Phase-matching loci and angular acceptance of non-collinear optical parametric amplification

doi:10.1364/OE.20.026176

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Phase-matching loci and angular acceptance of non-collinear optical parametric amplification

Benoît Trophème, Benoit Boulanger, and Gabriel Mennerat »View Author Affiliations

Optics Express, Vol. 20, Issue 24, pp. 26176-26183 (2012)
http://dx.doi.org/10.1364/OE.20.026176

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▸ Article Outline
– Abstract
1. Introduction
2. Methods of calculation of phase-matching directions and angular acceptances
3. Experimental setup
4. Measurements and calculations analysis in BBO and LBO
5. Conclusion
– References and links

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Abstract

A general study of phase-matching loci and associated angular acceptances is performed in the case of non-collinear parametric amplification. Numerical and analytical calculations, as well as measurements, are described for the uniaxial BBO crystal and the biaxial LBO crystal.

1. Introduction

Non-Collinear Optical Parametric Amplification (OPA) in birefringent nonlinear crystals using one or several pump beams is a promising way to amplify femtosecond pulses [1

D. Herrmann, R. Tautz, F. Tavella, F. Krausz, and L. Veisz, “Investigation of two-beam-pumped noncollinear optical parametric chirped-pulse amplification for the generation of few-cycle light pulses,” Opt. Express 18(5), 4170–4183 (2010). [CrossRef] [PubMed]

, 2

T. Kurita, K. Sueda, K. Tsubakimoto, and N. Miyanaga, “Experimental demonstration of spatially coherent beam combining using optical parametric amplification,” Opt. Express 18(14), 14541–14546 (2010). [CrossRef] [PubMed]

]. As in the collinear case, the phase-matching has to be fulfilled in order to obtain a large parametric gain. The associated angular tolerances set limits to the pump and signal beams diameters and divergences as well as to the length of the nonlinear crystal. Since angular acceptances are usually much tighter in non-collinear than in collinear geometry, this parameter is therefore of prime importance for the design of the whole system, especially when pumping by multiple fibers lasers. In that case, a critical tradeoff has to be found between a stronger focusing, in order to increase the pump intensity, and a correlated increasing of the beam divergence, that would lead to a poor angular overlap.

In addition, when pumping with mutually incoherent pumps, it is necessary to master and optimize the angular acceptances of the nonlinear coupling in order to minimize the transfer of incoherence toward the signal [1

, 2

]. In this paper we perform numerical and analytical calculations, as well as measurements of phase-matching loci and associated angular acceptances of non-collinear single-beam pumped OPA. We consider two important nonlinear crystals widely used in the framework of OPA: β BaB₂0₄ (BBO) [3

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB₂O₄,” IEEE J. Quantum Electron. 22(7), 1013–1014 (1986). [CrossRef]

] and LiB₃O₅ (LBO) [4

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]

2. Methods of calculation of phase-matching directions and angular acceptances

OPA corresponds to the amplification of a beam, called the signal, by an energetic pump beam through a travelling wave interaction involving the second order electric susceptibility of a nonlinear crystal. During this process a third wave (the idler) at wavelength λ_i is generated. According to the energy conservation, λ_i is given by:

\frac{1}{λ_{p}} = \frac{1}{λ_{s}} + \frac{1}{λ_{i}}

(1)

(λ_p, λ_s, λ_i) are the wavelengths of the three interacting waves where the indices (p), (s) and (i) denote the pump, signal and idler respectively.

The energy transfer between the pump and signal is maximum when the interference between the induced nonlinear polarization at λ_s and the corresponding radiated electric field at λ_s is constructive. This condition is achieved when the phase-mismatch

\vec{Δ k}

between the wave vectors verify the momentum conservation:

\vec{{Δ k}_{}} = \vec{k_{p}} - \vec{k_{s}} - \vec{k_{i}} = \vec{0}

(2)

The vectorial wave vectors configurations corresponding to the phase-matching and non-phase-matching cases are depicted in Fig. 1 .

