E n+1 = E(x + x ) = E(x) + E (x) x + 1 2! E (x) 2 x + 1 3! E (x) 3 x +... = 0. δ x E = 1 2 δ. δ 2 xe n = δ x. Paraxial Diffraction

(1)

Scalar Wave Eqn in 1+1D expanded as envelope and carrier

(∇²+ k²₀)E = 0 with E =E(x, z)e^ik⁰^z

∂²E

∂z² +∂²E

∂x² + k₀²= 0 = e^ik⁰^z



_✓^✓^✓^✓

✓

✼0^{SV EA}

∂²E

∂z²+ i2k0

∂E

∂z −^❙k❙❙²₀+∂²E

∂x²+^❙k❙❙²₀



 = 0

leads to evolution eqn

i2k0∂E

∂z +∂²E

∂x² = 0

If we have sampled representation of the transverse field E(x) → Enwe will need to evaluate derivatives along x to advance the field along z.

Which side should we take difference on?right or left?

dE

dz ≡En+1− Eⁿ

∆x

= δ_x^r dE

dz ≡En− Eⁿ−1

∆x

= δ^l_x

simple, but not very accurate. Errors build up with the number of numerical integra- tions.

Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 464

Numerically Evaluating Derivatives

En+1= E(x + ∆x) = E(x) + E^′(x)∆x+_2!¹E^′′(x)∆²_x+_3!¹E^′′′(x)∆³_x+ . . .

⇒ δ^rx= E^′(x) +∆x

2 E^′′(x) + . . . δ^l_x= E^′(x)−∆x

2 E^′′(x) + . . . Both approximations have error at 2nd order (are thus 1st order accurate).

How about averaging these two estimates δxE =¹₂ δ_x^r+ δ^l_x

E =En+1− En−1

2∆x

= E^′(x) +∆²_x

3 E^′′′(x) + . . . ∼= dE dx Much better: 2nd order accurate!

Second derivative δ_x²En= δx

En+1− Eⁿ−1

2∆x

=

E_n+2−En

2∆_x −^Eⁿ^−E2∆_xⁿ⁻²

2∆x

=En+2− 2Eⁿ+ En−2

(2∆x)² But better to use adjacent points with same 2nd order difference scheme

d²E

dx² ∼= δ^l_xδ_x^rEn≡En+1− 2Eⁿ+ En−1

(∆x)²

Runge-Kutta

Consider an evolution eqn y^′= f (x, y)

eg [x → z, y → E] inhomogeneous absorption or 2-photon nonlinear absorption

dE

dz =−α(z)E dE

dz =−β2|E|²E Euler method has O(∆x)error

yn+1= yn+ ∆xy_n^′ = yn+ ∆xf (xn, yn) For O(∆²x)error use 2nd order Runge-Kutta

yn+1= yn+¹₂[∆xf (xn, yn)

+∆xf (xn+ ∆x, yn+ ∆xf (xn, yn)]

y

x yⁿ

yⁿ⁺¹ y(x)

n∆x (n+1)∆x

y

x yn

yn+1 y(x)

n∆x (n+1)∆x

}

Q Q Q

Q Q Q Q

Q Q Q Q Q Q

Q Q

Q

10⁻⁶ 10⁻⁴ 10⁻² 10⁰

relative error

errors in Runge−Kutta First order

Second order

PDEs

Paraxial Diffraction

∂E

∂z + 1 i2k0

∂²E

∂x² = 0 Normalize variables x ← xk⁰

∂

∂zu(x, z) + i∂²

∂x²u(x, z) = 0

Represent sampled field at x = j∆x, z = n∆zas uⁿ_j. Simple discrete evolution eqn uⁿ⁺¹_j − uⁿj

∆z

= iuⁿ_j₋₁− 2uⁿj + uⁿ_j+1

(∆x)² ⇒ uⁿ⁺¹_j = uⁿ_j + ∆ziuⁿ_j₋₁− 2uⁿj + uⁿ_j+1 (∆x)² Canexplicitly advance field in z using known uⁿj. Unfortunately this isunstable.

implicit scheme calculates transverse spatial derivative_∂x2^∂2 after evolution step in z uⁿ⁺¹_j − uⁿj

∆ = iuⁿ⁺¹_j₋₁ − 2uⁿ⁺¹j + uⁿ⁺¹_j+1 (∆ )²

(2)

Tridiagonal System

uⁿ_j =−

a_j

z}|{

i^∆_∆^z₂

x uⁿ⁺¹_j−1+

b_j

z }| {

hi2^∆_∆^z₂

x− 1i uⁿ⁺¹_j −

c_j

z}|{

i^∆_∆^z₂

x uⁿ⁺¹_j+1





 uⁿ₁ uⁿ₂ ...

uⁿ_j₋₁ uⁿ_j uⁿ_j+1

...

