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