## The spatially varying polarization of a focused Gaussian beam in quasi-phase-matched superlattice under electro-optic effect |

Optics Express, Vol. 18, Issue 24, pp. 25000-25007 (2010)

http://dx.doi.org/10.1364/OE.18.025000

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### Abstract

We present in this paper a wave coupling theory of linear electro-optic (EO) effect for quasi-phase matched (QPM) of focused Gaussian beam in an optical superlattice (OSL). The numerical results indicate that, due to the EO effect of an appropriate applied electric field, the output beam will form spatially inhomogeneous polarization, changing continuously in transverse section of beam; the confocal parameter has a significant impact on the output polarization of Gaussian beam and determines the half-wave voltage.

© 2010 OSA

## 1. Introduction

## 2. Theory and Analyses

*y*-axis of the OSL and a monochromatic light wave propagates along the

*x*-axis of the OSL. In a cylindrical coordinate system, the total electric field participating in the process of linear EO effect can be expressed as [29

29. W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. **195**(1-4), 303–311 (2001). [CrossRef]

*r*is the radial distance from the propagation axis;

**E**(0) is the dc electric field or slow varying electric field; [

**E**(

*r*,

*x*)exp(-

*iωt*)/2 +

**c.c.]**is the light field with frequency

*ω*;

**c.c.**denotes the complex conjugate. According to Ref [30

30. A. Ciattoni, B. Crosignani, and P. Porto, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. **177**, 9–13 (2000). [CrossRef]

31. A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A **19**(7), 1422–1431 (2002). [CrossRef]

*g*= 1/(

*k*

_{0}/

*W*

_{0}), where

*k*

_{0}is the wave number of the light field in vacuum and

*W*

_{0}is the waist radius at the input surface. For a wavelength λ = 632.8 nm, when

*g*= 1/(

*k*

_{0}/

*W*

_{0})≤0.01, namely

*W*

_{0}≥10.07µm, the paraxial approximation condition holds. And the

*x*component (longitudinal component) of light field is too small so that it can be neglected. But, there exist two independent electromagnetic wave components of a monochromatic light wave propagating in the OSL, i.e.,where

**E**

_{1}(

*r*,

*x*) and

**E**

_{2}(

*r*,

*x*) denote the complex amplitudes of two perpendicular components of the light field when

*k*

_{1}=

*k*

_{2}, or those of two independent electric field components experiencing different refractive indices when

*k*

_{1}≠

*k*

_{2}.

**E**

*(*

_{j}*r*,

*x*) =

**G**

*(*

_{j}*x*)

*u*(

_{j}*r*,

*x*) (

*j*= 1,2), where

**G**

*(*

_{j}*x*) are the expansion coefficients of the Laguerre-Gaussian modes of zero order, and

*u*(

_{j}*r*,

*x*) are the Gaussian modes [10

10. V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. **184**(1-4), 245–255 (2000). [CrossRef]

11. G. Xu, T. Ren, Y. Wang, Y. Zhu, S. Zhu, and N. Ming, “Third-harmonic generation by use of focused Gaussian beams in an optical superlattice,” J. Opt. Soc. Am. B **20**(2), 360–365 (2003). [CrossRef]

*W*

_{01}=

*W*

_{02}=

*W*

_{0}. Therefore,

*u*(

_{j}*r*,

*x*) (

*j*= 1,2) read [10

10. V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. **184**(1-4), 245–255 (2000). [CrossRef]

12. C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett. **33**(7), 720–722 (2008). [CrossRef] [PubMed]

*b*=

_{j}*k*

_{j}W_{0}

^{2}are the confocal parameters and

*b*

_{2}=

*n*

_{2}/

*n*

_{1}

*b*

_{1}, with

*n*

_{1}and

*n*

_{2}being the unperturbed refractive indices of two wave components of different polarizations.

