## Dark surface waves in self-focusing media with diffusion and photovoltaic nonlinearities |

Optics Express, Vol. 21, Issue 13, pp. 15075-15080 (2013)

http://dx.doi.org/10.1364/OE.21.015075

Acrobat PDF (1137 KB)

### Abstract

Dark surface waves with photorefractive diffusion and photovoltaic nonlinearities are predicted for the first time. We find it is extraordinary that this type of dark surface waves should be in self-focusing media, which is very different from the surface dark solitons or other nonlinear dark surface waves. An oscillator model is proposed by which the above extraordinary phenomenon is demonstrated. In this model an equivalent force function is established, whose form determines the varieties of surface waves (bright surface waves, dark surface waves or others).

© 2013 OSA

## 1. Introduction

1. H. Z. Kang, T. H. Zhang, H. H. Ma, C. B. Lou, S. M. Liu, J. G. Tian, and J. J. Xu, “Giant enhancement of surface second-harmonic generation using photorefractive surface waves with diffusion and drift nonlinearities,” Opt. Lett. **35**(10), 1605–1607 (2010). [CrossRef] [PubMed]

3. P. Varatharajah, A. Aceves, J. V. Moloney, D. R. Heatley, and E. M. Wright, “Stationary nonlinear surface waves and their stability in diffusive Kerr media,” Opt. Lett. **13**(8), 690–692 (1988). [CrossRef] [PubMed]

6. I. I. Smolyaninov, C. H. Lee, and C. C. Davis, “Giant enhancement of surface second harmonic generation in BaTiO3 due to photorefractive surface wave excitation,” Phys. Rev. Lett. **83**(12), 2429–2432 (1999). [CrossRef]

1. H. Z. Kang, T. H. Zhang, H. H. Ma, C. B. Lou, S. M. Liu, J. G. Tian, and J. J. Xu, “Giant enhancement of surface second-harmonic generation using photorefractive surface waves with diffusion and drift nonlinearities,” Opt. Lett. **35**(10), 1605–1607 (2010). [CrossRef] [PubMed]

7. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. **21**(18), 1460–1462 (1996). [CrossRef] [PubMed]

8. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. **30**(18), 2466–2468 (2005). [CrossRef] [PubMed]

11. X. S. Wang, A. Bezryadina, Z. G. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. **98**(12), 123903 (2007). [CrossRef] [PubMed]

12. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. **98**(21), 213901 (2007). [CrossRef] [PubMed]

13. H. Z. Kang, T. H. Zhang, B. H. Wang, C. B. Lou, B. G. Zhu, H. H. Ma, S. M. Liu, J. G. Tian, and J. J. Xu, “(2+1)D surface solitons in virtue of the cooperation of nonlocal and local nonlinearities,” Opt. Lett. **34**(21), 3298–3300 (2009). [CrossRef] [PubMed]

_{3}[14

14. J. Safioui, E. Fazio, F. Devaux, and M. Chauvet, “Surface-wave pyroelectric photorefractive solitons,” Opt. Lett. **35**(8), 1254–1256 (2010). [CrossRef] [PubMed]

15. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. **73**(24), 3211–3214 (1994). [CrossRef] [PubMed]

18. D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. B **12**(9), 1628–1633 (1995). [CrossRef]

12. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. **98**(21), 213901 (2007). [CrossRef] [PubMed]

14. J. Safioui, E. Fazio, F. Devaux, and M. Chauvet, “Surface-wave pyroelectric photorefractive solitons,” Opt. Lett. **35**(8), 1254–1256 (2010). [CrossRef] [PubMed]

19. W. Q. Chen, X. Yang, S. Y. Zhong, Z. Yan, T. H. Zhang, J. G. Tian, and J. J. Xu, “Surface dark screening solitons,” Opt. Lett. **36**(19), 3801–3803 (2011). [CrossRef] [PubMed]

20. Y. V. Kartashov, F. W. Ye, V. A. Vysloukh, and L. Torner, “Surface waves in defocusing thermal media,” Opt. Lett. **32**(15), 2260–2262 (2007). [CrossRef] [PubMed]

