K-space Beam Propagation - Electrical, Computer &...

ECE 6006 Numerical Methods in Photonics

Robert R. McLeod, University of Colorado 218

Beam Propagation Statement of problem

x,y

z ( )r!ε( )0,,, =zyxtE!

!E t, x, y, z( )?

As long as each small slice obeys certain conditions (one of which is again the Born approximation) we can accommodate strong scattering in the volume. The penalty is that the solution is not perfectly accurate for non-paraxial rays. There are modifications that solve this limit.

Known incident field

Unknown field in volume

• K-space – Derivation

Now let the inhomogeneity be strong such that the Born approximation is not valid. We will now use an iterative solution that propagates through the space in small steps Δz


Robert R. McLeod, University of Colorado

Beam propagation Derivation of scalar solution (1/3)

219

E t, x, y, z( ) = Ftxy

−1 ε ω ,kx ,ky , z( ) e− jkz ω ,kx ,kx( )z{ }

Let the step size z be sufficiently small that exp(-j kz z) can be treated as invariant with kx and ky. Physically, this means the incident (inside brackets) and diffracted (outside) envelopes all propagate with the same phase and thus no diffraction occurs.

2 jkzddzε = k0

2 ε IH ω ,Kx ,Ky , z( )∗kx ,ky E ω ,kx ,ky , z( )⎡⎣ ⎤⎦e+ jkzz

Substitute the Fourier transform of the first into the second

2 jkzddzε = k0

2 ε IH ω ,Kx ,Ky , z( )∗kx ,ky ε ω ,kx ,ky , z( ) e− jkz ω ,kx ,kx( )z{ }⎡⎣

⎤⎦e

+ jkzz

ddzε ω ,Kx ,Ky , z( ) = k0

2

2 jkzε IH ω ,Kx ,Ky , z( )∗kx ,ky ε ω ,kx ,ky , z( ) e− jnH k0z{ }⎡⎣

⎤⎦e

+ jnH k0z

= k02

2 jkzε IH ω ,Kx ,Ky , z( )∗kx ,ky ε ω ,kx ,ky , z( )⎡⎣ ⎤⎦

ddzE t, !r( ) = − j k0

2nHε IH t, !r( )E t, !r( )

From the method of undetermined coefficients and the SVEA, we have

Assume for the kz term in the denominator. Physically, which is a projection factor that accounts for extra path length of angled rays, thus non-paraxial rays interact with the material as if they were normal incidence. Finally, inverse transform in the transverse coordinates:

kz ω ,kx ,ky( ) ≈ nHk0nHk0 kz = 1 cosθ

• Fourier beam propagation – Derivation

Approximations in red, numerical constraints in blue.




220

ddzE t, !r( ) = − jk0 nIH t, !r( )E t, !r( )

Change from dielectric to index perturbation:

εH + ε IH t, !r( ) = nH + nIH t, !r( )⎡⎣ ⎤⎦2= nH

2 + 2nHnIH t, !r( ) + nIH2 t, !r( )∴ε IH t, !r( ) ≈ 2nHnIH t, !r( )

to get a DE for the fields

E t, x, y, z + Δz( ) = E t, x, y, z( )exp − jk0 nIH t, !r( )dzz

z+Δz

∫⎡

⎣⎢

⎤

⎦⎥

≈ E t, x, y, z( )exp − jk0 nIH t, x, y, z + Δz2

⎛⎝⎜

⎞⎠⎟ Δz

⎡⎣⎢

⎤⎦⎥

Physically, this treats the inhomogeneous index as a thin transmission function via projection in a small step Δz. If the perturbation is zero, we obtain the solution that the field is invariant in z, as expected from the approximations made. This is equivalent to the homogeneous solution in which envelopes do not change with z and that we have left diffraction out of the solution above by assuming that there is no diffraction via exp(-j kz z) = constant. The evolution of the envelopes can be found by Fourier transforming the solution above:


small

whose solution is

ε ω ,kx ,ky , z + Δz( ) = Ftxy E t, x, y, z( )e

− jk0 nIH t ,x,y,z+Δz2

⎛⎝⎜

⎞⎠⎟Δz⎡

⎣⎢

⎤

⎦⎥




221


E t, x, y, z + Δz( ) = Ftxy−1 E ω ,kx ,ky , z + Δz( ) e− jkz ω ,kx ,kx( )Δz{ }

= Ftxy−1 Ftxy E t, x, y, z( )e

− jk0 nIH t ,x,y,z+Δz2

⎛⎝⎜

⎞⎠⎟Δz⎡

⎣⎢

⎤

⎦⎥e

− jkz ω ,kx ,kx( )Δz⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

E t, x, y, z( ) = Ftxy

−1 ε ω ,kx ,ky , z( ) e− jkz ω ,kx ,kx( )z{ }

Diffraction results when we insert the solution for the evolution of the envelopes back into the solution assumed in the method of undetermined coefficients:

To apply this formula, we must now assume that the envelopes don’t change significantly over Δz. That is, the refraction must not change the fields significantly while we are doing the diffraction step. Plugging the previous solution into the above, we get an iterative solution that advances the fields by one Δz

which is the final result.



Beam propagation Basic algorithm

This iterative solution switches back and forth between real space where refraction is applied and Fourier space where diffraction is calculated. It is thus often referred to as the Fourier split-step method

Refraction Diffraction

Boundary condition E(xi ,yj ,0)

Diffract one Δz via FFT

Refract one Δz

Optionally, save E (xi ,yj zk) for later plotting

For all zk

Post-process, plot


E t, x, y, z + Δz( ) = Ftxy

−1 Ftxy E t, x, y, z( )e− jk0 nIH t ,x,y,z+Δz

2⎛⎝⎜

⎞⎠⎟Δz⎡

⎣⎢

⎤

⎦⎥e

− jkz ω ,kx ,kx( )Δz⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪



Grin lens Half pitch 1 x 2.4 mm

0 50 100 150 200 250 300

0

100

200

300

400

500

0 50 100 150 200 250 300

0

100

200

300

400

500

Transverse sampling 512 steps of 2 λ0 = 2 µm

Longitudinal sampling 300 steps of 10 µm

( ) ⎟⎠⎞⎜

⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛=

24001300rect2cos07.0, zx

Lzxn

x

πδ

δn

|E| ( )2

3000,⎟⎠⎞⎜

⎝⎛−

=x

exE

• Fourier beam propagation – Passive optics


Robert R. McLeod, University of Colorado 224 0 50 100 150 200 250 300

0

100

200

300

400

500

0 50 100 150 200 250 300

0

100

200

300

400

500

Linear (Gradium) glass Simple way to implement prisms



( ) ⎟⎠⎞⎜

⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛=

19001050rect07.0, z

Lxzxnx

δ

δn

|E| ( )2

600,⎟⎠⎞⎜

⎝⎛−

=x

exE




0 50 100 150 200 250 300

0

100

200

300

400

500

0 50 100 150 200 250 300

0

100

200

300

400

500

Thin phase grating



δn

|E| ( )2

600,⎟⎠⎞⎜

⎝⎛−

=x

exE

( )2

1102cos

200500rect07.0,

+⎟⎠⎞⎜

⎝⎛

⎟⎠⎞⎜

⎝⎛ −=

xzzxn

π

δ




Thin amplitude grating

0 100 200 300 400 500

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300

0

100

200

300

400

500



Amplitude mask at z=200

|E| ( )2

600,⎟⎠⎞⎜

⎝⎛−

=x

exE

( )2

1102cos

,+⎟

⎠⎞⎜

⎝⎛

=x

zxA

π

Zero order since DC component to mask




Gratings implemented in Fourier-space

0 50 100 150 200 250 300

0

100

200

300

400

500

|E| ( )2

600,⎟⎠⎞⎜

⎝⎛−

=x

exE

Jaram and Banerjee use this to simulate an AO device on page 253 of their textbook.