**a**,

**b**, and

**c**are three unit vectors and

*A*

_{1}(

*x*) and

*A*

_{2}(

*x*) are the normalized amplitudes of the two wave components. Similarly to Ref [10

10. V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. **184**(1-4), 245–255 (2000). [CrossRef]

12. C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett. **33**(7), 720–722 (2008). [CrossRef] [PubMed]

29. W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. **195**(1-4), 303–311 (2001). [CrossRef]

32. G. L. Zheng, H. C. Wang, and W. L. She, “Wave coupling theory of Quasi-Phase-Matched linear electro-optic effect,” Opt. Express **14**(12), 5535–5540 (2006). [CrossRef] [PubMed]

*f*(

*x*) = 1 and −1 correspond to the positive and negative domains of OSL, respectively;

*i*= 1, 2, 3) being the same as those in Ref. [29

29. W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. **195**(1-4), 303–311 (2001). [CrossRef]

12. C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett. **33**(7), 720–722 (2008). [CrossRef] [PubMed]

*f*(

*x*) = sgn(Re{[1 + i(

*x*/

*b*

_{1})(1-

*n*

_{1}/

*n*

_{2})]

^{−1}exp(

*i*Δ

*kx*)}),where Re represents the real part; sgn is the sign function, sgn(

*x*) = 1 when

*x*≥0, sgn(

*x*) = −1 when

*x*<0. Under the condition of QPM, Eqs. (4) can be simplified as where

*L*is the length of OSL;

*R*is the reciprocal vector provided by the OSL; and

*φ*(

*x*) = arg{[1 ±

*i*(

*x*/

*b*

_{1})(1-

*n*

_{1}/

*n*

_{2})]

^{−1}}. For plane-wave interactions,

*φ*(

*x*) becomes a constant, and the OSL will degenerate to a periodic one [33

33. T. Kartaloğlu, Z. G. Figen, and O. Aytür, “Simultaneous phase matching of optical parametric oscillation and second-harmonic generation in aperiodically poled lithium niobate,” J. Opt. Soc. Am. B **20**(2), 343–350 (2003). [CrossRef]

32. G. L. Zheng, H. C. Wang, and W. L. She, “Wave coupling theory of Quasi-Phase-Matched linear electro-optic effect,” Opt. Express **14**(12), 5535–5540 (2006). [CrossRef] [PubMed]

*i*(

*x*/

*b*

_{1}) (1-

*n*

_{1}/

*n*

_{2})]

^{−1}for each term on the right side of Eqs. (5). The factor [1 ±

*i*(

*x*/

*b*

_{1})(1-

*n*

_{1}/

*n*

_{2})]

^{−1}depends on

*x*, which causes a continuously phase variation, so-called Gouy phase shift. When

*x*<<

*b*

_{1}, Eqs. (5) reduce to the familiar wave coupling equations under the plane-wave approximation [32

32. G. L. Zheng, H. C. Wang, and W. L. She, “Wave coupling theory of Quasi-Phase-Matched linear electro-optic effect,” Opt. Express **14**(12), 5535–5540 (2006). [CrossRef] [PubMed]

*ψ*∈ [-90°, 90°] and ellipticity

*e*∈ [-1, 1] (the positive and negative correspond to right- and left-handed polarizations, respectively).

*ψ*and

*e*can be obtained by the relations [34]where

**184**(1-4), 245–255 (2000). [CrossRef]

**33**(7), 720–722 (2008). [CrossRef] [PubMed]

*x*, the output beam will form a spatially inhomogeneous polarization, changing continuously in the transverse section of beam. It is obviously different from the EO effect of plane wave, for which the output beam has a polarization with homogeneous distribution transversely in space. The reason is that, two independent polarization components of Gaussian beam have different confocal parameters, i.e.,

*b*

_{1}≠

*b*

_{2}, which result in a phase difference between two independent wave components in OSL. The following numerical results will illustrate this further.

