Optimal vector phase-matching conditions in biaxial crystalline materials determined by the extreme surfaces method: the case of orthorhombic crystals

Oleh Buryy; Oleh Buryy; Dmytro Shulha; Nazariy Andrushchak; Andriy Danylov; Bouchta Sahraoui; Anatoliy Andrushchak

doi:10.1364/AO.518690

Applied Optics
Vol. 63,
Issue 13,
pp. 3725-3735
(2024)
•https://doi.org/10.1364/AO.518690

Optimal vector phase-matching conditions in biaxial crystalline materials determined by the extreme surfaces method: the case of orthorhombic crystals

Oleh Buryy, Dmytro Shulha, Nazariy Andrushchak, Andriy Danylov, Bouchta Sahraoui, and Anatoliy Andrushchak

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Oleh Buryy, Dmytro Shulha, Nazariy Andrushchak, Andriy Danylov, Bouchta Sahraoui, and Anatoliy Andrushchak, "Optimal vector phase-matching conditions in biaxial crystalline materials determined by the extreme surfaces method: the case of orthorhombic crystals," Appl. Opt. 63, 3725-3735 (2024)

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Abstract

The optimal geometries of vector phase matching are determined for the cases of second harmonic generation, sum frequency generation, and difference frequency generation in a number of orthorhombic nonlinear optical crystals: KTP, KTA, KB5, ${\rm{KNb}}{{\rm{O}}_3}$, LBO, CBO, and LRB4. The extreme surface method was used to define the wave vector directions of highest possible generation efficiency. As shown, in a significant number of cases vector phase matching ensures higher efficiencies than the scalar one.

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Crystal	Correspondence between the Crystal-Physics and Crystal-Optics Coordinate Systems	Nonlinear Susceptibilities $d_{ij}$ , pm/V
Point group mm2
KTP	$a b c \leftrightarrow X Y Z$	$d_{15} = 1.9$ ; $d_{24} = 3.7$ ; $d_{31} = 2.2$ ; $d_{32} = 3.7$ ; $d_{33} = 14.6$ [15]
KTA	$a b c \leftrightarrow X Y Z$	$d_{15} = 2.5$ ; $d_{24} = 4.4$ ; $d_{31} = 2.9$ ; $d_{32} = 5.1$ ; $d_{33} = 16.2$ [15]
KB5	$a b c \leftrightarrow X Y Z$	$d_{15} = d_{31} = 0.04$ ; $d_{24} = d_{32} = 0.003$ ; $d_{33} = 0.05$ [15]
$K N b O_{3}$	$b a c \leftrightarrow X Y Z$	$d_{15} = - 12.4$ ; $d_{24} = - 12.8$ ; $d_{31} = - 11.9$ ; $d_{32} = - 13.7$ ; $d_{33} = - 20.6$ [15]
LBO	$a c b \leftrightarrow X Y Z$	$d_{15} = d_{31} = - 0.67$ ; $d_{24} = d_{32} = 0.85$ ; $d_{33} = 0.04$ [14]
Point group 222
CBO	$c a b \leftrightarrow X Y Z$	$d_{14} = d_{36} = d_{25} = 1.49$ [14]
LRB4	$b c a \leftrightarrow X Y Z$	$d_{14} = d_{36} = d_{25} = 0.45$ [15]

	Scalar PM			Vector PM
	Angle Deg.			Angle (Pump Beam) Deg.		Angle (SH Beam) Deg.
Crystal	$θ$	$φ$	$η_{scal}^{extr}$ , ${p m}^{2} / V^{2}$	$θ_{p}$	$φ_{p}$	$θ$	$φ$	$η_{vect}^{extr}$ , ${p m}^{2} / V^{2}$
Type I Phase Matching
KTP	134.7	218.4	0.055	116.0	36.3	122.7	41.7	0.082 (49%)
KTP	134.7	218.4	0.055	129.3	48.1			0.082 (49%)
KTA	131.6	42.1	0.081	118.7	141.2	123.5	136.8	0.095 (17.3%)
KTA	131.6	42.1	0.081	128.2	131.7			0.095 (17.3%)
KB5	121.4	133.4	$2.0 \cdot 10^{- 4}$	100.6	90	90	90	$4.9 \cdot 10^{- 4}$ (145%)
KB5	121.4	133.4	$2.0 \cdot 10^{- 4}$	79.4	90			$4.9 \cdot 10^{- 4}$ (145%)
$K N b O_{3}$	71.4	0	12.0	102.6	90	90	90	16.7 (39.2%)
$K N b O_{3}$	71.4	0	12.0	77.4	90			16.7 (39.2%)
LBO	89.9	168.4	0.17	Coincides with scalar PM
CBO	51.4	337.9	0.67	Coincides with scalar PM
LRB4	53.3	215.4	0.032	Coincides with scalar PM
Type II Phase Matching
KTP	90	156.6	2.0	Coincides with scalar PM
LBO	20.5	90	0.10	Coincides with scalar PM
CBO	13.1	325.5	0.60	Almost coincides with scalar PM^a
LRB4	69.1	109.7	0.054	89.3	91.4	86.2	90.7	0.058
LRB4	69.1	109.7	0.054	83.1	90.0			(7.4%)

