Abstract

Theory of second-harmonic generation (SHG) in reflection from the (0001) face of hexagonal centrosymmetric (6/mmm) crystals is developed in a phenomenological approach. The bulk (electric-quadrupole) contribution to azimuthal dependences of SHG response is completely isotropic, whereas the surface (electric-dipole) contribution consists of isotropic and anisotropic parts for the 3m surface symmetry. Anisotropy entirely depends on only one component of the surface nonlinear susceptibility. This allows one to extract it directly by simultaneous measurement of the amplitude and phase of the SH wave in the (s,S) or (p,S) experimental setup. The feature is very important, if the observed anisotropy is strong and makes the surface SHG probing for these crystals completely different from the cases of cubic, rhombohedral, and tetragonal centrosymmetric crystals. In the case of weak anisotropy, the estimation of other surface components from azimuthal or polarization measurements is analyzed in detail.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  8. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
    [CrossRef]
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  15. K. R. Allakhverdiev, M. O. Yetis, S. Ozbek, T. K. Baykara, and E. Y. Salaev, “Effective nonlinear GaSe crystal. Optical properties and applications,” Laser Phys. 19, 1092–1104 (2009).
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    [CrossRef]
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  23. L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401(R) (2013).
  24. Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation,” Nano Lett. 13, 3329–3333 (2013).
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    [CrossRef]
  26. C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5, 722–726 (2010).
    [CrossRef]
  27. G. F. Smith, “The crystallographic property tensors of orders 1 to 8,” Ann. N.Y. Acad. Sci. 172, 57–106 (1970).
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    [CrossRef]
  29. R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurements in nonlinear optics,” Appl. Phys. B 63, 491–498 (1996).
    [CrossRef]
  30. J. Chen, S. Machida, and Y. Yamanoto, “Simultaneous measurement of amplitude and phase in surface second-harmonic generation,” Opt. Lett. 23, 676–678 (1998).
    [CrossRef]
  31. T. V. Dolgova, D. Shuhmacher, G. Marowsky, A. A. Fedyanin, and O. A. Aktsipetrov, “Second-harmonic interferometric spectroscopy of buried interfaces of column IV semiconductors,” Appl. Phys. B 74, 653–659 (2002).
    [CrossRef]
  32. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021–5026 (1997).
  33. F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, “Single-valued determination of second-order nonlinear susceptibilities by quarter-wave-plate rotation,” Appl. Phys. B 61, 135–141 (1995).
    [CrossRef]

2013 (5)

H. Zeng, G.-B. Liu, J. Dai, Y. Yan, B. Zhu, R. He, L. Xie, S. Xu, X. Chen, W. Yao, and X. Cui, “Optical signature of symmetry variations and spin-valley coupling in atomically thin tungsten dichalcogenides,” Sci. Rep. 3, 1608 (2013).

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Observation of strong second harmonic generation in monolayer MoS2,” Phys. Rev. B 87, 161403(R) (2013).

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401(R) (2013).

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation,” Nano Lett. 13, 3329–3333 (2013).

L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, and K. S. Novoselov, “Strong light-matter interactions in heterostructures of atomically thin films,” Science 340, 1311–1314 (2013).
[CrossRef]

2012 (1)

O. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7, 699–705 (2012).
[CrossRef]

2011 (1)

J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B. J. LeRoy, “Scanning tunneling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride,” Nat. Mater. 10, 282–285 (2011).
[CrossRef]

2010 (2)

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5, 722–726 (2010).
[CrossRef]

J. J. Dean and H. M. van Driel, “Graphene and a few-layer graphite probed by second-harmonic generation. Theory and experiment,” Phys. Rev. B82, 125411 (2010).
[CrossRef]

2009 (3)

J. J. Dean and H. M. van Driel, “Second harmonic generation from graphene and graphitic films,” Appl. Phys. Lett. 95, 261910 (2009).
[CrossRef]

K. R. Allakhverdiev, M. O. Yetis, S. Ozbek, T. K. Baykara, and E. Y. Salaev, “Effective nonlinear GaSe crystal. Optical properties and applications,” Laser Phys. 19, 1092–1104 (2009).
[CrossRef]

S. A. Yang, X. Li, A. D. Bristow, and J. E. Sipe, “Second harmonic generation from tetragonal crystals,” Phys. Rev. B 80, 165306 (2009).

