1. 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

2. 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

3. 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

4. Coupled equations Fields Intensity d: effective nonlinear coefficient

5. Manley-Rowe relations Integrals of the coupled equations n 3 |E 3 (z)| 2 /ω 3 + n 2 |E 2 (z)| 2 /ω 2 = const n 3 |E 3 (z)| 2 /ω 3 + n 1 |E 1 (z)| 2 /ω 1 = const n 2 |E 2 (z)| 2 /ω 2 – n 1 |E 1 (z)| 2 /ω 1 = const Number of pump photons annihilated in NL medium equals the number of signal photons created, which also equals the number of idler photons created These imply n 3 |E 3 (z)| 2 + n 2 |E 2 (z)| 2 + n 1 |E 1 (z)| 2 = const i.e. conservation of power flow in propagation direction

6. Solution to coupled equations: (1) where and

7. Solution to coupled equations: (2) If only one input E 2, (E 1 (0) = 0)[amplifier or SRO] Single-pass power gain (increment) is, ( Corresponding multiplicative power gain, ) For exact phase-match, g = Γ, so

8. Plane-wave, phase-matched, parametric gain If gain is small, (G 2 (L) << 1), gain increment is Note: incremental gain proportional to pump intensity ~ proportional to ω 3 2 proportional to d 2 / n 3 (widely quoted as NL Figure Of Merit)

9. Plane-wave, phase-matched parametric gain (multiplicative) Note: since Γ 2  pump power P the gain exponent depends on √P (unlike Raman gain, where exponent  P) For high gain, ΓL >> 1 Very high gain is possible with ultra-short pump pulses, since gain is exponentially dependent on peak pump intensity

10. Phase relation between pump, signal, idler Suppose both signal and idler are input. Assuming Δk = 0, then Adds, maximally, to gain if Gain maximised if phase of nonlinear polarisation at ω 2 leads (by  / 2 ) the phase of e.m. wave at ω 2 Note: Fields are

11. OPO threshold: SRO vs DRO (1) If Δk = 0, threshold condition (assuming pump, signal & idler phases Φ 3 – Φ 2 – Φ 1 = -  / 2 at input to crystal) Represent round-trip power loss by one cavity mirror having reflectance R 1 (idler), R 2 (signal) Threshold → round-trip gain = round-trip loss (for signal only, SRO, for signal and idler, DRO) R 1,2

12. OPO threshold: SRO vs DRO (2) For SRO, R 1 = 0 SRO DRO Advantage of DRO is low threshold If 1- R 1,2 << 1 SRO threshold DRO threshold = 200 for 1 – R 1 = 0.02 (2%)

13. Parametric gain bandwidth For plane waves, max parametric gain is for frequencies ω 30 = ω 20 + ω 10 that achieve exact phase-match, k 3 = k 2 + k 1 If the signal frequency ω 2 is offset by there is a phase-mismatch For small gain, the signal gain is reduced to ~ ½ max for ΔkL~π Δk = 0, ω 2 = ω 20 δω2δω2 Gain δω2-δω2- δω2+δω2+ 0 Solve for δω 2 +, δω 2 - Hence gain bandwidth δω 2 + - δω 2 - Bandwidth reduces with greater L

14. 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 >>Γ 2 L 2 Power gain (increment) vs Δk sinh 2 ΓL (ΓL) 2 0Δk=2Γ |Δk|=  /L g'L = , hence ΔkΔk

15. 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) sinh 2 ΓL 0Δk=2Γ half max ΔkΔk Γ2L2Γ2L2 For ΓL>>1, Gain is:

16. Pump acceptance bandwidth (Assumes first term in Taylor series dominates) What range of pump frequencies can pump a single signal frequency? Low gain case: half-width,

17. Signal gain bandwidth (1) Gain peak: phase-matched ω 30 = ω 20 + ω 10, k 30 – k 20 – k 10 = 0 For same pump, ω 30, calculate corresponding to signal ω 20 + δ ω 2 (idler ω 10 - δ ω 2 ) Taylor series: Solve for δω 2

18. Signal gain bandwidth (2) 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 δ ω  Δk ½  L -½ )  For accuracy, use Sellmeier equn. rather than Taylor series  For high gain find Δk bandwidth via Half-width

19. SRO tuning range within gain profile Zero gain (incremental) for If ΓL >> 1 then |Δk|, hence tuning range,  [ I ] ½ A more exact treatment calculates the Δk that makes If ΓL << 1 then |Δk|, and hence tuning range, independent of Γ' sinh 2 ΓL 0 ΔkΔk

20. 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 φ 3 -φ 2 -φ 1, 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.

21. 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.

22. 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

23. 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

24. 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

25. Calculation of conversion efficiency (2) For SRO, with Δk = 0 and plane wave, find for pump When N = (  / 2 ) 2 ~ 2.5, find E 3 (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% !)

26. 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

27. 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)

28. Input (X), output (Y) relation for phase matched SROPO If 1-R s <<1 then: If, also, X-1<<1, then: Exact; given X, R s, find Y ( Rosencher and Fabre JOSA B,19, 1107, 2002 )

29. Normalised signal output versus normalised pump input (p s is normalised pump threshold intensity) Rosencher & Fabre, JOSA B, 19, 1107, 2002

30. 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)

31. Optimum Gaussian Beam Focussing to Maximise parametric gain/pump power Confocal parameter b=2  w 0 2 n/ 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 w0w0 w0w0 w0w0 b Boyd&Kleinman, J. Appl. Phys. 39, 3597, (1968)

32. Effect of tight focus on  kL value for optimum gain  k Ξ k 3 -k 2 -k 1 is phase-mismatch for colinear waves. Focussed beam introduces non-colinearity. k2k2 k1k1 k3k3 k2k2 k1k1 k3k3 Closure of k vector triangle, to maximise parametric gain, requires k 2 +k 1 >k 3, negative  k Tighter focus, or higher-order pump-mode (greater non-colinearity) needs more negative  k

33. TEM 00 to TEM 01 mode change via tuning over the parametric gain band Hanna et al, J. Phys. D, 34, 2440, (2001)

34. 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

35. 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