*T*, the length of the OSL

*L*and the beam waist

*W*

_{0}are 632.8 nm, 298 K, 2.5 cm and 15 µm, respectively, which satisfy the paraxial approximation; the nonvanishing EO coefficients of lithium niobate used are

*r*

_{22}= 3.4 and

*r*

_{51}= 3.4 (in 10

^{−12}m/V) [29

**195**(1-4), 303–311 (2001). [CrossRef]

35. M. V. Hobden and J. Warner, “The temperature dependence of the refractive indices of pure lithium niobate,” Phys. Lett. **22**(3), 243–244 (1966). [CrossRef]

*A*

_{1}(0) = 0,

*A*

_{2}(0) = 1, we obtain the numerical results shown in Fig. 2 , which demonstrates the spatial distribution of polarization of output beam for different applied electric field

*E*

_{0}. One sees from Figs. 2(a) and 2(e) that, when

*E*

_{0}= 0 or 64 V/mm, the output beam is linearly polarized. This is because that when

*E*

_{0}= 0, it has no EO effect and the output beam is an extraordinary one; and when

*E*

_{0}= 64 V/mm,

*A*

_{1}(

*L*)|

^{2}= 1 the output beam has become an ordinary one fully. More interesting is that, when

*E*

_{0}takes other values, for example,

*E*

_{0}= 15, 30 or 45 V/mm, the polarization of output beam becomes spatially inhomogeneous. To further identify the relative change of polarization for output beam, we plot the dependence of

*ψ*and

*e*on

*r*at different

*E*

_{0}, as shown in Fig. 3 . One sees from Fig. 3 that, when

*E*

_{0}= 15 V/mm [corresponding to Fig. 2(b)],

*ψ*varies from −0.10° to −8.63° (Δ

*ψ*= 8.53°) and

*e*from −0.40 to −0.34 (Δ

*e*= 0.06) with

*r*increasing from 0 to 150 µm; when

*E*

_{0}= 45 V/mm [corresponding to Fig. 2(d)],

*ψ*varies from 0.14° to 13.82° (Δ

*ψ*= 13.68°) and

*e*from −0.48 to −0.46 (Δ

*e*= 0.02) with

*r*(Note that in Figs. 2(b) and 2(d), though Δ

*e*are small, Δ

*ψ*are great, the spatial inhomogeneity of polarization of output beam is still evident); and when

*E*

_{0}= 30 V/mm [corresponding to Fig. 2(c)],

*ψ*varies from −1.60° to −34.23° (Δ

*ψ*= 32.63°) and

*e*from −0.93 to −0.66 (Δ

*e*= 0.27) with

*r*. Δ

*e*and Δ

*ψ*are both great, so the spatial inhomogeneity of polarization of output beam is very evident.

*E*

_{0}, but also affected by the confocal parameters

*b*

_{1}and

*b*

_{2}. To demonstrate this, we fix

*E*

_{0}at 30 V/mm, and change

*b*

_{1}(

*b*

_{2}=

*n*

_{2}/

*n*

_{1}

*b*

_{1}). The numerical results are shown in Fig. 4 . It is found that, when

*b*

_{1}= 5.11 mm (

*W*

_{0}= 15µm), the spatial inhomogeneity of polarization of output beam is very evident. With the increase of

*b*

_{1}, however, the transverse polarization of output beam varies gradually from spatial inhomogeneity to spatial homogeneity. It can be understood by Fig. 5 . One sees that, when

*b*

_{1}= 4 × 5.11 mm,

*ψ*varies from −6.62° to −20.40°

**(**Δ

*ψ*= 13.78°) and

*e*from 0.94 to 0.68

**(**Δ

*e*= 0.26

**)**with

*r*. Δ

*ψ*is much smaller than that at

*b*

_{1}= 5.11 mm. And when

*b*

_{1}= 16 × 5.11 mm,

*ψ*varies from −6.25° to 4.83° (Δ

*ψ*= 11.08°

**)**and

*e*from −0.92 to −0.84 (Δ

*e*= 0.08

**)**with

*r*. Both of Δ

*ψ*and Δ

*e*are much smaller than those at

*b*

_{1}= 5.11 mm. Further, when

*b*

_{1}= 64 × 5.11 mm,

*ψ*varies from −1.91° to 0.87° (Δ

*ψ*= 2.78°

**)**and

*e*from −0.916 to −0.913

**(**Δ

*e*= 0.003

**)**with

*r*. Compared with that for

*b*

_{1}= 5.11 mm, the Δ

*ψ*here is very small and Δ

*e*is almost unchanged, meaning that the polarization is almost spatially homogeneous.

