Download - Optical-phase conjugation in difference-frequency generation A. Andreoni, M. Bondani, F. Paleari
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)