Michael Scalora U.S. Army Research, Development, and Engineering Center

Michael Scalora

U.S. Army Research, Development, and Engineering CenterRedstone Arsenal, Alabama, 35898-5000

&Universita' di Roma "La Sapienza"

Dipartimento di Energetica

OPTICS BY THE NUMBERS

L’Ottica Attraverso i Numeri

Rome, April-May 2004

BPM:Propagation in Planar Waveguides

Retarded Coordinate trasformation: time dependence, Raman scattering, self-phase modulation in PCFs

1.0

1.1

1.2

1.3

1.4

1.5

-15 -10 -5 0 5 10 15

Transverse Coordinate (m)

Ind

ex o

f R

efra

ctio

n

air core

14 m

5 m

fig.(4)

Study the transmissive properties of guided modes.

Propagation into the page

2 2 2 2 22 2 2

2 2 2 2 2

22(3) 2

2 2

( ) 2 ( )2 ( )

42

E E n x E i n x EE ik k n x E

z z c t c t c

i E Ec t t

22 22

2 2 2 2

4 nlPn EE

c t c t

2(3)nlP E E

( )( , , ) ( , , ) . .i kz tE z x t E z x t e c c

Assuming steady state conditions…

2 2 (3)202 0 0

0 0

( ) 4

in in

n x nE iE i E i E E

F n n

2 2 2 22

2 2 2 2

2 222 2 (3) 2

2 2 2

( ) 2 ( )2

4( ) 2

E E n x E i n x EE ik

z z c t c t

k n x E i E Ec c t t

2 2 (3)202 0 0

0 0

0 0

0 0

( ) 4

4

/

in in

in

n x nE iE i E i E E

F n n

nF Fresnel Number

z k nc

F s all

F

m

Wave front does not distort:Plane Wave propagation

Diffraction is very important

2 2 (3)202 0 0

0 0

( ) 4

in in

n x nE iE i E i E E

F n n

This equation is of the form:

Where:

EHE

2 2 (3)20 02 0

0 0

( ) 4

in in

n x ni i E V

n

iD

nFH

Using the split-step BPM algorithm

0

( ', ) '(0, )

/ 2(0, ) (0, ) / 2

( , ) (0, ) (0, )

(0, )D

H xH

V x x

x

V

E x e E x e E x

e e e E x

0.75

1.00

1.25

1.50

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.4

0.8

1.2airair

gla

ss;

n=

1.4

2

gla

ss;

n=

1.4

2

gla

ss;

n=

1.4

2

gla

ss;

n=

1.42

air guide ~ 5m

Transverse Coordinate (m)

Ind

ex o

f R

efra

ctio

n

Inte

nsi

ty

a=1.4m b=1m

Example: Incident angle is 5 degrees

Assume =0

xThe cross section along x renders the problem one-dimensional in nature

1.0

1.1

1.2

1.3

1.4

1.5

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Transverse Position (microns)

Tra

nsv

erse

Ind

ex P

rofile

0

0.2

0.4

0.6

0.8

1.0

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

14-micron core5-micron core

(m)

Nor

mal

ized

Tra

nsm

itta

nce

Transmissive properties in the linear (low intensity) regimeFor two different fibers. We set =0

0.75

1.00

1.25

1.50

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1.0

1.5Field bouncing back and forth from structure's wallsInpout Field Profile

Transverse Coordinate (m)

Ind

ex o

f R

efra

ctio

n

Inte

nsi

ty

-12 -10 -8 -6 -4 -2 -0 2 4 6 8 10 12

x (m)

0

100

200

300

400

500z(

m)

0.5

0.5

0.5

0.7

0.7

0.7

0.7

0.7

0.7

0.8

0.8 0.8

0.8

0.8

0.8

0.8

0.8

0.8

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.1

1.1

1.1

1.1

1.1

1.1

1.1

1.1 Field tuning corresponds toHigh transmission state.

0

0.2

0.4

0.6

0.8

1.0

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

14-micron core5-micron core

(m)

Nor

mal

ized

Tra

nsm

itta

nce

Direction of propagation

Same as previous figure.

