Lecture 2
Parametric amplification and oscillation:
Basic principles
David Hanna
Optoelectronics Research Centre
University of Southampton
Lectures at Friedrich Schiller University, Jena
July/August 2006
Outline of lecture
•
How to calculate parametric gain via the coupled wave equations
•
Expressions for small-gain and large-gain cases
•
Effect of phase-mismatch on gain, hence find signal gain-bandwidth
•
Comparison of threshold of SRO and DRO
•
Comparison of longitudinal mode behaviour of SRO and DRO
•
Calculation of slope-efficiency
•
Focussing considerations
Calculation of parametric gain
Assume plane waves
Assume cw fields
Neglect pump depletion
Coupled-wave equations for signal and idler are then soluble,
calculate output signal and idler fields for
given input pump, signal and idler fields
Coupled equations
Fields
Intensity
E(r, t )  1/ 2E(r, ) exp i(k.r  t )  cc
I  1 / 2nc 0 E (r ,  )
2
dE1
 i1 E3 E2* exp( ikz)
dz
dE2
 i 2 E3 E1* exp( ikz)
dz
 j   jd / n jc
k  k3  k 2  k1
d: effective nonlinear coefficient
Manley-Rowe relations
Integrals of the coupled equations
n3|E3(z)|2/ω3 + n2|E2(z)|2/ω2 = const
n3|E3(z)|2/ω3 + n1|E1(z)|2/ω1 = const
n2|E2(z)|2/ω2 – n1|E1(z)|2/ω1 = const
These imply
n3|E3(z)|2 + n2|E2(z)|2 + n1|E1(z)|2 = const
i.e. conservation of power flow in propagation direction
Number of pump photons annihilated in NL medium equals the
number of signal photons created, which also equals the
number of idler photons created
Solution to coupled equations:
(1)
E3 E 2* (0)


k
E1 ( L)  E1 (0) exp( ikL / 2) cosh gL  i
sinh gL  i 1
exp( ikL / 2) sinh gL
g
g


E3 E1* (0)


k
E 2 ( L)  E 2 (0) exp( ikL / 2) cosh gL  i
sinh gL  i 2
exp( ikL / 2) sinh gL
g
g



where g    (kL / 2)
2

2 1/ 2
and
   1 2 E 3
2
2
Solution to coupled equations:
If only one input E2, (E1(0) = 0)
(2)
[amplifier or SRO]
Single-pass power gain (increment) is,
G2 ( L) 
E 2 ( L)
E 2 (0)
For exact phase-match, g = Γ , so
2
1   L
2
2
2
sinh 2 gL
gL 2
G2 ( L)  sinh 2 L
(Corresponding multiplicative power gain, G2x ( L)  1  G2 ( L)  cosh 2 L )
Plane-wave, phase-matched, parametric gain
If gain is small, (G2(L) << 1) , gain increment is
21 2 d I 3
2
 2 L2 
n1 n2 n3 0 c 3
Note: incremental gain proportional to pump intensity
~ proportional to ω32
proportional to d2 / n3
(widely quoted as NL Figure Of Merit)
Plane-wave, phase-matched parametric gain
(multiplicative)
For high gain, ΓL >> 1
G2 ( L)  cosh 2 L  1 / 4 exp( 2L)
Very high gain is possible with ultra-short pump pulses,
since gain is exponentially dependent on peak pump intensity
Note: since Γ2  pump power P
the gain exponent depends on √P
(unlike Raman gain, where exponent  P)
Phase relation between pump, signal, idler
Suppose both signal and idler are input.
Assuming Δk = 0 , then
i 2 E3 E1* (0)
E 2 ( L)
 cosh L 
sinh L
E 2 (0)
E 2 (0)
Adds, maximally, to gain if
i exp i(3  2  1 )  1  3  2  1   / 2
Note: Fields are
E(r, t )  1 / 2 E(r, ) exp i(k.r  t   )
Gain maximised if phase of nonlinear polarisation at ω2
leads (by /2) the phase of e.m. wave at ω2
OPO threshold: SRO vs DRO
(1)
Represent round-trip power loss by one cavity mirror
having reflectance R1 (idler), R2 (signal)
R1,2
Threshold → round-trip gain = round-trip loss
(for signal only, SRO, for signal and idler, DRO)
If Δk = 0 , threshold condition
(assuming pump, signal & idler phases Φ3 – Φ2 – Φ1 = - /2 at input to crystal)
1  ( R1 R2 )1 / 2
cosh L  1 / 2
R1  R21 / 2
OPO threshold: SRO vs DRO
For SRO,
R1 = 0
(2)
R2 cosh 2 L  1
If 1- R1,2 << 1
SRO
 2 L2  1  R2
DRO
 2 L2  (1  R2 )(1  R1 ) / 4
Advantage of DRO is low threshold
SROthreshold
DROthreshold
= 200 for 1 – R1 = 0.02 (2%)
Parametric gain bandwidth
For plane waves, max parametric gain is for frequencies
ω30 = ω20 + ω10 that achieve exact phase-match, k3 = k2 + k1
If the signal frequency ω2 is offset by
there is a phase-mismatch
2  2  20
k (2 )  k 3  k 2  k 1
For small gain, the signal gain is reduced to ~ ½ max for ΔkL~π
Δk = 0 , ω2 = ω20
Gain
Solve for δω2+ , δω2-
k (2 ) L  
δω2
-
0
δω2
+
δω2
Hence gain bandwidth δω2+ - δω2Bandwidth reduces with greater L
Parametric gain bandwidth: small gain
Power gain (increment) vs Δk
(L) sinh gL
( gL) 2
2
sinh2ΓL
(ΓL)2
2
(L) 2 sin c 2 g L
g'L =  , hence
0
Δk=2Γ
g   2  (k / 2) 2
g   (k / 2) 2   2
k  2[ 2  ( / L) 2 ]1 / 2
Δk
|Δk|= /L
For small gain (ΓL << 1), gain-half-maximum is approximately given
by |Δk| = /L , hence independent of Γ (& therefore of intensity).
For high gain (ΓL >> 1), power gain is ~ ¼ exp(2ΓL), hence >>Γ2L2
Parametric gain bandwidth: large gain
For ΓL>>1, Gain is:
sinh2ΓL
~ 1 / 4 exp( 2 gL)  1 / 4 exp( 2  2  (k / 2) 2 L)
 exp[ 2L  (( k ) 2 / 4) L],
half max
(Δk << Γ)
Γ2L2
0
Δk=2Γ
Δk
3dB gain reduction for (ΔkL)2 / 4ΓL = ln 2 ; Δk = 2(Γln2/L)½
Δk bandwidth (high gain)
Δk bandwidth (low gain)
≈
(4 ln 2 ΓL)½