2008 (1)

J. J. Saarinen and J. E. Sipe, “A Green function approach to surface optics in anisotropic media,” J. Mod. Opt. 55, 13–32 (2008).
[CrossRef]

2007 (1)

W. Daum, “Optical studies of Si/SiO2 interfaces by second-harmonic generation spectroscopy of silicon interband transitions,” Appl. Phys. A 87, 451–460 (2007).
[CrossRef]

2006 (1)

M. A. Marquardt, N. A. Ashmore, and D. P. Cann, “Crystal chemistry and electrical properties of the delafossite structure,” Thin Solid Films 496, 146–156 (2006).
[CrossRef]

2002 (1)

T. V. Dolgova, D. Shuhmacher, G. Marowsky, A. A. Fedyanin, and O. A. Aktsipetrov, “Second-harmonic interferometric spectroscopy of buried interfaces of column IV semiconductors,” Appl. Phys. B 74, 653–659 (2002).
[CrossRef]

2001 (1)

J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, “Superconductivity at 39 K in magnesium diboride,” Nature 410, 63–64 (2001).
[CrossRef]

1999 (2)

S. K. Andersson, M. C. Schanne-Klein, and F. Hache, “Symmetry and phase determination of second-harmonic reflection from calcite surfaces,” Phys. Rev. B 59, 3210–3217 (1999).

G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
[CrossRef]

1998 (1)

1997 (1)

J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021–5026 (1997).

1996 (1)

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurements in nonlinear optics,” Appl. Phys. B 63, 491–498 (1996).
[CrossRef]

1995 (1)

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, “Single-valued determination of second-order nonlinear susceptibilities by quarter-wave-plate rotation,” Appl. Phys. B 61, 135–141 (1995).
[CrossRef]

1994 (1)

Y. R. Shen, “Surfaces probed by nonlinear optics,” Surf. Sci. 299–300, 551–562 (1994).
[CrossRef]

1988 (1)

1987 (2)

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second-and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).

1986 (1)

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation,” Phys. Rev. B 33, 8254–8263 (1986).

1985 (1)

J. A. Litvin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B31, 5543–5546, (1985).

1983 (1)

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-harmonic reflection from silicon surfaces and its relation to structural symmetry,” Phys. Rev. Lett. 51, 1983–1986 (1983).
[CrossRef]

1970 (1)

G. F. Smith, “The crystallographic property tensors of orders 1 to 8,” Ann. N.Y. Acad. Sci. 172, 57–106 (1970).

1969 (1)

J. A. Wilson and A. D. Yoffe, “The transition metal dichalcogenides. Discussion and interpretation of the observed optical, electrical and structural properties,” Adv. Phys. 18, 193–335 (1969).
[CrossRef]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Ajayan, P. M.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Observation of strong second harmonic generation in monolayer MoS2,” Phys. Rev. B 87, 161403(R) (2013).

Akimitsu, J.

J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, “Superconductivity at 39 K in magnesium diboride,” Nature 410, 63–64 (2001).
[CrossRef]

Aktsipetrov, O. A.

T. V. Dolgova, D. Shuhmacher, G. Marowsky, A. A. Fedyanin, and O. A. Aktsipetrov, “Second-harmonic interferometric spectroscopy of buried interfaces of column IV semiconductors,” Appl. Phys. B 74, 653–659 (2002).
[CrossRef]

Alencar, T. V.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401(R) (2013).

Allakhverdiev, K. R.

K. R. Allakhverdiev, M. O. Yetis, S. Ozbek, T. K. Baykara, and E. Y. Salaev, “Effective nonlinear GaSe crystal. Optical properties and applications,” Laser Phys. 19, 1092–1104 (2009).
[CrossRef]

Andersson, S. K.

S. K. Andersson, M. C. Schanne-Klein, and F. Hache, “Symmetry and phase determination of second-harmonic reflection from calcite surfaces,” Phys. Rev. B 59, 3210–3217 (1999).

Ashmore, N. A.