*b*

_{1}on the half-wave voltage

*V*

_{π}=

*E*

_{0}'

*d*, where

*E*

_{0}' is the applied electric field for turning an extraordinary light into an ordinary one fully;

*d*is the thickness of OSL along the direction of applied electric field. The dependence of

*E*

_{0}' on

*b*

_{1}for the output intensity of o-ray reaching at its maximum value is shown in Fig. 6 , from which one sees that,

*E*

_{0}' (or

*V*

_{π}) continually decreases as

*b*

_{1}increases from 2.3 to 21.14 mm. And when

*b*

_{1}≥21.14 mm,

*E*

_{0}' (or

*V*

_{π}) almost keeps a constant since (

*x*/

*b*

_{1})

^{2}(1-

*n*

_{1}/

*n*

_{2})

^{2}in Eq. (5) is close to zero in this case. Then, it is easy to obtain

**195**(1-4), 303–311 (2001). [CrossRef]

**14**(12), 5535–5540 (2006). [CrossRef] [PubMed]

*A*

_{1}(

*L*)|

^{2}as a function of

*b*

_{1}and

*E*

_{0}is shown in Fig. 7 , which exhibits a recurrence of o-ray to its maximum intensity with

*E*

_{0}for a fixed

*b*

_{1}. It can be understood as follows: according to Eq. (6), the ratio

*A*

_{2}(

*L*)/

*A*

_{1}(

*L*) determines the polarization of light field. And Fig. 7 shows that

*A*

_{1}(

*L*) and

*A*

_{2}(

*L*) have some periodicity vs

*E*

_{0}for a fixed

*b*

_{1}, which means that the dependence of the polarization on

*E*

_{0}has some periodicity. The phenomenon is slightly different from the recent work of J.W. Zhao

*et al*. [36

36. J. W. Zhao, C. P. Huang, Z. Q. Shen, Y. H. Liu, L. Fan, and Y. Y. Zhu, “Simultaneous harmonic generation and polarization control in an optical superlattice,” Appl. Phys. B **99**(4), 673–677 (2010). [CrossRef]

37. C. P. Huang, Q. J. Wang, and Y. Y. Zhu, “Cascaded frequency doubling and electro-optic coupling in a single optical superlattice,” Appl. Phys. B **80**(6), 741–744 (2005). [CrossRef]

39. H. Tang, L. Chen, G. Zheng, D. Huang, and W. She, “Electrically controlled second harmonic generation of circular polarization in a single LiNbO3 optical superlattice,” Appl. Phys. B **94**(4), 661–666 (2009). [CrossRef]

*E*

_{0}. A similar mechanism was found from acousto-optic tunable SHG [40

40. Z. Y. Yu, F. Xu, F. Leng, X. S. Qian, X. F. Chen, and Y. Q. Lu, “Acousto-optic tunable second harmonic generation in periodically poled LiNbO3,” Opt. Express **17**(14), 11965–11971 (2009). [CrossRef] [PubMed]

## 3. Conclusion

## Acknowledgments

## References and links

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2. | M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched Second Harmonic Generation: Tuning and Tolerances,” IEEE J. Quantum Electron. |

3. | S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science |

4. | A. Bahabad, M. Murnane, and H. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photonics |

5. | Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. |

6. | X. Chen, J. Shi, Y. Chen, Y. Zhu, Y. Xia, and Y. Chen, “Electro-optic Solc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. |

7. | Y. Q. Lu, M. Xiao, and G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. |

8. | K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, and J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. |