Fig. 1 Wave vector configurations and corresponding relevant angles of phase-matched (a) and non-phase-matched (b) non-collinear three-wave parametric interactions.

According to the notations defined in Fig. 1, the vectorial Eq. (2) leads to the two following scalar equations:

\frac{n_{p}}{λ_{p}} \cos (α_{p}) = \frac{n_{s}}{λ_{s}} + \frac{n_{i}}{λ_{i}} \cos (α_{i}) \frac{n_{p}}{λ_{p}} \sin (α_{p}) = \frac{n_{i}}{λ_{i}} \sin (α_{i})

(3)

where n_p,_s,_i are the refractive indices of the interacting waves in the considered phase-matching direction.

Parametric amplification is present even if

\vec{Δ k}

is not strictly equal to

\vec{0}

. In the undepleted pump approximation, the parametric gain behaves as sinc²

(\vec{Δ k} . \vec{L} / 2)

where

L = | \vec{L} |

is the crystal length taken along the signal wave vector according to our experiments. Then the quantity that is generally used for characterizing the mismatch effect on the conversion efficiency is the acceptance, which is defined as the deviation of

\vec{Δ k}

due to variations of angle, wavelength or temperature [5

B. Boulanger and J. Zyss, Non-linear Optical properties, Chapter 1.8 in International Tables for Crystallography, Vol. D: Physical Properties of Crystals, (A. Authier Ed., 2006) International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 178–219.

]. Here we focus our interest on the acceptance according to angle α_p between the pump and signal wave vectors as defined in Fig. 1, the signal direction being kept fixed and perpendicular to the entrance and exit faces of the nonlinear crystal contrary to the directions of the pump and idler. The angular acceptance Δα_p is then defined by the quantity (α_p^+π - α_p^-π) where α_p^+π and α_p^-π are the solutions of the equation

\vec{Δ k} . \vec{L} = \pm 2 π

corresponding to the first zeros of the sinc function from either side of the phase-matching value of α_p. So L.Δα_p is a characteristic of the wave-vector configuration in the considered phase-matching direction.

We consider here the uniaxial optical class as well as the biaxial one. In a given direction of propagation

\vec{u}

, there are two possible refractive index values according to the chosen eigenpolarization state ( + ) or (-), which are the solutions n ^± of the Fresnel equation [6

J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun. 99(3-4), 284–302 (1993). [CrossRef]

\begin{array}{l} n^{\pm} = {[\frac{2}{- B \mp {(B^{2} - 4 C)}^{1 / 2}}]}^{1 / 2} \\ B = - u_{x}^{2} (b + c) - u_{y}^{2} (a + c) - u_{z}^{2} (a + b) \\ C = u_{x}^{2} \underset{}{} b c + u_{y}^{2} \underset{}{} a c + u_{z}^{2} \underset{}{} a b \\ a = n_{x}^{- 2}, \underset{}{} b = n_{y}^{- 2}, \underset{}{} c = n_{z}^{- 2} \end{array}

(4)

n_x,_y,_z are the principal refractive indices at the considered wavelength, where n_x ≠ n_y ≠ n_z in the biaxial case and n_x = n_y ≠ n_z for uniaxial crystals; the Cartesian indices x, y and z refer to the dielectric frame depicted in Fig. 2(a) ; (u_x, u_y, u_z) are the Cartesian coordinates of the unit wave vector shown in Fig. 1(a) that can be expressed as a function of the angles of spherical coordinates (θ, φ) by:

Fig. 2 (a) Relative orientation of the dielectric frame (x, y, z) and of the laboratory frame (x”, y”, z”) linked to the signal wave vector

\vec{k_{s}}

; (θ_s, φ_s) are the angles of spherical coordinates of

\vec{k_{s}}

in the dielectric frame; the parallelepiped stands for the nonlinear crystal ; (θ, φ) are the spherical coordinates angles of an arbitrary direction