21. S. R. Skinner and D. R. Andersen, “Stationary fundamental dark surface waves,” J. Opt. Soc. Am. B **8**(4), 759–764 (1991). [CrossRef]

23. Y. J. Chen, “Stability of bright and dark surface waves,” J. Opt. Soc. B **10**(6), 1077–1080 (1993). [CrossRef]

24. T. H. Zhang, X. K. Ren, B. H. Wang, C. B. Lou, Z. J. Hu, W. W. Shao, Y. H. Xu, H. Z. Kang, J. Yang, D. P. Yang, L. Feng, and J. J. Xu, “Surface waves with photorefractive nonlinearity,” Phys. Rev. A **76**(1), 013827 (2007). [CrossRef]

## 2. Theory and model

*e*-polarized light beam with intensity

*I*(

*x*) propagating along the interface between air and a PR crystal (PRC), the complex amplitude

*E*(

*x*,

*z*) satisfies the nonlinear scalar wave equation:

*x*<0),

*k*=

*k*

_{0}

*n*

_{0}= 2

*π*/

*λ*

_{0},

*n*

_{0}= 1and

*λ*

_{0}is the wavelength in vacuum. In PRC (

*x*>0),

*k*=

*k*

_{0}(

*n*+ ∆

*n*),

*n*is the refractive index of

*e*polarized beam in the PRC, ∆

*n*is the disturbed refractive index induced by nonlinearity, (

*n*+ ∆

*n*)

^{2}=

*n*

^{2}−

*n*

^{4}

*r*

_{eff}

*E*

_{sc},

*r*

_{eff}is the effective electro-optical coefficient,

*E*

_{sc}is the space-charge field. With an

*o*-polarized coherent uniform background illumination, under open-circuit conditions

*E*

_{sc}can be written as [25

25. C. Anastassiou, M. F. Shih, M. Mitchell, Z. G. Chen, and M. Segev, “Optically induced photovoltaic self-defocusing-to-self-focusing transition,” Opt. Lett. **23**(12), 924–926 (1998). [CrossRef] [PubMed]

*k*is the Boltzman constant,

_{B}*T*is the temperature,

*q*is the charge of carriers, (negative for the electrons and positive for the holes),

*E*is the the photovoltaic field,

_{p}*I*is the equivalent dark irradiance,

_{d}*I*is the background illumination normalized by

_{b}*I*,

_{d}*κ*=

*β*

_{31}

*/β*

_{33},

*β*

_{33}and

*β*

_{31}are the photovoltaic constant for

*e*-polarized light and

*o*-polarized light, respectively. The first and the second terms in the right side of Eq. (2) describe the effects of the diffusion and the photovoltaic components of PR nonlinearity, respectively. For photovoltaic medium, such as LiNbO

_{3}a typical photovoltaic crystal, the effect of photovoltaic component is self-defocusing. Here the

*o*-polarized coherent uniform background illumination is used for self-defocusing-to-self-focusing transition [25

25. C. Anastassiou, M. F. Shih, M. Mitchell, Z. G. Chen, and M. Segev, “Optically induced photovoltaic self-defocusing-to-self-focusing transition,” Opt. Lett. **23**(12), 924–926 (1998). [CrossRef] [PubMed]

*E*(

*x*,

*z*) =

*A*(

*x*)exp(

*ißz*), where

*β*is the propagation constant and

*A*(

*x*) = [

*I*(

*x*)/(

*I*)]

_{d}^{1/2}is the normalized amplitude. Equation (1) can be rewritten as: where

*γ*= −2

*k*

_{0}

^{2}

*n*

_{e}^{4}

*r*

_{eff}

*k*/

_{B}T*q*,

*b*=

*k*

_{0}

^{2}

*n*

_{e}^{4}

*r*

_{eff}

*E*,

_{P}*g*=

*k*

_{0}

^{2}

*n*

_{e}^{2}–

*β*

^{2},

*β*is the propagation constant. In Eq. (3a), the first term indicates the diffraction spreading of the light beam, the second term describes the effect of diffusion mechanism, and the third term states the influence of photovoltaic component of the photorefractive nonlinearity.

