## Adiabatic optical parametric oscillators: steady-state and dynamical behavior |

Optics Express, Vol. 20, Issue 3, pp. 2466-2482 (2012)

http://dx.doi.org/10.1364/OE.20.002466

Acrobat PDF (1369 KB)

### Abstract

We study singly-resonant optical parametric oscillators with chirped quasi-phasematching gratings as the gain medium, for which adiabatic optical parametric amplification has the potential to enhance conversion efficiency. This configuration, however, has a modulation instability which must be suppressed in order to yield narrowband output signal pulses. We show that high conversion efficiency can be achieved by using either a narrowband seed or a high-finesse intracavity etalon.

© 2012 OSA

## 1. Introduction

1. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**, 12731–12740 (2009). [CrossRef] [PubMed]

2. C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. **35**, 3093–3095 (2010). [CrossRef] [PubMed]

*L*

*; after*

_{NL}*L*

*, back-conversion occurs, transferring energy back to the pump from the signal and idler waves [11*

_{NL}11. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. **127**, 1918–1939 (1962). [CrossRef]

12. W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-w continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. **21**, 1336–1338 (1996). [CrossRef] [PubMed]

*L*

*that is independent of transverse spatial position. The use of chirped QPM gratings offers a way of removing the above limitations on conversion efficiency of pulsed beams (in both the spatial and temporal domains) without the need for small beam areas, short pulse durations, or beam shaping.*

_{NL}12. W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-w continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. **21**, 1336–1338 (1996). [CrossRef] [PubMed]

19. R. Sowade, I. Breunig, I. Cmara Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express **17**, 22303–22310 (2009). [CrossRef]

20. A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B **16**, 609–619 (1999). [CrossRef]

25. R. White, Y. He, B. Orr, M. Kono, and K. Baldwin, “Transition from single-mode to multimode operation of an injection-seeded pulsed optical parametric oscillator,” Opt. Express **12**, 5655–5660 (2004). [CrossRef] [PubMed]

9. K. A. Tillman and D. T. Reid, “Monolithic optical parametric oscillator using chirped quasi-phase matching,” Opt. Lett. **32**, 1548–1550 (2007). [CrossRef] [PubMed]

10. K. A. Tillman, D. T. Reid, D. Artigas, J. Hellstrm, V. Pasiskevicius, and F. Laurell, “Low-threshold femtosecond optical parametric oscillator based on chirped-pulse frequency conversion,” Opt. Lett. **28**, 543–545 (2003). [CrossRef] [PubMed]

13. C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B **27**, 2687–2699 (2010). [CrossRef]

## 2. Coupled wave equations

14. C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Continuous wave monolithic quasi-phase-matched optical parametric oscillator in periodically poled lithium niobate,” Opt. Lett. **36**, 2973–2975 (2011). [CrossRef] [PubMed]

18. J. Kiessling, R. Sowade, I. Breunig, K. Buse, and V. Dierolf, “Cascaded optical parametric oscillations generating tunable terahertz waves in periodically poled lithium niobate crystals,” Opt. Express **17**, 87–91 (2009). [CrossRef] [PubMed]

19. R. Sowade, I. Breunig, I. Cmara Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express **17**, 22303–22310 (2009). [CrossRef]

*χ*

^{(2)}media [13

13. C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B **27**, 2687–2699 (2010). [CrossRef]

28. C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express **19**, 18754–18773 (2011) [CrossRef] [PubMed]

29. C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, J. Jiang, M. E. Fermann, and I. Hartl, “Supercontinuum generation in quasi-phasematched LiNbO_{3} waveguide pumped by a Tm-doped fiber laser system,” Opt. Lett. **36**, 3912–3914 (2011) [CrossRef] [PubMed]

13. C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B **27**, 2687–2699 (2010). [CrossRef]

*k*and

_{x}*k*, and reference velocity

_{y}*v*

_{ref}, which we choose to be the group velocity at

*ω*. The form of

_{s}*L̂*in Eq. (4) assumes paraxial diffraction in an isotropic medium. For propagation normal to the optical axis of an anisotropic medium, minor modifications are required for

_{j}*L*, but these do not significantly change the results of this analysis. For propagation at finite angles to the optical axis, first-order terms in