				Scalar PM			Vector PM
				Angle Deg.			Angle (Pump Beam) Deg.		Angle (SFG Beam) Deg.
Crystal	$λ_{1}$ , µm	$λ_{2}$ , µm	$λ_{3}$ , µm	$θ$	$φ$	$η_{scal}^{extr}$ , $p m^{2} / V^{2}$	$θ_{p}$	$φ_{p}$	$θ$	$φ$	$η_{vect}^{extr}$ , $p m^{2} / V^{2}$
Type I Phase Matching (ssf)
KTP	1.0642	1.5918	0. 6378	37.6	214	0.044	63.6	33	57	39	0.14 (218%)
							47.5	50
	0.7600	1.7700	0.5321	40.9	215.7	0.044	65.4	146.9	60.3	142	0.17 (286%)
							48.7	127.6
	1.9077	2.4069	1.0642	32.8	31.1	0.033	62.7	45.8	55	39.2	0.14 (324%)
						32.5	50
KTA	0.6594	1.3188	0.4396	121	44.2	0.054	67.9	138.3	65.2	135.3	0.075 (38.9%)
							60.1	128.6
	1.0642	1.9079	0.6831	36.7	143.9	0.059	62.7	47.4	57	39	0.20 (239%)
							33.2	51.9
KB5	0.3599	0.5398	0.2159	97.1	75.9	$1 . 3 \cdot 1 0^{- 6}$	Coincide with scalar ssf
	0.1900	1.3142	0.1660	99.4	71	$3 . 5 \cdot 1 0^{- 6}$	98.8	108.6	99	108.7	$3.9 \cdot 10^{- 6}$ (11%)
							100.8	110.1
$K N b O_{3}$	0.6594	1.3188	0.4396	89.9	54.8	14.7	83.9	90	90	90	15.9 (8%)
							102.8	90
	3.5303	5.2955	2.1182	70.8	252.4	18.0	83.3	90	90	90	20.4 (13%)
							100.6	90
	1.0642	1.3188	0.5889	62.9	0	10.9	77.3	90	90	90	17.0 (56%)
							106.0	90
LBO	0.5321	1.0642	0.3547	90.1	142.8	0.11	Coincide with scalar ssf
	0.2661	1.3188	0.2214	90.2	70.1	0.017	Coincide with scalar ssf
CBO	0.5321	1.0642	0.3547	35.3	60.0	0.41	52.6	90	43.5	90	0.58 (41%)
							39.0	90
	0.3547	1.0642	0.2661	127.7	90	0.53	Almost coincide with scalar $ssf$
LRB4	1.0642	1.9079	0.6831	36.2	144.8	0.041	52.9	90	47.7	90	0.058 (41%)
							38.3	90
	0.6594	1.3188	0.4396	44.2	37.9	0.038	52.5	90	47.7	90	0.057 (50%)
							38.1	90
Type II Phase Matching (sff)
KTP	1.5918	1.0642	0.6378	74.75	0	2.25	90.0	0	86.25	0	2.44 (8%)
							83.7	0
	1.0642	1.5918	0.6378	53.2	0	1.47	89.9	0	83.0	0	2.44 (66%)
							72.0	0
	0.7600	1.7700	0.5321	53.3	0	1.44	90.1	0	84.0	0	2.39 (66%)
							68.7	0
	2.4069	1.9077	1.0642	54.7	0	1.60	89.9	0	82.25	0	2.53 (58%)
							76.0	0
	1.9077	2.4069	1.0642	132.7	0	1.27	90.0	0	82.0	0	2.52 (98%)
							71.4	0
KTA	0.6594	1.3188	0.4396	90.1	60.1	1.40	Coincide with scalar sff
	1.0642	1.9079	0.6831	49.2	0	1.74	90.1	0	83.25	0	3.24 (86%)
							70.2	0
$K N b O_{3}$	1.0642	1.3188	0.5889	99.0	79.8	$1.1 \cdot 0^{- 6}$	103.7	95.4	105.2	92	0.023 ( $2.3 \cdot 10^{6} %$ )
							107.3	87.4
LBO	0.5321	1.0642	0.3547	42.3	90	0.062	Coincide with scalar sff
CBO	0.5321	1.0642	0.3547	90.1	40.3	0.57	Almost coincide with scalar sff
	0.3547	1.0642	0.2661	90.1	75.1	0.13	90	75.6	90	75.25	0.14 (8%)
							90	74.2
LRB4	0.6594	1.3188	0.4396	89.5	61.3	0.039	Almost coincide with scalar sff
	1.0642	1.9079	0.6831	89.5	53.5	0.053	Almost coincide with scalar sff

