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

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

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OPTICS BY THE NUMBERS L’Ottica Attraverso i Numeri. Michael Scalora U.S. Army Research, Development, and Engineering Center Redstone Arsenal, Alabama, 35898-5000 & Universita' di Roma "La Sapienza" Dipartimento di Energetica. Rome, April-May 2004. BPM:Propagation in Planar Waveguides - PowerPoint PPT Presentation

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Page 1: 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

Page 2: Michael Scalora U.S. Army Research, Development, and Engineering Center

BPM:Propagation in Planar Waveguides

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

Page 3: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 4: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 5: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 6: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 7: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 8: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 9: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 10: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 11: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 12: Michael Scalora U.S. Army Research, Development, and Engineering Center

-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

Page 13: Michael Scalora U.S. Army Research, Development, and Engineering Center

Same as previous figure.

Page 14: Michael Scalora U.S. Army Research, Development, and Engineering Center

-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

Page 15: Michael Scalora U.S. Army Research, Development, and Engineering Center

Same as previous figure.

Page 16: Michael Scalora U.S. Army Research, Development, and Engineering Center

-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

Page 17: Michael Scalora U.S. Army Research, Development, and Engineering Center

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.

Page 18: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 19: Michael Scalora U.S. Army Research, Development, and Engineering Center

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.

Page 20: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 21: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 22: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 23: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 24: Michael Scalora U.S. Army Research, Development, and Engineering Center

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.

Page 25: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 26: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 27: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 28: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 29: Michael Scalora U.S. Army Research, Development, and Engineering Center

Output Field Profile in the case Light is Guided.

Page 30: Michael Scalora U.S. Army Research, Development, and Engineering Center

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.

Page 31: Michael Scalora U.S. Army Research, Development, and Engineering Center

Output Field Profile in the case Light is Not Guided.

Page 32: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 33: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 34: Michael Scalora U.S. Army Research, Development, and Engineering Center

Initial pulse profile

Final profile

Page 35: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 36: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 37: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 38: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 39: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 40: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 41: Michael Scalora U.S. Army Research, Development, and Engineering Center

PUMP FIELD z

z=0

z,=L,0

Page 42: Michael Scalora U.S. Army Research, Development, and Engineering Center

PUMP FIELD

Page 43: Michael Scalora U.S. Army Research, Development, and Engineering Center

STOKES FIELD

Page 44: Michael Scalora U.S. Army Research, Development, and Engineering Center

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

PUMP FIELD

Page 45: Michael Scalora U.S. Army Research, Development, and Engineering Center

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

STOKES FIELD

Page 46: Michael Scalora U.S. Army Research, Development, and Engineering Center

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…

Page 47: Michael Scalora U.S. Army Research, Development, and Engineering Center

… 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

Page 48: Michael Scalora U.S. Army Research, Development, and Engineering Center

ON-AXIS INTENSITY PROFILE

PUMP FIELD

Page 49: Michael Scalora U.S. Army Research, Development, and Engineering Center

ON-AXIS INTENSITY PROFILE

STOKES FIELD

Page 50: Michael Scalora U.S. Army Research, Development, and Engineering Center

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

PUMP FIELD

Page 51: Michael Scalora U.S. Army Research, Development, and Engineering Center

TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT

STOKES FIELD

Poisson Spot like effect

Page 52: Michael Scalora U.S. Army Research, Development, and Engineering Center

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

Page 53: Michael Scalora U.S. Army Research, Development, and Engineering Center

Direction of Propagation