EFFICIENT PARAMETRIC AMPLIFICATION IN

DOUBLE SYSTEMS IN THE ABSENCE OF TWO-PHOTON

MAXIMAL COHERENCE

A.D. Wilson-Gordon, H. Shpaisman and H. Friedmann,Department of Chemistry, Bar-Ilan University,

Ramat Gan 52900, Israel

2

Two-level system: pump and probe

1

2

1 2

•If self-focusing and diffraction balanced, Gaussian pump propagates as a spatial soliton

•If pump-induced cross focusing of probe balanced by diffraction, the weak probe propagates as if it is a spatial soliton

•If pump is sufficiently intense, radiation generated at FWM frequency

•Parametric amplification between probe and FWM via the pump

•Process occurs over many diffraction lengths: EIPM important

3

Self-focusing

Nonlinear refractive index

Thus

Focusing obtained when

Self-focusing obtained when laser is detuned to the blue!

0 2n n n I

2n dn dI

0dn dI

4

Coherent Population Trapping(CPT)

1 2

3

31 32

• Fields are equally intense

• Population trapped in lower levels

• Two-photon coherence is maximal:

• Zero absorption

2 121 11 22 4| |

5

Electromagnetically Induced Transparency( EIT)

• Pump and probe fields

• Population optically pumped into state |2>

• Two-photon coherence is small

1 2

3

3132

6

Double System

• Several possible configurations

• Highly efficient FWM when CPT occurs (Harris)

1 2

3

41 3231 42

4

7

Model

• Three or four beams with Gaussian transverse intensity profile (GTIP)

• Beams copropagate• Assume steady-state• Compare results with plane-wave beams• Cases studied:• CPT with maximal coherence , either initially or on propagation• Four identical beams with 0 and phase• Three strong fields• Two strong fields • Incoherent pumping from state 2 to 4 (not shown here)• Raman detuning (not shown here)

8

Maxwell-Bloch Equations2' ' '

4ij ij ijTijD

i iV VLz L

2 2 2 2 2 2/ (1/ ) / (1/ ) /T

'ijV

DL

ijL

'ij

Transverse radial coordinate

Direction of propagationInteraction length

Diffraction length

Density matrix element

Rabi frequency for transition

z

j i

9

Bloch Equations

11 13 31 14 41 31 13 41 14 12 11 21 22 31 33 41 44

22 23 32 24 42 32 23 42 24 12 11 21 22 32 33 42 44

33 31 13 32 23 13 31 23 32 31 32 33 43 44

44

( ' ' ' ' ) ,

( ' ' ' ' ) ,

( ' ' ' ' ) ( ) ,

i V V V V

i V V V V

i V V V V

41 14 42 24 14 41 24 42 41 42 43 44 24 44 22

21 23 31 24 41 31 23 41 24 21 21 21

31 31 11 32 21 31 33 41 34 31 31 31

32 32 22 31 12

( ' ' ' ' ) ( ) ( ),

' ( ' ' ' ' ) ( ) ' ,

' ( ' ' ) ( ) ' ,

' ( '

i V V V V r

i V aV V aV i

i V V V V i

i V V

32 33 42 34 32 32 32

41 41 11 42 21 31 43 41 44 41 41 41

42 42 22 41 12 32 43 42 44 42 42 42

43 41 13 42 23 13 41 23 42 43 43

* ' ) ( ) ' ,

' ( * ' ' ) ( ) ' ,

' ( ' ' ) ( ) ' ,

' ( ' * ' ' * ' ) (

V a V i

i V a V V V i

i V aV aV V i

i V a V V a V i

43) '

10

Notation

31 32 42 41

*24 4 2

*

exp( ), is initial relative phase,

is longitudinal decay rate from state ,

is total decay rate from state ,

0.5( ) is transverse decay rate,

is r

kl

i

kl k l kl k l

kl

a i

k l

i

r

24

21 21 3

ate of phase-changing collisions,

is rate of incoherent pumping from state 2 4 ,

' is one-photon detuning from resonance; (3,4), (1,2),

' exp[ ( )],

' exp{ [(

ij ij ij

ij ij ij ij ij

r

i j

i t k z

i

1 32 31 32 31 32

43 43 41 31 41 31 41 31

) ( ) ( )]},

' exp{ [( ) ( ) ( )]}.

t k k z

i t k k z

11

Multiphoton Resonance Condition

31 32 42 41

31 32 41 42 21

41 31 42 32 43

0 31 32 42 41

0

, two-photon detuning

or ,

0, initial phase mismatchk k k k k

12

Analytical Solution

Real part – refraction Imag part – absorption FWM

Effect of phase: when =0, a=1; when =, a=-1

(1) (3)

