Lecture 2 Parametric amplification and oscillation: Basic principles

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Lecture 2Parametric amplification and
oscillation Basic principles

• David Hanna
• Optoelectronics Research Centre
• University of Southampton
• Lectures at Friedrich Schiller University, Jena
• July/August 2006

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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

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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

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Coupled equations
Fields
Intensity
d effective nonlinear coefficient
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Manley-Rowe relations
Integrals of the coupled equations
n3E3(z)2/?3 n2E2(z)2/?2 const
n3E3(z)2/?3 n1E1(z)2/?1 const
n2E2(z)2/?2 n1E1(z)2/?1 const
These imply n3E3(z)2 n2E2(z)2 n1E1(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
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Solution to coupled equations (1)
and
where
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Solution to coupled equations (2)
If only one input E2, (E1(0) 0)
amplifier or SRO
Single-pass power gain (increment) is,
For exact phase-match, g G , so
(Corresponding multiplicative power gain,

)
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Plane-wave, phase-matched, parametric gain
If gain is small, (G2(L) ltlt 1) , gain increment
is
Note incremental gain proportional to pump
intensity
proportional to ?32
proportional to d2 / n3
(widely quoted as NL Figure Of Merit)
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Plane-wave, phase-matched parametric gain
(multiplicative)
For high gain, GL gtgt 1
Very high gain is possible with ultra-short pump
pulses, since gain is exponentially dependent on
peak pump intensity
Note since G2 ? pump power P the gain
exponent depends on vP (unlike Raman gain, where
exponent ? P)
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Phase relation between pump, signal, idler
Suppose both signal and idler are input.
Assuming ?k 0 , then
Adds, maximally, to gain if
Note Fields are
Gain maximised if phase of nonlinear polarisation
at ?2 leads (by ?/2) the phase of e.m. wave at ?2
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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 ?k 0 , threshold condition (assuming
pump, signal idler phases F3 F2 F1 - ?/2
at input to crystal)
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OPO threshold SRO vs DRO (2)
For SRO, R1 0
If 1- R1,2 ltlt 1
SRO
DRO
Advantage of DRO is low threshold
SROthreshold
200 for 1 R1 0.02 (2)
DROthreshold
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Parametric gain bandwidth
For plane waves, max parametric gain is for
frequencies ?30 ?20 ?10 that achieve exact
phase-match, k3 k2 k1
If the signal frequency ?2 is offset by there is
a phase-mismatch
For small gain, the signal gain is reduced to
½ max for ?kLp
Gain
?k 0 , ?2 ?20
Solve for d?2 , d?2-
Hence gain bandwidth d?2 - d?2-
Bandwidth reduces with greater L
d?2
d?2-
d?2
0
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Parametric gain bandwidth small gain
Power gain (increment) vs ?k
sinh2GL
(GL)2
g'L ? , hence
?k
0
?k2G
?k ?/L
For small gain (GL ltlt 1), gain-half-maximum is
approximately given by ?k ?/L , hence
independent of G ( therefore of intensity).
For high gain (GL gtgt 1), power gain is ¼
exp(2GL), hence gtgtG2L2
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Parametric gain bandwidth large gain
For GLgtgt1, Gain is
sinh2GL
half max
(?k ltlt G)
G2L2
?k
0
?k2G
3dB gain reduction for (?kL)2 / 4GL ln 2 ?k
2(Gln2/L)½
?k bandwidth (high gain)
(4 ln 2 GL)½
0.53 (GL)½

?
?k bandwidth (low gain)
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Pump acceptance bandwidth
What range of pump frequencies can pump a single
signal frequency?
Low gain case half-width,
(Assumes first term in Taylor series dominates)
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Signal gain bandwidth (1)
Gain peak phase-matched ?30 ?20 ?10 , k30
k20 k10 0
For same pump, ?30 , calculate
corresponding to signal ?20 d ?2 (idler ?10 -
d ?2)
Taylor series
Solve for d?2
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Signal gain bandwidth (2)
For small gain, ?kL/2 ?/2 defines the
half-max. gain condition
provided 1st. Taylor series term
dominates
Half-width

• At degeneracy, use second Taylor term (note d?
  ? ?k½ ?L-½ )
• For accuracy, use Sellmeier equn. rather than
  Taylor series
• For high gain find ?k bandwidth via

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SRO tuning range within gain profile
sinh2GL
Zero gain (incremental) for
If GL ltlt 1 then ?k, and hence tuning range,
independent of G'
If GL gtgt 1 then ?k, hence tuning range, ?
I½
?k
0
A more exact treatment calculates the ?k that
makes
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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 f3-f2-f1, 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.

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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.
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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

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CW singly-resonant OPOs in PPLN

• First cw SRO Bosenberg et al. O.L., 21, 713
  (1996)
• 13w NdYAG pumped 50mm XL, 3w threshold, gt1.2w _at_
  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 _at_ 2.1µm
  (2.5w pump)
• Fibre-laser-pumped Gross et al. O.L., 27, 418
  (2002)
• 1.9w idler _at_ 3.2µm for 8.3w pump

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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

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Calculation of conversion efficiency (2)
For SRO, with ?k 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 !)
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Typical OPO conversion efficiencies

• Generally high conversion efficiency (gt 50)
• is observed at 2-3 x threshold
• Initial slope efficiency gt 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

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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)

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Input (X), output (Y) relation for phase matched
SROPO
Exact given X, Rs, find Y
If 1-Rsltlt1 then
If, also, X-1ltlt1, then
( Rosencher and Fabre JOSA B,19, 1107, 2002 )
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Normalised signal output versus normalised pump
input
(ps is normalised pump threshold
intensity) Rosencher Fabre, JOSA B, 19, 1107,
2002
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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)

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Optimum Gaussian Beam Focussing to Maximise
parametric gain/pump power
Confocal parameter b2pw02n/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.
BoydKleinman, J. Appl. Phys. 39, 3597, (1968)
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Effect of tight focus on DkL value for optimum
gain
Dk ? k3-k2-k1 is phase-mismatch for colinear
waves. Focussed beam introduces non-colinearity.
Closure of k vector triangle, to maximise
parametric gain, requires k2k1gtk3, negative Dk
Tighter focus, or higher-order pump-mode (greater
non-colinearity) needs more negative Dk
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TEM00 to TEM01 mode change via tuning over the
parametric gain band
Hanna et al, J. Phys. D, 34, 2440, (2001)
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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

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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