uⁿ_J







=





 b1 c1

a2 b2 c2

... ... ...

aj−1 bj−1 cj−1

aj bj cj

aj+1 bj+1 cj+1

... ... ...

aJ bJ











 uⁿ⁺¹₁ uⁿ⁺¹₂ ...

uⁿ⁺¹_j₋₁ uⁿ⁺¹_j uⁿ⁺¹_j+1 ...

uⁿ⁺¹_J





 soln

start with β1= b1, γ1= uⁿ₁/b1then iterate to j = 2, ..., J βj= bj−ajcj−1

βj−1

γj=uⁿ_j − a^jγj−1

βj

uⁿ⁺¹_j = γj−cjuⁿ⁺¹_j+1 βj

and uⁿ⁺¹_J = γJ

Stability Analysis

Consider a plane wave u(x) = eîkx≡ eîk∆^x^j= eîκj Solution with z (eg n) will be of the form

u(n, j) = ξⁿe^iκj

The stability is determined by |ξ(κ)| > 1 unstable for some κ. For explicit scheme δ_z^ru(n, j) = iδ²_xu(n, j) = ieîκ(j⁻¹⁾− 2eîκj+ eîκ(j+1)

(∆x)² ξⁿ=−iξⁿ e^iκj

(∆x)² e^−iκ− 2 + e^iκ uⁿ⁺¹_j − uⁿj =−iξⁿe^iκj 4

(∆x)²sin²(κ/2) uⁿ⁺¹_j = ξⁿe^iκj− i∆^zξⁿe^iκj 4

(∆x)²sin²(κ/2)

= ξⁿe^iκj

1− i 4∆z

(∆x)²sin²(κ/2)

| {z }

= ξⁿ⁺¹e^iκj

|ξ| > 1 for ALL κ ⇒ unstable For κ ≪ 1 |ξ| ∼= 1and OK accuracy but for κ ∼= 1, get rapid growth

Noise associated with high frequencies ( that we are not interested in) blows up

Lax method for improving stability

Replace uⁿ_j →¹2(uⁿ_j+1+ uⁿ_j₋₁)with smoothed average in longitudinal derivative uⁿ⁺¹_j − uⁿj

∆z

= iuⁿ_j−1− 2uⁿj+ uⁿ_j+1

(∆x)² ⇒ uⁿ⁺¹_j −¹2(uⁿ_j+1+ uⁿ_j₋₁)

∆z

= iuⁿ_j−1− 2uⁿj + uⁿ_j+1 (∆x)²

uⁿ⁺¹_j =¹₂(uⁿ_j+1+uⁿ_j₋₁)+i∆zδ²_xuⁿ_j=¹₂ξⁿ

e^iκ(j+1)+e^iκ(j⁻¹⁾ +i∆z

eîκ(j⁻¹⁾−2eîκj+eîκ(j+1) (∆x)² ξⁿ

=¹₂ξⁿe^iκj2 cos κ− i∆z

∆²_xe^iκjξⁿ4 sin²(κ/2)

Stability Conditioncos κ − i4^∆∆^z²_xsin²(κ/2) ≤ 1 ⇒ cos²κ+ 4_∆^∆^z₂

xsin²(κ/2)2

≤ 1

1−κ²

2

+ 4²

∆z

∆²_x

2κ

✓✓2

4

≤ 1 ⇒ ∆z

∆²_x<

√2 4 Why? rewrite

uⁿ⁺¹_j − uⁿj =¹₂ uⁿ_j+1− 2uⁿj + uⁿ_j₋₁

+ i∆zδ²_xuⁿ_j ⇒ i∂

∂zu +

1− i∆z

∆²_x

∂²

∂x²u again for κ ≪ 1 (low freq) |ξ| ∼= 1and good accuracy. Dissipitave term added to eqn for κ ∼= 1 dz|ξ| < 1 and exponential decay (high frequency noise decays away).

Crank-Nicholson

Why use uⁿ⁺¹_j after propagation for_∂x^∂²2derivative instead of values of uⁿ_j before?

How about averaging these two approaches uⁿ⁺¹_j − uⁿj

∆z

= i 2

"

uⁿ⁺¹_j₋₁− 2uⁿ⁺¹j + uⁿ⁺¹_j+1

(∆x)² +uⁿ_j₋₁− 2uⁿj + uⁿ_j+1 (∆x)²

#

unconditionally stable! second order accurate in both x and z implicit, requires tridiagonal solution

a b c d e f

− z}|{i 2∆²_xuⁿ⁺¹_j₋₁+

z }| {

i

∆²_x+ 1

∆z

uⁿ⁺¹_j −

z}|{i 2∆²_xuⁿ⁺¹_j+1 =

z}|{i 2∆²_xuⁿ_j₋₁−

z }| {

i

∆²_x− 1

∆z

uⁿ_j +

z}|{i 2∆²_xuⁿ_j+1







b ca b c

... ... ...