M. A. Marquardt, N. A. Ashmore, and D. P. Cann, “Crystal chemistry and electrical properties of the delafossite structure,” Thin Solid Films 496, 146–156 (2006).
[CrossRef]

Barboza, A. P. M.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401(R) (2013).

Baykara, T. K.

K. R. Allakhverdiev, M. O. Yetis, S. Ozbek, T. K. Baykara, and E. Y. Salaev, “Effective nonlinear GaSe crystal. Optical properties and applications,” Laser Phys. 19, 1092–1104 (2009).
[CrossRef]

Belle, B. D.

L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, and K. S. Novoselov, “Strong light-matter interactions in heterostructures of atomically thin films,” Science 340, 1311–1314 (2013).
[CrossRef]

Berkovic, G.

R. Stolle, G. Marowsky, E. Schwarzberg, and G. Berkovic, “Phase measurements in nonlinear optics,” Appl. Phys. B 63, 491–498 (1996).
[CrossRef]

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Bristow, A. D.

S. A. Yang, X. Li, A. D. Bristow, and J. E. Sipe, “Second harmonic generation from tetragonal crystals,” Phys. Rev. B 80, 165306 (2009).

Britnell, L.

L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, and K. S. Novoselov, “Strong light-matter interactions in heterostructures of atomically thin films,” Science 340, 1311–1314 (2013).
[CrossRef]

Bulmash, D.

J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B. J. LeRoy, “Scanning tunneling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride,” Nat. Mater. 10, 282–285 (2011).
[CrossRef]

Cann, D. P.

M. A. Marquardt, N. A. Ashmore, and D. P. Cann, “Crystal chemistry and electrical properties of the delafossite structure,” Thin Solid Films 496, 146–156 (2006).
[CrossRef]

Casiraghi, C.

L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, and K. S. Novoselov, “Strong light-matter interactions in heterostructures of atomically thin films,” Science 340, 1311–1314 (2013).
[CrossRef]

Castro Neto, A. H.

L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, and K. S. Novoselov, “Strong light-matter interactions in heterostructures of atomically thin films,” Science 340, 1311–1314 (2013).
[CrossRef]

Ceballos, F.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Observation of strong second harmonic generation in monolayer MoS2,” Phys. Rev. B 87, 161403(R) (2013).

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Chen, J.

Chen, W.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation,” Phys. Rev. B 33, 8254–8263 (1986).

Chen, X.

H. Zeng, G.-B. Liu, J. Dai, Y. Yan, B. Zhu, R. He, L. Xie, S. Xu, X. Chen, W. Yao, and X. Cui, “Optical signature of symmetry variations and spin-valley coupling in atomically thin tungsten dichalcogenides,” Sci. Rep. 3, 1608 (2013).

Coleman, J. N.

O. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,” Nat. Nanotechnol. 7, 699–705 (2012).
[CrossRef]

Cui, Q.

N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Observation of strong second harmonic generation in monolayer MoS2,” Phys. Rev. B 87, 161403(R) (2013).

Cui, X.

H. Zeng, G.-B. Liu, J. Dai, Y. Yan, B. Zhu, R. He, L. Xie, S. Xu, X. Chen, W. Yao, and X. Cui, “Optical signature of symmetry variations and spin-valley coupling in atomically thin tungsten dichalcogenides,” Sci. Rep. 3, 1608 (2013).

Dai, J.

H. Zeng, G.-B. Liu, J. Dai, Y. Yan, B. Zhu, R. He, L. Xie, S. Xu, X. Chen, W. Yao, and X. Cui, “Optical signature of symmetry variations and spin-valley coupling in atomically thin tungsten dichalcogenides,” Sci. Rep. 3, 1608 (2013).

Daum, W.

W. Daum, “Optical studies of Si/SiO2 interfaces by second-harmonic generation spectroscopy of silicon interband transitions,” Appl. Phys. A 87, 451–460 (2007).
[CrossRef]

de Paula, A. M.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401(R) (2013).

Dean, C. R.

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation,” Nano Lett. 13, 3329–3333 (2013).

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5, 722–726 (2010).
[CrossRef]

Dean, J. J.

