## Design of quasi-phasematching gratings via convex optimization |

Optics Express, Vol. 21, Issue 8, pp. 10139-10159 (2013)

http://dx.doi.org/10.1364/OE.21.010139

Acrobat PDF (1132 KB)

### Abstract

We propose a new approach to quasi-phasematching (QPM) design based on convex optimization. We show that with this approach, globally optimum solutions to several important QPM design problems can be determined. The optimization framework is highly versatile, enabling the user to trade-off different objectives and constraints according to the particular application. The convex problems presented consist of simple objective and constraint functions involving a few thousand variables, and can therefore be solved quite straightforwardly. We consider three examples: (1) synthesis of a target pulse profile via difference frequency generation (DFG) from two ultrashort input pulses, (2) the design of a custom DFG transfer function, and (3) a new approach enabling the suppression of spectral gain narrowing in chirped-QPM-based optical parametric chirped pulse amplification (OPCPA). These examples illustrate the power and versatility of convex optimization in the context of QPM devices.

© 2013 OSA

## 1. Introduction

1. 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] .

8. C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express **20**, 2466–2482 (2012) [CrossRef] [PubMed] .

9. J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasi-phase-matched interactions,” Opt. Lett. **31**, 604–606 (2006) [CrossRef] [PubMed] .

10. S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical super-lattice,” Science **278**, 843–846 (1997) [CrossRef] .

14. G. Porat, Y. Silberberg, A. Arie, and H. Suchowski, “Two photon frequency conversion,” Opt. Express **20**, 3613–3619 (2012) [CrossRef] [PubMed] .

2. 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] .

15. 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 **18**, 534–539 (2001) [CrossRef] .

18. Ł. Kornaszewski, M. Kohler, U. K. Sapaev, and D. T. Reid, “Designer femtosecond pulse shaping using grating-engineered quasi-phase-matching in lithium niobate,” Opt. Lett. **33**, 378–380 (2008) [CrossRef] [PubMed] .

19. M. A. Albota and F. C. Wong, “Efficient single-photon counting at 1.55 *μ*m by means of frequency upconversion,” Opt. Lett. **29**, 1449–1451 (2004) [CrossRef] [PubMed] .

21. J. S. Pelc, Q. Zhang, C. R. Phillips, L. Yu, Y. Yamamoto, and M. M. Fejer, “Cascaded frequency upconversion for high-speed single-photon detection at 1550 nm,” Opt. Lett. **37**, 476–478 (2012) [CrossRef] [PubMed] .

22. T. Fuji, J. Rauschenberger, A. Apolonski, V. S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T. W. Haensch, W. Lehnert, M. Scherer, and F. Krausz, “Monolithic carrier-envelope phase-stabilization scheme,” Opt. Lett. **30**, 332–334 (2005) [CrossRef] [PubMed] .

27. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. **31**, 1881–1883 (2006) [CrossRef] [PubMed] .

28. M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature **432**, 374–376 (2004) [CrossRef] [PubMed] .

29. R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. **95**, 133901–133904 (2005) [CrossRef] [PubMed] .

*d̄*(

*z*) ≡

*d*(

*z*)/

*d*

_{0}= ±1, where

*d*(

*z*) and

*d*

_{0}are the relevant nonlinear coefficients in the grating and in the unperturbed material, respectively, (2) the maximum grating length available (typically based on size of the wafer), and (3) the minimum QPM period (typically around 5

*μ*m in MgO:LiNbO

_{3}, although much shorter periods have been implemented [30

30. C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Photon. **1**, 459–462 (2007) [CrossRef] .

*d̄*= ±1 constraint inherent in QPM gratings, combined with the fact that thousands of QPM domains are typically present in a single device, means that optimal QPM design of three-wave mixing interactions is challenging, even if one can assume only one envelope (e.g. a generated idler wave) is changing within the device. To overcome this issue, one can work in the first-order-QPM approximation: the grating is written in terms of an arbitrary but smooth phase function

*ϕ*(

*z*) and duty cycle function

*D*(

*z*) as where Eq. (1a) implies

*d̄*= ±1, and Eq. (1b) follows from Eq. (1a) as an identity. This identity could be derived, for example, by assuming a constant duty cycle

*D*and linear phase

*ϕ*(

*z*), expressing the resulting periodic grating as a Fourier series, and then noting the point-wise convergence of that series to Eq. (1a) for arbitrary phase and duty cycle. Equation (1c) defines the Fourier coefficients

*d̄*

*, and in particular*

_{m}*d̄*

_{1}, which is given by Rather than dealing with

*d̄*, terms other than

*d̄*

_{1}and

*d̄*

_{−1}are neglected, and

*d̄*= ±1 is replaced by the constraint |

*d̄*

_{±1}| ≤ 2/

*π*(or, with a constant 50% duty cycle, |

*d̄*

_{±1}| = 2/

*π*) [31

31. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron. **28**, 2631–2654 (1992) [CrossRef] .

*d̄*

_{−1}, while those driving the lower-frequency signal and idler waves involve

*d̄*

_{+1}(or vice versa, depending on the envelope convention used). Since

*d̄*is real,

*ϕ*(

*z*) in terms of the (continuous) grating k-vector

*K*(

_{g}*z*),

*K*), relative to their center or carrier spatial frequency, are required. Even if few-cycle pulses are involved, this narrow-grating-bandwidth condition can still hold very well (as discussed in section 4). For chirped gratings, this condition means that the range of periods required is small compared to the average period; for periodic gratings, it means that many periods are present in the device.