9. | D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. |

10. | V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. |

11. | G. Xu, T. Ren, Y. Wang, Y. Zhu, S. Zhu, and N. Ming, “Third-harmonic generation by use of focused Gaussian beams in an optical superlattice,” J. Opt. Soc. Am. B |

12. | C. Zhang, Y. Q. Qin, and Y. Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett. |

13. | L. X. Chen and W. L. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. |

14. | Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

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17. | T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, “The Pancharatnam-Berry phase for non-cyclic polarization changes,” Opt. Express |

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29. | W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. |

30. | A. Ciattoni, B. Crosignani, and P. Porto, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. |

31. | A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A |

32. | G. L. Zheng, H. C. Wang, and W. L. She, “Wave coupling theory of Quasi-Phase-Matched linear electro-optic effect,” Opt. Express |

33. | T. Kartaloğlu, Z. G. Figen, and O. Aytür, “Simultaneous phase matching of optical parametric oscillation and second-harmonic generation in aperiodically poled lithium niobate,” J. Opt. Soc. Am. B |

34. | R. Azzam, and N. Bashara, Ellipsometry and Polarized Light (Amsterdam: North-Holland, 1977). |

35. | M. V. Hobden and J. Warner, “The temperature dependence of the refractive indices of pure lithium niobate,” Phys. Lett. |

36. | J. W. Zhao, C. P. Huang, Z. Q. Shen, Y. H. Liu, L. Fan, and Y. Y. Zhu, “Simultaneous harmonic generation and polarization control in an optical superlattice,” Appl. Phys. B |

37. | C. P. Huang, Q. J. Wang, and Y. Y. Zhu, “Cascaded frequency doubling and electro-optic coupling in a single optical superlattice,” Appl. Phys. B |

38. | Y. Kong, X. Chen, and Y. Xia, “Competition of frequency conversion and polarization coupling in periodically poled lithium niobate,” Appl. Phys. B |

39. | H. Tang, L. Chen, G. Zheng, D. Huang, and W. She, “Electrically controlled second harmonic generation of circular polarization in a single LiNbO3 optical superlattice,” Appl. Phys. B |

40. | Z. Y. Yu, F. Xu, F. Leng, X. S. Qian, X. F. Chen, and Y. Q. Lu, “Acousto-optic tunable second harmonic generation in periodically poled LiNbO3,” Opt. Express |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(230.2090) Optical devices : Electro-optical devices

(260.5430) Physical optics : Polarization

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 1, 2010

Manuscript Accepted: November 4, 2010

Published: November 16, 2010

**Citation**

Haibo Tang, Lixiang Chen, and Weilong She, "The spatially varying polarization of a focused Gaussian beam in quasi-phase-matched superlattice under electro-optic effect," Opt. Express **18**, 25000-25007 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25000

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### References

- J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]
- M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched Second Harmonic Generation: Tuning and Tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]
- S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997). [CrossRef]
- A. Bahabad, M. Murnane, and H. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photonics 4(8), 571–575 (2010). [CrossRef]
- Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77(23), 3719–3721 (2000). [CrossRef]
- X. Chen, J. Shi, Y. Chen, Y. Zhu, Y. Xia, and Y. Chen, “Electro-optic Solc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. 28(21), 2115–2117 (2003). [CrossRef] [PubMed]
- Y. Q. Lu, M. Xiao, and G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. 78(8), 1035–1037 (2001). [CrossRef]
- K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, and J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. 40(31), 5638–5642 (2001). [CrossRef]
- D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, R. Sander, J. M. Robinson, F. Muhammad, P. Chandramani, and F. Kiamilev, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81(17), 3140–3142 (2002). [CrossRef]
- V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184(1-4), 245–255 (2000). [CrossRef]
- G. Xu, T. Ren, Y. Wang, Y. Zhu, S. Zhu, and N. Ming, “Third-harmonic generation by use of focused Gaussian beams in an optical superlattice,” J. Opt. Soc. Am. B 20(2), 360–365 (2003). [CrossRef]
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