J. J. Dean and H. M. van Driel, “Graphene and a few-layer graphite probed by second-harmonic generation. Theory and experiment,” Phys. Rev. B82, 125411 (2010).
[CrossRef]

J. J. Dean and H. M. van Driel, “Second harmonic generation from graphene and graphitic films,” Appl. Phys. Lett. 95, 261910 (2009).
[CrossRef]

Deshpande, A.

J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B. J. LeRoy, “Scanning tunneling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride,” Nat. Mater. 10, 282–285 (2011).
[CrossRef]

Dolgova, T. V.

T. V. Dolgova, D. Shuhmacher, G. Marowsky, A. A. Fedyanin, and O. A. Aktsipetrov, “Second-harmonic interferometric spectroscopy of buried interfaces of column IV semiconductors,” Appl. Phys. B 74, 653–659 (2002).
[CrossRef]

Eckmann, A.

L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, and K. S. Novoselov, “Strong light-matter interactions in heterostructures of atomically thin films,” Science 340, 1311–1314 (2013).
[CrossRef]

Fedyanin, A. A.

T. V. Dolgova, D. Shuhmacher, G. Marowsky, A. A. Fedyanin, and O. A. Aktsipetrov, “Second-harmonic interferometric spectroscopy of buried interfaces of column IV semiconductors,” Appl. Phys. B 74, 653–659 (2002).
[CrossRef]

Felderhof, B. U.

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, “Single-valued determination of second-order nonlinear susceptibilities by quarter-wave-plate rotation,” Appl. Phys. B 61, 135–141 (1995).
[CrossRef]

Geiger, F.

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, “Single-valued determination of second-order nonlinear susceptibilities by quarter-wave-plate rotation,” Appl. Phys. B 61, 135–141 (1995).
[CrossRef]

Geim, A. K.

L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. Castro Neto, and K. S. Novoselov, “Strong light-matter interactions in heterostructures of atomically thin films,” Science 340, 1311–1314 (2013).
[CrossRef]

Georgiou, T.

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Figures (5)

Fig. 1.
Fig. 1.

Mutual orientation of crystallographic (X,Y,Z), local (X1,X2,X3), and laboratory (x,y,z) frames. The sample is rotated on an azimuthal angle ϕ with respect to the laboratory frame. One of the vertical mirror planes m (dashed lines) of the 3m group of surface symmetry is chosen normal to the X2 axis, for specificity.

Fig. 2.
Fig. 2.

Idealized scheme of the fundamental and SH wave propagation near the (0001) face of a hexagonal crystal. Wave vectors k1(o) and k1(e) of ordinary and extraordinary waves together with the corresponding s and p components of the electric field are shown in the main section of a negative uniaxial crystal (such as 2H-polytype of TX2), for specificity. The angle of incident wave polarization is denoted by φ. The S and P components of the output SH wave in air are also shown.

Fig. 3.
Fig. 3.

(a) SHG intensities IsS(2ω)(ϕ) and IpS(2ω)(ϕ) versus azimuthal angle for αa=αc=1. SHG intensities IsP(2ω)(ϕ) versus azimuthal angle for γa1=1 and different sets of other parameters: (b) αc1=3, Δγa1,αc1=π/3; (c) ×4, αc1=1, Δγa1,αc1=2π/3; (d) ×4; αc1=0.5, Δγa1,αc1=π/2; (e) αc1=1, Δγa1,αc1=0; (f) ×2, αc1=0.5, Δγa1,αc1=2π/3. Designations ×4; ×2 indicate multiplication factors for intensities.

Fig. 4.
Fig. 4.

