Abstract

We describe an infrared interferometric technique based on a two-dimensional spatial fringe analysis Fourier method for investigating the characteristic ring diffraction pattern generated by the self-phase-modulation effect induced in nematic liquid crystals (NLCs) by an infrared laser beam and for measuring the nonlinear refractive index of the NLCs. The experimental setup employs a Mach-Zehnder interferometer with a cw CO2 laser emitting at 10.6 µm and a pyroelectric optoelectronic sensor matrix to detect the modulated ring-pattern intensity distribution formed in the far field by a nematic E7 sample. A Fourier-transform-based analysis of the interference fringe pattern allows comparison of the measurements with the theoretical ring-pattern intensity distribution. We show that accurate determination of the nonlinear refractive index can be obtained by analyzing the two-dimensional phase distribution of the modulated ring pattern.

© 2003 Optical Society of America

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References

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  1. C. Khoo, S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).
    [CrossRef]
  2. F. Simoni, Nonlinear Optical Properties of Liquid Crystals and Polymer Dispersed Liquid Crystals (World Scientific, Singapore, 1997).
    [CrossRef]
  3. A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).
  4. J. C. Khoo, S. L. Zhuang, S. Shepard, “Self-focusing of a low power cw laser beam via optically induced birefringence in a nematic liquid-crystal film,” Appl. Phys. Lett. 39, 937–940 (1981).
    [CrossRef]
  5. C. Khoo, T. H. Liu, P. Y. Yan, “Nonlocal radial dependence of laser-induced molecular reorientation in a nematic liquid crystal: theory and experiment,” J. Opt. Soc. Am. B 4, 115–120 (1987).
    [CrossRef]
  6. S. D. Durbin, S. M. Arakelian, Y. R. Shen, “Laser-induced diffraction rings from a nematic-liquid-crystal film,” Opt. Lett. 6, 411–413 (1981).
    [PubMed]
  7. C. Khoo, J. Y. Hou, T. H. Liu, P. Y. Yan, R. R. Michael, G. M. Finn, “Transverse self-phase modulation and bistability in the transmission of a laser beam through a nonlinear thin film,” J. Opt. Soc. Am. B 4, 886–891 (1987).
    [CrossRef]
  8. W. K. Bajdecki, L. Calero, R. Meucci, “Nonlinear infrared optical measurements of elastic constants in a nematic liquid crystal,” Opt. Commun. 176, 473–477 (2000).
    [CrossRef]
  9. L. Calero, W. K. Bajdecki, R. Meucci, “Reorientation effect induced by a CW CO2 laser in nematic liquid crystals,” Opt. Commun. 168, 201–206 (1999).
    [CrossRef]
  10. S. Brugioni, R. Meucci, “Self-phase modulation in a nematic liquid crystal film induced by a low-power CO2 laser,” Opt. Commun. 206, 445–451 (2002).
    [CrossRef]
  11. D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
    [CrossRef]
  12. E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring the inclination angle of parallel fringe patterns,” Opt. Laser Technol. 30, 167–173 (1998).
    [CrossRef]
  15. I. H. Takeda, S. Kobayashy, “Fourier-transform method of fringe-pattern analysis for computed-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  16. T. Kreiss, Computer-Aided Evaluation of Holographic Interferograms Holographic Interferometry, P. K. Rastogy, eds. (Springer, Berlin, 1994).

2003

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

2002

S. Brugioni, R. Meucci, “Self-phase modulation in a nematic liquid crystal film induced by a low-power CO2 laser,” Opt. Commun. 206, 445–451 (2002).
[CrossRef]

2000

W. K. Bajdecki, L. Calero, R. Meucci, “Nonlinear infrared optical measurements of elastic constants in a nematic liquid crystal,” Opt. Commun. 176, 473–477 (2000).
[CrossRef]

1999

L. Calero, W. K. Bajdecki, R. Meucci, “Reorientation effect induced by a CW CO2 laser in nematic liquid crystals,” Opt. Commun. 168, 201–206 (1999).
[CrossRef]

1998

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring the inclination angle of parallel fringe patterns,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
[CrossRef]

1987

1982

1981

S. D. Durbin, S. M. Arakelian, Y. R. Shen, “Laser-induced diffraction rings from a nematic-liquid-crystal film,” Opt. Lett. 6, 411–413 (1981).
[PubMed]

A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).

J. C. Khoo, S. L. Zhuang, S. Shepard, “Self-focusing of a low power cw laser beam via optically induced birefringence in a nematic liquid-crystal film,” Appl. Phys. Lett. 39, 937–940 (1981).
[CrossRef]

Allaria, E.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

Arakelian, S. M.

Bajdecki, W. K.