At z=200 µm shift Fourier spectrum by to implement a single side-band grating

xkδπ102




0 50 100 150 200 250 300

0

100

200

300

400

500

Waveguide

0 50 100 150 200 250 300

0

100

200

300

400

500

Transverse sampling 512 steps of λ0 = 1 µm


δn

|E|

( )2

300,⎟⎠⎞⎜

⎝⎛−

=x

exE

( ) ( )2

54001.0,⎟⎠⎞⎜

⎝⎛−

>=x

ezzxnδ

0 50 100 150 200 250 300

0

100

200

300

400

500

|E| Δn=.01 Δn=.04




0 50 100 150 200 250 300

0

100

200

300

400

500

Curved surfaces

0 50 100 150 200 250 300

0

100

200

300

400

500



δn x, z( ) = 0.2Rect x300

⎛⎝⎜

⎞⎠⎟ x2 + z −1500( )2 ≤ 6002 & z <1800⎡⎣ ⎤⎦

δn

|E| ( )2

2000,⎟⎠⎞⎜

⎝⎛−

=x

exE

Curved surface anti-aliased

Even with averaging to reduce discontinuities, discretized surface acts as random diffraction grating




Smooth surfaces in BPM Sample problem – tilted interface

230

• Fourier beam propagation – Smooth surfaces

Index for an air/glass interface with a surface normal = 30o

The difference is the DC-free δn(x), used in the refraction operator.

n=1 n=1.5

δn=0 δn=0

Subtract the average index at each x. The average index is used in the diffraction propagator.

δn= .5

δn= -.5



Smooth surfaces in BPM Wrong: discretize n

231


Diff

ract

ion

from

surf

ace



Smooth surfaces in BPM Right: integrate n in z

232


Diff

ract

ion

from

surf

ace



Smooth surfaces in BPM E.g. lenses (1/2)

233




Smooth surfaces in BPM E.g. lenses (2/2)

234

Shows expected aberration behavior including spherical, coma, and field curvature.




Smooth surfaces in BPM Limitation

235


OPL for previous lens calculated by projection of (n-1) in z.

The change in OPL between adjacent transverse samples must be < λ. If it is not, the refracted wave will suffer spatial-frequency aliasing and will propagate at an unintended angle. The Nyquist frequency is a built-in limit of transverse sampling . If the x cell size in the previous case is increased by a factor of 4, this limit is violated at the edges of the beam:

Low angles within Nyquist.

Aliasing of high angles.

Aliasing of high angles.



Absorbing boundary condition

0 50 100 150 200 250m

0.5

0.6

0.7

0.8

0.9

1

T

0 50 100 150 200 250m

0.5

0.6

0.7

0.8

0.9

1

T

z [cells] z [cells]

x [c

ells

]

x [cells] x [cells]

T

( ) ( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=⎟⎠⎞⎜

⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡ −+⎟⎠⎞⎜

⎝⎛ −

+−+=

=⎟⎠⎞⎜

⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡ −+⎟⎠⎞⎜

⎝⎛ −

==

xwxw

xw

wxw

ww

w

NNNmN

NmNNNNm

NmN

mN

zxmxT

…

…

…

21cos

21

111

1211cos

21

,γπγ

γπγ

δβ

β

Adjustable parameters: γ the absorber strength, Nw the absorber width in units of δx.

40,5.0 == wNγ 5,5.0 == wNγ

x: 256 cells of λ/2 z: 500 cells of 2 λ

x: 256 cells of λ/2 z: 500 cells of 2 λ

( )( )xjkx

exE!10sin

30

2

0,−−⎟

⎠⎞⎜

⎝⎛−

=

40 cells 5 cells

• Fourier beam propagation – Enhancements to method

–  ABC



Example of conservation checking Conservation of E in FFT BPM

µm0.15.1005.µm4

====

λ

δ

cladnnd

Problem: Verify guided mode of step-index slab waveguide by invariance to propagation. Step 1: Calculate mode with slab mode solver:

Step 2: Set dn in the BPM and launch this field

Note use of average index at guide edges.