= 0.53 (ΓL)½
Pump acceptance bandwidth
What range of pump frequencies can pump a single signal frequency?
Low gain case: half-width,
3 

k 3 /   k1 /  

v

1
g3
 v g11
(Assumes first term in Taylor series dominates)

Signal gain bandwidth
(1)
Gain peak: phase-matched ω30 = ω20 + ω10 , k30 – k20 – k10 = 0
For same pump, ω30 , calculate
k  k 30  k 2  k1  (k 20  k 2 )  (k10  k1 )
corresponding to signal ω20 + δ ω2 (idler ω10 - δ ω2)
Taylor series:
1   2 k1  2 k2 
 k1 k2 
2




k  





 ...
 2
2
2
2 

2  
 
   
Solve for δω2
Signal gain bandwidth
(2)
For small gain, ΔkL/2 = /2 defines the ~ half-max. gain condition
 2 
Half-width

L(v g11  v g12 )
provided 1st. Taylor series term
dominates
 At degeneracy, use second Taylor term (note δω  Δk½ L-½ )
 For accuracy, use Sellmeier equn. rather than Taylor series
 For high gain find Δk bandwidth via
 


R 1   2 / g 2 sinh 2 gL  1
SRO tuning range within gain profile
sinh2ΓL
Zero gain (incremental) for
k  2( 2  ( / L) 2 )1 / 2
If ΓL << 1 then |Δk|, and hence
tuning range, independent of Γ'
If ΓL >> 1 then |Δk|, hence
0
Δk
tuning range,  [I]½
A more exact treatment calculates the Δk that makes

  2 sinh 2  2  (k / 2) 2
R 1 
2
2


(

k
/
2
)




1/ 2

L 
 1

Consequences of phase relation between
pump, signal, idler.

If more than one wave is fed back in an OPO,
then phases may be over constrained

Double- or multiple pass amplifiers can also suffer similar problems

The fixed value of relative phase φ3-φ2-φ1, can be exploited to achieve selfstabilisation of carrier envelope phase (CEP)

In a SRO, relative phase of pump and signal is not determined, hence
signal selects a cavity resonance frequency.
Stability: comparison of SRO and DRO
SRO:
No idler input. Gain does not depend on pump/signal relative phase.
Signal frequency free to choose a cavity resonance;
Idler free to take up appropriate frequency and phase.
Signal frequency stability depends on cavity stability
and pump frequency stability.
DRO:
Cavity resonance for both signal & idler generally not achieved;
Overconstrained.
Signal/idler pair seeks compromise between cavity resonance and
phase-mismatch;
large fluctuation of frequency result.
OPO: Spectral behaviour of cw SRO

No analogue of spatial hole-burning in a laser

Oscillation only on the signal cavity mode closest to gain maximum

Use of a single-frequency pump typically results in single frequency
operation (signal & idler).