W. K. Bajdecki, L. Calero, R. Meucci, “Nonlinear infrared optical measurements of elastic constants in a nematic liquid crystal,” Opt. Commun. 176, 473–477 (2000).
[CrossRef]

L. Calero, W. K. Bajdecki, R. Meucci, “Reorientation effect induced by a CW CO2 laser in nematic liquid crystals,” Opt. Commun. 168, 201–206 (1999).
[CrossRef]

Brugioni, S.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

S. Brugioni, R. Meucci, “Self-phase modulation in a nematic liquid crystal film induced by a low-power CO2 laser,” Opt. Commun. 206, 445–451 (2002).
[CrossRef]

Calero, L.

W. K. Bajdecki, L. Calero, R. Meucci, “Nonlinear infrared optical measurements of elastic constants in a nematic liquid crystal,” Opt. Commun. 176, 473–477 (2000).
[CrossRef]

L. Calero, W. K. Bajdecki, R. Meucci, “Reorientation effect induced by a CW CO2 laser in nematic liquid crystals,” Opt. Commun. 168, 201–206 (1999).
[CrossRef]

Csillag, L.

A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).

De Feo, D.

D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
[CrossRef]

De Nicola, S.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
[CrossRef]

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring the inclination angle of parallel fringe patterns,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

Durbin, S. D.

Ferraro, P.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring the inclination angle of parallel fringe patterns,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
[CrossRef]

Finn, G. M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grilli, S.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

Hou, J. Y.

Khoo, C.

Khoo, J. C.

J. C. Khoo, S. L. Zhuang, S. Shepard, “Self-focusing of a low power cw laser beam via optically induced birefringence in a nematic liquid-crystal film,” Appl. Phys. Lett. 39, 937–940 (1981).
[CrossRef]

Kitaeva, V. F.

A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).

Kobayashy, S.

Kreiss, T.

T. Kreiss, Computer-Aided Evaluation of Holographic Interferograms Holographic Interferometry, P. K. Rastogy, eds. (Springer, Berlin, 1994).

Kroo, N.

A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).

Liu, T. H.

Maddalena, P.

D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
[CrossRef]

Meucci, R.

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

S. Brugioni, R. Meucci, “Self-phase modulation in a nematic liquid crystal film induced by a low-power CO2 laser,” Opt. Commun. 206, 445–451 (2002).
[CrossRef]

W. K. Bajdecki, L. Calero, R. Meucci, “Nonlinear infrared optical measurements of elastic constants in a nematic liquid crystal,” Opt. Commun. 176, 473–477 (2000).
[CrossRef]

L. Calero, W. K. Bajdecki, R. Meucci, “Reorientation effect induced by a CW CO2 laser in nematic liquid crystals,” Opt. Commun. 168, 201–206 (1999).
[CrossRef]

Michael, R. R.

Pierattini, G.

D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
[CrossRef]

Shen, Y. R.

Shepard, S.

J. C. Khoo, S. L. Zhuang, S. Shepard, “Self-focusing of a low power cw laser beam via optically induced birefringence in a nematic liquid-crystal film,” Appl. Phys. Lett. 39, 937–940 (1981).
[CrossRef]

Simoni, F.

F. Simoni, Nonlinear Optical Properties of Liquid Crystals and Polymer Dispersed Liquid Crystals (World Scientific, Singapore, 1997).
[CrossRef]

Sobolev, N. N.

A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).

Takeda, I. H.

Wu, S. T.

C. Khoo, S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).
[CrossRef]

Yan, P. Y.

Zhuang, S. L.

J. C. Khoo, S. L. Zhuang, S. Shepard, “Self-focusing of a low power cw laser beam via optically induced birefringence in a nematic liquid-crystal film,” Appl. Phys. Lett. 39, 937–940 (1981).
[CrossRef]

Zolot’ko, A. S.

A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).

Appl. Phys. Lett.

J. C. Khoo, S. L. Zhuang, S. Shepard, “Self-focusing of a low power cw laser beam via optically induced birefringence in a nematic liquid-crystal film,” Appl. Phys. Lett. 39, 937–940 (1981).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

JETP Lett.

A. S. Zolot’ko, V. F. Kitaeva, N. Kroo, N. N. Sobolev, L. Csillag, “Freedericsz transition in MBBA crystal due to a light wave field,” JETP Lett. 34, 250–254 (1981).

Opt. Commun.

W. K. Bajdecki, L. Calero, R. Meucci, “Nonlinear infrared optical measurements of elastic constants in a nematic liquid crystal,” Opt. Commun. 176, 473–477 (2000).
[CrossRef]

L. Calero, W. K. Bajdecki, R. Meucci, “Reorientation effect induced by a CW CO2 laser in nematic liquid crystals,” Opt. Commun. 168, 201–206 (1999).
[CrossRef]

S. Brugioni, R. Meucci, “Self-phase modulation in a nematic liquid crystal film induced by a low-power CO2 laser,” Opt. Commun. 206, 445–451 (2002).
[CrossRef]

E. Allaria, S. Brugioni, S. De Nicola, P. Ferraro, S. Grilli, R. Meucci, “Digital holography at 10.6 µm,” Opt. Commun. 215, 257–262 (2003).
[CrossRef]

Opt. Laser Technol.