• Fourier beam propagation – Validation & step size choice

x [mm] -10 -5 5 10

E [V

/m]

0

1

x [mm] -2 2

dn

0

0.005



X=128 x 0.5 µm, Z = 10K x 10 µm no ABC

0 2000 4000 6000 8000 10000iz

20.02

20.04

20.06

20.08

20.1

P

0 20 40 60 80 100 120i

0

0.25

0.5

0.75

1

1.25

1.5

1.75

»E»

iz

ix

|E|

Power conserved

But clear reflections


|E(x,0)| |E(x,zmax)|



X=128 x 0.5 µm, Z = 10K x 10 µm 5 cell ABC each side

0 10 20 30 40 50 60i

0

0.25

0.5

0.75

1

1.25

1.5

»E»

ix

But no reflections

|E|

0 2000 4000 6000 8000 10000iz

10

12

14

16

18

20

P

iz

Power NOT conserved





Largest possible Δz

zk

xkzkΔ

xk

When applying refraction operator, we assume that diffraction is negligible. This requires that the phase acquired by different transverse spatial frequency components be nearly the same:

( ) ( ) ( )zkkjx

zkj zeke Δ−−− + EE 0

π=Δ≈Δ−−=ΔΔ⎥⎥⎦

⎤

⎢⎢⎣

⎡ zkkzkkkzk x

xz 2

222

22

xk

kz π=Δ

For an object of finite size L

Lkx

π2=λ

2

nDiffractioLz =Δ

245 250 255 260 265 270i

0

0.001

0.002

0.003

0.004

0.005

dn

Similarly, during application of the diffraction operator we assume refraction is negligible. This requires phase at different positions be the same:

( ) ( ) ( ) ( )zxnkjzxnkj exEexE 201021

δδ −− +π=ΔΔ znk0

nz

Δ=Δ20

Refractionλ

We would expect these to be the correct magnitude but too large.

=14.6 µm

=100 µm




Try twice maximum Δz step z=500 steps of 2*14.6 µm

0 100 200 300 400 500iz

0

5

10

15

20

P

0 100 200 300 400 500i

0

0.25

0.5

0.75

1

1.25

1.5

»E»

iz

ix

|E|

Power really not conserved! Thus the extra-long step in z is breaking the guiding, allowing light to leak and be absorbed by the ABC.

Almost nothing left at end





Try maximum Δz step z=1000 steps of 14.6 mm

0 200 400 600 800 1000iz

2.5

5

7.5

10

12.5

15

17.5

20

P

0 100 200 300 400 500i

0

0.25

0.5

0.75

1

1.25

1.5

»E»

iz

ix

Better but not great

|E|





Half maximum Δz step z=2000 steps of 14.6/2 µm

0 500 1000 1500 2000iz

10

12

14

16

18

20

P

0 100 200 300 400 500i

0

0.25

0.5

0.75

1

1.25

1.5

»E»

Field at end of guide beginning to look like launch

iz

ix

Only ½ of power lost

|E|





1/4 maximum Δz step z=4000 steps of 14.6/4 µm

0 1000 2000 3000 4000iz

19.2

19.4

19.6

19.8

20

P

Electric fields at entry and exit overlap.

0 100 200 300 400 500i

0

0.25

0.5

0.75

1

1.25

1.5

»E»

iz

ix

Pretty good. Only 4.5% of power lost

|E|

Therefore

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

≤Δλ

λ 20 ,

2min41 L

nz





Acoustooptics (1/3)

• Fourier beam propagation – Active optics

Typically grating is quite weak so can approximate the refraction step as

( )( ) ( )( )xKtjxKtj

znkj

eeCzk

xKtCzkjznkje

−Ω−−Ω

Δ−

−Δ−=

−ΩΔ−=Δ−≈

21

sin11

0

0

00 δδ

Jarem and Banerjee present an algorithm that simultaneously simulates both the positive and negative side-bands using shifts in the Fourier space to represent the grating, but there’s a problem with this. Each side band is positive or negative Doppler shifted, which we can not simulate in a one single-frequency simulation. Thus the two terms will appear to interfere, but should not. Solution 1: Ignore problem but examine fields only where diffraction off of individual side-bands don’t overlap Solution 2: Only simulate one side-band since Bragg matching typically makes only one important.



Acoustooptics (2/3)

246

Focused incident field

Diffracted field

Propagation of acoustics

Perfect Fourier transform lens

Undiffracted field

FT Image plane

va (t)


Portion of index structure at one time instant



Acoustooptics (3/3)

247


240 MHz

280 MHz

320 MHz

360 MHz

400 MHz

440 MHz

01020304050607080

100 150 200 250 300 350 400 450 500Freqeuncy (MHz)

Diff

ract

ion

Effic

ienc

y (%

)

Experiment

Numerical Modeling

∆ F

•  “Hole” carved in angular spectrum of incident beam.