_{j}*k*or

_{x}*k*will appear and substantially alter the results; this case is beyond the scope of this paper. The linear propagation operator for the DC envelope,

_{y}*L̂*, has a similar form to

_{T}*L̂*, modified for a backwards-propagating wave, where

_{j}*α*(

*ω*) is the frequency-dependent power attenuation coefficient. We will neglect

*α*(

*ω*) at optical frequencies but not at THz frequencies. Finally, the coupling coefficients are given by

*γ*

_{opt}(

*ω*) = (

*ω*/

*c*)

^{2}(2

*d*

_{opt})/(

*πk*(

*ω*)) and

*γ*

_{THz}(

*ω*) = (

*ω*/

*c*)

^{2}(2

*d*

_{THz})/(

*πk*(

*ω*)), where

*d*

_{opt}and

*d*

_{THz}are the second-order nonlinear coefficients for the optical-optical and optical-THz mixing processes, respectively, and are determined by the relevant tensor elements and QPM orders. We assume that

*d*

_{opt}and

*d*

_{THz}are non-dispersive over the frequency ranges of interest.

20. A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B **16**, 609–619 (1999). [CrossRef]

## 3. Steady-state solution for chirped QPM OPOs

*z*-dependent field profiles found by solving Eq. (1) and imposing self-consistency after a cavity round-trip. These steady-state solutions are greatly simplified by using the asymptotic expressions for the chirped-QPM signal gain and pump depletion which apply for sufficiently strongly chirped QPM profiles [1

1. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**, 12731–12740 (2009). [CrossRef] [PubMed]

2. C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. **35**, 3093–3095 (2010). [CrossRef] [PubMed]

4. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B **25**, 463–480 (2008). [CrossRef]

4. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B **25**, 463–480 (2008). [CrossRef]

*k*′ =

*∂*Δ

*k*/

*∂z*is the QPM chirp rate (units of m

^{−2}), assumed to be constant near Δ

*k*(

*z*) ≈ 0. To understand this relation, we introduce a peak gain rate

*z*[31

31. R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. **15** (6), 432 (1979) [CrossRef]

*z*-dependent here due to the QPM chirp. To find the total gain, this gain rate is integrated over the region for which Re[

*g*] > 0 (i.e. the region of the grating for which the phase mismatch is small enough to allow OPA). For a linear chirp rate, this integration yields Eq. (6); this procedure can be made more formal via WKB analysis [4

4. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B **25**, 463–480 (2008). [CrossRef]

_{p}_{,}

*is the threshold signal gain coefficient and*

_{th}*R*= 1 –

_{s}*a*is the effective net round-trip reflectance (defined in terms of the round-trip signal power loss,

_{s}*a*).

_{s}1. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**, 12731–12740 (2009). [CrossRef] [PubMed]

*η*= |

_{p}*A*(

_{p}*L*)/

*A*(0)|

_{p}^{2}. Again assuming low losses, Eq. (6) and (7) can be used to express the OPO operating point, by equating the decrement to the number of signal photons lost due to the round trip losses and the increment from the depletion of the pump, yielding an implicit equation for the pump depletion coefficient Λ

*, where the pump ratio*

_{s}*N*is the ratio of pump intensity to threshold pump intensity (i.e. the number of times above oscillation threshold). Based on this equation, the circulating signal intensity (which is proportional to Λ

*by definition), and hence the conversion efficiency, are predicted to increase monotonically with pump intensity. This behavior is illustrated in Fig. 1, which shows the pump depletion predicted by solving Eq. (8) for Λ*

_{s}*as a function of*

_{s}*N*. Since conversion efficiency increases monotonically with signal and pump power in regimes with high gain and high pump depletion [2

2. C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. **35**, 3093–3095 (2010). [CrossRef] [PubMed]

*I*(

_{j}*z*) for the three waves [

*j*= (

*i*,

*s*,

*p*)], normalized to the input pump intensity

*I*

_{0}; for this example, we assume

*N*= 6 and

*R*= 95%. The pump is converted primarily near the center of the QPM grating, where phasematching is satisfied. Each of the fields is almost

_{s}*π*/2 out of phase with its nonlinear source term throughout the grating (as opposed to in phase in the case of phasematched interactions in periodic QPM gratings), corresponding to adiabatic following of local nonlinear eigenmodes [32

32. M. D Crisp, “Adiabatic-following approximation,” Phys. Rev. A **8**, 2128–2135 (1973). [CrossRef]

33. G. Luther, M. Alber, J. Marsden, and J. Robbins, “Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media,” J. Opt. Soc. Am. B **17**, 932–941 (2000). [CrossRef]

*N*(

*x*,

*y*) ≫ 1 across most of the spatiotemporal profile of the pump (positions

*x*and

*y*). By using a signal cavity mode somewhat larger than the pump beam and a cavity lifetime comparable to or longer than the duration of the pump pulse, this condition is more easily attained.