_{g}*d̄*

*and*

_{i}*d̄*

*for*

_{j}*i*≠

*j*, and hence only a single term

*d̄*

*must be considered for a particular nonlinear-optical interaction. It should be noted that this non-overlapping-spectra property may not hold when the grating structure has errors arising from the fabrication process [31*

_{i}31. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron. **28**, 2631–2654 (1992) [CrossRef] .

*d̄*

*with |*

_{j}*j*| ≫ 1); we do not consider these issues here. The first-order QPM approximation has been utilized extensively in both modeling and design of QPM devices.

15. 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 **18**, 534–539 (2001) [CrossRef] .

15. 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 **18**, 534–539 (2001) [CrossRef] .

## 2. Coupled wave equations

### 2.1. Difference frequency generation

**18**, 534–539 (2001) [CrossRef] .

*κ*(

_{j}*ω*) =

*ωd*

_{0}/(

*n*(

_{j}*ω*)

*c*),

*d̄*(

*z*) =

*d*(

*z*)/

*d*

_{0}is the normalized nonlinear coefficient discussed in section 1, and

*n*(

_{j}*ω*) is the refractive index of wave

*j*. Subscripts

*i*,

*s*, and

*p*represent the idler, signal, and pump, respectively. Tilde denotes a frequency-domain field quantity. The frequency-dependent phase mismatch Δ

*k*(

*ω*,

*ω*′) is given by where

*k*(

_{j}*ω*) =

*n*(

_{j}*ω*)

*ω*/

*c*is the wavevector associated with the polarization of envelope

*j*, evaluated at frequency

*ω*. The envelopes

*A*are defined to contain only positive optical frequency components, and are such that the total electric field satisfies where

_{j}*u*(

*ω*) is the Heaviside step function, and Eq. (6b) implicitly defines envelopes

*B*. This analytic signal approach leads to the [0, ∞) integration limits in Eq. (4). Note that analytic signals were not used in Ref. [15

_{j}**18**, 534–539 (2001) [CrossRef] .

*χ*

^{(2)}devices [25

25. 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] .

35. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**, 053841–053844 (2010) [CrossRef] .

*Ã*are constant under linear propagation. We assume that the signal and pump waves are unperturbed by the nonlinear process, and hence

_{j}*Ã*(

_{j}*z*,

*ω*) =

*Ã*(0,

_{j}*ω*) for

*j*=

*s*and

*j*=

*p*. We have also introduced the envelopes

*B*: these envelopes are more conventional time-domain envelopes, i.e. envelopes whose inverse Fourier transform directly yields a component of the electric field at position

_{j}*z*and time

*t*. For notational simplicity, the definition of these envelopes [Eq. (6b)] does not include a

*z*-dependent phase factor; note that this is in contrast to the “

*B*” envelopes in [15

**18**, 534–539 (2001) [CrossRef] .

*ik*(

_{j}*ω*)

_{j}*z*) for carrier frequency

*ω*of wave

_{j}*j*[compare Eqs. (5) and (6) of [15

**18**, 534–539 (2001) [CrossRef] .

*g*(

*k*) is the spatial Fourier transform of the normalized nonlinear coefficient, Note that for a grating of length

*L*,

*d̄*= 0 for

*z*< 0 and

*z*>

*L*.

*k*(

_{j}*ω*)

*L*, which is not accounted for explicitly in the envelopes

*Ã*but is included in the envelopes

_{j}*B̃*

*. The full input-output relation for the device is thus given by with*

_{j}*Ã*(

_{s}*ω*) =

*B̃*

*(0,*

_{s}*ω*) and

*Ã*(

_{p}*ω*) =

*B̃*

*(0,*

_{p}*ω*). Equation (9) shows that there is a linear relationship between the spatial Fourier transform of the QPM grating,

*g*(

*k*), and the generated idler wave

*B̃*

*(*

_{i}*L*,

*ω*). To take advantage of this linearity, in subsection 2.2 we express Eq. (9) as a linear system.

### 2.2. General matrix formulation of DFG

### 2.3. Form of the DFG coupling matrix

**T**. The form of

**T**is the main factor that determines which pulse designs are possible. Its structure originates from the functional forms of the phase mismatch and the nonlinear polarization. If the pump is narrow-bandwidth, then for each output idler frequency there is only a small range of signal frequencies that are able to contribute. There is thus a correspondingly small range of spatial frequencies which contribute, governed by the excursions of Δ

*k*over this frequency range.

**T**will depend mostly on group velocity dispersion and will have a “crescent” like shape. This shape in fact persists even in the presence of GVM, but is tilted in

*ω*−

*K*space. These general properties are illustrated in Fig. 1. The parameters for the figure are as follows. The pump center wavelength is 1064 nm. The signal center wavelength is either 1550 nm [Figs. 1(a) and 1(c)] or 2000 nm [Figs. 1(b) and 1(d)]. Both the signal and pump pulses are Gaussian with no chirp. The signal 1/

_{g}*e*

^{2}duration is 40 fs, while the pump 1/

*e*

^{2}duration is either 2 ps [for Figs. 1(a) and 1(b)] or 200 fs [Figs. 1(c) and 1(d)]. We assume a MgO:LiNbO

_{3}crystal at room temperature [36

36. 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 Opt. **91**, 343–348 (2008) [CrossRef] .