•  Bragg matching causes “hole” to change in location

•  Efficiency of diffraction depends on distribution in K-space

•  Matches well to experiment

•  Offset likely due to small rotation of crystal around acoustic propagation direction

•  Accurately predicts bandwidth of device

Sarah Walter, 2004 NMiP class project



Integrating nonlinearity e.g. photopolymer

0 50 100 150 200 250 300

0

100

200

300

400

500

0 50 100 150 200 250 300

0

100

200

300

400

500

0 50 100 150 200 250 300

0

100

200

300

400

500

0 50 100 150 200 250 300

0

100

200

300

400

500

0 50 100 150 200 250 300

0

100

200

300

400

500

|E1| |E2|

|E4| |E3|

|E5| ( ) ∫ −= dtEzxn 2310,δ

Filament forms. No saturation in this model, which might stabilize




Kerr nonlinearities

( ) 220 EnnEn +=

Simple scalar, instantaneous χ3 nonlinearity

In the mid 80’s, amplifiers on the Nova laser had the habit of developing tiny pinholes in the direction of propagation. This was identified as nonlinear focusing. Mike Feit and Joe Fleck successfully predicted the formation of high-intensity filaments using nonparaxial beam propagation. Conservation of energy in the simulation is essential (several early works used a difference scheme that gave nonphysical results due to numerical nonlinear absorption).

0 50 100 150 200 250 300

0

100

200

300

400

500

|E| Transverse sampling 512 steps of 1 λ0 = 1 µm


( )2

300,⎟⎠⎞⎜

⎝⎛−

=x

exE

( ) 2310, Ezxn −=δ




Solitons

( ) ( )kxkn

x κκε hsec02

=

Substitute the equation for n into the wave equation to get the nonlinear Schrodinger equation.

where κ is a free parameter

Transverse sampling 512 steps of λ0 / n0 = 1/1.5 µm


κ = 1/ 50 1µm

⎡⎣⎢

⎤⎦⎥

( ) 2310, Ezxn −=δ

εεε 220

2

21 Enjkjkz

−∇=∂∂

⊥

In one dimension this has a stable solution:


k ≡ k0n0



Solitons in multiple dimensions (x,y,t)

Write the permittivity as a sum of frequency dependent dispersive linear and instantaneous nondispersive nonlinear parts

or, taking ε = n2 = (n0 + δn)2 ≈ n02 + 2 n0 δn

Shift into the GV coordinate system and scale the field and coordinates

to give the unitless, 3+1 dimensional nonlinear Schrodinger equation (NLS)


R. Mcleod, K. Wagner and S. Blair, “3+1 dimensional optical soliton dragging logic”, Phys Rev A, vol. 52(4), pp. 3254-78, Oct 1995.



Eigen-solutions of NLS Assume a stable (eigen) solution that propagates only with a change of phase:

which reduces the d-dimensional NLS to an ordinary DE in the radial coordinate, r

We can deduce a scaling relationship from this equation

Solutions for d=3, a=1:




Solution in various coordinate systems

To solve the diffraction portion of the d+1 dimensional NLS, we need to express the field u as a sum of the eigenfunctions of the Laplacian:

Since Γ is an orthogonal, complete basis in d dimensions, we can write the transform pair:

In rectangular, cylindrical, and spherical coordinates, Γ has the following forms

where Jm and Ym are Bessel functions, jl and yl are the spherical Bessel functions and Yl

m is the spherical harmonic




Soliton stability by BPM in spherical coordinates

Solitons are stable to propagation when dP / dβ > 0. Using the scaling relationship, we can find

To verify this prediction, let us assume only a radial dependence to the field u and propagate in this reduced spherical coordinate system. The eigenfunctions of the Laplacian reduce to:

Stable

Marginally stable

Unstable

The transformation into the radial k-space can then be written with Fourier transforms:

Forward transform

Reverse transform


A. A. Kolokolov, Journal of Applied Mechanics and Technical Physics, V 3, p 426, 1973



Results of radial-spherical BPM with saturating n2

Replace n2 |E|2 with n2 |E|2 /(1 + |E|2 /E2sat). Previous stability test now predicts

stable 3+1 light-bullets when E(0) ≥ Esat

Distance of stable propagation vs strength of saturation

Three radial-spherical BPM results for linear medium, E(0) = 95% Esat and E(0) = 1.01 Esat




Vector nonlinear propagation

As usual, write the linear material properties for the ordinary and extra-ordinary polarization

but now let them be coupled by the fourth-order nonlinear susceptibility tensor. This is assumed to be instantaneous and thus nondispersive.