## 4. Temporal MI for chirped QPM OPO

*N*> 1, chirped QPM OPOs exhibit a temporal modulation instability. In order to yield a single-mode or narrow-band signal, this MI must be suppressed (by an intracavity etalon, for example).

**27**, 2687–2699 (2010). [CrossRef]

*z*-dependent steady-state solutions of Eq. (1), and then use Eq. (1) to calculate the single-pass amplification of small sidebands, detuned from the respective carrier frequencies by an amount ±Ω, superposed on each of the envelopes. We assume that Ω is positive (without loss of generality), so these sidebands have absolute optical frequencies

*ω*=

*ω*± Ω (for

_{j}*j*=

*i*,

*s*,

*p*, corresponding to the idler, signal, and pump envelopes) and

*ω*= Ω (corresponding to the DC envelope,

*A*). While the formalism we develop can address the general case of non-collinear sidebands, in this section, we use that formalism to evaluate the MI of chirped QPM, plane-wave OPOs with collinear sidebands, i.e. for sideband spatial frequency

_{T}*k*=

_{x}*k*= 0. For simplicity we consider the profile given by Eq. (9). Different grating profiles will exhibit comparable behavior provided that the phase mismatch is swept smoothly, monotonically, and slowly through zero, and has sufficiently large magnitude at

_{y}*z*= 0 and

*z*=

*L*.

*λ*= 1.064

_{p}*μ*m,

*λ*= 1.55

_{s}*μ*m, and the temperature

*T*= 150° C, and use the nonlinear coefficients and dispersion relation of MgO:LiNbO

_{3}[34

34. O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO_{3},” Appl. Phys. B: Lasers and Optics **91**, 343–348 (2008). [CrossRef]

35. K. L. Vodopyanov, “Optical THz-wave generation with periodically-inverted GaAs” Laser Photon. Rev. **2**, No. 1–2, 11–25 (2008) [CrossRef]

*N*, there is a range of frequencies for which there is MI gain (

*G*> 0). Therefore, this OPO would not operate in a single mode. The MI is not an artifact of the particular parameters used for Fig. 3(a), but occurs for many resonant wavelengths and chirp rates, provided that there is a sufficient grating chirp for adiabatic conversion to occur. However, the structure of the net sideband gain does depend on the material dispersion, as with conventional OPOs [13

**27**, 2687–2699 (2010). [CrossRef]

**27**, 2687–2699 (2010). [CrossRef]

*χ*

^{(2)}). Consider the interaction between an idler sideband

*ã*(∓Ω), a signal sideband

_{i}*ã*(±Ω), and the pump wave carrier

_{s}*n*

_{g,j}is the group index of wave

*j*, and

*ϕ*is the phase of carrier wave

_{j}*k*(

*z*) is given by Eq. (3). Similarly, for the interaction between a signal sideband

*ã*(±Ω), a pump sideband

_{s}*ã*(±Ω), and the idler carrier wave

_{p}*ã*(±Ω), a pump sideband

_{i}*ã*(±Ω), and the signal carrier wave

_{p}*k*

_{is,eff}(

*z*,+Ω) = 0 at

*z*=

*z*

_{pm}_{,}

*(+Ω)≈ −0.3*

_{is}*L*, where the pump is undepleted [see Fig (2)]. For the interaction between idler and pump sidebands, Δ

*k*

_{ip}_{,eff}(

*z*,±Ω) = 0 at

*z*=

*z*

_{pm}_{,}

*(±Ω) ≈ ∓0.01*

_{ip}*L*, which involves the strong steady-state signal field. For the interaction between signal and pump sidebands, Δ

*k*

_{sp}_{,eff}(

*z*,−Ω) = 0 at

*z*=

*z*

_{pm}_{,}

*(−Ω)≈ 0.31*

_{sp}*L*; this interaction involves the steady-state idler field, which is strong at

*z*

_{pm}_{,}

*(−Ω).*

_{sp}*k*(

*z*) = 0 (and in which each of the waves is nearly

*π*/2 out of phase with its nonlinear source term due to the adiabatic following process).

3. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO_{3},” Opt. Lett. **35**, 2340–2342 (2010). [CrossRef] [PubMed]

7. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B **25**, 1402–1413 (2008). [CrossRef]

28. C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express **19**, 18754–18773 (2011) [CrossRef] [PubMed]

*G*(Ω) < 1 for all sideband frequency detunings Ω. Thus, if an etalon is used, the free spectral range should be comparable to the MI gain bandwidth [as shown in Fig. 3(a) for a particular example], and the finesse must introduce sufficient loss at sideband frequencies within this spectral region that

*G*(Ω) < 1. The required etalon facet reflectance

*R*to fully suppress the MI can be estimated as [(1 −

*R*)/(1 +

*R*)]

^{2}= (

*G*

_{0})

^{–1}, where

*G*

_{0}denotes the peak MI gain in the absence of the etalon. In cases where the design parameters of a single intracavity etalon would be too constrained, multiple etalons or the combination of an etalon and a diffraction grating could be used.

*G*< 1 is necessary to avoid sideband amplification in the steady-state. Based on Fig. 3(a), a high reflectance etalon (e.g. with

*R*> 75%) is needed. This reflectance is significantly higher than that required to yield stable operation of CW-pumped OPOs using periodic QPM gratings, for which Fresnel reflections from uncoated optics are sufficient [13

**27**, 2687–2699 (2010). [CrossRef]

*G*is reduced. Conversely, it may not be necessary to fully suppress the MI within the etalon peak corresponding to the signal carrier frequency in order to achieve adiabatic conversion. The design issues associated with nanosecond-pumped chirped QPM OPOs are discussed in Section 5.

## 5. Numerical simulation of chirped QPM OPOs

### 5.1. Design considerations

*N*=

_{rt}*τ*/

_{p}*t*. For large

_{rt}*N*, the time-dependent signal intensity during the build-up stage can be expressed approximately as where the 2

_{rt}*π*Λ

_{p}_{,}

*is the signal gain coefficient associated with the peak of the pump pulse, and*

_{pk}*f*(

_{p}*t*) is the pump intensity profile normalized to its peak value, and takes values between 0 and 1.

*t*

_{0}is the time at which gain exceeds the cavity loss (i.e. where the integrand crosses zero), and

*I*

_{0}is an effective input noise intensity. The 2

*π*Λ

_{p}_{,}

*factor originates from Eq. (6). A second OPO constraint is that the signal should have a cavity lifetime comparable to the pump duration, in order to ensure depletion of the trailing edge of the pump. This constraint is described in terms of*

_{pk}*N*, the ratio of pump duration to the signal-cavity lifetime, given by

_{s}*N*should be of order unity. A third OPO constraint is that the pump should be intense enough to support adiabatic conversion on its leading edge, based on Eq. (8), and hence

_{s}*N*, the ratio of the signal gain at the peak of the pump pulse to the round trip cavity losses, should satisfy

_{pk}*N*by choosing appropriate values of Λ

_{rt}

_{R}_{,}

*and*

_{pk}*R*, although a more careful analysis is needed for cases when

_{s}*N*≫̸ 1. Assuming

_{rt}*N*≫ 1, the next consideration is the signal linewidth. For OPOs with a large

_{rt}*N*, the MI might not be suppressed for cavity modes near the relevant etalon peak (due to the finite etalon finesse), and in this case the signal bandwidth is comparable to the etalon bandwidth. Only one etalon peak should lie within the OPO acceptance bandwidth. This acceptance bandwidth can be approximated as 2

_{pk}*π*Δ

*f*≈ |(Δ

_{BW}*k*′

*L*)/(

*δn*/

_{g}*c*)|, where

*δn*=

_{g}*n*(

_{g}*ω*) –

_{s}*n*(

_{g}*ω*) for group index

_{i}*n*(

_{g}*ω*). Thus, the etalon free spectral range

*f*

_{fsr}can be constrained according to

*k*′

*L*

^{2}| ≈ 10

^{2}, as discussed in Section 3 (although the required grating k-space bandwidth can be reduced slightly by using nonlinear chirp profiles). Hence, in typical cases, the required free spectral range of the etalon satisfies