## 3. Numerical simulation

### 3.1 Modes of PR SWs

*(*1*)* g > b, bκI_{b} –g(I_{b} + 1) > 0

*F*as a function of amplitude

*A*(

*x*) is sketched in Fig. 1(a1)-1(c1).

*F*has three points of intersection with

*x*-axis at

*A*

_{1}(

*x*) = 0 and

*A*

_{2,3}(

*x*) = ± {[

*bκI*–

_{b}*g*(

*I*+ 1)]/(

_{b}*g*−

*b*)}

^{1/2}.

*F*always exhibits repulsive force around

*A*

_{1}(

*x*) while

*F*always exhibits attractive force around

*A*

_{2,3}(

*x*). That is to say

*A*

_{1}(

*x*) is not a stable balance position and

*A*

_{2,3}(

*x*) are two stable balance positions. So the oscillation can only converge to the couple of nonzero values

*A*

_{2,3}(

*x*) rather than at

*A*

_{1}(

*x*) = 0, which indicates the existence of PR DSWs.

*F*always exhibits attractive force. That means

*F*= 0 and

*dF*/

*dA*(

*x*) < 0 should be satisfied at these positions, consequently based on Eq. (4) one can getSo

*κI*> (

_{b}*I*+ 1) should be satisfied. From Eq. (3a) one can see that in this case the effect of photovoltaic component is self-focusing. That is to say PR DSWs should be supported by self-focusing nonlinearities. When

_{b}*κI*< (

_{b}*I*+ 1), the effect of photovoltaic component is self-defocusing and PR DSWs cannot exist; instead, bright PR SWs may occur.

_{b}*g*>

*b*and

*bκI*–

_{b}*g*(

*I*+ 1) > 0 also means

_{b}*κI*> (

_{b}*I*+ 1).

_{b}*g*(higher

*β*), moderate

*g*(moderate

*β*) and higher

*g*(lower

*β*), respectively. All the modes behave like damped oscillation and converge to the nonzero values corresponding to

*A*

_{2,3}(

*x*) in Fig. 1(a1)-1(c1). From Eq. (3a) one can see that the spatial frequency of the PR DSW modes mainly depends on

*F*and diminishes with increasing of

*g*, as shown in Figs. 1(a2)-1(c2) and 1(a5)-1(c5). Base on the damping oscillation model, larger amplitude of PR DSWs means higher energy of the oscillator, and more intense oscillation will occur. So the decaying oscillation of PR DSWs for same

*g*larger amplitude of PR DSW responds to more intense oscillation, and out-of phase profile will occur more possible.

#### (2) g > b, bκI_{b} –g(I_{b} + 1) < 0

#### (3) g < b, bκI_{b} –g(I_{b} + 1) < 0

*F*has three points of intersection with

*x*-axis at

*A*

_{1}(

*x*) = 0 and

*A*

_{2,3}(

*x*) = ± {[

*bκI*–

_{b}*g*(

*I*+ 1)]/(

_{b}*g*−

*b*)}

^{1/2}, as shown in Fig. 2(b1).

*A*

_{1}(

*x*) is a stable balance position, while

*A*

_{2,3}(

*x*) are two unstable balance positions. So the oscillation can converge at 0 and the oscillator is steady only in a limited range of amplitude. That indicates bright PR SWs with lower amplitude, as shown in Fig. 2(b2). In this case,

*g*<

*b*,

*bκI*–

_{b}*g*(

*I*+ 1) < 0 means

_{b}*κI*> (

_{b}*I*+ 1) and the effect of photovoltaic component is self-defocusing.

_{b}#### (4) g < b, bκI_{b} –g(I_{b} + 1) > 0

*F*has only one point of intersection with

*x*-axis at

*A*(

*x*) = 0, where

*F*always exhibits repulsive force. In this case, the solutions are corresponding to evanescent waves, as shown in Fig. 2(c2).