\vec{u}

. (b) Orientation of phase-matched signal, pump and idler wave vectors

\vec{k_{s}}

\vec{k_{p}}

and

\vec{k_{i}}

resp. in the laboratory frame.

u_{x} = \cos φ \sin θ_{}_{} u_{y} = \sin φ \sin θ_{}_{} u_{z} = \cos θ

(5)

It is useful to define a laboratory frame (x”, y”, z”) that is linked to the signal wave [7

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng. 39(4), 1016–1039 (2000). [CrossRef]

], the z”- axis being along the signal wave vector as depicted in Fig. 2(a). The pump and idler wave vectors can be then expressed in this frame with the angle of spherical coordinates (α_p, ψ_p) and (α_i, ψ_i) respectively as shown in Fig. 2(b).

In the laboratory frame, the phase-mismatch expression becomes:

\vec{Δ k} = 2 π (\begin{array}{l} \frac{n_{p}}{λ_{p}} \sin (α_{p}) . \cos (ψ_{p}) - \frac{n_{1}}{λ_{1}} \sin (α_{1}) . \cos (ψ_{1}) \\ \frac{n_{p}}{λ_{p}} \sin (α_{p}) . \sin (ψ_{p}) - \frac{n_{1}}{λ_{1}} \sin (α_{1}) . \sin (ψ_{1}) \\ \frac{n_{p}}{λ_{p}} \cos (α_{p}) - \frac{n_{s}}{λ_{s}} - \frac{n_{1}}{λ_{1}} \cos (α_{1}) \end{array})

(6)

The angles (α_p, ψ_p) and (α_i, ψ_i) are linked to the angles (θ_p, φ_p) and (θ_i, φ_i) by:

\begin{array}{l} \cos α_{p, i} = \sin θ_{s} \cos φ_{s} . \sin θ_{p, i} \cos φ_{p, i} + \sin θ_{s} \sin φ_{s} . \sin θ_{p, i} \sin φ_{p, i} + \cos θ_{s} . \cos θ_{p, i} \\ \sin ψ_{p, i} = \frac{\cos φ_{s} . \sin θ_{p, i} \sin φ_{p, i} - \sin φ_{s} . \sin θ_{p, i} \cos φ_{p, i}}{\sin α_{p, i}} \end{array}

(7)

Note that the phase-matching case

\vec{Δ k} = \vec{0}

implies ψ_p = ψ_i so that Eq. (6) reduces to Eqs. (3).

We developed a numerical code to calculate, for any signal wave vector direction expressed in the dielectric frame, all the corresponding possible non-collinear phase-matching directions (α_p, ψ_p). Equation (6) is solved numerically where the refractive indices n_p,s,i are given by Eq. (4) written at λ_p,s,i resp. according to the corresponding polarization state ( + ) or (-) and to the respective directions of propagation. These calculations can be performed for the three possible phase-matching types defined by the following refractive index combinations [6

]: type I {n_p^-, n_s⁺, n_i⁺}, type II {n_p^-, n_s⁺, n_i^-} and type III {n_p^-, n_s^-, n_i⁺}. The phase-matching directions being determined, the code calculates the associated angular acceptance L.Δα_p: for each considered couple (α_p, ψ_p) taken out of the phase-matching conditions, the corresponding value of Δα_p is calculated using the couple (α_i, ψ_i) that minimizes

\vec{Δ k}

in Eq. (6).

L.Δα_p can be also analytically expressed using the following first-order approximation:

\vec{Δ k} . \vec{L} \approx L \cdot Δ α_{p} \frac{\partial}{\partial α_{p}} (\vec{Δ k} \cdot \frac{{\vec{k}}_{s}}{‖ {\vec{k}}_{s} ‖}) = 4 π

(8)

Starting from Eq. (6) and using Eq. (8), we generally end up with complicated expressions, except when the three wave vectors are located in the same principal plane where the differentiation with respect to α_p reduces to ∂/∂α_p = cosψ_p∂/∂θ_p. Note that the principal planes correspond to any plane containing the optical axis in the case of uniaxial crystals, and to the three planes xy, xz and yz in the case of biaxial crystals.

