Abstract

Third-harmonic-generation processes in crystals are governed by the fourth-rank tensor χijkl(3), which reflects the crystal symmetry. In this case, the third-order nonlinear susceptibility tensor can be contracted to the compact matrix form χim(3). The matrices χim(3) for isotropic media and all 32 crystallo-graphic point groups are presented. With these matrices, the analytic expressions of third-order effective nonlinear susceptibility can be easily derived.

© 1995 Optical Society of America

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References

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  1. A. Penzkofer, P. Qiu, “Picosecond third-harmonic light generation in calcite,” Appl. Phys. B 47, 71–81 (1988).
    [Crossref]
  2. H. Kobayashi, K. Kubodera, “Analysis of asymmetric fringe patterns of third-harmonic light generation in a molecular crystal,” J. Appl. Phys. 69, 3807–3810 (1991).
    [Crossref]
  3. J. E. Midwinter, J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1665–1674 (1965).
  4. P. N. Butcher, “Nonlinear optical phenomena,” Bulletin 200 (Engineer Experiment Station, Ohio State University, Columbus, Ohio, 1965).
  5. C. C. Shang, H. Hsu, “The spatial symmetric forms of the third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
    [Crossref]
  6. P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
    [Crossref]
  7. M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
    [Crossref]
  8. P. Qiu, A. Penzkofer, “Picosecond third-harmonic light generation in β–BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
    [Crossref]

1993 (1)

M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
[Crossref]

1991 (1)

H. Kobayashi, K. Kubodera, “Analysis of asymmetric fringe patterns of third-harmonic light generation in a molecular crystal,” J. Appl. Phys. 69, 3807–3810 (1991).
[Crossref]

1988 (2)

A. Penzkofer, P. Qiu, “Picosecond third-harmonic light generation in calcite,” Appl. Phys. B 47, 71–81 (1988).
[Crossref]

P. Qiu, A. Penzkofer, “Picosecond third-harmonic light generation in β–BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[Crossref]

1987 (1)

C. C. Shang, H. Hsu, “The spatial symmetric forms of the third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[Crossref]

1965 (1)

J. E. Midwinter, J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1665–1674 (1965).

Butcher, P. N.

P. N. Butcher, “Nonlinear optical phenomena,” Bulletin 200 (Engineer Experiment Station, Ohio State University, Columbus, Ohio, 1965).

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
[Crossref]

Cotter, D.

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
[Crossref]

Coutaz, J. L.

M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
[Crossref]

Ehooman, J. B.

M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
[Crossref]

Hsu, H.

C. C. Shang, H. Hsu, “The spatial symmetric forms of the third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[Crossref]

Kobayashi, H.

H. Kobayashi, K. Kubodera, “Analysis of asymmetric fringe patterns of third-harmonic light generation in a molecular crystal,” J. Appl. Phys. 69, 3807–3810 (1991).
[Crossref]

Kubodera, K.

H. Kobayashi, K. Kubodera, “Analysis of asymmetric fringe patterns of third-harmonic light generation in a molecular crystal,” J. Appl. Phys. 69, 3807–3810 (1991).
[Crossref]

Ledoux, I.

M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
[Crossref]

Midwinter, J. E.

J. E. Midwinter, J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1665–1674 (1965).

Neergaard, G. R.

M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
[Crossref]

Penzkofer, A.

P. Qiu, A. Penzkofer, “Picosecond third-harmonic light generation in β–BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[Crossref]

A. Penzkofer, P. Qiu, “Picosecond third-harmonic light generation in calcite,” Appl. Phys. B 47, 71–81 (1988).
[Crossref]

Pierre, M.

M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
[Crossref]

Qiu, P.

P. Qiu, A. Penzkofer, “Picosecond third-harmonic light generation in β–BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[Crossref]

A. Penzkofer, P. Qiu, “Picosecond third-harmonic light generation in calcite,” Appl. Phys. B 47, 71–81 (1988).
[Crossref]

Shang, C. C.

C. C. Shang, H. Hsu, “The spatial symmetric forms of the third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[Crossref]

Warner, J.

J. E. Midwinter, J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1665–1674 (1965).

Appl. Phys. B (2)

A. Penzkofer, P. Qiu, “Picosecond third-harmonic light generation in calcite,” Appl. Phys. B 47, 71–81 (1988).
[Crossref]

P. Qiu, A. Penzkofer, “Picosecond third-harmonic light generation in β–BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[Crossref]

Br. J. Appl. Phys. (1)

J. E. Midwinter, J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1665–1674 (1965).

IEEE J. Quantum Electron. (1)

C. C. Shang, H. Hsu, “The spatial symmetric forms of the third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[Crossref]

J. Appl. Phys. (1)

H. Kobayashi, K. Kubodera, “Analysis of asymmetric fringe patterns of third-harmonic light generation in a molecular crystal,” J. Appl. Phys. 69, 3807–3810 (1991).
[Crossref]

Opt. Commun. (1)

M. Pierre, J. B. Ehooman, G. R. Neergaard, J. L. Coutaz, I. Ledoux, “Phase matched third harmonic generation from organic crystal of 4-aminobenzonitrile,” Opt. Commun. 98, 209–216 (1993).
[Crossref]

Other (2)

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1990).
[Crossref]

P. N. Butcher, “Nonlinear optical phenomena,” Bulletin 200 (Engineer Experiment Station, Ohio State University, Columbus, Ohio, 1965).