SHG intensities IφP(2ω)(φ,ϕP(m)) versus polarization angle for γa1=δa2=1 and different sets of other parameters. Curves with 2B1>B3, 2B5>B3: (a) ×8 αc1=αc2=1, Δγa1,αc1=Δδa2,αc2=2π/3, ϕP(m)=π/6, Δδa2,γa1=π/2; (b) αc1=3, αc2=0.5, Δγa1,αc1=2π/3, Δδa2,αc2=π/3, ϕP(m)=π/6, Δδa2,γa1=7π/12; and (c) αc1=1, αc2=3, Δγa1,αc1=2π/3, Δδa2,αc2=π/3, ϕP(m)=π/2, Δδa2,γa1=7π/12. Curves with 2B1<B3, 2B5>B3: (d) ×7, αc1=0.5, αc2=1, Δγa1,αc1=Δδa2,αc2=π/2, ϕP(m)=π/6, Δδa2,γa1=π/4; and (e) αc1=1, αc2=3, Δγa1,αc1=2π/3, Δδa2,αc2=π/3, ϕP(m)=π/6, Δδa2,γa1=π/3. Curves with 2B1>B3, 2B5<B3: (f) ×7, αc1=1, αc2=0.5, Δγa1,αc1=Δδa2,αc2=π/2, ϕP(m)=π/2, Δδa2,γa1=π/4; and (g) αc1=3, αc2=1, Δγa1,αc1=π/3, Δδa2,αc2=2π/3, ϕP(m)=π/6, Δδa2,γa1=π/3. Designations ×7, ×8 indicate multiplication factors for intensities. (h) SHG intensity IφS(2ω)(φ,ϕS(m)) versus polarization angle for αb˜=1, βb=3, Δb˜α,bβ=π/4, and ϕS(m)=π/6.

Fig. 5.
Fig. 5.

SHG intensities IφS(2ω)(φ,ϕS(m)) versus polarization angle for αa=1 and different sets of other parameters. Curves with 2B˜1>B˜3, 2B˜5>B˜3: (a) βb=1, αc=0.5, Δαa,βb=0, Δβb,αc=π, ϕS(m)=π/3, B˜2=B˜4=0; and (d) βb=3, αc=0.5 Δαa,βb=2π/3, Δβb,αc=π/3, ϕS(m)=0, B˜2=B˜40. Curves with 2B˜1<B˜3, 2B˜5>B˜3: (b) βb=αc=3, Δαa,βb=Δβb,αc=π, ϕS(m)=0, B˜2=B˜4=0; and (e) βb=1, αc=2, Δαa,βb=Δβb,αc=π/4, ϕS(m)=π/3,B˜2=B˜40. Curves with 2B˜1>B˜3, 2B˜5<B˜3: (c) βb=αc=0.5, Δαa,βb=Δβb,αc=π, ϕS(m)=π/3, B˜2=B˜4=0; and (f) βb=αc=0.5, Δαa,βb=0, Δβb,αc=π/3, ϕS(m)=0, B˜2=B˜40.

Equations (57)