*f*

_{fsr}

*t*≈ 10

_{rt}^{3}. If the minimum etalon length is constrained (e.g. to tens of

*μ*m), then Eq. (16) also limits the minimum length of the QPM grating. Therefore, in the following numerical example, we will use a relatively long QPM grating (5 cm) and a pump pulse duration of 15 ns, long enough that

*N*≫ 1. The final design parameter is the etalon finesse, the effects of which can be explored numerically.

_{rt}### 5.2. Numerical example

*e*

^{2}duration of 15 ns and a peak intensity of 32 MW/cm

^{2}. The grating length is 5 cm with a chirp rate of 4 × 10

^{4}m

^{−2}(except in the apodization regions) and a QPM period chosen to phasematch a 1550-nm-wavelength signal in the middle of the grating; the grating has a similar profile to the example shown in Fig. 1(b). The round-trip losses were taken to be 19%. With these parameters,

*N*≈ 20.5, Λ

_{rt}*≈ 0.95, and hence the peak times above threshold*

_{p}*N*≈ 28. For cases when an intracavity etalon is included, the etalon has a free spectral range of 2 THz and a power reflectance of 64% on each facet. The corresponding etalon bandwidth is Δ

_{pk}*f*

_{et}≈ 120 GHz, and the product

*f*

_{fsr}

*t*≈ 1450. The value |(Δ

_{rt}*k*′

*L*

^{2}

*n*

_{g}_{,}

*)/(2*

_{s}*πδn*)| ≈ 1200, as discussed in relation to Eq. (16). The simulations use a method similar to the one described in Ref. [20

_{g}20. A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B **16**, 609–619 (1999). [CrossRef]

*N*> 1, noise-seeded signal sidebands are amplified over most of the pump pulse. A typical signal spectrum corresponding to a single simulation with a white-noise-seeded signal is shown in Fig. 5(b). The spectrum fills the OPO acceptance bandwidth, which corresponds to several THz. The corresponding transmitted signal and pump intensities

*I*(

_{j}*t*) are shown in Fig. 5(a). The conversion efficiency associated with this example is significantly reduced compared to Eq. (4) because the smooth signal phase profile required for adiabatic following is no longer present due to seeding with noise rather than a single frequency. This reduction in efficiency can be seen in the inset of Fig. 5(a), which plots

*W*(

*t*).

*W*(

*t*), which is plotted in the inset of Fig. 6(a): after saturation, the (average) slope

*dW*(

*t*)/

*dt*is positive but small compared to the slope shown in Fig. 5(a). The structure of the output pump pulses is shown in Fig. 6(c), where

*I*(

_{p}*t*)/

*I*

_{0}is plotted over a limited temporal region. These output pump spikes correspond to the finite bandwidth of the signal shown in Fig. 6(b), and repeat (approximately) every round-trip time. Conventional nanosecond OPOs can exhibit a similar self-pulsing behavior [20

**16**, 609–619 (1999). [CrossRef]

23. G. Arisholm, “General analysis of group velocity effects in collinear optical parametric amplifiers and generators,” Opt. Express **15**, 6513–6527 (2007). [CrossRef] [PubMed]

*N*, the suppression of spectral sidebands will scale with this finesse and with

*N*, and the MI gain bandwidth (and hence the etalon’s required free spectral range) will scale with the OPO signal-idler acceptance bandwidth. If the etalon has a large enough free spectral range that only one etalon peak lies within the OPO acceptance bandwidth but has an insufficient finesse to fully suppress the MI within that peak, the signal bandwidth is comparable to the etalon bandwidth Δ

_{rt}*f*

_{et}. To suppress the pump pulsing, a higher finesse etalon could be used; to yield a single- or few-mode signal via intracavity filters with realistic parameters, multiple filters (e.g. a grating and an etalon) or injection seeding might be required.

## 6. Conclusions

7. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B **25**, 1402–1413 (2008). [CrossRef]

36. G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A **49**, 2028–2032 (1994) [CrossRef]

*χ*

^{(3)}self phase modulation effects could lead to new and interesting types of OPO-based frequency combs.