a b c

... ... ...

a b













uⁿ⁺¹₁ uⁿ⁺¹₂

...

uⁿ⁺¹_j−1 uⁿ⁺¹_j uⁿ⁺¹_j+1

...

uⁿ⁺¹_J







=







e f d e f

... ... ...

d e f

... ... ...

d e













uⁿ₁ uⁿ₂ uⁿ_j−1...

uⁿ_j uⁿ_j+1

u...ⁿ_J







(3)

Split-Step

i∂

∂zE + L{E} + f(|E|²)E = 0

Where L{E} is a linear operator (eg _∂x^∂²2, or nonparaxial diffraction) and f() is an operator describing the inhomogeneities and/or nonlinearities

integration of this type of first order equation results in exponential. In the case of just the linear operator

E(z + ∆z) = e^i∆^z^LE(z) = e^i∆^z^∂x2^∂2E(z) =X

m

i∆z∂²

∂x²

m

m! E(z)

While just in the presence of nonlinearity/inhomogeneity E(z + ∆z) = e^i∆^z^{f (∆}^z^/2)E(z)

In the presence of both

E(z + ∆z) ∼= eî∆^z^Leî∆^z^{f (z)}E(z) = eî∆^z^{f (z+∆}^z⁾eî∆^z^LE(z)

= eî∆^z^L/2eî∆^z^{f (z+∆}^z^/2)eî∆^z^L/2E(z)

Symmetrized split-step approach gives 2nd order accurate result

∆z 2∆z

2

∆z Evaluate inhomogeneity in middle of step

Beam Propagation, sampling, and Fourier Space

Suppose we know a sampled version of our field at some transvers plane z = 0, and wish to propagate it to subsequent planes z = m∆z, while obeying the differential equations containing terms such as diffraction, index inhomogeneities and lenses, absorption variations, and other inhomogeneous and nonlinear effects.

∆x=2λ

∆x=λ

∆x=λ/2

∆x

∆z E(n∆x)

x

∆k= 2π Ν∆x

Real Space

Fourier

Space k-space

Advance Sampled Field

2-wavelengths

Beam Propagation: Paraxial vs Nonparaxial

Given input on planar boundary, find FT

E(k^x, ky) =F^xy{E(x, y)}

Each transverse component propagates with its own phase factor kz=q

k₀²− kx²− k²y= k0

s

1−k_x²+ k_y²

k²₀ =≈ k⁰−k_x²+ k²_y 2k0

Transfer function (phase only for propagating kx, kysuch that k_x²+ k²_y≤ k²0) Hz(kx, ky) = e^−ik^z^(k^x^,k^y^)z= e^−iz√

k₀²−k²x−ky²≈ e^−ik⁰^zeⁱ

k2x+k2y 2k0 z

Fourier spectrum at a distance z

E(k^x, ky; z) =E(k^x, ky; 0)Hz(kx, ky)

k-space version of transfer function vs Feit and Fleck

Phase rate of the transfer function e^ik^z^z= eⁱ√

n²k²₀−kx²−k²yzwith z can be written as kz=

r n²ω²

c² +∇²_⊥=q

n²k₀²− kx²− k²y= nk0+q

n²k²₀− k²x− k²y− nk0

Where the second form separates out the fast varying and slower varying part, but when used over very wide angular ranges would be superflous.

Feit and Fleck use a less intutive version when introducing Fourier beam propagation

e^iβz= e

iz

"

k+ ∇2⊥

k+√_k2+

∇2T

#

= e

iz

"

nk₀− ^k2^x+k²^y nk0+√

n2k20−k2x−ky2

#

Equivalence can be shown by multiplying thru by the denominator q

n²k²₀− kx²− ky²

h

nk0+q

n²k²₀− k²x− ky²

i

= nk0

h

nk0+q

n²k²₀− k²x− k²y

i−kx²+k_y²

nk0

qn²k²₀− k²x− ky²+ nk₀²− kx²− k²y= n²k²₀+ nk0

qn²k₀²− k²x− k²y− k²x− ky²

(4)

Beam Propagation: Sampling requirement

Consider homogeneous rectangular region of length L and width W (we will do 1-D analysis).

Transversely sample the field at a spacing of ∆x(∆y) being sure to obey the Nyquist condition for a field of bandwidth B.

∆x≤ 1 2Bx

# samples N = W

∆x

This gives a sampled transverse field En= E(n∆x).

Now divide up the longitudinal propagation distance L into M steps of spacing ∆z. How big can we choose ∆z– As big as we want!

(only true for homogeneous case, and can result in periodic wrap around) Typically make phase factor < π for highest frequency component

B²_x 2k0

∆z< π say ∆z< 2πk0

10B_x² or increase spacing ∆zuntil you notice change of the final result.