*ω*′ (in particular, this is necessary if Δ

*k*(

*ω*,

*ω*′) passes through a minimum or maximum with respect to

*ω*′).

*K*.

_{g}### 2.4. Transfer function limit

*ω*, we have

_{p}*A*(

_{p}*ω*) = 2

*πA*

_{p}_{0}

*δ*(

*ω*−

*ω*). Therefore, Eq. (9) becomes where

_{p}*H*(

*ω*) is a transfer function. As discussed in [15

**18**, 534–539 (2001) [CrossRef] .

**T**are given, in the case of a cw pump, by If the optical frequency and spatial frequency grids coincide, i.e. if for all elements

*ω*of the frequency grid, Δ

_{n}*k*(

*ω*,

_{n}*ω*−

_{p}*ω*) =

_{n}*k*, where

_{m}*k*is one of the elements of the spatial frequency grid, then

_{m}*v*(Δ

_{m}*k*(

*ω*,

_{n}*ω*−

_{p}*ω*)) = 1 for such pairs of indices and is equal to zero for all other pairs. In other cases, the general form of

_{n}*v*(Δ

*k*(

*ω*,

_{n}*ω*−

_{p}*ω*)) defined in Eq. (8) can be used. Coinciding grids can be obtained in this transfer function case simply by choosing a non-uniformly spaced spatial frequency grid.

_{n}## 3. Optimal QPM DFG design examples

### 3.1. Optimization framework

*g*(

*k*) in Eq. (8) can be accurately evaluated as a spatial Fourier transform of

*d̄*

_{+1}(

*z*) rather than

*d̄*(

*z*). Importantly, the equality constraint |

*d̄*| = 1 from Eq. (1a) is not affine, while the relaxed constraint |

*d̄*

_{1}| < 2/

*π*from Eq. (2) is a convex inequality. Furthermore, as shown above, linear relationships can be established between all of the important variables [for example, Eq. (11)]. Therefore, these quantities can be constrained to have specific values (linear equality constraints), or substituted into convex functions (convex inequality and objective functions). Additionally,

*d̄*

_{1}can typically be resolved with a grid having of order a thousand points, which is convenient for optimization purposes.

*d̄*

_{1}(

*z*), expressed on a grid of

*z*points.

*d̄*

_{1}(

*z*) is assumed to be zero outside this grid.

*g*(

*k*) is found by a Fourier transform. The equality constraint |

*d̄*

_{1}| = 2/

*π*is not affine (and hence not compatible with our approach), and therefore solutions typically imply a spatially varying QPM duty cycle [see Eq. (2)]. Nonetheless, we can impose the necessary constraint that |

*d̄*

_{1}| < 2/

*π*. We also show in subsection 3.3 that designs having |

*d̄*

_{1}| = 2/

*π*can be obtained by an appropriate choice of objective function. A smooth and slowly-varying

*d̄*

_{1}(

*z*) can be achieved straightforwardly by expressing its spatial derivatives as matrix multiplications and imposing additional constraints. Another possible constraint useful in some situations, but which we do not exploit here, is that particular regions of the grating can be constrained to not deviate substantially from a particular target spatial profile

*d̄*

*(*

_{T}*z*).

*g*(

*k*) can be obtained from

*d̄*

_{1}by matrix multiplication, similar constraints can be applied to it as well. In particular,

*g*(

*k*) can be constrained to be close to a target spectrum, |

*g*(

*k*) −

*g*(

_{T}*k*)| <

*ε*(for some

*ε*). The closeness to this target can be treated as an inequality constraint (fixed

*ε*) or as the optimization variable (minimize

*ε*while satisfying the specified constraints). Since

*B̃*

*(*

_{i}*ω*) and

*B*(

_{i}*t*) are also obtained by matrix multiplications, the same holds for generating a pulse with a target temporal or spectral profile.

`CVX`, a package for specifying and solving convex programs [37].

**F**is a discrete spatial Fourier transform,

_{k}**F**is a discrete inverse temporal Fourier transform,

_{t}**B**

*is a discrete approximation to*

_{t}*B*(

_{i}*L*,

*t*),

**d**is a sampled version of

*d̄*

_{1}(

*z*), and

**D**is a square matrix which approximates the relevant

_{n}*n*derivative (e.g. a spatial derivative if applied to

^{th}**d**). The

*l*norm is denoted ||.||

_{n}*, and |*

_{n}**f**| denotes the point-wise magnitude of the elements of

**f**.

### 3.2. Generation of target pulse profiles

**B**

_{target}. If a particular pulse profile is needed for an experiment, generating that pulse via DFG provides a particularly simple experimental approach, especially if the desired pulse is in a new wavelength range where pulse synthesis is not convenient, such as the mid-IR. Rather than employ complicated optical arrangements such as pulse shapers, DFG-based pulse shaping can be employed in a purely collinear geometry without spatially or temporally dispersing the input pulses [15

**18**, 534–539 (2001) [CrossRef] .

5. 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] .

**d**as a complex variable, and minimize ||

**B**

*−*

_{t}**B**

_{target}||

_{2}with no constraints. However, this approach is unlikely to yield a “clean” grating profile. Instead, we solve the following convex optimization problem: where

**d**is the (complex-valued) optimization variable. Minimizing ||

**D**||

_{2}d_{2}leads to a grating profile which is as smooth as possible, without the need to specify an explicit upper bound (a suitable value for which might not be known

*a priori*). In this problem,

*ε*is chosen according to the tolerable deviations of the pulse from the target. Note that the energy of the target pulse, i.e.

