Lecture 2 Parametric amplification and oscillation: Basic principles David Hanna Optoelectronics...
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Transcript of Lecture 2 Parametric amplification and oscillation: Basic principles David Hanna Optoelectronics...
Lecture 2Parametric amplification and oscillation:
Basic principles
David HannaOptoelectronics Research Centre
University of Southampton
Lectures at Friedrich Schiller University, JenaJuly/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
d: effective nonlinear coefficient
cctrkirEtrE ).(exp),(2/1),( 2
0 ),(2/1 rEncI
)exp(*231
1 kziEEidz
dE
)exp(*132
2 kziEEidz
dE
cnd jjj /
123 kkkk
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
Number of pump photons annihilated in NL medium equals thenumber of signal photons created, which also equals the number of idler photons created
These imply
n3|E3(z)|2 + n2|E2(z)|2 + n1|E1(z)|2 = consti.e. conservation of power flow in propagation direction
Solution to coupled equations: (1)
2/122 )2/( kLg 2
3212 E
gLkLig
EEigL
g
kigLkLiELE sinh)2/exp(
)0(sinhcosh)2/exp()0()(
*23
111
where and
gLkLig
EEigL
g
kigLkLiELE sinh)2/exp(
)0(sinhcosh)2/exp()0()(
*13
222
Solution to coupled equations: (2)
If only one input E2, (E1(0) = 0) [amplifier or SRO]
Single-pass power gain (increment) is,
(Corresponding multiplicative power gain, )
22
222
2
2
22
sinh1
)0(
)()(
gL
gLL
E
LELG
LLG 22 sinh)(
LLGLG x 222 cosh)(1)(
For exact phase-match, g = Γ , so
Plane-wave, phase-matched, parametric gain
If gain is small, (G2(L) << 1) , gain increment is
Note: incremental gain proportional to pump intensity
~ proportional to ω32
30321
3
2
2122 2
cnnn
IdL
proportional to d2 / n3
(widely quoted as NL Figure Of Merit)
Plane-wave, phase-matched parametric gain (multiplicative)
Note: since Γ2 pump power P
the gain exponent depends on √P
(unlike Raman gain, where exponent P)
)2exp(4/1cosh)( 22 LLLG
For high gain, ΓL >> 1
Very high gain is possible with ultra-short pump pulses,since gain is exponentially dependent on peak pump intensity
Phase relation between pump, signal, idler
2/1)(exp 123123 ii
).(exp),(2/1),( trkirEtrE
Suppose both signal and idler are input.
Assuming Δk = 0 , then
Adds, maximally, to gain if
Gain maximised if phase of nonlinear polarisation at ω2 leads (by /2) the phase of e.m. wave at ω2
Note: Fields are
LE
EEiL
E
LE
sinh
)0(
)0(cosh
)0(
)(
2
*132
2
2
OPO threshold: SRO vs DRO (1)
If Δk = 0 , threshold condition (assuming pump, signal & idler phases Φ3 – Φ2 – Φ1 = - /2 at input to crystal)
2/12
2/11
2/121 )(1
coshRR
RRL
Represent round-trip power loss by one cavity mirror having reflectance R1 (idler), R2 (signal)
Threshold → round-trip gain = round-trip loss
(for signal only, SRO, for signal and idler, DRO)
R1,2
OPO threshold: SRO vs DRO (2)
1cosh 22 LR
222 1 RL
4/)1)(1( 1222 RRL
For SRO, R1 = 0
SRO
DRO
Advantage of DRO is low threshold
If 1- R1,2 << 1
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
For small gain, the signal gain is reduced to ~ ½ max for ΔkL~π
2022
1232 )( kkkk
Δk = 0 , ω2 = ω20
Lk )( 2
δω2
Gain
δω2- δω2
+0
Solve for δω2+ , δω2
-
Hence gain bandwidth δω2+ - δω2
-
Bandwidth reduces with greater L
Parametric gain bandwidth: small gain
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
22 )2/( kg
22)2/( kg
Power gain (increment) vs Δk
2/122 ])/([2 Lk
2
22
)(
sinh)(
gL
gLL
LgcL 22 sin)(
sinh2ΓL
(ΓL)2
0 Δk=2Γ|Δk|= /L
g'L = , hence
Δk
Parametric gain bandwidth: large gain
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
(Δk << Γ)
sinh2ΓL
0 Δk=2Γ
half max ],)4/)((2exp[
))2/(2exp(4/1)2exp(4/1~2
22
LkL
LkgL
Δk
Γ2L2
For ΓL>>1, Gain is:
Pump acceptance bandwidth
111313
3 //
gg vvkk
(Assumes first term in Taylor series dominates)
What range of pump frequencies can pump a single signal frequency?
Low gain case: half-width,
Signal gain bandwidth (1)
Gain peak: phase-matched ω30 = ω20 + ω10 , k30 – k20 – k10 = 0
For same pump, ω30 , calculate
corresponding to signal ω20 + δ ω2 (idler ω10 - δ ω2)
...2
1 222
22
21
2
221
kkkkk
Taylor series:
)()( 1102201230 kkkkkkkk
Solve for δω2
Signal gain bandwidth (2)
)( 12
11
2
gg vvL
For small gain, ΔkL/2 = /2 defines the ~ half-max. gain condition
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 1sinh/1 222 gLgR
Half-width
SRO tuning range within gain profile
2/122 ))/((2 Lk Zero gain (incremental) for
If ΓL >> 1 then |Δk|, hence
tuning range, [I]½
1
)2/(
)2/(sinh1
22
2/12222
k
LkR
A more exact treatment calculates the Δk that makes
If ΓL << 1 then |Δk|, and hence
tuning range, independent of Γ'
sinh2ΓL
0Δk
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 self-stabilisation 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 (1)
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
Calculation of conversion efficiency (2)
)(sincos)0(
)( 2/1122
3
2
3 NcE
LE
2
,3
2
3 )0(/)0( thresholdEEN
For SRO, with Δk = 0 and plane wave, find for pump
When N = (/2)2 ~ 2.5 , find E3(L) = 0 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 (back-conversion 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)
YRY s )1(
ppth
ss
P
PY
/
/ pthp PPX /
sss RXYRXRisn /11//1cosh 12
Y
YX
2sin
16 XY
Input (X), output (Y) relation for phase matched SROPO
If 1-Rs<<1 then:
If, also, X-1<<1, then:
Exact; given X, Rs, find Y
( 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 toMaximise parametric gain/pump power
Confocal parameterb=2w0
2n/
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.
L
n
w0
w02w02
b
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.
k2 k1
k3
k2 k1
k3
Closure of k vector triangle, to maximise parametric gain,requires k2+k1>k3, negative k
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