Beam Propagation Algorithm

Propagate sample field in Fourier domain

FFT {E(n)} = F (n; 0)

F (n; 1) = F (n∆k; 1· ∆^z) = F (n; 0)H1(n)

∆k=^2π_W =_{N ∆}^2π

Transfer function over a distance 1 · ∆x ^z

H1(n) = e⁻ⁱ√

k²₀−(n∆k)²∆_z

≈ e^−ik⁰^∆^zeⁱ^n2(∆k)

2 2k0 ∆_z

Invert transform to get field back in real space.

IFFT {F (n : 1)} = E(n∆^x; 1· ∆^z) Repeat to step through all M slices sequentially.

E(n;m) FFT

F(n;m)

F(n;m+1) E(n;m+1)

IFFT

Repeat Propagate

H (n)₁

Beam Jumping: Only for homogeneous media and without boundary absorptionb

However each time you go around this loop you can accumulate numerical noise Why? isnt digital computing perfect? Roundoff errors

How could you minimize or alleviate the accumulation of such numerical noise? Use double precision complex.

E(n;0) FFT F(n;0)

H (n)1

H (n)2

H (n)3

H (n)4

H (n)m

IFFT E(n;1) IFFT E(n;2) IFFT E(n;3) IFFT E(n;4) IFFT E(n;m) log RMS[E(n;m)-E (n;m)]actual

Avoid any accumulated noise by beam jumping.

FT input field F (n; 0) = FFT {E(n; 0)}

multiply by transfer function to get directly to m∆z

Hm(n) = e⁻ⁱ√

k²₀−(n∆k)²m∆_z

Inverse transform and repeat for all m

E(n; m) =IFFT {F (n; 0)H^m(n)}

Inhomogeneous Beamprop

Inhomogeneous and nonlinear index effects n = n0+ δn(~r, E) k²=

π λ0

2

(n²₀+ 2nδn + δn²)≈ k²0(1 + 2δn/n0) Start with scalar wave eqation

∂²E

∂z² +∂²E

∂x² + k²E = 0 Expand E as envelope and carrier E(x, z) = a(x, z)e^−ik⁰^z Insert into wave eqn and use SVEA^∂_∂z²^a2 ≈ 0

∂²E

∂z² = e^−ik⁰^z

∂²a

∂z²− 2ik⁰∂a

∂z− k²0a

e^−ik⁰^z

−2ik⁰∂a

∂z− k0²a +∂²a

∂x²+ k²₀(1 + 2δn/n0)a

= 0 Evolution equation

∂a

∂z = −i 2k0

∂²a

∂x²+ k²₀2δn n0

a

(5)

Inhomogeneous Beamprop

Formal solution: integrate in z starting with B.C. E(x, 0) = a(x, 0) a(x, z) = e⁻^2k0ⁱ

´_z 0

h∂2a

∂x2+k₀²^2δn_n0ai

dza(x, 0) Seperate into Index step followed by diffraction step

Index step in real space

small step in z, δz – define average inhomogeneous index δn = 1

∆z

ˆ z+∆_z z

δn(x, z)dz 2nd term of operator is multiplication by complex phase factor

N^∆za(x, z) = e⁻ⁱ^2π^λ0^δn(x,z)∆^za(x, z) Diffraction step in Fourier space

FT envelope in transverse x coordinate a(x, z) =

ˆ

A(k^x; z)e^ik^x^xdkx

A(k^x; z) = ˆ

a(x, z)e^−ik^x^xdx

E(x, z) = ˆ

E(k^x; z)e^ik^x^xdkx

E(k^x; z) = ˆ

E(x, z)e^−ik^x^xdx

Inhomogeneous Beamprop

Partial derivative easily evaluated as multiplication in Fourier space_∂x^∂ ⇐⇒ (ik^x)

∂²a

∂x² ⇐⇒ (ik^x)²A = −kx²A A(k^x; z+∆z) =A(k^x; z)e⁻ⁱ⁽√

k²₀−k_x²−k0)∆_z

≈ A(k^x; z)eⁱ^2k0^k2^x^∆^z ⇐⇒ M^∆za(x, z) E(k^x; z+∆z) =E(k^x; z)e⁻ⁱ√

k₀²−k²x∆z≈ E(k^x; z)e^−ik⁰^∆^zeⁱ^2k0^k2^x^∆^z ⇐⇒ M^∆zE(x, z) Complete Propagation step – only 1st order accurate (M∆zN∆z: Why not N∆zM∆z?)

a(x, z + ∆z) =Fx⁻¹

ne^−ik^z^(k^x^)∆^zF^x

a(x, z)e^−ik⁰^δn∆^z o

=M^∆zN^∆za(x, z) Symmetrized split-step operator – 2nd order accurate

a(x, z + ∆z) = M^∆z₂ N^∆zM^∆z₂ a(x, z)