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Tables (3)

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Table 1 Matrices of the Third-Order Nonlinear Susceptibility χ i m ( 3 ) for Crystals and Isotropic Mediaa

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Table 2 [em] Forms of the Negative and Positive Uniaxial Crystals when the Wave Vector K Is along the (θ ϕ) Direction, where a = sin ϕ, b = cos ϕ, c = sin θ, d = cos θ

Tables Icon

Table 3 [em] Forms of the Triclinic Biaxial Crystals when the Wave Vector K Lies in the Principal Planes, where a = sin ϕ, b = cos ϕ, c = sin θ, d = cos θ

Equations (23)

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j k l x x x y y y z z z y z z y y z x z z x x z x y y x x y x y z m 1 2 3 4 5 6 7 8 9 0 .
[ χ i m ( 3 ) ] = [ χ 11 χ 12 χ 13 χ 14 χ 15 χ 16 χ 17 χ 18 χ 19 χ 10 χ 21 χ 22 χ 23 χ 24 χ 25 χ 26 χ 27 χ 28 χ 29 χ 20 χ 31 χ 32 χ 33 χ 34 χ 35 χ 36 χ 37 χ 38 χ 39 χ 30 ] .
χ eff ( 3 ) = e 4 · [ χ i j k l ( 3 ) ] · [ e j k l ] = e 4 · [ χ i m ( 3 ) ] · [ e m ] ,
[ e m ] = { L x x x L y y y L z z z L y z z + L z y z + L z z y L y y z + L y z y + L z y y L x z z + L z x z + L z z x L x x z + L x z x + L z x x L x y y + L y x y + L y y x L x x y + L x y x + L y x x L x y z + L x z y + L y x z + L y z x + L z x y + L z y x e 4 = [ e 4 x , e 4 y , e 4 z ] , L u υ w = e u ( ω 1 ) e υ ( ω 1 ) e w ( ω 1 ) ,
e ( 0 ) = [ sin ϕ , cos ϕ , 0 ] ,
e ( e ) = [ cos θ cos ϕ , cos θ sin ϕ , sin θ ] ,
e ( e 1 ) = [ cos θ cos ϕ cos δ sin ϕ sin δ , cos θ sin ϕ × cos δ + cos ϕ sin δ, sin θ cos δ ] ,
e ( e 2 ) = [ cos θ cos ϕ sin δ sin ϕ cos δ, cos θ sin ϕ × sin δ + cos ϕ cos δ, sin θ sin δ ] ,
cot 2 δ = cot 2 Ω sin 2 θ cos 2 θ cos 2 ϕ + sin 2 ϕ cos θ sin 2 ϕ ,
tan Ω = n z n x [ n y 2 n x 2 n z 2 n y 2 ] 1 / 2 .
[ e m ] = [ sin 3 ϕ , cos 3 ϕ , 0 , 0 , 0 , 0 , 0 , 3 sin ϕ cos 2 ϕ , 3 sin 2 ϕ cos ϕ , 0 ] .
e 4 = [ cos θ cos ϕ , cos θ sin ϕ , sin θ ] .
χ eff , I ( 3 ) = [ χ 15 sin ( 3 ϕ ) + χ 10 cos ( 3 ϕ ) ] sin θ .
χ eff , II ( 3 ) = 1 / 3 χ 11 cos θ cos θ + [ χ 10 sin ( 3 ϕ ) χ 15 cos ( 3 ϕ ) ] sin ( 2 θ ) + χ 16 sin 2 θ,
χ eff , III ( 3 ) = 3 / 2 [ χ 10 cos ( 3 ϕ ) + χ 15 sin ( 3 ϕ ) cos θ sin ( 2 θ ) ] .
e ( e 1 ) = [ 0 , 1 , 0 ] , e ( e 2 ) = e 4 = [ cos θ , 0 , sin θ ] .
χ eff , I ( 3 ) = χ 12 cos θ + χ 25 sin θ, χ eff , II ( 3 ) = χ 18 cos 2 θ + χ 24 sin 2 θ χ 15 sin 2 θ, χ eff , III ( 3 ) = 3 χ 10 sin θ cos 2 θ 3 χ 14 sin 2 θ cos θ χ 19 cos 3 θ + χ 23 sin 3 θ .
e ( e 1 ) = [ cos θ , 0 , sin θ ] , e ( e 2 ) = e 4 = [ 0 , 1 , 0 ] .
χ eff , I ( 3 ) = χ 19 cos 3 θ χ 23 sin 3 θ + χ 14 sin 2 θ cos θ 3 χ 10 sin θ cos 2 θ , χ eff , II ( 3 ) = χ 18 cos 2 θ + χ 24 sin 2 θ χ 15 sin 2 θ, χ eff , III ( 3 ) = χ 12 cos θ χ 25 sin θ .
e ( e 1 ) = [ 0 , cos θ , sin θ ] , e ( e 2 ) = e 4 = [ 1 , 0 , 0 ] .
χ eff , I ( 3 ) = χ 13 sin 3 θ χ 12 cos 3 θ + 3 χ 15 sin θ cos 2 θ 3 χ 14 sin 2 θ cos θ , χ eff , II ( 3 ) = χ 18 cos 2 θ + χ 16 sin 2 θ χ 10 sin 2 θ, χ eff , III ( 3 ) = χ 17 sin θ χ 19 cos θ .
e ( e 1 ) = [ 0 , 0 , 1 ] , e ( e 2 ) = e 4 = [ sin ϕ , cos ϕ , 0 ] .
χ eff , I ( 3 ) = χ 13 sin ϕ χ 23 cos ϕ , χ eff , II ( 3 ) = χ 24 cos 2 ϕ + χ 16 sin 2 ϕ χ 14 sin 2 ϕ , χ eff , III ( 3 ) = 3 χ 15 sin ϕ cos 2 ϕ 3 χ 10 sin 2 ϕ cos ϕ + χ 17 sin 3 ϕ χ 25 cos 3 ϕ .

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