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E0(ω)(r,t)=E0(ω)ei(k0zz+k0yyωt)+c.c.
k0=(ω/c),k0yky=k0sinϑ,k0z=k01sin2ϑ.
E0s(ω)(z=+0)=E0(ω)sinφ,E0p(ω)(z=+0)=E0(ω)cosφ.
E1(ω)(r,t)=sE1s(ω)ei(k1(o)rωt)+pE1p(ω)ei(k1(e)rωt)+c.c.,
p=fcy^+fsz^,fc=(nω(o)k1z(e))/(k0ε(ω)),fs=(nω(o)ky)/(k0ε||(ω)).
k1(o)=k1y(o)y^k1z(o)z^,k1(o)=(ω/c)nω(o),k1y(o)=ky,k1z(o)=k0ε(ω)sin2ϑ,
k1(e)=k1y(e)y^k1z(e)z^,k1y(e)=ky,k1z(e)=k0ε(ω)(ε(ω)/ε||(ω))sin2ϑ.
E1s(ω)(z=0)=t01sE0s(ω)(z=+0),E1p(ω)(z=0)=t01pE0p(ω)(z=+0),
t01s=(2k0z)/(k0z+k1z(o)),t01p=(2nω(o)k0z)/(k0zε(ω)+k1z(e)),
Pi(2ω),C=χijklEj(ω),CkCEl(ω),C;i,j,k,l=X1,X2,X3,
χ1=χxxxx=χyyyy,χ2=χzzzz,χ3=χxyxy=χyxyx,χ4=χzxzx=χzyzy,χ5=χxzxz=χyzyz,χ6=χzzxx=χzzyy=χzyyz=χzxxz,χ7=χxxzz=χyyzz=χxzzx=χyzzy,χ8=χxxyy=χyyxx=χyxxy=χxyyx.
E1x(ω)(z)=E1s(ω)(z=0)ei(k1z(o)zk0yy),E1y(ω)(z)=fcE1p(ω)(z=0)ei(k1z(e)zk0yy),E1z(ω)(z)=fsE1p(ω)(z=0)ei(k1z(e)zk0yy).
Px(2ω)(z)=i[2χ8fck0yχ7fs(k1z(e)+k1z(o))]E1s(ω)E1p(ω)ei(k1z(e)+k1z(o))z,Py(2ω)(z)=iχ3k0y(E1s(ω))2ei2k1z(o)z+i[(χ1fc2+χ5fs2)k0y2χ7fcfsk1z(e)](E1p(ω))2ei2k1z(e)z,Pz(2ω)(z)=iχ4k1z(o)(E1s(ω))2ei2k1z(o)z+i[2χ6fcfsk0y(χ4fc2+χ2fs2)k1z(e)](E1p(ω))2ei2k1z(e)z.
E1S(2ω),V(z=0)=i(2ω/c)22ε0K1z(o)ei(Kyy2ωt)0eiK1z(o)zPx(2ω)(z)dz
E1P(2ω),V(z=0)=i(2ω/c)22ε0K1z(e)ei(Kyy2ωt)0eiK1z(e)z[FSPz(2ω)(z)FCPy(2ω)(z)]dz,
K1(o)=(2ω/c)n2ω(o),K1y(o)Ky=2ky,K1z(o)=K1(o)ε(2ω)sin2ϑ,n2ω(o)=ε(2ω)
K1y(e)=Ky,K1z(e)=2k0ε(2ω)(ε(2ω)/ε||(2ω))sin2ϑ
FS=n2ω(o)Ky/[(2ω/c)ε||(2ω)],FC=n2ω(o)K1z(e)/[(2ω/c)ε(2ω)],
E0S(2ω),V(z=+0)=T10SE1S(2ω),V(z=0),E0P(2ω),V(z=+0)=T10PE1P(2ω),V(z=0),
T10S=2K1z(o)/(K0z+K1z(o)),T10P=2n2ω(o)K1z(e)/(K0zε(2ω)+K1z(e)).
E0S(2ω),V(z=+0)=ASVE1s(ω)E1p(ω),E0P(2ω),V(z=+0)=APV,1(E1s(ω))2+APV,2(E1p(ω))2,
ASV=i(2ω/c)2T10S[χ82kyfcχ7(k1z(o)+k1z(e))fs]/[2ε0(K1z(o)+k1z(o)+k1z(e))K1z(o)],APV,1=i(2ω/c)2T10P(χ3kyFC+χ4k1z(o)FS)/[2ε0(K1z(e)+2k1z(o))K1z(e)],APV,2={[i(2ω/c)2T10P/[2ε0(K1z(e)+2k1z(e))K1z(e)]]}×{FS[(χ4fc2+χ2fs2)k1z(e)2χ6fsfcky]+FC[(χ1fc2+χ5fs2)ky2χ7fsfck1z(e)]}.