= Fx⁻¹

(

e^−ik^z^(k^x⁾^∆z² F^x

e^−ik⁰^δn∆^zFx⁻¹

ne^−ik^z^(k^x⁾^∆z²F^x

a(x, z) o)

Rectangular aperture diffraction

dark=bright aperture

dark=dim obstruction

Solid:

Exact RS Dotted:

Paraxial Dashed:

BPM Formal Derivation

Evolution Equation to be solved in the form of operator Q = L+N consisting of linear and inhomogeneous parts

∂A

∂z = iQA Formal solution

A(z) = eⁱ^´⁰^z^Q(z^′^)dz^′A(0)

over a small propagation distance ∆zthe linear part, which is constant can be integrated A(∆z) = eⁱ

h´_∆z

0 L(z^′)dz^′+´_∆z 0 N (z^′)dz^′i

= eⁱ

h

∆_zL+´_∆z 0 N (z^′)dz^′i

A(0) Approximate integral over the spatially varying nonlinear/inhomogeneous part ˆ ∆_z

0 N (z^′)dz^′= ˆ ∆_z

0

N

∆z

2

+

z^′−∆z

2

∂

∂zN

∆z

2

+O(∆²z)

dz^′

=N

∆z

2

∆z+1 2

z^′−∆z

2

∆_z

∂

∂zN + O(∆³z) =N

∆z

2

∆z+O(∆³z)

(6)

Symmetrical Split Step

Approximation to integral of nonlinear operator is 2nd order accurate, but we dont know self consistent field at^∆₂^z. BPM approximates this field after linear step of ^∆₂^z neglecting NL contribution. Results in symmetrized split step that is 2nd order accurate in ∆z

A(∆z) = eⁱ^∆z²^Le^i∆z^{N (∆}^z^/2)eⁱ^∆z²^LA(0)

Where the linear propagation is most easily applied in the Fourier domain eⁱ^∆z²^LA(0) =F⁻¹n

eⁱ^∆z²√

k²₀−k²_TF{A(0)}o

Note when applying these steps, we can coalesce adjacent linear half steps into a full step unless we need to know field at full step locations.

A(2∆z) = eⁱ^∆z²^Le^{i∆zN (∆}^z^/2)e|ⁱ^∆z²^L{zeⁱ^∆z²^L} e^{i∆zN (∆}^z^/2)eⁱ^∆z²^LA(0)

= eⁱ^∆z²^Leî∆z^{N (∆}^z^/2)eî∆^z^Leî∆z^{N (∆}^z^/2)eⁱ^∆z²^LA(0)

Numerical Evaluation of BPM Error

ǫ =

pP||A^test| − |A^ref||²

|A^ref|²

Comparison versus minimum step size Comparison versus theoretical solution

Shows second order accuracy up to large steps where it becomes first order because of the innacuracy of the estimation of the field at the half step. Roll-off at small ∆zis due to discretization in the transverse dimension with step size ∆x.

2+1D Beam propagation

Circular Aperture 3+1D Beam Propagation Crossection

(7)

Imaging with Fresnel Zone Plate

✓z✓✓_o²+ h²= (zo+ δ)²=_✓z^✓✓_o²+ 2zoδ + ^✒0

δ² _✓z^✓✓²_i + h²= (zi+ δ^′)²=_✓z^✓✓_i²+ 2ziδ^′+ ^✒0 δ^′2 δ = h²

2zo

δ^′= h² 2zi

∆ = δ + δ^′= h² 2zo

+ h² 2zi

is OPD Successive zones with an aditional half wavelength OPD are

labeled as successive fresnel zones with radial boundaries hm

∆m=mλ 2 =h²_m

2

1 zo

+ 1 zi

⇒ h^m= smλ

1/f =p mλf Area of mth annulus bounded by hm−1and hm

Am= πh²_m− πh²m−1= π(mλf− (m − 1)λf) = πλf circular apertures that consist of N zones will sum on-axis fields out of phase with equal amplitude contributions

h1 h2 h3 h4 h5 h6 h7 h8

zo zi

hm δ δ’

AT OT = A1− A²+ A3− A⁴+· · · ± A^N =

(N odd ≈ A¹⇒ I^{T OT}= A²₁ N even ≈ 0 ⇒ I^{T OT} = 0

2-D crosssections every 8 ^λ from a

D = 16λ Circular Aperture BPM

Circular disk diffraction: Fresnel/Arago’s

Bright Spot Babinet’s principle and quadrature

combination with incident plane wave

u⁰

ua

u0 = +u ua d

u0

ua

ud = −

(8)

No absorbing boundary conditions:

wraparound can interfere producing unwanted fringes

Absorbing boundary conditions: Smoothly taper to avoid reflection at impedance

discontinuities

Circulant wraparound due to FFT computation Abrupt absorbing edge causes reflection due to impedance mismatch