Which results in a coupled vector non-paraxial wave equation:




Small perturbation and phase matching

Assume perturbations are small, which reduces previous equation to two coupled scalar wave equations

In an anisotropic material, only a few terms of the nonlinear perturbation are phase matched

allowing us to collapse the 4th order tensor expression to self and cross-phase modulation indices:

021

021

2

,2

,222

2

,2

,222

=⎟⎟⎠

⎞⎜⎜⎝

⎛++∇+

∂∂−

=⎟⎟⎠

⎞⎜⎜⎝

⎛++∇+

∂∂−

extordselfext

crossext

extextext

ordextselford

crossord

ordordord

uunn

uuuj

uunn

uuuj

ξητ

ξητ

ζ

ζ

resulting in two coupled NLS equations




Interaction of two light-bullets Small angle




Interaction of two light-bullets Large angle




Spatio-temporal BPM in anisotropic, nonlinear media

1.  Transform into group-velocity and anisotropic walk-off coordinate system

a)  Removes first-order terms in kz

2.  Propagate two polarizations in independent spaces

a)  Polarization basis should be eigenpolarizations of unperturbed medium

3.  Couple polarizations by nonlinearity calculated in real-space for each grid




Energy conservation

• Fourier beam propagation – Conservation of energy

( ) ( )[ ] ( ){ } ( ) ( )22,1 0 Δz/x,y,zTeex,y,zEΔzx,y,zE ΔzΔz/x,y,zδnkjΔzkω,kjkxyxy

zxz +=+ +−−− FF

Diffraction Refraction

I prefer to add an explicit absorber at each plane rather than a complex δn, but they are equivalent.

Transmission

Using Parseval’s theorem and the fact that δn and kz are real. Note kz real only for propagating fields (not evanescent) – power is lost for evanescent modes.

( ) ( )[ ]

( )[ ] ( )

( )[ ] ( ){ }

( )[ ] ( ){ }( )( )

dydxΔz/x,y,zTΔzx,y,zE

dydxeex,y,zEF

dydxex,y,zEF

dkdkex,y,zE

dkdkx,y,zEdydxx,y,zE

ΔzδnkjΔzkω,kjkxyxy

Δzkω,kjkxyxy

yxΔzkω,kjk

xy

yxxy

yxz

yxz

yxz

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫∫ ∫

∞

∞−

∞

∞−

−−−

∞

∞−

−−

∞

∞−

−

∞

∞−

∞

∞−

+

+=

=

=

=

=

2

2

2,1

2,1

2,

22

2

0F

F

F

F Parseval

kz real

Parseval

dn real

FFT-BPM algorithm (from above)

( ) ( ) ( ) dydxzyxEzyxTdydxzzyxE ∫ ∫∫ ∫∞

∞−

∞

∞−

=Δ+′ 222 ,,,,,,

Let the field just after the transmission mask be E’: E’(x,y,z) = E(x,y,z) T(x,y)



1+1 dimensional TM propagation

011

02

2

2

222

02

220

2

=∂∂

∂∂−

∂∂

∂∂−+∇

=+∇

xH

xn

nzH

zn

nHnkH

EnkE

yyyy

yy

Exact equations for TE (top) and TM (bottom) fields:

Let us make a substitution in the TM wave equation

( ) ( ) ( )zxFzxnzxH y ,,, =

We now find that the “field” F satisfies the following wave equation:

0220

2 =+∇ FnkF equivwhere

( ) ( ) ( )( )⎥⎦

⎤⎢⎣

⎡∇−=zxnk

zxnzxnzxnequiv ,

1,,, 2

20

222

Which can now be propagated in our regular scalar/TE BPM. Note that the equivalent index is singular at material discontinuities. I usually handle this by smoothing my waveguides slightly (typically with a second-order interpolation to the step).

