(x,y,zk), first find intersections of rays with plane normal to C and through P:

( )[ ] CCCCC vvhPhh ˆˆ•−+=′!!!!

( )[ ] Δ••−

ΔΔ Δ

Δ+=′ vhhC

CvvvhP ˆˆˆˆ

!!!!

Δ′h!

y y′

z′

Δh!

Ch!Ch′

!

Then write rays in new coordinates:

( ) ( )( ) ( )zuuyhh

zuuyhh

C

CC

CC

−+′•′−′=Ω

−+′•′−′=Δ′

=′

ΩΩ

ΔΔ

ˆ

ˆ

0

!!

!!

( )[ ] Ω••−

ΩΩ Ω

Ω+=′ vhhC

CvvvhP ˆˆˆˆ

!!!!

Note that in 3D must also allow for rotation of orientation of Gaussian. This is in Greynolds.





Off-axis examples: 1) λ = 1 µm, w0 = 10 λ, f = 500 µm, 1 f – 1 f system.

Notes •  In (1), waist is centered at zero (as expected of Fourier trransform) •  In (2), image is at -10 µm, expected from magnification M=-1. •  Note that rays do not need to stay confined to evaluation space = no BCs!


2) λ = 1 µm, w0 = 10 λ, f = 500 µm, 2 f – 2 f system.

3) λ = 1 µm, w0 = 10 λ, f = 500 µm, 3 f – 3/2 f system.



Gaussian beam superposition Algorithm

ASAP Technical Guide: Wave Optics in ASAP

• Gaussian beam superposition – Concept

1.  Synthesize input electric field as superposition of paraxial Gaussian beams.

a.  There are almost no constraints on the Gaussian beams. They need not all have the same waist, be parallel to the paraxial axis, and they do not need to be of the same size (amplitude or waist) or regularly spaced.

b.  There is no optimal set of Gaussian beams to use or any nice way to decide what set of Gaussian beams give us the most efficient simulation.

2.  Represent each Gaussian via its chief, waist and divergence rays. a.  In two dimensions, two of each. Need to consider rotation of

coordinates 3.  Propagate rays through system via ABCD, FD, or full non-paraxial trace

a.  Individual Gaussians must be paraxial, but not ray trace. Thus method is non-paraxial!

b.  Must accumulate phase along chief ray since not in q(z). 4.  Sum Gaussians represented by rays at any plane to calculate E field.



Choice of Gaussian set Regular sampling


Assume a perfect Fourier transform geometry (case 2 above) and a rectangular incident electric field.

-‐ 75 -‐ 50 -‐ 25 0 25 50 75 1000

0.2

0.4

0.6

0.8

1

300 400 500 600 7000

0.25

0.5

0.75

1

1.25

1.5Perfect Rect Sum of 10 Gaussians Single Gaussian

E in

cide

nt

E in

Fou

rier p

lane

Side lobe is diffraction order of regular array of Gaussians



Examples Fourier transform of a rect (1)

• Gaussian beam superposition – Examples

1f-1f transform of a 100 µm rect represented by 11 Gaussian beams of w0 = 5 µm separated by 2 w0.

Note the interference between Gaussians that can be observed in the Fresnel planes.

Hung Loui NMiP HW, 2004

A diffraction side lobe appears in the Fourier plane corresponding to the 2 w0 ripple in the object.



Examples Fourier transform of a rect (2)


Note the interference between Gaussians is virutually absent.

The diffraction side-lobe has disappeared and the solution is accurate for the main and first lobe. Higher-order lobes are weak due to the smooth edges of the rect.

Same as previous but now 21 beams separated by 2 w0.



Examples Imaging a rect

y [um]

2f-2f imaging of a 100 µm rect represented by 21 Gaussian beams of w0 = 5 µm separated by w0.

In this case the method is as accurate as the input representation.

Rays superimposed over field color plot.




Gaussian beam superposition Extra goodies


( ) ( )∑=i

i rErE !!

For completely spatially coherent sources, fields add, while for incoherent sources, intensities add:

( ) ( )∑=i

i rIrI !!

Coherence and interference

Transmit and reflect rays at each intersection to form tree of rays. Terminate when ray intensity falls below a defined threshold.

Non-sequential

Where polarizations are decoupled, trace rays for each polarization. Where they couple (e.g. a waveplate), decompose input polarizations into eigenstates and propagate both. Similar to non-sequential tracing, this will generate a tree of rays.

Polarization

Curved surfaces and real aberrations Unlike Fourier BPM, can handle real lenses, although this takes some geometry.

ece 6006 numerical methods in photonics ray and gaussian...

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