Absorbing Boundary condition to avoid wraparound, smooth to avoid reflection

xi=IndGen(nx)-(nx/2) abwdth=nx/128 boundabsx=(1-exp(-(nx/2-abs(xi+.5))/abwdth)) xphase=EXP((-j*2*!PI*delz*(conj(SQRT(DCOMPLEX((n0)ˆ2-((xind)/xsize)ˆ2))) )))

FOR z = 1, nz-1 DO BEGIN field = FFT(FFT(field,-1)*xphase,1)*boundabsx ENDFOR

Absorbing boundary conditions: power may not be conserved

Beam Propagation in 1-D without Additional Aberrations

(9)

Beam Propagation in 1-D with 1 wave Spherical Aberration

Beam Propagation in 1-D with 2 wave Spherical Aberration

Beamprop through Lens Systems

Double slit diffraction and Fourier Transform Comparison of BPM with theory

Thick Lenses

Thick Singlet Lens with Aberations showing spherical aberration

Multiple Incoherent Input Beam showing field curvature, 1D coma

(10)

BeamPropagation through 4F lens system

4F lens system with Schlieren filter Converts Phase Modulation to Amplitude

(11)

4F lens system with Zernike Phase contrast Converts Phase Modulation to Amplitude dot

Beamprop in coupled singlemode waveguides

Waveguide separation .5 λ Waveguide Coupled Modes

Waveguide separation 2 λ Waveguide Coupled Modes

Beamprop in wide multimode waveguides

Waveguide separation 0 λ Waveguide Coupled Modes

Wide multimode waveguide Waveguide Coupled Modes

(12)

Arc in k-space from Fourier transform of BPM of Gaussian Beam

2+1D Beam Propagation of Nonparaxial Spot array: Aberration of Free Space

spot array through focus showing off-axis aberrations of free space

3D Fourier space of nonparaxial spot array

(13)

k-space propagation

Projection of 3-D k-sphere onto ^k

x

− k

z

plane to get FT of x-z crosssection of circular

aperture

Comparing k-space (optics centric) vs

the Ewald sphere (Material centric)

for X-ray discrete lattice diffraction

(14)

k-space and the McCutchen Theorem

s

P

O W

Q

f

q R

k-sphere

2π λ

k^z kx

Aperture

R

k

Axial Converging Spherical wave Cut out by aperture

s− f = ˆq · ~Rfor P near geometric focus dS = f²dΩ≈ s²dΩ

McCutchen Theorem

C.W. McCutchen, Generalized Aperture and the Three-Dimensional Diffraction Image, JOSA Vol 54, pg 240, 1960.

U ( ~R) =Ae^−ikf iλf

¨

W

e^iksi

s dS = A iλ

¨

Ω_W

e^{−ikˆq· ~}^RdΩ = 1 iλ

˚

e^{−ik ~}^{Q· ~}^RdVk

ˆ

qis a unit vector over all angles

Q~ is not a unit vector, so must enforce A(Q) = Q(q)δ(|Q| − 1).

McCutchen Theorem

C.W. McCutchen, Generalized Aperture and the Three-Dimensional Diffraction Image, JOSA Vol 54, pg 240, 1960.

3D Annular aperture is 2D projected onto the z-axis as a rectangular projection so 1-D slice through z-axis in region of focus is sinc

Gaussian beam illuminating aperture has one sided exponential projection so z-axis slice is Lorentzian

Wide Angular diffraction and Fourier space projection- slice : .4

(15)

Wide Angular diffraction and Fourier space projection- slice : .512

Wide Angular diffraction and Fourier space projection- slice : .8

Wide Angular diffraction and Fourier space

projection- slice : .96 Comparison of conventional BPM with FT

of circular arc in Fourier space

(16)

Apodization of the aperture

Min Gu, Advanced Optical Imaging Theory, Springer, (2000) ch 6

r

f = g(θ)is ray projection function

dS = 2πrdr = 2πf²g(θ)g^′(θ)dθand dΩ = 2πf²sin θdθ p²(r)dS = P²(θ)dΩ Conservation of energy p²(r)2πrdr = 2πf²g(θ)g^′(θ)dθ = P²(θ)2πf²sin θdθ P (θ) = p(r)^g(θ)gsin θ^′^(θ)

Sine Condition Ray in image space meets the focal sphere at same height at which corresponding ray in object space enters system

g(θ) = sin θ r = f sin θ P (θ) = p(r)√ cos θ Herschel Condition Ray density constant over wavefront

g(θ) = 2 sinθ

2 r = 2f sinθ

2 P (θ) = p(r)

r ray density p(r)

P(θ) θ Angular ray density

Fresnel Transmisson

dS

dΩ

Uniform Projection Condition Equal radial distances converted to equal angular in- tervals on the reference surface

g(θ) = θ r = f θ P (θ) = p(r)

r θ

sin θ Helmholtz Condition Distortion free imaging

g(θ) = tan θ r = f tan θ P (θ) = p(r) cos^−3/2θ

Dispersion in Lenses and femtosecond pulses

Propagation Time Difference Group Velocity Dispersion

Can analyze polychromatic case by coherently summing up contributions from each spectral component, appropriately weighted by complex amplitude spectra.