Poladian and Ladouceur, IEEE PTL, V10, 105-107


–  TM propagation



BPM of di/converging waves Required spatial sampling


Consider the problem of modeling a converging wave:

To avoid aliasing of phase fronts at the edge of the converging beam, let the phase difference between adjacent cells at edge of beam be < π

θ 0 w o

z = 0

NAzx ⋅=Δ 2

-z

Binomial

Keep term of order N

(δx N)/(2z) = NA

Nyquist sampling

Sampling must be ~w0 IN ALL SPACE

[ ] ( )[ ] ( )[ ]{ }( )[ ] ( )[ ]

00

0

2

0

22

0

2222

02/12/

0

22

2

22122

21222

wNA

x

NAx

Nxz

zNxNx

zNxzNxrr NN

πλδ

π

δλπ

δλπ

δδλπ

δδλπ

λπ

=<

<

=

≈

−+≈

+−++=−+



BPM of di/converging waves Number of samples required

( )

0

0

0

2

0

4

42

2

zzzz

zNANA

NAzxxN x

≈

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

⎟⎠⎞⎜

⎝⎛

⋅>

Δ=

π

λ

λ

δ

Number of cells along one side is equal to distance from focus in units of Rayleigh range:

2

0⎟⎟⎠

⎞⎜⎜⎝

⎛>zzNN yx

Example: 0.6 NA beam 4 mm from focus

Nx Ny > (4000/0.47)2 = 85002 = 72M cells = ½ GB single precision complex for single z plane = 1 GB min unless using in-place FFT

Fine sampling and large beam size result in excessive memory requirements.

# of cells in one transverse direction.

From previous page.

Using expression for z0




Sziklas and Siegman algorithm Mapping of converging to regular propagation

265

02 2 =+∇∂∂− EEz

jk xy

( ) ( ) ( )zyxEezzyxEyx

zkj

,,,,22

2++

=′′′′

( ) ( )( ) ( ) ( )zj

zRyxjkjkz

zwyx

ezwwArE

ζ++−−+−

= 200

22

2

22

! ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞⎜

⎝⎛+≡

201zzzzR

The Gaussian beam

is a solution to the homogeneous paraxial wave equation

This motivates a coordinate transform in which the parabolic phase curvature is removed from the field:

Origin

Origin

zzzz

zzyy

zxx

−≡′≡′≡′ 2,, ααα

The new field E´ obeys the paraxial wave equation

02 2 =+ ′∇′′∂

∂− ′′ EEz

jk yx

if


z coordinate here is measured from focus, so this implies that an invariant coordinate grid x´ would converge approaching focus and diverge leaving focus like

xzx ′=α



Sziklas and Siegman algorithm Mapping of converging to regular propagation

θ 0

z = 0

NAzx ⋅=Δ 2

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

incinc z

zxzx δδ

Region near focus must use regular propagation and thus is subject to phase curvature & aliasing restriction. As cell size decreases and distance to focus decreases, sampling density decreases.


Divide field by Gaussian phase matched in curvature to field Propagate in uniform (not converging space). Cell size dynamically changes proportional to distance from focus:



Sziklas and Siegman algorithm An adaptive grid for spherical waves

Regular FFT propagation

Sziklas FFT propagation

Non-paraxial imaging of a rectangular aperture in 1+1 dimensions


E. A. Sziklas, A. E. Siegman Applied Optics 14, pp. 1874-1889, 1975



Tilted planes Fourier-space rotation

N. Delen, B. Hooker, JOSA A, V 15, N 4, 857-867, April 1998

kx

kz

k!

x

z

Real space Fourier space

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

′′′

zyx

zyx

Rzyx

ψψ

ψψ

cos0sin010sin0cos

ψ ψ

kRk!!