### 3.2 Stabilities of PR SWs

_{3}is taken as sample and the material parameters at

*λ*= 633 nm are

*n*= 2.2,

_{e}*r*

_{eff}=

*r*

_{33}= 31 × 10

^{−12}m/V,

*κ*=

*β*

_{31}/

*β*

_{33}= 1.2,

*E*= −1.67 × 10

_{P}^{4}V/m. At room temperature, there are

*γ*= 3.7 × 10

^{3}m

^{−1}, and

*b*= 1.1949 × 10

^{9}m

^{−2},

*q*= −1.6 × 10

^{−19}C.

## 4. Conclusion

## Acknowledgments

## References and links

1. | H. Z. Kang, T. H. Zhang, H. H. Ma, C. B. Lou, S. M. Liu, J. G. Tian, and J. J. Xu, “Giant enhancement of surface second-harmonic generation using photorefractive surface waves with diffusion and drift nonlinearities,” Opt. Lett. |

2. | D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” P. Prog. Opt. |

3. | P. Varatharajah, A. Aceves, J. V. Moloney, D. R. Heatley, and E. M. Wright, “Stationary nonlinear surface waves and their stability in diffusive Kerr media,” Opt. Lett. |

4. | G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A |

5. | M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. |

6. | I. I. Smolyaninov, C. H. Lee, and C. C. Davis, “Giant enhancement of surface second harmonic generation in BaTiO3 due to photorefractive surface wave excitation,” Phys. Rev. Lett. |

7. | D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. |

8. | K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. |

9. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. |

10. | A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. |

11. | X. S. Wang, A. Bezryadina, Z. G. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. |

12. | B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. |

13. | H. Z. Kang, T. H. Zhang, B. H. Wang, C. B. Lou, B. G. Zhu, H. H. Ma, S. M. Liu, J. G. Tian, and J. J. Xu, “(2+1)D surface solitons in virtue of the cooperation of nonlocal and local nonlinearities,” Opt. Lett. |

14. | J. Safioui, E. Fazio, F. Devaux, and M. Chauvet, “Surface-wave pyroelectric photorefractive solitons,” Opt. Lett. |

15. | M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. |

16. | Z. G. Chen, M. Mitchell, M. F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett. |

17. | G. C. Valley, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A |

18. | D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. B |

19. | W. Q. Chen, X. Yang, S. Y. Zhong, Z. Yan, T. H. Zhang, J. G. Tian, and J. J. Xu, “Surface dark screening solitons,” Opt. Lett. |

20. | Y. V. Kartashov, F. W. Ye, V. A. Vysloukh, and L. Torner, “Surface waves in defocusing thermal media,” Opt. Lett. |

21. | S. R. Skinner and D. R. Andersen, “Stationary fundamental dark surface waves,” J. Opt. Soc. Am. B |

22. | Y. J. Chen, “Bright and dark surface waves at a nonlinear interface,” Phys. Rev. A |

23. | Y. J. Chen, “Stability of bright and dark surface waves,” J. Opt. Soc. B |

24. | T. H. Zhang, X. K. Ren, B. H. Wang, C. B. Lou, Z. J. Hu, W. W. Shao, Y. H. Xu, H. Z. Kang, J. Yang, D. P. Yang, L. Feng, and J. J. Xu, “Surface waves with photorefractive nonlinearity,” Phys. Rev. A |

25. | C. Anastassiou, M. F. Shih, M. Mitchell, Z. G. Chen, and M. Segev, “Optically induced photovoltaic self-defocusing-to-self-focusing transition,” Opt. Lett. |

**OCIS Codes**

(190.4350) Nonlinear optics : Nonlinear optics at surfaces

(190.5330) Nonlinear optics : Photorefractive optics

(240.6690) Optics at surfaces : Surface waves

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 1, 2013

Revised Manuscript: May 30, 2013

Manuscript Accepted: May 30, 2013

Published: June 17, 2013

**Citation**

Zhonghao Luo, Fangli Liu, Yuhui Xu, Haoyu Liu, Tianhao Zhang, Jingjun Xu, and Jianguo Tian, "Dark surface waves in self-focusing media with diffusion and photovoltaic nonlinearities," Opt. Express **21**, 15075-15080 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15075

