Abstract

Noncollinear second-harmonic-generation (SHG) processes involving an unfocused laser beam and its scattered radiation can act to produce one or more cones of phase-matched second harmonic, resulting in rings on an observation screen. We calculate expressions for the ring parameters (center and radius) for types I and II uniaxial crystals in terms of the crystal indices of refraction and geometry. Measurement of the ring parameters for several crystal tilt angles can yield very accurate relative values for the indices of refraction of a uniaxial SHG crystal. A method for the automatic maintenance of optimum SHG which does not require the placement of optics in the beam is also suggested.

© 1981 Optical Society of America

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References

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  1. P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
    [Crossref]
  2. D. A. Kleinman, A. Ashkin, G. D. Boyd, Phys. Rev. 145, 338 (1966).
    [Crossref]
  3. J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).
    [Crossref]
  4. R. A. Baumgartner, R. L. Byer, IEEE J. Quantum Electron. QE-15, 432 (1979).
    [Crossref]

1979 (1)

R. A. Baumgartner, R. L. Byer, IEEE J. Quantum Electron. QE-15, 432 (1979).
[Crossref]

1966 (1)

D. A. Kleinman, A. Ashkin, G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

1962 (2)

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).
[Crossref]

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Ashkin, A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

Baumgartner, R. A.

R. A. Baumgartner, R. L. Byer, IEEE J. Quantum Electron. QE-15, 432 (1979).
[Crossref]

Boyd, G. D.

D. A. Kleinman, A. Ashkin, G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

Byer, R. L.

R. A. Baumgartner, R. L. Byer, IEEE J. Quantum Electron. QE-15, 432 (1979).
[Crossref]

Giordmaine, J. A.

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).
[Crossref]

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

IEEE J. Quantum Electron. (1)

R. A. Baumgartner, R. L. Byer, IEEE J. Quantum Electron. QE-15, 432 (1979).
[Crossref]

Phys. Rev. (1)

D. A. Kleinman, A. Ashkin, G. D. Boyd, Phys. Rev. 145, 338 (1966).
[Crossref]

Phys. Rev. Lett. (2)

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).
[Crossref]

P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

k0 and k1 are fundamental beam k vectors. k2 is the second harmonic. (a) Collinear process is not phase-matched, (b) Non-collinear process in which k0 is the main laser beam, k1 is a scattered off-axis fundamental wave, and k2 is the phase-matched off-axis second-harmonic k vector.

Fig. 2
Fig. 2

Crystal and k vector geometry, ĉ lies in the x-z plane at angle θ0 to the z axis. Direction of k2 is obtained by rotating z ̂ by a small angle α about the y axis and then rotating the obtained vector further by an angle β about the z axis.

Fig. 3
Fig. 3

Angular polar coordinates α and β and a cone of rays centered about a ray at an angle αc from the z axis (toward the c axis) with angular radius αR.

Fig. 4
Fig. 4

Predicted ring patterns of the two type II (0.532-μm) rings obtained with 1.06-μm radiation in a type II KD*P crystal. The spot at the origin of each coordinate system corresponds to the unscattered laser beam. The crystal c axis intersects the x axis in each figure at the value θ0. The calculated phase-matching angle is θ0 = 55.4°. (The index of refraction data used were taken from Ref. 4.) The polarization of the scattered light giving rise to each ring is indicated by E or 0: (a) Δn ≫ 0; (b) Δn > 0; (c) Δn slightly less than 0; (d) Δn < 0.

Fig. 5
Fig. 5

Photographs of the two type II (0.532-μm) rings obtained with 1.06-μm radiation in a type II KD*P crystal. The spot in each figure is that of the unscattered laser beam. Values of the crystal tilt angles are approximately those in the corresponding figures of Fig. 4. The very small predicted ring in Fig. 4(c) is not evident in Fig. 5(c) probably because of obscuration by the larger ring and main beam spot. The type I ring in each case was larger than the camera frame and too dim to photograph.

Tables (1)

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Table I Noncollinear Phase-Matching Processes

Equations (45)