(1) (3)31 31 31 31 32 24 41

(1) (3)32 32 32 32 31 14 42

(1) (3)41 41 41 41 42 23 31

(1) (3)42 31 42 42 41 13 32

' ' ' ,

' ,

' * ,

' * ,

' .

ij ij ij

V a V V V

V a V V V

V a V V V

V a V V V

13

CPT and Maximal Two-Photon Coherence

• When CPT exists, no absorption or focusing or defocusing occurs since (1)=0

• Phase-matching unimportant• Maximum FWM occurs within a

propagation distance less than diffraction length

• This is completely different from a two-level system

14

CPT and Maximal Two-Photon Coherence

31 32 42 41

31 32 41 423

8; V 8; V 1; V 0.001;

4;

1.66 10 ;NL D

V

L L

15

Transverse Intensity Profile

16

Comparison: GTIP’s and PW’s

17

CPT and Maximal Two-Photon Coherence

31 32 42 41

31 32 41 424

8; V 8; V 8; V 0.001;

4; 100;

1.66 10 ;NL D

V

L L

18

Transverse Intensity Profile

19

Comparison: GTIP’s and PW’s

20

Onset of CPT vs. Maximum Conversion

• For detuning , CPT exists at the outset. Maximum conversion of 87% occurs at 0.047.

• For detuning , CPT occurs at 0.1, whereas maximum conversion of 73% occurs at 0.009.

• Thus, it is possible to get efficient conversion before CPT, without focusing, defocusing or ring formation

41 42 100

41 42 10

21

CPT and Maximal Two-Photon Coherence

31 32 42 41

31 32 41 424

8; V 8; V 8; V 0.001;

4; 10;

1.66 10 ;NL D

V

L L

22

Focusing

• Focusing can occur before CPT established

• Beams blue-detuned

• Nonlinear length sufficiently long

• Maximum FWM can still occur within a short propagation distance

• Phase-matching still unimportant

23

Three strong lasers

31 42 32 41

31 32 41 423

8; V 8; V 8; V 0.001;

4;

1.52 10 ;L D

V

L L

24

Initial gain at FWM frequency

-50

5

00.05

0.10

5

10

z/Ld

|V31

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V32

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V41

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V42

| Am

plitu

de

gain

25

Focusing

-50

5

00.05

0.10

5

10

z/Ld

|V31

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V32

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V41

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V42

| Am

plitu

deFocusing

Focusing

26

Maximum focusing and conversion

-50

5

00.05

0.10

5

10

z/Ld

|V31

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V32

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V41

| Am

plitu

de

-50

5

00.05

0.10

5

10

z/Ld

|V42

| Am

plitu

de

Focusing

FocusingFocusing

Focusing

27

Comparison: GTIP’s and PW’s

0 0.1 0.20

5

10

15T

PW

on-

axis

am

plitu

de

Z/Ld

|V31||V32|

0 0.1 0.20

5

10

TP

W o

n-ax

is a

mpl

itude

Z/Ld

|V42||V41|

0 0.1 0.24

6

8

10

PW

am

plitu

de

Z/Ld

|V31||V32|

0 0.1 0.20

2

4

6

8

PW

am

plitu

de

Z/Ld

|V42||V41|

Max

Max

28

Two strong lasers: strong-weak-strong –weak configuration

31 42 32 41

31 32 41 423

4; V 4; V 0.1; V 0.001;

4;

1.66 10 ;L D

V

L L

29

Two strong lasers: strong-weak-strong –weak configuration

31 42 32 41

31 32 41 423

4; V 4; V 0.1; V 0.001;

4;

1.66 10 ;L D

V

L L

30

Two strong lasers: strong-weak-strong –weak configuration

31 42 32 41

31 32 41 423

4; V 4; V 0.1; V 0.001;

4;

1.66 10 ;NL D

V

L L

31

Focusing on propagation

32

Phase dependence

• When field at FWM frequency absent or small at outset, phase is unimportant

• When field at FWM frequency present at outset, dramatic phase effects can be obtained

33

Four Strong Fields: phase 0 vs. 31 42 32 41

31 32 41 423

4; V 4; V 4; V 4;

4;

1.11 10 ; NL D

V

L L

34

Zero PhaseNo Change on Propagation: CPT

35

Four strong fields: phase 31 42 32 41

31 32 41 423

4; V 4; V 4; V 4;

4;

1.11 10 ; = ;NL D

V

L L

36

Phase Focusing on propagation: no CPT

37

Conclusions

• Efficient FWM in double lambda systems can be obtained even before CPT occurs

• It is obtained at short propagation distances• Focusing can be obtained by blue one-photon

detuning• Often accompanied by ring formation