## A. Modulation instability for CW OPO

**27**, 2687–2699 (2010). [CrossRef]

*z*-dependent field profiles which satisfy self-consistency in amplitude and phase after a single cavity round-trip. We then assume weak time-dependent perturbations around these zeroth-order solutions and solve the linear system which results from neglecting products of the perturbations, or sidebands,

*a*. The envelopes are assumed to have the form for zeroth-order fields

_{j}*a*(

_{j}**r**,

*t*). The zeroth-order fields

*ω*and have spatial frequencies

_{j}*k*=

_{x}*k*= 0.

_{y}*R*; this relation determines

_{s}*N*, for a given system. We assume that the signal carrier frequency corresponds to an axial mode of the cavity [15

15. S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B **10**, 1684–1695 (1993). [CrossRef]

38. M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, and R. L. Byer, “Dynamic response of a Fabry-Perot interferometer,” J. Opt. Soc. Am. B **16**, 523–532 (1999). [CrossRef]

*Ẽ*(

_{s}*ω*), is also self-consistent after a cavity round-trip. Although

_{s}*a*are coupled to each other in the presence of the zeroth-order fields, we define spatial frequency vector

_{j}**k**

_{⊥}=

*k*

_{x}**x̂**+

*k*

_{y}**ŷ**and sideband frequency Ω = |

*ω*–

*ω*|. Because we assume

_{j}*k*=

_{x}*k*= 0 for

_{y}*a*can only be coupled together in a limited number of ways. To write down the coupling matrix, we first introduce a shorthand notation for the sidebands:

_{j}*k*and

_{x}*k*can be positive or negative. We now define a sideband vector where the dependencies of

_{y}*z*,

**k**

_{⊥}, and Ω have been suppressed. For the DC envelope, only

*ω*for

_{j}*j*= (

*i*,

*s*,

*p*) (an appropriate assumption when the OPO acceptance bandwidth is much less than the pump, signal and idler carrier frequencies, which is almost always the case). Propagation for this sideband vector is described by a linear system, where

*M*(

*z*) is a 7x7 coupling matrix which depends on the frequency-domain arguments as well as

*z*. Due to the assumed axial symmetry of the problem, the same coupling matrix applies to both

*ṽ*(+

**k**

_{⊥}, Ω) and

*ṽ*(−

**k**

_{⊥}, Ω) for arbitrary

**k**

_{⊥}. The coupling matrix

*M*(

*z*) has a form similar to the one introduced in Ref. [13

**27**, 2687–2699 (2010). [CrossRef]

*Ã*in Eq. (1),

_{T}*κ*=

_{T}*γ*

_{THz}(Ω),

*M*are determined by the linear differential operators

*L̂*[Eqs. (4) and (5)], while the off-diagonal elements determine coupling between the sideband vectors due to

_{j}*χ*

^{(2)}interactions. For chirped QPM gratings, all the non-zero elements of

*M*are

*z*-dependent.

*, denoted*

_{rt}*λ*

_{Φ}

_{,}

*for*

_{j}*j*= 1 and

*j*= 2. Modes of the “hot” cavity are those frequencies for which

*λ*

_{Φ}

_{,}

*are real; these frequencies can differ from the frequencies of the “cold” cavity modes as a result of phase shifts due to the three-wave interaction, and due to coupling between frequencies at +Ω and −Ω. Exponential growth (MI) of such a cavity mode occurs when*

_{j}*λ*

_{Φ}

_{,}

*(Ω) > 1, for some*

_{j}*j*and some Ω. We assume that the cavity modes are closely-spaced in frequency compared to the variation of the eigenvalues with sideband frequency Ω; therefore, we can define the MI condition as |

*λ*

_{Φ}

_{,}

*| > 1. A more detailed description of the MI calculation is given in Ref. [13*

_{j}**27**, 2687–2699 (2010). [CrossRef]

## Acknowledgments

## References and links

1. | H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express |

2. | C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. |

3. | C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO |

4. | M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B |

5. | G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B |

6. | L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings,” Opt. Lett. |

7. | M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B |

8. | M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan, “Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett. |

9. | K. A. Tillman and D. T. Reid, “Monolithic optical parametric oscillator using chirped quasi-phase matching,” Opt. Lett. |