-12 -10 -8 -6 -4 -2 -0 2 4 6 8 10 12

x (m)

0

100

200

300

400

500

z (

m)

0.1

0.1

0

0.2

0.4

0.6

0.8

1.0

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

14-micron core5-micron core

(m)

Nor

mal

ized

Tra

nsm

itta

nce

Same as previous figure.

-12 -10 -8 -6 -4 -2 -0 2 4 6 8 10 12

x (m)

0

100

200

300

400

500z(

m)

0.5

0.5

0.5

0.7

0.7

0.7

0.7

0.7

0.7

0.8

0.8 0.8

0.8

0.8

0.8

0.8

0.8

0.8

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.1

1.1

1.1

1.1

1.1

1.1

1.1

1.1

0/ 0.025z 0/ 0.0125x x

200000N 4096xN

For the example discussed:

5-mm guide~ 8 minutes on this laptop3.2GHz, 1Gbts RAM

2

2 2 (3)20 0 0

0 0

( ) 4

in in

nE iE

x n

n nFi E i E E

If (3) is non-zero, the refractive index

is a function of the local intensity.

Solutions are obtained using the same algorithm

but with a nonlinear potential.

1.0

1.1

1.2

1.3

1.4

1.5

-15 -10 -5 0 5 10 15

Transverse Coordinate (m)

Inde

x of

Ref

ract

ion air core

14 m

5 m

fig.(4)

Optical Switch

0

0.2

0.4

0.6

0.8

1.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

non-zero (3)

linear transmittance

scaled frequency (1/ where is in microns)

nor

mal

ized

tra

nsm

itta

nce

The band shifts because the location and the width of each gap depends on the exact values of n2 and n1, and on their local

difference.

fig.(5a)

0

0.2

0.4

0.6

0.8

1.0

0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10

(m)

Nor

mal

ized

Tra

nsm

itta

nce

0

0.2

0.4

0.6

0.8

1.0

0.715 0.720 0.725 0.730 0.735 0.740

Nonlinear Transmittance

Linear Transmittance

(m)

Tra

nsm

itta

nce

fig.(5b)

Optical Switch

on

off

0

0.2

0.4

0.6

0.8

1.0

0 1000 2000 3000 4000 5000

Longitudinal Position (m)

Cor

e E

nerg

y

fig.(6)

off

on

2 2 2 22

2 2 2 2

2 222 2 (3) 2

2 2 2

( ) 2 ( )2

4( ) 2

E E n x E i n x EE ik

z z c t c t

k n x E i E Ec c t t

0/ /z t z v v c n Retarded coordinateTransformation

1

zz z v

t tt

2 2 2

2 2 2 2

1 2

z v v

2 2

2 2t

2 22

2 2 2 2

2 222 (3) 2

2

2 2

22

2

( ) ( )

( )

22

42

E E E i EE ik

z z c t c t

k E i E Ec c t

n x

xt

n x

n

N.B.:An implicit and important assumption we have made is that one can go to a retarded coordinate provided the grating is shallow so that a group velocity can be defined

unumbiuosly and uniquely.

0/ /z t z v v c n

In other words, the effect of the grating on the group velocity is scaled away into an effective group velocity v. It is obvious that

care should be excercised at every step when reaching conclusions, in order to properly account for both material index

and modal dispersion, if the index discontinuity is large.

2 2

2 2 2

2

2

2

20

2 2 22 2

2 2 2

2(3) 22

0

2

2

2

2

4

1

2

E E i n Ec

n i nE E k n E

v v

c c

c

E Ec

n

c

i

Symplifying and

Dropping all Higher order Derivatives…

222 2 2 (3) 2

0 2 2

42 2E i n E k n E i E Ec c c

2 2 (3)202 0 0

0 0

2 (3) **0

0

( ) 4

42

in in

in

n x niH i i E

F n n

in

2 22 (3)20 0

20 0

22 (3)

0

0

( ) 4

4

in

in

n x nii i

F x n n

in

22 * * *

* 2 *2 2

Now we look at the linear regime, by injecting a beam inside the guide from the left and then from the right.