For type I phase-matching in BBO that is a negative uniaxial crystal, i.e. n_x = n_y (≡ n_o) > n_z (≡ n_e) where n_o and n_e are the ordinary and extraordinary refractive indices resp., the following expression is obtained:

\begin{matrix} L \cdot Δ α_{p} = \frac{2 λ_{p}}{n_{p}} [\frac{1}{2} \sin (2 θ_{p}) n_{p}^{2} (\frac{1}{n_{o}^{2} (λ_{p})} - \frac{1}{n_{e}^{2} (λ_{p})}) (\cos α_{p} + \sin (θ_{i} - θ_{s}) \sin (θ_{p} - θ_{i})) \\ _{}_{}_{}_{}_{}_{} {- \sin (θ_{p} - θ_{s}) + \sin (θ_{i} - θ_{s}) \cos (θ_{p} - θ_{i})]}^{- 1} \end{matrix}

(9)

The first term of Eq. (9) depends on birefringence and is predominant for geometries close to the collinear configuration, which corresponds to θ_s = θ_i = θ_p and thus α_p = 0. The second line becomes dominant with increasing values of α_p. Hence angular tolerances set practical limits to the possible “degree” of non-collinearity, especially for large high-energy beams when near-field spatial overlap is not critical.

We considered also the case of type I phase-matching in LBO that is a negative biaxial crystal, i.e. V_z = tan⁻¹(n_x⁻² - n_y⁻² / n_y⁻² – n_z⁻²)^1/2 > 45° where V_z is the angle between the optical axis and the z-axis. When the wave vectors are located in the xy plane (θ_p,s,i = 90°), the angular acceptance takes the same general form as above, i.e.:

\begin{matrix} L \cdot Δ α_{p} = \frac{2 λ_{p}}{n_{p}} [\frac{1}{2} \sin (2 φ_{p}) n_{p}^{2} (\frac{1}{n_{x}^{2} (λ_{p})} - \frac{1}{n_{y}^{2} (λ_{p})}) (\cos α_{p} + \sin (φ_{i} - φ_{s}) \sin (φ_{p} - φ_{i})) \\ _{}_{}_{}_{}_{}_{} {- \sin (φ_{p} - φ_{s}) + \sin (φ_{i} - φ_{s}) \cos (φ_{p} - φ_{i})]}^{- 1} \end{matrix}

(10)

For a beam propagating in the xz plane (φ_p,s,i = 0°), with angles θ_p,s,i > V_z(λ_p,s,i), we obtain the following formula:

\begin{array}{l} L . Δ α_{p} = \frac{2 λ}{n_{p}} {[\frac{(n_{i} \cos (θ_{i} - θ_{p}) + n_{i}' \sin (θ_{i} - θ_{p})) (n_{i} \sin (θ_{i} - θ_{s}) - n_{i}' \cos (θ_{i} - θ_{s}))}{n_{i} ² + n_{i}' ²} - \sin (θ_{p} - θ_{s})]}^{- 1} \\ n_{i}^{'} = \frac{1}{2} \sin (2 θ_{i}) n_{i}^{3} (\frac{1}{n_{x}^{2} (λ_{i})} - \frac{1}{n_{y}^{2} (λ_{i})}) ._{}_{}_{}_{}_{} \end{array}

(11)

In the case of a propagation in the yz plane (φ_p,s,i = 90°), the effective coefficient of type I in LBO is nil [8

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB₃O₅,” J. Opt. Soc. Am. B 6(4), 616–621 (1989). [CrossRef]

] so that it is irrelevant to calculate the corresponding acceptance.