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring the inclination angle of parallel fringe patterns,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

D. De Feo, S. De Nicola, P. Ferraro, P. Maddalena, G. Pierattini, “A Fourier-transform-based interferometric technique for measuring the elastic anisotropy of a nematic liquid crystal,” Pure Appl. Opt. 7, 1301–1308 (1998).
[CrossRef]

Other

C. Khoo, S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).
[CrossRef]

F. Simoni, Nonlinear Optical Properties of Liquid Crystals and Polymer Dispersed Liquid Crystals (World Scientific, Singapore, 1997).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

T. Kreiss, Computer-Aided Evaluation of Holographic Interferograms Holographic Interferometry, P. K. Rastogy, eds. (Springer, Berlin, 1994).

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Figures (7)

Fig. 1
Fig. 1

Mach-Zehnder interferometric setup employed to record the self-phase-modulated interference pattern due to the NLC.

Fig. 2
Fig. 2

Schematic geometry of the incident Gaussian beam and the recording apparatus. A sketch of the liquid crystal cell is also shown. LC, liquid crystal.

Fig. 3
Fig. 3

(a) Interference pattern recorded without the NLC sample. (b) Recorded ring pattern at incident beam power P = 80 mW. (c) Interference pattern with the sample inserted into the object arm of the interferometer.

Fig. 4
Fig. 4

Numerical simulations of the corresponding patterns shown in Fig. 3 according to the Fresnel diffraction model of the self-phase-modulated transmitted beam with nonlinear coefficient n2 = 0.82 × 10-4 mm2/mW.

Fig. 5
Fig. 5

Recorded interference patterns at increasing incident beam power: (a) P = 45 mW, (b) P = 55 mW, and (c) P = 75 mW. The recording distance is z = 535 mm.

Fig. 6
Fig. 6

(a) Unwrapped three-dimensional map of the phase change Δψz(x, y) corresponding to an incident beam power P = 80 mW. (b) The corresponding theoretically computed phase change.

Fig. 7
Fig. 7

Cross-section profile along the horizontal axis of the theoretical (solid curve) and experimental (dashed curve) phase-change distributions.

Equations (27)

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ψiξ, η=I0exp-ξ2+η2ω02+i k21R-1f×ξ2+η2,
ψfu, v=I0iλf-+-+ ψiξ, ηexpi k2fu-ξ2+v-η2dξdη.
ψfu, v=I0,fiexpikfexpi k21f-1Rfu2+v22+i tan-1zRR-u2+v22ωf2.
Rf=f2R1+RzR2,
ωf=2fkω01+zRR21/2,
Ifu, v=|ψfu, v|2=I0,f exp-2u2+v2ωf2.
τNLu, v=expiΦNLu, v.
hx, y; z=expikziλzexpik2zx2+y2,
ψzx, y=ψfu, vτNLx, yhx, y; f×expikx2+y22fhx, y; z.
ψzx, y=expikz+fiλf-+-+ ψfu, v×expiΦNLu, v+i k2zf2-1fu2+v2-i kfxu+yvdudv.
ΦNLu, v=kdn2Ifu, v,
τNLu, v=n=0ikdn2Ifnn!exp-2nu2+v2ωf2.
ψzx, y=I0,fiλfexpik2f+z+i arctanzRR-iπ/2n=0ikdn2Ifnn!-+-+×expi k2lu2+v2-u2+v2ωf,n2-i kfxu+yvdudv,
ωf,n2=ωf21+2n
1l=2f-zf2-1Rf.
ψzx, y=n=0 ψznx, y,
ψznx, y=2Pπ1+2nωz,n21/2 expik2f+z×ikdn2I0,fnn!expi arctanzf,nl-r2ωz,n2+i kr2Rz,n
Rz,n=-zRlzf,01+2n2+zf,0/l21+zf,0/l2,
ωz,n2=1+2nω021+zf,n/l21+zR/R2.
ψz0x, y=2Pπωz,021/2 expik2f+z×expi arctanzf,0l-r2ωz,02+i kr2Rz,0,
zf,n=zf,01+2n.
zf,0l=-z-2fzR1+zRR2-zRR,
Iobj,zx, y=|ψzx, y|2=n=0 ψznx, y2.
Δψz=argψz-argψz0,
Iz0x, y=az0x, y+bz0x, ycosargψz0-kxx+kyy,
ϕrx, y=argψz0-kxx+kyy,
Izx, y=azx, y+bzx, ycosΔψz+ϕrx, y,

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