10. | K. A. Tillman, D. T. Reid, D. Artigas, J. Hellstrm, V. Pasiskevicius, and F. Laurell, “Low-threshold femtosecond optical parametric oscillator based on chirped-pulse frequency conversion,” Opt. Lett. |

11. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

12. | W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-w continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. |

13. | C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B |

14. | C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Continuous wave monolithic quasi-phase-matched optical parametric oscillator in periodically poled lithium niobate,” Opt. Lett. |

15. | S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B |

16. | L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum. Electron. |

17. | A. Henderson and R. Stafford, “Spectral broadening and stimulated Raman conversion in a continuous-wave optical parametric oscillator,” Opt. Lett. |

18. | J. Kiessling, R. Sowade, I. Breunig, K. Buse, and V. Dierolf, “Cascaded optical parametric oscillations generating tunable terahertz waves in periodically poled lithium niobate crystals,” Opt. Express |

19. | R. Sowade, I. Breunig, I. Cmara Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express |

20. | A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B |

21. | A. V. Smith, “Bandwidth and group-velocity effects in nanosecond optical parametric amplifiers and oscillators,” J. Opt. Soc. Am. B |

22. | G. Arisholm, “Quantum noise initiation and macroscopic fluctuations in optical parametric oscillators,” J. Opt. Soc. Am. B |

23. | G. Arisholm, “General analysis of group velocity effects in collinear optical parametric amplifiers and generators,” Opt. Express |

24. | G. Arisholm, G. Rustad, and K. Stenersen, “Importance of pump-beam group velocity for backconversion in optical parametric oscillators,” J. Opt. Soc. Am. B |

25. | R. White, Y. He, B. Orr, M. Kono, and K. Baldwin, “Transition from single-mode to multimode operation of an injection-seeded pulsed optical parametric oscillator,” Opt. Express |

26. | A. Yariv, |

27. | G. Agrawal, |

28. | C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express |

29. | C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, J. Jiang, M. E. Fermann, and I. Hartl, “Supercontinuum generation in quasi-phasematched LiNbO |

30. | M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A |

31. | R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. |

32. | M. D Crisp, “Adiabatic-following approximation,” Phys. Rev. A |

33. | G. Luther, M. Alber, J. Marsden, and J. Robbins, “Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media,” J. Opt. Soc. Am. B |

34. | O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO |

35. | K. L. Vodopyanov, “Optical THz-wave generation with periodically-inverted GaAs” Laser Photon. Rev. |

36. | G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A |

37. | J. E. Schaar, “Terahertz Sources Based On Intracavity Parametric Frequency Down-Conversion Using Quasi-Phase-Matched Gallium Arsenide,” Ph.D. thesis, Stanford University (2009) |

38. | M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, and R. L. Byer, “Dynamic response of a Fabry-Perot interferometer,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(190.4360) Nonlinear optics : Nonlinear optics, devices

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 16, 2011

Revised Manuscript: December 23, 2011

Manuscript Accepted: December 27, 2011

Published: January 19, 2012

**Citation**

C. R. Phillips and M. M. Fejer, "Adiabatic optical parametric oscillators: steady-state and dynamical behavior," Opt. Express **20**, 2466-2482 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2466

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### References

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- C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett.35, 3093–3095 (2010). [CrossRef] [PubMed]
- C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO3,” Opt. Lett.35, 2340–2342 (2010). [CrossRef] [PubMed]
- M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B25, 463–480 (2008). [CrossRef]
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- L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings,” Opt. Lett.26, 614–616 (2001). [CrossRef]
- M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B25, 1402–1413 (2008). [CrossRef]
- M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan, “Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett.30, 634–636 (2005). [CrossRef] [PubMed]
- K. A. Tillman and D. T. Reid, “Monolithic optical parametric oscillator using chirped quasi-phase matching,” Opt. Lett.32, 1548–1550 (2007). [CrossRef] [PubMed]
- K. A. Tillman, D. T. Reid, D. Artigas, J. Hellstrm, V. Pasiskevicius, and F. Laurell, “Low-threshold femtosecond optical parametric oscillator based on chirped-pulse frequency conversion,” Opt. Lett.28, 543–545 (2003). [CrossRef] [PubMed]
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