The analytical expressions given above are valid for any direction of propagation of uniaxial crystals and in the principal planes of biaxial crystals. For the latest ones, it is much more complicated to establish the corresponding equations when the propagation is taken out of the principal planes, which would be the case as example of YCOB for which the effective coefficient is maximal out of the principal plane [9

G. Aka, A. Kahn-Harari, F. Mougel, D. Vivien, F. Salin, P. Coquelin, P. Colin, D. Pelenc, and J. P. Damelet, “Linear and nonlinear optical properties of a new gadolinium calcium oxoborate crystal Ca₄GdO(BO₃)₃,” J. Opt. Soc. Am. B 14(9), 2238–2247 (1997). [CrossRef]

]. However, our numerical code enables to perform the calculations in any direction of propagation.

3. Experimental setup

We set-up an experiment devoted to the measurement of the non-collinear phase-matching loci and the associated angular acceptances of any crystal (Fig. 3 ).

Fig. 3 Setup for the measurements of the non-collinear OPA phase-matching loci and angular acceptances.

The pump beam at λ_p = 532 nm is emitted by a GCR 150 QuantaRay laser delivering up to 200 mJ per pulse with a full width at half maximum (FWHM) duration of 8 ns. The laser output is imaged on the OPA crystal. Spatial filtering within the imaging telescope ensures a quasi-Gaussian beam profile (2 mm FWHM spot) and a pointing stability better than 50 µrad.

A part of the pump beam is used for pumping a home-made singly resonant optical parametric oscillator (OPO) based on walk-off compensated KTP crystal for the generation of the signal at λ_s = 720 nm. The energy can reach 3 mJ over a pulse duration of 6 ns (FWHM). The signal beam is also spatially filtered leading to a residual divergence of about 225 µrad. Temporal synchronization of the pump and signal pulses is achieved using a delay line. Polarizations are adjusted according the phase-matching type using half-wave plates.

The studied crystal are inserted in a temperature controlled oven (20°C to 200°C, ± 0.1°C) and mounted on several goniometric stages allowing a full rotation of the crystal with an accuracy better than 0.05°. The signal beam is kept at normal incidence to the entrance face of the crystal with an accuracy better that 100 µrad controlled by a HeNe laser. The direction of the pump beam can be tuned using the rotation of the crystal as well as of a mirror, which enables to access respectively to angles ψ_p and α_p. The overlap of the pump and signal beams inside the crystal is controlled by translating the mirror and monitored using a CCD camera. The OPA parametric gain is measured by two J25LP-1 calorimeters from Molectron.

4. Measurements and calculations analysis in BBO and LBO

We studied type I {n_p^-, n_s⁺, n_i⁺} OPA in two different crystals: the uniaxial BBO and the biaxial LBO. For these two crystals, we measured several non-collinear phase-matching loci (α_p,ψ_p) and the corresponding angular acceptances L.Δα_p using the experimental setup described in section 3, and we compared the results with calculations based on the equations given in section 2 and the Sellmeier equations of BBO [3

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB₂O₄,” IEEE J. Quantum Electron. 22(7), 1013–1014 (1986). [CrossRef]

] and LBO [4

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]

The 15-mm-long BBO crystal is cut at (θ = 23°, φ = 0°) in the dielectric frame (x, y, z) as shown in Fig. 4(a) . The signal beam is kept along this direction. The link between the dielectric frame and the laboratory frame (x”, y”, z”) is depicted in Fig. 4(a). The crystal was rotated on it-self around the z”-axis of the laboratory frame allowing the pump beam to propagate at different ψ_p angles. For each addressed ψ_p angle, the incidence mirror was rotated in order to find the corresponding phase-matching α_p angle corresponding to a maximum parametric gain. The experiments were performed at room temperature. The measured phase-matching angles (Fig.4b) are in very good agreement with our calculations using Eqs. (3), as well as with those given by the SNLO software in the principal planes [10

SNLO nonlinear optics code available from A. V. Smith, AS-Photonics, Albuquerque, NM.