This works for both incoherent sources and femtosecond laser pulses Alternatively can analyze the spatio-temporal pulse envelope evolution

Use differential equations and analytically solve for pulse evolution

Use 2+1D spatio-temporal beam propagation with dispersion like diffraction where reduced time in group velocity frame T = t − z/vgis transverse dimension

Fourier representation of Green’s function

A. Ba˜nos Jr, Dipole Radiation in the presence of a conducting half-space, Pergamon, 1966

Scalar waves expand according to a spherical wave Green’s function g(R) =e^ikR

R

Where the Green’s fnc satisfies Helmholtz eqn driven by point source (∇²+ k²)g =−4πδ(x)δ(y)δ(z)

Now represent this Green’s function in Fourier space G(kx, ky, kz) =

˚ _∞

−∞

g(x, y, z)e^−i(k^x^x+k^y^y+k^z^z)dx dy dz g(x, y, z) = 1

(2π)³

˚ _∞

−∞

G(kx, ky, kz)e^i(k^x^x+k^y^y+k^z^z)dx dy dz

where G(kx, ky, kz) represents an analytic function of the 3 transform variables kx, ky, kz.

To compute G(kx, ky, kz) multiply both sides of inhomogeneous Helmholtz eqn by e^−i(k^x^x+k^y^y+k^z^z)and integrate wrt x, y, z

3-D k-space

˚

e^−i(k^x^x+k^y^y+k^z^z)(∇²+ k²)g(x, y, z)dx dy dz

=

˚

e^−i(k^x^x+k^y^y+k^z^z)

∂²g

∂z²+∂²g

∂y²+∂²g

∂x²

dx dy dz + k²G(kx, ky, kz)

=

˚

e^−i(k^x^x+k^y^y+k^z^z)[−4πδ(x)δ(y)δ(z)] dx dy dz = −4π Derivative theorem turns derivatives to (ikj)so this becomes

(−kx²− k²y− k²z+ k²)G(kx, ky, kz) =−4π So the 3-D Fourier transform of the scalar Green’s function is

G(kx, ky, kz) = 4π k_x²+ k²_y+ k²_z− k² In 2-D this would be

G(kx, kz) = 4π k²_x+ k_z²− k²

(17)

Spherical Coordinates

Transforming spherically symmetric functions

x = R sin θ cos φ kx=K sin α cos β y = R sin θ sin φ ky=K sin α sin β

z = R cos θ kz=K cos α

kxx+kyy+kzz = R sin θ cos φK sin α cos β + R sin θ sin φK sin α sin β + R cos θK cos α

= RK cos α By rotating coordinates to align R with z ⇒ cos θ = 1 This spherically symmetric 3-D FT of the 3-D Green’s function can be represented as

G(K) = 4π K²− k²

Inverse transform must be independent of θ, φ can be represented as g(R) = 4π

(2π)³ ˆ _∞

0

ˆ π 0

ˆ 2π 0

e^{iKR cos α}

K²− k²K²dK sinα dα dβ =2π 2π²

ˆ _∞

0

K² K²−k²

ˆ π 0

e^{iKR cos α}sinα dα dK

= 1 π

ˆ _∞

0

K^✓^✓² K²− k²

e^{iKR cos α}

−i^✚KR^✚^✚

π

0

dK = 1

−iπR ˆ _∞

0

e^−iKR− e^iKR

K²−k² KdK K^′= K

= 1 iπR

ˆ _∞

−∞

e^iKR

K²− k²KdK =e^ikR R

2-D K-sphere as complex variable with pole

G(kx, kz) = 4π k_x²+ k²_z− k²

K-space Beam Propagation R.K. Kupka JOSA v12(2) p. 404 1995

2-D (x,z) propagation of e^iωtsideband results in TE and TM scalar equations

∇²+ k²₀n²(x, z)Ey(x, z) = 0

∇²+ 1 n²(x, z)

∂n²(x, z)

∂x

∂

∂x+ k₀²n²(x, z)

Hy(x, z) = 0 Both can be solved using BPM with matrix representation of operators.

Discretize fields across width W by m samples of spacing ∆x= W/m.