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### References

- H. Z. Kang, T. H. Zhang, H. H. Ma, C. B. Lou, S. M. Liu, J. G. Tian, and J. J. Xu, “Giant enhancement of surface second-harmonic generation using photorefractive surface waves with diffusion and drift nonlinearities,” Opt. Lett.35(10), 1605–1607 (2010). [CrossRef] [PubMed]
- D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” P. Prog. Opt.27, 229–313 (1989).
- P. Varatharajah, A. Aceves, J. V. Moloney, D. R. Heatley, and E. M. Wright, “Stationary nonlinear surface waves and their stability in diffusive Kerr media,” Opt. Lett.13(8), 690–692 (1988). [CrossRef] [PubMed]
- G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A51(2), 1571–1577 (1995).
- M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett.20(20), 2075–2077 (1995). [CrossRef] [PubMed]
- I. I. Smolyaninov, C. H. Lee, and C. C. Davis, “Giant enhancement of surface second harmonic generation in BaTiO3 due to photorefractive surface wave excitation,” Phys. Rev. Lett.83(12), 2429–2432 (1999). [CrossRef]
- D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett.21(18), 1460–1462 (1996). [CrossRef] [PubMed]
- K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett.30(18), 2466–2468 (2005). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett.96(7), 073901 (2006). [CrossRef] [PubMed]
- A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett.98(17), 173903 (2007). [CrossRef]
- X. S. Wang, A. Bezryadina, Z. G. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett.98(12), 123903 (2007). [CrossRef] [PubMed]
- B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett.98(21), 213901 (2007). [CrossRef] [PubMed]
- H. Z. Kang, T. H. Zhang, B. H. Wang, C. B. Lou, B. G. Zhu, H. H. Ma, S. M. Liu, J. G. Tian, and J. J. Xu, “(2+1)D surface solitons in virtue of the cooperation of nonlocal and local nonlinearities,” Opt. Lett.34(21), 3298–3300 (2009). [CrossRef] [PubMed]
- J. Safioui, E. Fazio, F. Devaux, and M. Chauvet, “Surface-wave pyroelectric photorefractive solitons,” Opt. Lett.35(8), 1254–1256 (2010). [CrossRef] [PubMed]
- M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett.73(24), 3211–3214 (1994). [CrossRef] [PubMed]
- Z. G. Chen, M. Mitchell, M. F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett.21(9), 629–631 (1996). [CrossRef] [PubMed]
- G. C. Valley, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A50(6), R4457–R4460 (1994). [CrossRef] [PubMed]
- D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. B12(9), 1628–1633 (1995). [CrossRef]
- W. Q. Chen, X. Yang, S. Y. Zhong, Z. Yan, T. H. Zhang, J. G. Tian, and J. J. Xu, “Surface dark screening solitons,” Opt. Lett.36(19), 3801–3803 (2011). [CrossRef] [PubMed]
- Y. V. Kartashov, F. W. Ye, V. A. Vysloukh, and L. Torner, “Surface waves in defocusing thermal media,” Opt. Lett.32(15), 2260–2262 (2007). [CrossRef] [PubMed]
- S. R. Skinner and D. R. Andersen, “Stationary fundamental dark surface waves,” J. Opt. Soc. Am. B8(4), 759–764 (1991). [CrossRef]
- Y. J. Chen, “Bright and dark surface waves at a nonlinear interface,” Phys. Rev. A45(7), 4974–4978 (1992). [CrossRef] [PubMed]
- Y. J. Chen, “Stability of bright and dark surface waves,” J. Opt. Soc. B10(6), 1077–1080 (1993). [CrossRef]
- T. H. Zhang, X. K. Ren, B. H. Wang, C. B. Lou, Z. J. Hu, W. W. Shao, Y. H. Xu, H. Z. Kang, J. Yang, D. P. Yang, L. Feng, and J. J. Xu, “Surface waves with photorefractive nonlinearity,” Phys. Rev. A76(1), 013827 (2007). [CrossRef]
- C. Anastassiou, M. F. Shih, M. Mitchell, Z. G. Chen, and M. Segev, “Optically induced photovoltaic self-defocusing-to-self-focusing transition,” Opt. Lett.23(12), 924–926 (1998). [CrossRef] [PubMed]

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