				Scalar PM			Vector PM
				Angle Deg.			Angle (Pump Beam) Deg.		Angle (SFG Beam) Deg.
Crystal	$λ_{1}$ , µm	$λ_{2}$ , µm	$λ_{3}$ , µm	$θ$	$φ$	$η_{scal}^{extr}$ , ${p m}^{2} / V^{2}$	$θ_{p}$	$φ_{p}$	$θ$	$φ$	$η_{vect}^{extr}$ , ${p m}^{2} / V^{2}$
Type I Phase Matching (ffs)
KTP	0. 6378	1.5918	1.0642	53.2	0	1.47	96.9	0	90	0	2.44 (66 %)
							107.9	0
	0.5321	1.7700	0.7600	53.3	0	1.44	96.1	0	90	0	2.39 (66%)
							111.4	0
	1.0642	2.4069	1.9077	132.7	0	1.27	98.0	0	90	0	2.52 (98.4%)
							108.6	0
KTA	0.4396	1.3188	0.6594	90.1	60.1	1.88	Almost coincide with scalar ffs
	0.6831	1.0642	1.9079	90.0	166.0	4.14	Coincide with scalar ffs
$K N b O_{3}$	0.5889	1.3188	1.0642	103.9	58.0	0.00095	76.1	103.6	78.3	106.5	0.028 (2847%)
							73.2	99.8
	1.9105	4.7762	3.1841	79.9	125.4	$5 \cdot 10^{- 5}$	76.8	97.1	78.3	99.8	0.021 (41900%)
							74.5	92.7
Type II Phase Matching (fsf, fss)
KTP	0.6378	1.5918(s)	1.0642(f)	74.8	0	2.25	93.7	0	96.3	0	2.44 (8%)
							89.9	0
	0.6378	1.5918(s)	1.0642(s)	37.7	33.3	0.033	57.1	140.9	63.5	147.3	0.12 (263.6%)
							48.0	129.2
	0.5321	1.7700(s)	0.7600(s)	139.0	144.9	0.034	119.5	141.6	114.5	146.8	0.15 (341%)
							130.7	126.8
	1.0642	2.4069(s)	1.9077(f)	54.7	0	1.60	82.3	0	76	0	2.53 (58%)
							89.9	0
	1.0642	2.4069(s)	1.9077(s)	147.1	30.4	0.025	55.0	39.4	62.5	32.3	0.12 (380%)
							46.2	50.7
KTA	0.4396	1.3188(s)	0.6594(s)	121.0	135.6	0.070	Coincide with scalar fss
	0.6831	1.0642(s)	1.9079(f)	49.2	0	1.74	83.3	0	70.3	0	3.24 (86%)
							90.1	0
	0.6831	1.0642(s)	1.9079(s)	36.6	36.7	0.078	56.2	140.4	46.8	127.8	0.22 (182%)
							61.8	146.1
$K N b O_{3}$	0.4396	1.3188(s)	0.6594(s)	90.1	141.5	13.7	Almost coincide with scalar fss
	2.1182	3.5303(s)	5.2955(s)	86.0	0	17.0	90.1	90	79.5	90	17.8 (4.7%)
							96.8	90
	0.5889	1.064(s)	1.3188(f)	110.2	71.8	0.00012	75.5	96	73	92	0.022 (18233%)
							77.4	99.6
	0.5889	1.064(s)	1.3188(s)	62.7	0	11.8	90	90	74	90	14.8 (25.4%)
							102.8	90
	1.9105	4.7762(s)	3.1841(s)	66.9	0	14.4	89.9	90	80.8	90	17.3 (20%)
							104.3	90

Crystal	Correspondence between the Crystal-Physics and Crystal-Optics Coordinate Systems	Nonlinear Susceptibilities $d_{ij}$ , pm/V
Point group mm2
KTP	$a b c \leftrightarrow X Y Z$	$d_{15} = 1.9$ ; $d_{24} = 3.7$ ; $d_{31} = 2.2$ ; $d_{32} = 3.7$ ; $d_{33} = 14.6$ [15]
KTA	$a b c \leftrightarrow X Y Z$	$d_{15} = 2.5$ ; $d_{24} = 4.4$ ; $d_{31} = 2.9$ ; $d_{32} = 5.1$ ; $d_{33} = 16.2$ [15]
KB5	$a b c \leftrightarrow X Y Z$	$d_{15} = d_{31} = 0.04$ ; $d_{24} = d_{32} = 0.003$ ; $d_{33} = 0.05$ [15]
$K N b O_{3}$	$b a c \leftrightarrow X Y Z$	$d_{15} = - 12.4$ ; $d_{24} = - 12.8$ ; $d_{31} = - 11.9$ ; $d_{32} = - 13.7$ ; $d_{33} = - 20.6$ [15]
LBO	$a c b \leftrightarrow X Y Z$	$d_{15} = d_{31} = - 0.67$ ; $d_{24} = d_{32} = 0.85$ ; $d_{33} = 0.04$ [14]
Point group 222
CBO	$c a b \leftrightarrow X Y Z$	$d_{14} = d_{36} = d_{25} = 1.49$ [14]
LRB4	$b c a \leftrightarrow X Y Z$	$d_{14} = d_{36} = d_{25} = 0.45$ [15]

Abstract

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Applied Optics