Ey(x)→ ~E(x) = [E(−m∆^x/2), . . . , E(0), . . . , E((m/2− 1)∆^x)]

FFT of field array gives k-space vector, E(k^x)with spaing ∆k= 2π/W multiplication by a function replaced by a m × m matrix

n(x)→ diag[n(−m∆^x/2), . . . , n(0), . . . , n((m/2− 1)∆^x)] n(x) = diag[n(x)]

Operators represented by matrices, eg Fourier transform

K-space Beam Propagation

k matrix k → k = diag[−m∆^k/2, . . . , 0, . . . , (m/2− 1)∆^k]allows Fourier transform of real space differentiation

∂ ~E(x)

∂x =−iF⁻¹kE(k^x) dual Convolution theorem F

n²(x)

∗ F{E(x)} = F

n²(x)E(x)

becomes NE(kx) = F n²(x)· ~E(x)

where N defined by the commutation n²(x)F⁻¹= F⁻¹N is equivalent to a convolution N → n²(kx)∗ Column vectors of N contain FT of n²(x)but they are shifted along the mian diagonal as banded diagonal Toeplitz matrix

F F T{n²(x)} = [a−∆km/2, . . . , a_−∆_k, a0, a∆_k, . . . , a∆_k(m−1)/2]







... ... ... ... ... . . . . . . a0 a₋₁ a₋₂ a₋₃ a₋₄ . . .







E n+1 = E(x + x ) = E(x) + E (x) x + 1 2! E (x) 2 x + 1 3! E (x) 3 x +... = 0. δ x E = 1 2 δ. δ 2 xe n = δ x. Paraxial Diffraction

Scalar Wave Eqn in 1+1D expanded as envelope and carrier

Numerically Evaluating Derivatives

Runge-Kutta

PDEs

Tridiagonal System

Stability Analysis

Lax method for improving stability

Crank-Nicholson

Split-Step

Beam Propagation, sampling, and Fourier Space

Beam Propagation: Paraxial vs Nonparaxial

k-space version of transfer function vs Feit and Fleck

Beam Propagation: Sampling requirement

Beam Propagation Algorithm

Beam Jumping: Only for homogeneous media and without boundary absorptionb

Inhomogeneous Beamprop

Inhomogeneous Beamprop

Inhomogeneous Beamprop

Rectangular aperture diffraction

BPM Formal Derivation

Symmetrical Split Step

Numerical Evaluation of BPM Error

2+1D Beam propagation

Circular Aperture 3+1D Beam Propagation Crossection

Imaging with Fresnel Zone Plate

2-D crosssections every 8 λ from a

D = 16λ Circular Aperture BPM

Circular disk diffraction: Fresnel/Arago’s

Bright Spot Babinet’s principle and quadrature

combination with incident plane wave

No absorbing boundary conditions:

wraparound can interfere producing unwanted fringes

Absorbing boundary conditions: Smoothly taper to avoid reflection at impedance

discontinuities

Absorbing boundary conditions: power may not be conserved

Beam Propagation in 1-D without Additional Aberrations

Beam Propagation in 1-D with 1 wave Spherical Aberration

Beam Propagation in 1-D with 2 wave Spherical Aberration

Beamprop through Lens Systems

Thick Lenses

BeamPropagation through 4F lens system

BeamPropagation through 4F lens system

4F lens system with Schlieren filter Converts Phase Modulation to Amplitude

4F lens system with Schlieren filter Converts Phase Modulation to Amplitude

4F lens system with Zernike Phase contrast Converts Phase Modulation to Amplitude dot

4F lens system with Zernike Phase contrast Converts Phase Modulation to Amplitude dot

Beamprop in coupled singlemode waveguides

Beamprop in wide multimode waveguides

Arc in k-space from Fourier transform of BPM of Gaussian Beam

2+1D Beam Propagation of Nonparaxial Spot array: Aberration of Free Space

spot array through focus showing off-axis aberrations of free space

3D Fourier space of nonparaxial spot array

k-space propagation

k-space propagation

Projection of 3-D k-sphere onto k

− k

plane to get FT of x-z crosssection of circular

aperture

Comparing k-space (optics centric) vs

the Ewald sphere (Material centric)

for X-ray discrete lattice diffraction

k-space and the McCutchen Theorem

McCutchen Theorem

McCutchen Theorem

Wide Angular diffraction and Fourier space projection- slice : .4

Wide Angular diffraction and Fourier space projection- slice : .512

Wide Angular diffraction and Fourier space projection- slice : .8

Wide Angular diffraction and Fourier space

projection- slice : .96 Comparison of conventional BPM with FT

of circular arc in Fourier space

Apodization of the aperture

Dispersion in Lenses and femtosecond pulses

Fourier representation of Green’s function

3-D k-space

Spherical Coordinates

Transforming spherically symmetric functions

2-D K-sphere as complex variable with pole

K-space Beam Propagation R.K. Kupka JOSA v12(2) p. 404 1995

K-space Beam Propagation

2-D crosssections every 8 ^λ from a

Projection of 3-D k-sphere onto ^k