Fig. 4 (a) Configuration of orientation of the BBO crystal with respect to the signal wave vector , the dielectric frame (x, y, z), and the laboratory frame (x”, y”, z”). (b) Non-collinear phase-matching angles (α_p, ψ_p) of the pump wave vector in the laboratory frame; the circular dots correspond to experimental data and the continuous line to numerical calculations; the encircled dot at (α_p = 0°, ψ_p = 0°) corresponds to z” and.

The angular acceptance L.Δα_p was measured for each ψ_p angle. An example of such a recording at ψ_p = 180° is shown in Fig. 5(a) : it appears a very good agreement between the calculation (blue curve) taking into account the sinc² behavior (red curve) convoluted by the pointing instability, i.e. 50 µrad for the pump and 225 µrad for the signal. All the measured angular acceptances, which have been deconvoluted from the pointing instability, are given in Fig. 5(b) as a function of ψ_p and are compared with the numerical calculation using Eq. (6) and the analytical calculations from Eq. (9) at ψ_p equal to 0° and 180°. Again the agreement is very satisfying.

Fig. 5 Angular acceptance properties of the non-collinear OPA of a signal at λ_s = 720 nm in a 15-mm-long BBO crystal cut at (θ = 23°, φ = 0°) and pumped at λ_p = 532 nm. (a) Parametric gain at ψ_p = 180° as a function of the phase-matching angle α_p; (b) angular acceptance L.Δα_p as a function of the phase-matching angle ψ_p.

The 22-mm-long LBO crystal is cut along the x-axis, i.e. (θ = 90°, φ = 0°) in the dielectric frame, which corresponds to the direction of the signal wave vector. Then the z”- axis of the laboratory frame is along the x-axis as shown in Fig. 6(a) . The crystal temperature was maintained at 193 ± 0.1°C. The measurements were carried out exactly as in the case of BBO. The phase-matching angles (α_p,ψ_p) are given in Fig. 6(b) where there is also a very good agreement between the measured and calculated data.

Fig. 6 (a) Orientation of the LBO crystal with respect to the signal wave vector, the dielectric frame (x, y, z) and the laboratory frame (x”, y”, z”). (b) Non-collinear phase-matching angles (α_p, ψ_p) of the pump wave vector in the laboratory frame ; the circular dots correspond to experimental data and the continuous line to numerical calculations; the encircled dot at (α_p = 0°, ψ_p = 0°) corresponds to z” and .

Figure 7(a) shows an example of a parametric gain curve as a function of α_p at ψ_p = 0°, and Fig. 7(b) gives the angular acceptance as a function of α_p. Figure 7(b) shows the very good agreement between experiments and calculations using our numerical code or the analytical calculations at ψ_p equal to 0°, 90° and 180° using Eqs. (7), (10) and (11).

Fig. 7 Angular acceptance properties of the non-collinear OPA of a signal at λ_s = 720 nm in a 22-mm-long LBO crystal cut at (θ = 90°, φ = 0°) and pumped at λ_p = 532 nm. (a) Parametric gain at ψ_p = 0° as a function of the phase-matching angle α_p; (b) angular acceptance L.Δα_p as a function of the phase-matching angle ψ_p.

5. Conclusion

In this work we performed calculations and measurements of phase-matching directions and the associated angular acceptances of non-collinear optical parametric amplification in uniaxial and biaxial crystals. These calculations were experimentally verified in BBO and LBO crystals that are widely used for frequency generation and amplification in the visible and near infrared ranges. The numerical code we developed can be applied to any other nonlinear crystals and any direction of propagation even out of the principal planes of the dielectric frame. Analytic formulae derived in principal planes show that the angular tolerance decreases by increasing the angles between the pump and the signal beams, and that quite independently of the “nature” of the crystal. It is then recommended to limit these angles to a few degrees for practical use. This conclusion is also relevant when propagation is considered out of the principal planes of biaxial crystals. Finally, this knowledge opens the way to further experiments of multi-pumps optical parametric amplification.