**d**| = (2/

*π*)

**1**). Further details are given in Fig. 2.

*L*is determined implicitly by the

*z*grid: while

**d**can be constrained to equal zero in the outer regions of the grid (or over any region of the grid), there is no computational advantage to doing so. However, if designs involving duty cycle apodization are favorable, then the first and last elements of

**d**can be constrained to equal zero: this, in combination with the objective function, will favor nonlinear coefficients which are turned on/off smoothly at the edges of the grating.

*L*can be chosen to equal a maximum length (determined for example by fabrication constraints), or estimated based on the dispersion of the crystal.

*e*

^{2}) Gaussian pump pulse centered at 1064 nm, and a 25-fs (1/

*e*

^{2}) Gaussian signal pulse centered at 1550 nm. We allow for a 10-mm MgO:LiNbO

_{3}grating length, and assume that the pulses are temporally overlapped at the center of the grating. Figure 2 shows an example mid-IR target pulse and corresponding solution to problem (16).

### 3.3. Design of custom transfer functions

38. V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express **21**, 196–203 (2013) [CrossRef] [PubMed] .

39. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B **17**, 304–318 (2000) [CrossRef] .

40. S. Yang, A. M. Weiner, K. R. Parameswaran, and M. M. Fejer, “Ultrasensitive second-harmonic generation frequency-resolved optical gating by aperiodically poled LiNbO_{3} waveguides at 1.5 *μ*m,” Opt. Lett. **30**, 2164–2166 (2005) [CrossRef] [PubMed] .

41. Z. Jiang, D. S. Seo, S. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, spectrally coded OCDMA system demonstration using Low-Power nonlinear processing,” J. Lightwave Technol. **23**, 143–158 (2005) [CrossRef] .

**18**, 534–539 (2001) [CrossRef] .

*K*(

_{g}*z*) profile and a cw (or narrowband) pump wave, the output idler group delay can be estimated as [5

5. 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] .

*z*(

_{pm}*ω*) is the phasematching point, satisfying Δ

*k*(

*ω*,

*ω*−

_{p}*ω*) −

*K*(

_{g}*z*(

_{pm}*ω*)) = 0, with Δ

*k*(

*ω*,

*ω*′) defined in Eq. (5).

*v*(

_{g,j}*ω*) denotes the group velocity of wave

*j*at frequency

*ω*, and

*L*is the crystal length.

*τ*(

_{s}*ω*) is the group delay of the signal at the input of the QPM grating.

5. 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] .

*ω*travels with the signal before its phasematching point, and hence accumulates delay according to the signal group velocity at the corresponding signal frequency (i.e.

*v*(

_{g,s}*ω*−

_{p}*ω*)). However, for the rest of the crystal length after

*z*(

_{pm}*ω*) (distance

*L*−

*z*(

_{pm}*ω*)), it travels at the idler group velocity

*v*(

_{g,i}*ω*).

*K*(

_{g}*z*) profile which will yield a target idler group delay

*τ*(

_{i}*ω*) (or to determine if a target is not possible with a monotonically chirped QPM grating). However, since our goal here is to illustrate an optimization method, we take a simpler approach to constructing the target transfer function. We first select a crystal length and a target bandwidth, and then choose a center spatial frequency and constant grating chirp rate sufficient to generate an idler across this bandwidth. We then substitute this

*K*(

_{g}*z*) profile into Eq. (17) and assume a transform limited input signal (

*τ*(

_{s}*ω*) = 0) to determine

*τ*(

_{i}*ω*).

*τ*(

_{i}*ω*) is then integrated to obtain a spectral phase

*ϕ*(

_{i}*ω*) (since

*τ*= −

_{i}*∂ϕ*/

_{i}*∂ω*). The target transfer function is then chosen to be

*H*(

_{T}*ω*) = |

*H*(

_{T}*ω*)|exp(

*iϕ*(

_{i}*ω*)), where |

*H*(

_{T}*ω*)| =

*H*

_{0}for

*ω*within the chosen idler bandwidth (

*H*

_{0}constant), and zero otherwise. The target amplitude and phases are shown in Fig. 3.

*H*(

_{T}*ω*) ≠ 0), and whose errors within this region are both small and slowly varying. We give little emphasis in the design to the regions outside the passband, since these regions are much less significant provided that the input spectrum is bandwidth-limited (as is often the case).

*δω*is the spacing of the angular optical frequency grid, [

*PB*] denotes the indices corresponding to the passband (the components for which

**H**

*≠ 0), and*

_{T}*λ*is a distance comparable to the average QPM period.

_{D}*τ*

_{GVM}= |

*v*(

_{g,s}*ω*) −

_{s}*v*(

_{g,i}*ω*)|

_{i}*L*is the group delay accumulated between the signal and idler over the length of the crystal. The parameters

*ε*

_{1}and

*ε*

_{2}are small and chosen according to the design goals (described below).

**d**|. Such solutions can be approximated as having constant duty cycle, and are hence easier to fabricate. Constraint (18b) ensures that the transfer function is close to the target. Constraint (18c) prevents

**d**from varying rapidly over a single QPM period, since such features may not be captured by the actual grating or can result in small QPM domains. Constraint (18d) is chosen to ensure that the deviations of the transfer function from the target are smooth. More specifically: the term

**D**((

_{2}**H**−

**H**

*)/ max(*

_{T}**H**

*)) yields a second derivative with respect to angular frequency; taking the 1-norm of the magnitude of this vector and multiplying by*

_{T}*δω*corresponds to integration, and hence yields an effective delay. We expect that in a high quality design having slowly-varying deviations from the target transfer function, this ‘accumulated’ group delay should be much less than the maximum delays supported by GVM within the crystal length, and hence we select a small value for

*ε*

_{2}.