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P ( 2 ω 0 ) = η P 1 ( ω 0 ) P 2 ( ω 0 ) ,
k 0 = 2 π λ 0 n i ( ω 0 , θ 0 ) z ̂
c ̂ = cos θ 0 z ̂ + sin θ 0 x ̂ ,
k 2 = 2 π ( 1 2 λ 0 ) n k ( 2 ω 0 , θ 2 ) ( sin α cos β x ̂ + sin α sin β y ̂ + cos α z ̂ ) ,
cos θ 2 = k 2 c ̂ | k 2 | | c ̂ | ,
= cos θ 0 cos α + sin θ 0 sin α cos β .
cos θ 1 cos θ 0 cos 2 α + sin θ 0 sin 2 α cos β .
1 2 [ n i ( ω 0 , θ 0 ) + n j ( ω 0 , θ 1 ) ] cos α = n k ( 2 ω 0 , θ 2 ) ,
n e ( ω , θ ) = { n e 2 ( ω ) [ n e 2 ( ω ) n o 2 ( ω ) ] cos 2 θ } 1 / 2
n e ( ω 0 , θ 1 ) n e ( ω 0 , θ 0 ) + 2 A 1 α cos β ,
n e ( 2 ω 0 , θ 2 ) n e ( 2 ω 0 , θ 0 ) + A 2 α cos β ,
A 1 = A 1 ( ω 0 , θ 0 ) = 1 2 b 1 n e 3 ( ω 0 , θ 0 ) sin 2 θ 0 ,
A 2 = A 2 ( 2 ω 0 , θ 0 ) = 1 2 b 2 n e 3 ( 2 ω 0 , θ 0 ) sin 2 θ 0 ,
b 1 = n e 2 ( ω 0 ) n 0 2 ( ω 0 ) ,
b 2 = n e 2 ( 2 ω 0 ) n 0 2 ( 2 ω 0 ) .
α 2 2 α α c cos β + α c 2 α R 2 = 0 .
α c = ( A 1 A 2 ) / n ,
α R = ( A 1 A 2 n ) 2 + 2 Δ n n ,
n = 1 2 [ n i ( ω 0 , θ 0 ) + n j ( ω 0 , θ 0 ) ] ,
Δ n = 1 2 [ n i ( ω 0 , θ 0 ) + n j ( ω 0 , θ 0 ) ] n k ( 2 ω 0 , θ 0 ) .
| k 1 | 2 = ( 4 π λ 0 ) 2 { n k 2 ( 2 ω 0 , θ 2 ) sin 2 α + [ n k ( 2 ω 0 , θ 2 ) cos α 1 2 n i ( ω 0 , θ 0 ) ] 2 } .
| k 1 | 2 = ( 2 π λ 0 ) 2 n j 2 ( ω 0 , θ 1 ) ,
cos θ 1 = k 1 c ̂ | k 1 | | c ̂ |
= cos θ 0 + n k ( 2 ω 0 , θ 0 ) sin θ 0 n k ( 2 ω 0 , θ 0 ) 1 2 n i ( ω 0 , θ 0 ) α cos β 2 A 2 n i ( ω 0 , θ 0 ) sin θ 0 [ 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) ] 2 α 2 cos 2 β 2 n k 2 ( 2 ω 0 , θ 0 ) cos θ 0 [ 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) ] 2 α 2
4 n k 2 ( 2 ω 0 , θ 2 ) + n i 2 ( ω 0 , θ 0 ) 4 n i ( ω 0 , θ 0 ) n k ( 2 ω 0 , θ 2 ) cos α = n j 2 ( ω 0 , θ 1 ) .
n k ( 2 ω 0 , θ 2 ) = n k ( 2 ω 0 , θ 0 ) + A 2 α cos β + 1 2 B 2 α 2 cos 2 β + 1 2 C 2 α 2 ,
B 2 = b 2 n e 3 ( 2 ω 0 , θ 0 ) sin 2 θ 0 [ 1 + 3 b 2 n e 2 ( 2 ω 0 , θ 0 ) cos 2 θ 0 ] ,
C 2 = b 2 n e 3 ( 2 ω 0 , θ 0 ) cos 2 θ 0 ,
n j 2 ( ω 0 , θ 1 ) n j 2 ( ω 0 , θ 0 ) + A 1 α cos β + B 1 α 2 cos 2 β + C 1 α 2 ,
A 1 = 2 b 1 n k ( 2 ω 0 , θ 0 ) n e 4 ( ω 0 , θ 0 ) 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) sin 2 θ 0
= 4 n k ( 2 ω 0 , θ 0 ) n e ( ω 0 , θ 0 ) 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) A 1
B 1 = 4 b 1 n k 2 ( 2 ω 0 , θ 0 ) n e 4 ( ω 0 , θ 0 ) [ 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) ] 2 sin 2 θ 0 + 4 b 1 2 n k 2 ( 2 ω 0 , θ 0 ) n e 6 ( ω 0 , θ 0 ) [ 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) ] 2 sin 2 2 θ 0 2 b 1 A 2 n i ( ω 0 , θ 0 ) n e 4 ( ω 0 , θ 0 ) [ 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) ] 2 sin 2 θ 0
C 1 = 4 b 2 n k 2 ( 2 ω 0 , θ 0 ) n e 4 ( ω 0 , θ 0 ) [ 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) ] 2 cos 2 θ 0
U α 2 + V α 2 cos 2 β 2 W α cos β + X = 0 ,
U = 2 n i ( ω 0 , θ 0 ) n k ( 2 ω 0 , θ 0 ) 2 C 2 n i ( ω 0 , θ 0 ) + 4 C 2 n k ( 2 ω 0 , θ 0 ) C 1 ,
V = 4 A 2 2 + 4 B 2 n k ( 2 ω 0 , θ 0 ) 2 B 2 n i ( ω 0 , θ 0 ) B 1 ,
W = 2 A 2 n i ( ω 0 , θ 0 ) 4 A 2 n k ( 2 ω 0 , θ 0 ) + 1 2 A 1 ,
X = [ 2 n k ( 2 ω 0 , θ 0 ) n i ( ω 0 , θ 0 ) ] 2 n j 2 ( ω 0 , θ 0 ) .
α c = ( W U + V ) ,
α x 2 = ( 1 U + V ) ( W 2 U + V X ) ,
α y 2 = ( 1 U ) ( W 2 U + V X ) .
α C = A 2 n ; α R = ( A 2 n ) 2 + 2 Δ n n
α C = A 1 n ; α R = ( A 1 n ) 2 + 2 Δ n n
α C = A 1 n ; α R = ( A 1 n ) 2 + 2 Δ n n
α R = 2 Δ n n

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