On-Axis Intensity as Beam Propagates Down the Guide. Beam is Guided.

Output Field Profile in the case Light is Guided.

On-Axis Intensity as Beam Propagates Down the Guide. Beam

is Tuned to a Minimum of Transmission, and is Not Guided, and energy

Quickly Dissipates Away.

Output Field Profile in the case Light is Not Guided.

0

0.2

0.4

0.6

0.8

1.0

0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00

Input Spectrum

OutputSpectrum

Propagating from left to right the pulse is tuned on the red curve, igniting self-phase modulation, and the spectral shifts indicated on the graph. A good portion of the input energy is transmitted. Spectra are to scale.

Fig. 4

2 max2 inn I L

c

I 1013 W/cm2

n2 510-19 cm2/WL 8 cm 100 fs

ON-AXIS

0

0.2

0.4

0.6

0.8

1.0

0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95

Output spectrumInput spectrum

scaled frequency (=1/ where is in m)

Tra

nsm

itta

nce

Linear transmittance for 2 slightly different guides

Propagation from right to left does not induce nonlinearities because the light quickly dissipates. The pulse is tuned with respect to the blue curve. Spectra are to scale.

Fig. 4

Initial pulse profile

Final profile

0

200

400

600

800

1000

0.65 0.70 0.75 0.80

scaled frequency (1/)

pow

er s

pec

tru

m

Spectrum of the pulse as it propagates. Note splitting.

Initial profile

Self-phase modulation

A process whereby new frequencies (or wavelengths) are generated such that:

2 max2 inn I L

c

400 500 600 700 800 9000,0

0,5

1,0

In

ten

sit

y, arb

. u

nit

s

, nm

maxI I

t

Example: input 100fs pulse at 800nm is broadened by ~30nm

Stimulated Raman Scattering

p stokesanti stokes

2

2*

si

Ap pE Ei

F xQ E QEe

2

2iA A

P

E EiQE e

F x

2*

2S S

P

E EiQ E

F x

* * iS P A P

QQ E E QE E e

p

stokes

2

2

p ps

E EiQE

F x

2*

2S S

P

E EiQ E

F x

*S P

QQ E E

The simplest case

Raman Soliton: A sudden relative phase shift between the pump and the Stokes at the input field generates a “phase wave”,or soliton, a temporary repletionof the pump at the expense of the Stokes intensity

stokes

2

2

p ps

E EiQE

F x

2*

2S S

P

E EiQ E

F x

*S P

QQ E E

The simplest case

00

00

(0, ) ss

s

EE

E

The gain changes sign temporarily,For times of order 1/

The soliton is the phase wave

The Input Stokes field undergoes a -phase shift

0

500

1000

1500

2000

2500

0 0.02 0.04 0.06 0.08

PUMP INTENSITYSTOKES INTENSITY

TIME

INT

EN

SIT

YIntensity at cell output

The Pump signal is temporarily repleted

The Stokes minimum is referred to as a Dark Soliton

PUMP FIELD z

z=0

z,=L,0

PUMP FIELD

STOKES FIELD

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

PUMP FIELD

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

STOKES FIELD

0

1000

2000

3000

0 0.02 0.04 0.06 0.08

F=F=20

TIME

INT

EN

SIT

Y

ON-AXIS INTENSITY AT CELL OUTPUT

The onset of diffraction causes the soliton to decay……almost as expected. Except that…

… the Stokes field undergoes significant replenishement on its axis,as a result of nonlinear self focusing

0

1000

2000

3000

4000

5000

0 0.02 0.04 0.06 0.08

F=F=20

TIME

STO

KE

S IN

TE

NSI

TY

ON-AXIS INTENSITY AT CELL OUTPUT

ON-AXIS INTENSITY PROFILE

PUMP FIELD

ON-AXIS INTENSITY PROFILE

STOKES FIELD

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

PUMP FIELD

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

STOKES FIELD

Poisson Spot like effect

Examples: single slit

0

0.2

0.4

0.6

0.8

-40 -30 -20 -10 0 10 20 30 40

transverse coordinate

inte

nsity

Direction of Propagation