Multi frequency pump can give multiple gain maxima, possibly multiple
signal frequencies, certainly multiple idler frequencies

Signal frequency will mode-hop if OPO cavity length varies, or if pump
frequency changes

Additional signal modes possible when pumping far above
threshold – due to back conversion of the phase-matched mode,
allowing phase-mismatched modes to oscillate
CW singly-resonant OPOs in PPLN
 First cw SRO: Bosenberg et al. O.L., 21, 713 (1996)
13w NdYAG pumped 50mm XL, ~3w threshold, >1.2w @ 3.3µm
 Cw single-frequency: van Herpen et al. O.L., 28, 2497 (2003)
Single-frequency idler, 3.7 → 4.7 µm, ~1w → 0.1w
 Direct diode-pumped: Klein et al. O.L., 24, 1142 (1999)
925nm MOPA diode, 1.5w thresh., 0.5w @ 2.1µm (2.5w pump)
 Fibre-laser-pumped: Gross et al. O.L., 27, 418 (2002)
1.9w idler @ 3.2µm for 8.3w pump
Calculation of conversion efficiency
Problem:
pump is depleted, hence need all three coupled
equations. (Threshold calculation avoids this).
Solve approx, assuming constant signal field
i.e. solve two coupled equations, for pump and idler.


Generated idler photons = generated signal photons
Increase (gain) in signal photons = loss of signal photons
Hence calculate pump depletion, and hence signal/idler o/p
(1)
Calculation of conversion efficiency
(2)
For SRO, with Δk = 0 and plane wave, find for pump
E3 ( L)
E3 (0)
2
2

 cos 2 sin c 1 ( N 1/ 2 )
2

N  E3 (0) / E3,threshold (0)
When N = (/2)2 ~ 2.5 , find E3(L) = 0
2
i.e. 100% pump depletion
Initial slope efficiency at threshold, defined as
d(signal photons generated)/d(pump photons annihilated),
is 3 (i.e. 300% !)
Typical OPO conversion efficiencies

Generally high conversion efficiency (> 50%)
is observed at 2-3 x threshold

Initial slope efficiency > 100% is typical

Pumping above 3-4 x threshold typically results in reduced efficiency (backconversion of signal/idler to pump)

Unlike lasers, OPOs do not have competing pathways for
loss of pump energy
Analytical treatment of OPO
with pump depletion
• Armstrong et al., Phys Rev ,127, 1918, (1962)
• Bey and Tang, IEEE J Quantum Electronics, QE 8, 361,
(1972)
• Rosencher and Fabre, JOSA B, 19, 1107, (2002)
Input (X), output (Y) relation for phase matched SROPO
Y 
Ps / s 
P
pth
X  Pp / Ppth
/ p 
Y   (1  Rs )Y


sn 2 i cosh 1 1 / Rs
 X  R Y / X   1 1/ R
s
s
Exact; given X, Rs, find Y
Y
If 1-Rs<<1 then:
X
If, also, X-1<<1, then:
Y 6
sin 2 Y 


X 1
( Rosencher and Fabre JOSA B,19, 1107, 2002 )
Normalised signal output versus
normalised pump input
(ps is normalised pump threshold intensity)
Rosencher & Fabre, JOSA B, 19, 1107, 2002
OPO with focussed Gaussian pump beam.
•
Seminal paper:
‘Parametric interaction of focussed Gaussian light beams’
Boyd and Kleinman, J. Appl. Phys. 39, 3597, (1968)
•
Extension to non-degenerate OPO. Relates treatments for plane-wave,
collimated Gaussian and focussed Gaussian:
‘Focussing dependence of the efficiency of a singly resonant OPO’
Guha, Appl. Phys. B, 66, 663, (1998)
Optimum Gaussian Beam Focussing to
Maximise parametric gain/pump power
b
2 w0
w0
n
L
2 w0
Confocal parameter
b=2w02n/l
Gain is maximised
(degenerate OPO,
no double-refraction)
for L/b = 2.8
Somewhat smaller L/b can be more convenient (1-1.5),
with only small gain reduction but a (usefully) significant
reduction of required pump intensity.
Boyd&Kleinman, J. Appl. Phys. 39, 3597, (1968)
Effect of tight focus on kL value for optimum gain
k Ξ k3-k2-k1 is phase-mismatch
for colinear waves.
Focussed beam
introduces non-colinearity.
Closure of k vector triangle,
to maximise parametric gain,
requires k2+k1>k3, negative k
k2
k1
k3
k2
k1
k3
Tighter focus, or higher-order pump-mode
(greater non-colinearity) needs more negative k
TEM00 to TEM01 mode change via tuning over the
parametric gain band
Hanna et al, J. Phys. D, 34, 2440, (2001)
Summary: Attractions of OPOs
• Very wide continuous tuning from a single device, via tuning
the phase-match condition
• High efficiency
• No heat input to the nonlinear medium
• No analogue of spatial-hole-burning as in a laser, hence
simplified single-frequency operation
• Very high gain capability
• Very large bandwidth capability
Demands posed by OPOs
• Signal frequency mode-hops caused by OPO cavity length change,
(as in a laser), AND by pump frequency shifts
• Single-frequency idler output requires single-frequency pump
• High pump brightness is required, (i.e. longitudinal laser-pumping);
no analogue of incoherent side-pumping of lasers
• Gain only when the pump is present
• Analytical description of OPO more complex than for a laser