=′

Coordinate transformation for a point P to a point P’

Then in Fourier space

xkδ

xk′δ

ψψψ

ψψ

sin1682

cossin

sincos

5

6

3

42

22

⎟⎟⎠

⎞⎜⎜⎝

⎛+++++−≈

−−=′

…kk

kk

kkkk

kkkk

xxxx

xxxSampling in new space is not regular

even to first order in ψ

Thus must interpolate back to regular grid in k’x which is not numerically accurate

Discard

Assum

e=0


–  Imaginary distance method



Tilted planes Real-space propagation version

( ) ( )[ ] ( ){ }( ) ( ){ }

( ) ( )[ ]

( )∑

∑⋅−

++−

−−

−−

==

==

==

=

yx

yx

yxzyx

yxz

yxz

,kk

rkjyx

,kk

zkω,kkykxkjyx

zkω,kjkyxxy

zkω,kjkxyxy

e,z,kkE

e,z,kkE

e,z,kkE

ex,y,Ex,y,zE

!!

0

0

0

0

,

,1

,1

F

FF

x

z ψ

Write the free-space propagation algorithm:

Define the new tilted plane as a set of vectors that regularly sample the space

yynxxmr nm ′′+′′= ˆˆ, δδ!

( ) ( )∑ ⋅−==′=′′=′yx

nm

kk

rkjyx ezkkEynyxmxE

,

,0,,,!!

δδ

And use the propagation formula to find E on this tilted, regularly sampled grid

Now use a regular FFT to find the spectrum for propagation in the tilted coordinates

( ) ( )[ ]00 ,y,xE,zk,kE yxyx ′′==′′ ′′F

This is a irregular, discrete inverse FT. The sum is over the regularly sampled spatial frequencies kx and ky. As shown on the previous page, those spatial frequencies in (x,y,z) that correspond to evanescent waves or backwards traveling waves in (x’,y’,z’) should be discarded.

Note that for E in the x,z plane there will also be a polarization projection.


–  Tilted planes



Propagating in curved spaces Heuristic derivation

Pollock and Lipson 8.6.1, pg. 179

z

x

y z

x

y θ

r

Straight waveguide Bent waveguide, radius R

The waves propagating in the curved space accumulate extra phase proportional to distance from the center of the waveguide.

( )

( )

( ) ( )

( )[ ]zxnjk

RzxRxnjk

rxnjk

zxnjk

Rx

eyxE

eyxEeyxEryxeyxEzyx

)1(

/)(

0

0

0

0

)0,,(

)0,,()0,,(),,(Ε)0,,(),,(Ε

+−

+−

−

−

=

=

=

=

δ

δ

θδ

δ

θ

Usual propagator

Equivalent in curved space

Map to prev. coordinates

Equivalent index

x x

ncl nco nco

0 0

( ) )1( Rxxn +δ( )xnδ

Thus we can simulate (and understand) curved waveguides via a scaled dn.


–  Curved propagation



Propagating in curved spaces Conformal transform



( )[ ] ( ) 0,,220

2 =+∇ yxEzxnkxz

Consider a field in a 2D (x,z) plane described by This formally applies only TE propagation of a field Ez given a physical structure that is invariant in y. With minor modifications, it can describe TM propagation or waveguides with confinement in the y direction via the effective index approximation. Apply a conformal transformation to a new set of coordinates u (radial) and v (azimuthal). The proper transform is:

22

22 lnln

RjzxR

RZRjvuW +==+≡

( ) ( )( )[ ]{ } ( ) 0,,,,22

02 2 =+∇ yxEvuzvuxnek Ruuv

Which gives the new wave equation:

( ) ( )

( ) ( )ρ

ρ

nuORuneun Ru

⎥⎦⎤

⎢⎣⎡ ++≈

=

21

2

Heilblum & Harris, IEEE J. Q. E., VOL. QE-11, pp. 75-83, 1975

22 ln RRu ρ=

2Rv θ=

which is the previous expression

Application:



BPM in straight guide

Launch mode from slab waveguide solver. Mode travels > 3mm without significant chage.

|E| vs. x,z

50068.1001.n5.1

m][ 1m][ 5

0

==Δ===

eff

co

N

nµµa

λ

Neff



δn

0

0.001



Launch mode from slab waveguide solver. Mode at exit is leaky, lower total power and shifted towards larger radius

|E| vs. x,z . The plotting coordinate system has been warped to match the curvature of the space. The original propagation, however, is simply rectangular BPM with the index above.

BPM in curved guide

Radius of curvature, R 20 mm



Neff Lossy Tail

xEscape

δn

0

0.001

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