_{3}grating at 150 °C. To calculate the group delay in Eq. (17) we assume a linear grating profile given by

*K*(

_{g}*z*) =

*K*

_{g}_{0}+

*K*

_{g}_{1}(

*z*−

*L*/2) for

*K*

_{g}_{0}= 208.1 mm

^{−1}and

*K*

_{g}_{1}= 3.3 mm

^{−2}. For the solution shown, we assume

*ε*

_{1}= 0.1 and

*ε*

_{2}= 10

^{−3}.

*K*(

_{g}*z*) profile based on a target group delay spectrum, our method could be applied to a wide variety of transfer function designs. Furthermore, we chose a constant-amplitude target function for illustration purposes. Such a profile is not usually obtained for two reasons: firstly, because ripples on the transfer function are usually partially (but not completely) suppressed by apodization techniques, and secondly because the driving term in Eq. (4) is proportional to the idler optical frequency, which changes substantially over the chosen passband; therefore, part of the phase modulation in the grating arises in order to compensate for this frequency dependence. As such, it is possible to accomodate non-uniform target amplitude profiles as well. Therefore, this design approach (and variations thereof) is likely to be applicable to a wide range of optical systems employing QPM gratings.

## 4. Optimal design of chirped QPM gratings for OPCPA

43. S. Witte and K. Eikema, “Ultrafast optical parametric Chirped-Pulse amplification,” Selected Topics in IEEE J. Quant. Electron. **18**, 296–307 (2012) [CrossRef] .

46. T. Fuji, N. Ishii, C. Y. Teisset, X. Gu, T. Metzger, A. Baltuska, N. Forget, D. Kaplan, A. Galvanauskas, and F. Krausz, “Parametric amplification of few-cycle carrier-envelope phase-stable pulses at 2.1 *μ*m,” Opt. Lett. **31**, 1103–1105 (2006) [CrossRef] [PubMed] .

45. A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. **74**, 2268–2270 (1999) [CrossRef] .

46. T. Fuji, N. Ishii, C. Y. Teisset, X. Gu, T. Metzger, A. Baltuska, N. Forget, D. Kaplan, A. Galvanauskas, and F. Krausz, “Parametric amplification of few-cycle carrier-envelope phase-stable pulses at 2.1 *μ*m,” Opt. Lett. **31**, 1103–1105 (2006) [CrossRef] [PubMed] .

47. Y. Deng, A. Schwarz, H. Fattahi, M. Ueffing, X. Gu, M. Ossiander, T. Metzger, V. Pervak, H. Ishizuki, T. Taira, T. Kobayashi, G. Marcus, F. Krausz, R. Kienberger, and N. Karpowicz, “Carrier-envelope-phase-stable, 1.2 mJ, 1.5 cycle laserpulses at 2.1 *μ*m,” Opt. Lett. **37**, 4973–4975 (2012) [CrossRef] [PubMed] .

1. 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] .

**30**, 634–636 (2005) [CrossRef] [PubMed] .

7. 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] .

44. C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express **20**, 26888–26894 (2012) [CrossRef] [PubMed] .

48. 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] .

51. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express **20**, 18066–18071 (2012) [CrossRef] [PubMed] .

1. 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] .

7. 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] .

44. C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express **20**, 26888–26894 (2012) [CrossRef] [PubMed] .

48. 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] .

50. 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] .

51. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express **20**, 18066–18071 (2012) [CrossRef] [PubMed] .

**25**, 463–480 (2008) [CrossRef] .

### 4.1. Approximate form of the signal gain

*τ*(

_{s}*ω*) and

*τ*(

_{i}*ω*) associated with spectral components of the signal and idler, respectively. By assuming that group velocity mismatch (GVM) is negligible compared to the duration of the pump pulse given the length of the device, these group delays satisfy, approximately, for pump frequency

*ω*. Hence mixing between signal frequency

_{p}*ω*and idler frequency

*ω*−

_{p}*ω*involves the pump intensity at delay

*τ*(

_{s}*ω*). Assuming a narrow-band (but not cw) pump, Eqs. (19a) and (19b) can be written as Note that these equations could be derived from Eqs. (19a) and (19b) via stationary phase approximations. Note also that Eq. (20) is a useful approximation for calculating the gain spectrum given a narrow-bandwidth pump, but is not sufficient for calculating the group delay and corresponding spectral phase of the idler for subsequent pulse compression [for such problems, Eq. (17) must be used].