Acknowledgment

The research leading to these results has received funding from LASERLAB-EUROPE II (grant agreement no. 228334, EC's Seventh Framework Programme).

References and links

1.	D. Herrmann, R. Tautz, F. Tavella, F. Krausz, and L. Veisz, “Investigation of two-beam-pumped noncollinear optical parametric chirped-pulse amplification for the generation of few-cycle light pulses,” Opt. Express 18(5), 4170–4183 (2010). [CrossRef] [PubMed]
2.	T. Kurita, K. Sueda, K. Tsubakimoto, and N. Miyanaga, “Experimental demonstration of spatially coherent beam combining using optical parametric amplification,” Opt. Express 18(14), 14541–14546 (2010). [CrossRef] [PubMed]
3.	K. Kato, “Second-harmonic generation to 2048 Å in β-BaB₂O₄,” IEEE J. Quantum Electron. 22(7), 1013–1014 (1986). [CrossRef]
4.	K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]
5.	B. Boulanger and J. Zyss, Non-linear Optical properties, Chapter 1.8 in International Tables for Crystallography, Vol. D: Physical Properties of Crystals, (A. Authier Ed., 2006) International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 178–219.
6.	J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun. 99(3-4), 284–302 (1993). [CrossRef]
7.	N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng. 39(4), 1016–1039 (2000). [CrossRef]
8.	C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB₃O₅,” J. Opt. Soc. Am. B 6(4), 616–621 (1989). [CrossRef]
9.	G. Aka, A. Kahn-Harari, F. Mougel, D. Vivien, F. Salin, P. Coquelin, P. Colin, D. Pelenc, and J. P. Damelet, “Linear and nonlinear optical properties of a new gadolinium calcium oxoborate crystal Ca₄GdO(BO₃)₃,” J. Opt. Soc. Am. B 14(9), 2238–2247 (1997). [CrossRef]
10.	SNLO nonlinear optics code available from A. V. Smith, AS-Photonics, Albuquerque, NM.

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4400) Nonlinear optics : Nonlinear optics, materials
(190.4975) Nonlinear optics : Parametric processes

ToC Category:
Nonlinear Optics

History
Original Manuscript: July 10, 2012
Revised Manuscript: August 31, 2012
Manuscript Accepted: September 10, 2012
Published: November 5, 2012

Citation
Benoît Trophème, Benoit Boulanger, and Gabriel Mennerat, "Phase-matching loci and angular acceptance of non-collinear optical parametric amplification," Opt. Express 20, 26176-26183 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26176

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References

D. Herrmann, R. Tautz, F. Tavella, F. Krausz, and L. Veisz, “Investigation of two-beam-pumped noncollinear optical parametric chirped-pulse amplification for the generation of few-cycle light pulses,” Opt. Express18(5), 4170–4183 (2010). [CrossRef] [PubMed]
T. Kurita, K. Sueda, K. Tsubakimoto, and N. Miyanaga, “Experimental demonstration of spatially coherent beam combining using optical parametric amplification,” Opt. Express18(14), 14541–14546 (2010). [CrossRef] [PubMed]
K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron.22(7), 1013–1014 (1986). [CrossRef]
K. Kato, “Temperature-tuned 90° phase-matching properties of LiB3O5,” IEEE J. Quantum Electron.30(12), 2950–2952 (1994). [CrossRef]
B. Boulanger and J. Zyss, Non-linear Optical properties, Chapter 1.8 in International Tables for Crystallography, Vol. D: Physical Properties of Crystals, (A. Authier Ed., 2006) International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 178–219.
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SNLO nonlinear optics code available from A. V. Smith, AS-Photonics, Albuquerque, NM.

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