Πi(2ω),C(z=+0)=ΔijkCEj(ω),C(z=0)Ek(ω),C(z=0);i,j,k=X1,X2,X3,
(ΔiμC)=(Δ11C0Δ31CΔ11C0Δ31C00Δ33C0Δ15C0Δ15C000Δ11C0).
(Πx(2ω)Πy(2ω)Πz(2ω))=(Δi,μ)((E1x(ω))2(E1y(ω))2(E1z(ω))22E1y(ω)E1z(ω)2E1x(ω)E1z(ω)2E1x(ω)E1y(ω)),
Δiμ=giigjjgkkΔiμC,μ=(j,k),(k,j),
(xy)=(cosϕsinϕsinϕcosϕ)(X1X2),z=X3.
Πx(2ω)=Δ11Ccos(3ϕ)[(E1s(ω))2fc2(E1p(ω))2]+2[Δ15Cfs+Δ11Csin(3ϕ)fc]E1s(ω)E1p(ω),Πy(2ω)=Δ11Csin(3ϕ)(E1s(ω))22Δ11Cfccos(3ϕ)E1s(ω)E1p(ω)+[2Δ15CfcfsΔ11Csin(3ϕ)fc2](E1p(ω))2,Πz(2ω)=Δ31C(E1s(ω))2+[Δ31Cfc2+Δ33Cfs2](E1p(ω))2.
E0S(2ω),S(z=+0)=i(2ω/c)22ε0K0z(1+R01S)Πx(2ω),E0P(2ω),S(z=+0)=i(2ω/c)2ε0K0z[K0zΠy(2ω)(R01P1)+K0yΠz(2ω)(R01P+1)],
R01S=(K0zK1z(o))/(K0z+K1z(o)),R01P=(K0zε(2ω)K1z(e))/(K0zε(2ω)+K1z(e))
E0S(2ω),S(z=+0)=ASSΠx(2ω),E0P(2ω),S(z=+0)=APSΠz(2ω)+BPSΠy(2ω),
ASS=i(2ω/c)2/[ε0(K0z+K1z(o))],APS=i(2ω/c)Kyε(2ω)/[ε0(K0zε(2ω)+K1z(e))],BPS=APSK1z(e)/Ky.
Eψ(2ω)=E0P(2ω)cosψ+E0S(2ω)sinψ,
E0S,P(2ω)(ϕ,φ,z=+0)=E0S,P(2ω),V(φ,z=+0)+E0S,P(2ω),S(ϕ,φ,z=+0).
E0S(2ω)(ϕ,φ,z=+0)={a1SSScos(3ϕ)sin2φ+[a1SSP+a2SSPsin(3ϕ)]sinφcosφ+a1SPPcos(3ϕ)cos2φ}E02.
α^αeiΔα=Δ11C,
β^βeiΔβ=Δ15C+ASV/(2ASSfs),
a1SSS=ASS(t01s)2Δ11CαaeiΔαa,a1SSP=2ASSt01st01pfs[Δ15C+ASV/(2ASSfs)]βbeiΔβb,a2SSP=2ASSt01st01pfcΔ11Cαb˜eiΔαb˜,a1SPP=ASS(t01p)2fc2Δ11CαceiΔαc,
E0P(2ω)(ϕ,φ,z=+0)={[a1PSS+a2PSSsin(3ϕ)]sin2φ+a1PSPcos(3ϕ)sinφcosφ+[a1PPP+a2PPPsin(3ϕ)]cos2φ}E02,
a1PSS=APS(t01s)2(Δ31C+APV,1/APS)γa1eiΔγa1,a2PSS=BPS(t01s)2Δ11Cαc1eiΔαc1,a1PPP=2BPS(t01p)2fcfs{Δ15C+[Δ31C(fc/fs)+Δ33C(fs/fc)][APS/(2BPS)]+APV,2/(2BPSfcfs)}δa2eiΔαα2,a2PPP=BPS(t01p)2fc2Δ11Cαc2eiΔαc2.
γ^γeiΔγ=Δ31C+APV,1/APS,
δ^δeiΔδ=Δ15C+[Δ31C(fc/fs)+Δ33C(fs/fc)][APS/(2BPS)]+APV,2/(2BPSfcfs).
E0S(2ω)(ϕ)IsS(2ω)(ϕ)exp(iΦsS)=a1SSScos(3ϕ)E02.
E0P(2ω)(ϕ)IsP(2ω)(ϕ)exp(iΦsP)=a1PSS+a2PSSsin(3ϕ).
E0P(2ω)(ϕ)IpP(2ω)(ϕ)exp(iΦpP)=a1PPP+a2PPPsin(3ϕ).
