Optical-phase conjugation in difference-frequency generation

A. Andreoni, M. Bondani, F. PaleariDept. Sciences, Univ. Insubria

andIstituto Nazionale di Fisica della Materia, I.N.F.M.

Como, Italy

V. N. MikhailovPhotophysics of Holographic Processes Department,

S.I.Vavilov State Optical InstituteSt. Petersburg, Russia

k1

E1E2

optical axis

y

z0

0, ,y zr

x

NL crystalentrance face

E3

k3

k2

E3 pump field (Nd SH) at 2

E1 seed field (Nd fund) at

E2 difference-freq. generated (DFG) field at

Space-dependent phase reversal

Aim:

E2 optical phase conjugate

(OPC) of E1, being E1 strongly phase/amplitude modulated

Theory of DFG in conditions of phase-mismatch, for seed and generated fields non-collinearly propagating (non-depleted plane-wave pump). Experiments showing that the pump-field wave-fronts behave as efficient phase-conjugating mirrors.

Outline:

Equations that describe the interaction:

02 2 2

2ˆ, cos

( )t a t

n

E r x r r

01 1 1

2ˆ, cos

( )t a t

n

E r x r r

Interacting fields:

AMP seed

DFG

k3

k2

k1

E1E2

E3

O.A.

y

z0

0, ,y zr

x

*1 1 3 2

ˆ 0 exp effa ig a a i k r r k r

*2 2 3 1

ˆ 0 exp effa ig a a i k r r k r Error in phase-matching

Coupling coefficient (type I)

3 1 2 k k k k

3 3

022 31

4cos sin

( ) (2 )effg d dn n

*1 1 3 2

ˆ 0 exp effa ig a a i k r r k r

*2 2 3 1

ˆ 0 exp effa ig a a i k r r k r

Property of the equations:

2 2

1 1 2 2ˆ ˆa a r k r k

conservation law

Boundary conditions

1 0 0a 2 0 0a 3 30 0a a rSolutions

21 32 2

1

1 2

ˆ0 4 sinh

ˆ ˆ ˆ ˆ 2

a Aa Q Q

Q

kr r

k k k k

3

2 1

2

ˆ2 0 sinh

ˆ ˆ 2

Aa a Q

Q

kr r

k k

11 1 1

ˆ0 tan tanh

2 2

kQ

Q

k kr k r r r

2 1 3 20 02 2

kr k r r

2

3 2

1 2

4

ˆ ˆ ˆ ˆ

AQ k

k k k k

3 3 0effA g a

2

1,2ˆa r k

Field amplitudes

Direction of propagation of constant-phase surfaces

Poynting vectors (energy propagation)

Properties of the solutions:

2 22 21 1 2

1

ˆ ˆ(0)

ˆ ˆa a a

k k

r rk k

2

11 1 2

1

0

2 2

a

a

k k

r = kr

2 2 2

k

r = k

22 11

1 1 1 21 1

0

2 2

aa

k a

k kS r r k

r

222 2 2

2 2a

k

kS r r k

Increase in photon fluxdensity of the seed field

E1 due to AMP

Photon fluxdensity of the

DFG field E2

Direction of k Continuity with the solution in PM

3 1 2 1 22 1

1 2

0 sinh sin cos2 2cos

2

PM Aa a y z

r

312 1

2 1 2

ˆ ˆˆ0 sinh

ˆ ˆ ˆ ˆ ˆ ˆ

Aa a

k kr k r

k k k k k k

Bondani et al.,Phys. Rev. A 66 (2002)

1 2ˆ ˆ ˆ ˆ k k k k

k3

k1

1 3

k1-surface

3 2

k2-surface

k2

C

k

k1-k2 bisector

Conservation of photon flux densities

21 221

1 21

ˆ ˆ0

ˆ ˆa

S r S rk k

k k

Theoretical conclusion (relevant to OPC experiment)

2 2 2 1 30 0 0 2 2 = r + k r = k r

3 zkTake and a plane mirror at exit face reflecting E2 back to .0z

3k̂

P

Mir

ror

(P)r

z

y

Diff

usin

g p

late

11 2 3

ˆtan tanh

2 2

kQ

Q

kr r r r

1 2 3 2

r r r

On planes parallel to the E3

wavefronts, E2 OPC of E1 .

ˆ0

2Q

kr

In a regime of linear amplification

and .

If then:

1 1tan tan 2 k r

1tan 2 2 k r

2 1O O .const =

O

3 P 2 =

2 P 1 P

2 1P P =

Lens

BBO I

CCDsensor

DichroicMirror

Pump E3

Seed E1

PD2Pump

monitor

L6M3

PH2

PH1L3DMCCD camera

F3 M1

BS1 (wedged)

L2

L4M2

L1

F1

PD1Seed

monitor

F2 Nd:YLF source

Seed/AMP1 = 1053 nm

Pump = 526.5 nm

DFG2 = 1053 nm

BBO

L5

BBO II

Mirror

CCD sensor

DiffusingPlate

DM BS

Polarizer

Pump E3

Seed E1

DiffusingPlate

Experimental setup Nd:YLF passive Q-switchring oscillator; Nd:glass double-pass amplifier(Brillouin scattering phase-conjugating mirror) Frequency doubling KTP.Nd

Long coherence length (>3m)Pulse duration 20 nsEnergy per pulse 1 J at =526.5 nm

●BBO I ●Cut for collinear SHG of Nd:YAG●Fujian Castech Crystals Inc., China

35 5 2 mm

0 10000 20000 30000 40000 500000.00

0.05

0.10

0.15

0.20

0 50 100 150 200

2 23 photons/mA

2Pump intensity MW/cm

Fig. 5

0100

2000

100

200

m

m

efficiency: 10% per mmVERYFIED

Linear regime ofAmpl./DFG.

OPC of the co-propagatingfields and beyond the crystal

EE11 EE22 EE11 EE22

Lens

BBO I

CCDsensor

DichroicMirror

Pump E3

Seed E1

BBO II

Mirror

CCD sensor

DiffusingPlate

DM BS

Polarizer

Pump E3

Seed E1

DiffusingPlate

2 mm

55 mmCCD sensor

BBO II Distance D

●BBO II 310 10 3 mm

The diffusing glass plate introduces a seed beam divergence of 0.5 deg and produces the intensity distribution

mm0 1 2 3 4

0

1

2

3

4

mm

Speckle pattern of measured at 6 cm from diffusing plate

EE11 EE11

Measurements of back-reflected by mirrorand at same distance D(D = 51 cm, 291 cm)

EE11 EE11

EE22 EE22

Fig. 7

mm

mm

0

1

2

3

4

0 1 2 3 40

1

2

3

4

mm

mm

a)

b)

0 1 2 3 4

Fig. 9

0

1

2

3

4

0 1 2 3 40

1

2

3

4b)

mm

mm

mm

mm

a)

b)

0 1 2 3 4

51 cm

291 cm

Seed beam after removal of diffusingplate and in the absence of pump.

Reflected DFG beam after back-propagation through diffusing plate.

Viktor N. Mikhailov

Maria Bondani

Fabio Paleari

THANKSTO CO-AUTHORS

ANDTO AUDIENCE

Paper in press:J. Opt. Soc. Am. B (Aug. 2003)