*ω*+ Ω mixes only with idler frequency

_{s}*ω*− Ω, where

_{i}*ω*+

_{i}*ω*=

_{s}*ω*. Therefore, it is convenient to define frequency-shifted envelopes

_{p}*ã*(Ω) =

_{j}*Ã*(

_{j}*ω*+ Ω). We also neglect higher orders of the QPM grating at this stage, so that

_{j}*d̄*∼

*d̄*

_{±1}[see Eq. (1)]. As mentioned in section 1, for an energy-conserving first-order QPM process, the nonlinear polarization at the pump frequency (which is not included here since we are neglecting pump depletion) involves

*d̄*

_{−1}while the nonlinear polarizations for the signal and idler waves involve

*d̄*

_{+1}. Thus, combining Eq. (20) with Eqs. (21a) and (21b) yields where the single-frequency-argument phase mismatch is given by Equations (22a) and (22b) have forms analogous to Eqs. (3) and (4) of Ref. [1

**25**, 463–480 (2008) [CrossRef] .

*d̄*

_{1}| = 2/

*π*within the structure), as where

*z*(Ω) are frequency-dependent, real-valued turning points: they are the two positions at which the integrand in Eq. (24) is zero. Equation (24) can only be applied for a frequency Ω when both

_{tp,j}*z*(Ω) are within the structure (and are not too near to an edge). These turning points are introduced in appendix F of [1

_{tp,j}**25**, 463–480 (2008) [CrossRef] .

*γ*(Ω) satisfies where

*I*(

_{p}*t*) is the pump intensity.

*dK*/

_{g}*dz*= −

*∂*Δ

*k/∂z*constant). Integrating Eq. (24) for this case yields the following gain for the signal intensity: where Λ(Ω) =

*γ*(Ω)

^{2}/|

*∂*Δ

*k/∂z*|. Equation (26) corresponds to the Rosenbluth amplification formula [52

52. M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. **29**, 565–567 (1972) [CrossRef] .

*γ*(Ω) is strongly frequency dependent in an OPCPA system utilizing signal pulses of comparable duration to the pump, high gain across the whole spectrum can only be obtained by over-saturating the peak of the pump; operating in such a regime may be undesirable for many reasons [7

7. 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] .

### 4.2. Optimization framework

*K*(

_{g}*z*) which yields a target signal gain spectrum while satisfying several constraints. This problem can be put into convex form by a change of variables in Eq. (24). Our main assumption is that

*K*(

_{g}*z*) is monotonic: this assumption means that the integration variable in Eq. (24) can be changed to

*K*instead of

_{g}*z*. The limits of integration in this case are the frequency-dependent k-space turning points

*K*, which satisfy |Δ

_{tp,j}*k*(Ω) −

*K*(Ω)| = 2

_{tp,j}*γ*(Ω). For spatial frequencies outside the interval [

*K*

_{tp,}_{1}(Ω),

*K*

_{tp}_{,2}(Ω)], the integrand in Eq. (24) is imaginary. Provided that both

*K*(Ω) lie within the range of

_{tp,j}*K*(i.e. provided that the amplification region of a particular spectral component is fully contained within the grating), the integration limits can be extended to cover the entire range of

_{g}*K*by taking the real part of the integrand (so that the imaginary components outside the turning points do not contribute). We therefore obtain where

_{g}*z*(

_{K}*K*) =

_{g}*dz/dK*is the reciprocal of the local chirp rate, expressed as a function of spatial frequency.

_{g}*K*and

_{i}*K*are the grating k-vectors at the input and output ends of the device, respectively. The local gain rate Γ is given by Analogously to Eq. (24), Eq. (27) is valid for spectral components for which Γ(Ω,

_{f}*K*) = Γ(Ω,

_{i}*K*) = 0, provided that the chirp rate (

_{f}*dz/dK*)

_{g}^{−1}remains sufficiently slow within the [

*K*,

_{i}*K*] interval.

_{f}*k*and

*γ*are known functions of frequency, values of

*K*and

_{i}*K*can be chosen in order to contain the amplification regions of all the spectral components of interest. Based on Eq. (27), each spectral component Ω is amplified over the interval

_{f}*K*∈ [Δ

_{g}*k*(Ω) − 2

*γ*(Ω), Δ

*k*(Ω) + 2

*γ*(Ω)]; within this region, the local signal-idler coupling rate

*γ*(Ω) sufficiently exceeds the local phase mismatch, since |Δ

*k*(Ω)−

*K*| < 2

_{g}*γ*(Ω). If gain is required over a particular spectral range, the (nominal) range of grating k-vectors should thus be extended beyond the range of Δ

*k*(Ω) so that the constraint Γ(Ω,

*K*) = Γ(Ω,

_{i}*K*) = 0 is met for all Ω of interest. A simple and sufficient condition can be obtained by imposing this constraint while neglecting the frequency dependence of

_{f}*γ*(Ω): where

*K*

_{max}= max

_{Ω}(Δ

*k*(Ω)),

*K*

_{min}= min

_{Ω}(Δ

*k*(Ω)), and

*γ*

_{0}= max

_{Ω}(

*γ*) (the peak signal-idler coupling rate). For a constant chirp rate Δ

*k*′ =

*∂*Δ

*k/∂z*, the grating length required to support this range of k-vectors can be written as where

*G*) = 2

_{s}*π*Λ

_{0}[Eq. (26)]. Equation (30) can be used as a rough estimate for the required grating length. In an optimized device where all spectral components achieve the same gain, the length will usually be longer than predicted by Eq. (30), since

*γ*(Ω) ≤

*γ*

_{0}.

*z*(

_{K}*K*) =

_{g}*dz/dK*. For this reason, we will take

_{g}*z*to be the optimization variable. The nominal grating profile

_{K}*K*(

_{g}*z*) can be constructed from

*z*by noting that the position is given, as a function of

_{K}*K*, by and this function can be inverted to find

_{g}*K*(

_{g}*z*). Note also that since the approximate model of Eq. (27) does not account for gain ripples that arise from the hard edges of the grating, a grating profile determined by an optimization algorithm based on Eq. (27) alone will yield a gain spectrum with strong ripples in amplitude and phase. However, it is possible to append apodization regions to both ends of the device in order to almost completely suppress such ripples [1

**25**, 463–480 (2008) [CrossRef] .

8. C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express **20**, 2466–2482 (2012) [CrossRef] [PubMed] .

9. J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasi-phase-matched interactions,” Opt. Lett. **31**, 604–606 (2006) [CrossRef] [PubMed] .

51. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express **20**, 18066–18071 (2012) [CrossRef] [PubMed] .

_{1},..., Ω

*], and spatial frequency*

_{N}*K*, [

_{g}*K*

_{1},...,

*K*]. We define a matrix

_{N}**Γ**as Γ(Ω,

*K*) evaluated on these grids, appropriately weighted by the

_{g}*K*grid spacing. The log of the signal gain

_{g}**G**is thus given simply by matrix multiplication: For a uniform grid spacing

_{s}*δK*, the (nominal) grating length is given by

*L*=

*δK*(

**1**

^{T}**z**). Since the (logarithmic) gain is linear in

_{K}**z**, it can be readily optimized.

_{K}### 4.3. Suppression of gain narrowing

44. C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express **20**, 26888–26894 (2012) [CrossRef] [PubMed] .

50. 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] .

**20**, 18066–18071 (2012) [CrossRef] [PubMed] .

*μ*m, a FWHM duration of 10 ps, and a peak intensity such that

*γ*

_{0}=

*γ*(Ω = 0) = 3 mm

^{−1}. The Gaussian input signal pulse has center wavelength 3.5

*μ*m, bandwidth supporting a 35-fs FWHM duration, and for simplicity has a purely quadratic spectral phase such that the actual FWHM duration is 3.5 ps. We consider an undepleted-pump example, so the amplified signal intensity is negligible compared with the pump. Note however that due to the properties of adiabatic frequency conversion [6

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

8. C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express **20**, 2466–2482 (2012) [CrossRef] [PubMed] .

*e*

^{2}signal pulse bandwidth, which corresponds to spectral components between 3.1- and 4.0-

*μ*m. In MgO:LiNbO

_{3}at 150 °C, the grating k-vectors phasematching the ends of this spectrum are

*K*

_{min}= 2.037 × 10

^{5}m

^{−1}and

*K*

_{max}= 2.189 × 10

^{5}m

^{−1}. A linear

*K*grid is defined with end-points

_{g}*K*

_{min}− 2

*γ*

_{0}and

*K*

_{max}+ 2

*γ*

_{0}, to ensure that the entire amplification region of each spectral component is contained within the grating [see Eq. (28)]. A frequency-independent target gain of

*G*= exp(2

_{T}*π*Λ

*) with Λ*

_{T}*= 2.2 is chosen, which corresponds to approximately 60 dB power gain. For the value of*

_{T}*γ*

_{0}= 3 mm

^{−1}above, a constant chirp rate of

*κ*′= ±4.1 mm

^{−2}would be required to obtain

**z**: which is a quadratic program (QP). The chosen objective function ensures smoothness of the solutions. The parameters Λ

_{K}_{min}= 1 and Λ

_{max}= 10 indicate bounds on the local chirp rate, and are chosen to ensure a sensible solution. In particular, these parameters were selected such that 0 ≪ Λ

_{min}≤ Λ

*≪ Λ*

_{T}_{max}; as long as these conditions are met, the solution is not too sensitive to these parameters. Note also that the Λ

_{min}> 0 constraint ensures monotonicity of the

*z*and hence

_{K}*K*(

_{g}*z*).

*δG*= 0.05, since this tolerance will typically be satisfactory.

*δG*can be made smaller, but eventually the problem may become infeasible. In general, the length

*L*

_{max}can be treated as a parameter: we solve Eq. (33) for a range of choices of

*L*

_{max}, numerically simulate the OPCPA process in the resulting grating with the full coupled-wave equations, and select a feasible value of

*L*

_{max}which yields a useful output spectrum from these simulations. For the present example, we chose

*L*

_{max}= 12 mm [twice the minimum length indicated by Eq. (30)]. In the solution shown below, both this maximum length inequality (33c) and the chirp rate bounds (33d) are not active, which emphasizes that very little information, and no scanning of

*L*

_{max}, is needed in order to arrive at a good solution. Finally, we apply nonlinear-chirp apodization regions to the grating profile returned by the optimization routine [1

**25**, 463–480 (2008) [CrossRef] .

*L*≈ 9.55 mm.

*i*and

*j*. For a pump wavelength of 1.064

*μ*m and signal wavelength of 1.45

*μ*m (1.65

*μ*m), the resulting GVM values are given by

*δν*= −182 fs/mm (−55 fs/mm);

_{si}*δν*= −88 fs/mm (−105 fs/mm); and

_{sp}*δν*= 93 fs/mm (−49 fs/mm). Thus, given the optimized grating length of 9.55 mm, the range of walk-offs involved is significantly smaller than the duration of the pulses. Nonetheless, GVM still leads to slight modifications in the signal gain spectrum compared with the WKB-predicted spectrum.

_{ip}**z**is a good approximation to an underlying smooth variable. Second, recall that this approach relies on the validity of Eq. (24), which in turn relies on having a sufficiently slowly-varying grating chirp: by using smoothness as an objective, we maximize the validity of the approximation upon which the optimization relies.