E0S(2ω)(ϕP(m),φ=π/4)=Iπ/4S(2ω)(ϕP(m))exp(iΦπ/4S)=(1/2)[a1SSP+a2SSPsin(3ϕP(m))].
Iψ(2ω)(φ)=B1sin4φ+B2sin3φcosφ+B3sin2φcos2φ+B4sinφcos3φ+B5cos4φ,
B1=α2c12sin2(3ϕ)+γ2a12+2αγc1a1cosΔγa1,αc1sin(3ϕ),B2=2[α2c1b˜1cosΔc1b˜1sin(3ϕ)+αγa1b˜1cosΔγa1,αb˜1]cos(3ϕ),B3=α2b˜12cos2(3ϕ)+2α2c1c2cosΔc1c2sin2(3ϕ)+2γδa1a2cosΔγa1,δa2+2αδc1a2cosΔαc1,δa2sin(3ϕ)+2αγc2a1cosΔγa1,αc2sin(3ϕ),B4=2[α2c2b˜1cosΔc2b˜1sin(3ϕ)+αδa2b˜1cosΔδa2,αb˜1]cos(3ϕ),B5=α2c22sin2(3ϕ)+δ2a22+2αc2δa2cosΔδa2,αc2sin(3ϕ).
B1=α2a2cos2(3ϕ),B2=2[α2ab˜cosΔab˜sin(3ϕ)+αβabcosΔαa,βb]cos(3ϕ),B3=α2b˜2sin2(3ϕ)+β2b2+2α2accosΔαa,αccos2(3ϕ)+2αβbb˜cosΔαb˜,βbsin(3ϕ),B4=2[α2cb˜cos(Δcb˜)sin(3ϕ)+αβcbcosΔβb,αc]cos(3ϕ),B5=α2c2cos2(3ϕ).
IS(2ω)(ϕS(m),φφ0)=B˜1sin4(φφ0)+B˜3sin2(φφ0)cos2(φφ0)+B˜5cos4(φφ0)+sin(φφ0)cos(φφ0)(B˜2(ϕ(m))sin2(φφ0)+B˜4(ϕ(m))cos2(φφ0)),
Δ15C+a7(ϑ)χ7+a8(ϑ)χ8=β^(ϑ),
Δ31C+a3(ϑ)χ3+a4(ϑ)χ4=γ^(ϑ),
a33(ϑ)Δ33C+a2(ϑ)χ2+a5(ϑ)χ5+a6(ϑ)χ6=δ^(ϑ)Δ15Ca31(ϑ)Δ31Ca1(ϑ)χ1a˜4(ϑ)χ4a˜7(ϑ)χ7.
ξ=ξcosφ0ηsinφ0,η=ξsinφ0+ηcosφ0.
B˜1=B1cos4φ0B2cos3φ0sinφ0+B3cos2φ0sin2φ0B4cosφ0sin3φ0+B5sin4φ0,B˜2=B2cos4φ0+2(2B1B3)cos3φ0sinφ0+3(B4B2)cos2φ0sin2φ02(2B5B3)cosφ0sin3φ0B4sin4φ0,B˜3=B3sin4φ0+3(B4B2)sin3φ0cosφ0+2(3B12B3+3B5)sin2φ0cos2φ03(B4B2)sinφ0cos3φ0+B3cos4φ0,B˜4=B2sin4φ0+2(2B1B3)sin3φ0cosφ03(B4B2)sin2φ0cos2φ02(2B5B3)sinφ0cos3φ0+B4cos4φ0,B˜5=B1sin4φ0+B2sin3φ0cosφ0+B3sin2φ0cos2φ0+B4sinφ0cos3φ0+B5cos4φ0.
a1=(k1z(e)ε||(ω))/(ε(ω)2g2),a2=ky2ε(ω)ε(2ω)/[K1z(e)ε||(ω)ε||(2ω)g2],a3=K1z(e)/(2ε(2ω)g1),a4=k1z(o)/(ε||(2ω)g1),a˜4=[(k1z(e))2ε||(ω)ε(2ω)]/(g2K1z(e)ε(ω)ε||(2ω)),a5=(ky2ε(ω))/(2g2k1z(e)ε||(ω)),a6=(2ky2ε(2ω))/(K1z(e)g2ε||(2ω)),a7=(k0z(o)+k1z(e))/(2g3),a˜7=k1z(e)/g2,a8=(k1z(e)ε||(ω))/(ε(ω)g3),a31=ε(2ω)(k1z(e)ε||(ω))/(K1z(e)ε(ω)),a33=ε(2ω)(ky2ε(ω))/(K1z(e)k1z(e)ε||(ω)),
g1=[K1z(e)+2k1z(o)],g2=[K1z(e)+2k1z(e)],g3=[K1z(o)+k1z(e)+k1z(o)].

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