_{K}**r**on the pump, a matrix

**Γ**(

**r**) can be calculated and used to specify additional constraints. Another set of constraints could be obtained by using Eq. (17) to obtain an estimate of the frequency-dependent group delay. It would then be possible to trade off gain narrowing and group delay effects in one or more OPCPA stages.

## 5. Discussion

### 5.1. Diffraction and transverse beam effects

54. G. D. Boyd, “Parametric interaction of focused gaussian light beams,” Journal of Applied Physics **39**, 3597–3639 (1968) [CrossRef] .

### 5.2. Nonlinear interactions with depletion

38. V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express **21**, 196–203 (2013) [CrossRef] [PubMed] .

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

**20**, 2466–2482 (2012) [CrossRef] [PubMed] .

**20**, 2466–2482 (2012) [CrossRef] [PubMed] .

56. 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] .

### 5.3. Robustness of the designs

31. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron. **28**, 2631–2654 (1992) [CrossRef] .

## 6. Conclusions

`CVX`[37], which make use of fast and reliable interior-point methods (see, for example, chapter 11 of [34]).

**d**|) tends to result in a solution with |

*d*(

*z*)| = max(|

**d**|), which corresponds to a purely phase-modulated grating. Since fabricating QPM gratings with a varying duty cycle is typically quite challenging, this objective function yields a device which can be easily fabricated. Another useful objective function is ||

**D**||

_{2}d_{2}, which yields a smooth grating profile but typically involves some duty cycle modulation, except for certain target pulses or transfer functions.

## Acknowledgments

## References and links

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34. | S. P. Boyd and L. Vandenberghe, |

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

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

37. | M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 1.21 (2011). |

38. | V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express |

39. | G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B |

40. | S. Yang, A. M. Weiner, K. R. Parameswaran, and M. M. Fejer, “Ultrasensitive second-harmonic generation frequency-resolved optical gating by aperiodically poled LiNbO |

41. | Z. Jiang, D. S. Seo, S. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, spectrally coded OCDMA system demonstration using Low-Power nonlinear processing,” J. Lightwave Technol. |

42. | G. D. Miller, “Periodically poled lithium niobate: modeling, fabrication, and nonlinear-optical performance,” PhD dissertation, Stanford University, Stanford, CA (1998). |

43. | S. Witte and K. Eikema, “Ultrafast optical parametric Chirped-Pulse amplification,” Selected Topics in IEEE J. Quant. Electron. |

44. | C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express |

45. | A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. |

46. | T. Fuji, N. Ishii, C. Y. Teisset, X. Gu, T. Metzger, A. Baltuska, N. Forget, D. Kaplan, A. Galvanauskas, and F. Krausz, “Parametric amplification of few-cycle carrier-envelope phase-stable pulses at 2.1 |

47. | Y. Deng, A. Schwarz, H. Fattahi, M. Ueffing, X. Gu, M. Ossiander, T. Metzger, V. Pervak, H. Ishizuki, T. Taira, T. Kobayashi, G. Marcus, F. Krausz, R. Kienberger, and N. Karpowicz, “Carrier-envelope-phase-stable, 1.2 mJ, 1.5 cycle laserpulses at 2.1 |

48. | 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 |

49. | M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Theory and simulation of gain-guided noncollinear modes in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B |

50. | 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 |

51. | C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express |

52. | M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. |

53. | C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B. |

54. | G. D. Boyd, “Parametric interaction of focused gaussian light beams,” Journal of Applied Physics |

55. | J. E. Schaar, “Terahertz sources based on intracavity parametric frequency down-conversion using quasi-phase-matched gallium arsenide,” PhD dissertation, Stanford University, Stanford, CA (2009). |

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

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(320.7080) Ultrafast optics : Ultrafast devices

(230.7405) Optical devices : Wavelength conversion devices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 8, 2013

Revised Manuscript: March 1, 2013

Manuscript Accepted: March 1, 2013

Published: April 16, 2013

**Citation**

C. R. Phillips, L. Gallmann, and M. M. Fejer, "Design of quasi-phasematching gratings via convex optimization," Opt. Express **21**, 10139-10159 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-10139

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### References

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- V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express21, 196–203 (2013). [CrossRef] [PubMed]
- G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B17, 304–318 (2000). [CrossRef]
- S. Yang, A. M. Weiner, K. R. Parameswaran, and M. M. Fejer, “Ultrasensitive second-harmonic generation frequency-resolved optical gating by aperiodically poled LiNbO3 waveguides at 1.5 μm,” Opt. Lett.30, 2164–2166 (2005). [CrossRef] [PubMed]
- Z. Jiang, D. S. Seo, S. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, spectrally coded OCDMA system demonstration using Low-Power nonlinear processing,” J. Lightwave Technol.23, 143–158 (2005). [CrossRef]
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- 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, B. Afeyan, and M. M. Fejer, “Theory and simulation of gain-guided noncollinear modes in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B27, 824–841 (2010).
- 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]
- C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express20, 18066–18071 (2012). [CrossRef] [PubMed]
- M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett.29, 565–567 (1972). [CrossRef]
- C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B.
- G. D. Boyd, “Parametric interaction of focused gaussian light beams,” Journal of Applied Physics39, 3597–3639 (1968). [CrossRef]
- J. E. Schaar, “Terahertz sources based on intracavity parametric frequency down-conversion using quasi-phase-matched gallium arsenide,” PhD dissertation, Stanford University, Stanford, CA (2009).
- C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B27, 2687–2699 (2010). [CrossRef]

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