Abstract

In this paper the response purely refractive of a thin nonlinear material, in the z-scan technique experiment, is modeled as a lens with a focal length that is a function of some integer power of the incident beam radius. We demonstrate that different functional dependences of the photoinduced lens of a thin nonlinear material give typical z-scan curves with special features. The analysis is based on the propagation of Gaussian beams in the approximation of thin lens and small distortion for the nonlinear sample. We obtain that the position of the peak and valley, the transmittance near the focus and the transmittance far from the Rayleigh range depend on the functional dependence of the focal length. Special values of the power reproduce the results obtained for some materials under cw excitation.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Sheik Bahae, A. A. Said, and E. W. Van Stryland, "High sensitivity single beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
    [CrossRef]
  2. M. Sheik-Bahae, A. A: Said, T. Wei, D. Hagan, E. W. Van Stryland, and E. W. Van Stryland, "Sensitive measurement of Optical Nonlinearities using a single Beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  3. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of λ/10 wave-front distortion," Opt. Lett. 19, 317-319 (1994).
    [CrossRef] [PubMed]
  4. W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
    [CrossRef]
  5. H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurement of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666 (1991).
    [CrossRef]
  6. D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, "Reflection Z-scan technique for measurements of optical properties surfaces," Appl. Phys. Lett. 65, 1067 (1994).
    [CrossRef]
  7. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
    [CrossRef]
  8. L. C. Oliveira and S. C. Zilio, "Single beam time-resolved Z-scan measurements of slow absorbers," Appl. Phys. Lett. 65, 2121-2123 (1994).
    [CrossRef]
  9. S. J. Sheldon, L. V Knight, and J. M. Thorne, "Laser-induced thermal lens effect: a new theoretical model," Appl. Opt. 21, 1663-1669 (1982).
    [CrossRef] [PubMed]
  10. L. Pálfalvi and J. Hebling "Z-scan study of the thermo-optical effect," Appl. Phys. B 78, 775-780 (2004)
    [CrossRef]
  11. B. Gu, X. C. Peng, T. Jia,J. P. Ding, J. L. He, and H. T. Wang, "Determinations of third- and fifth-order nonlinearities by the use of the top-hat-beam Z scan: theory and experiment," J. Opt. Soc. Am. B 22,446-452 (2005).
    [CrossRef]
  12. M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, "Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime)," Optik 100, 49-56 (1995).
  13. H. Kogelnik and T. Li, "Laser beams and Resonators," Appl. Opt. 5,1550-1567 (1966).
    [CrossRef] [PubMed]
  14. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
    [CrossRef]
  15. P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995).
    [CrossRef]
  16. M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991).
    [CrossRef]
  17. R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis,  46, 586-592 (2000).
  18. C. Hu and J. R. Whinnery, "New thermo optical measurement method and comparison with other methods," Appl. Opt. 12, 72-79 (1973).
    [CrossRef] [PubMed]
  19. F. L. S. Cuppo, A. M. F. Neto, S. L. Gómez and P. Palffy-Muhoray, "Termal-lens model compared with the Sheik-Bahae formalism in interpreting Z-scan experiments on lyotropic liquid crystals," J. Opt. Soc. Am. B 19,1342-1348 (2002).
    [CrossRef]
  20. C. H. Kwak, Y. L. Lee and S. G. Kim, "Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As2S3 thin film," J. Opt. Soc. Am. B 16,600-604 (1999).
    [CrossRef]

2005

2004

L. Pálfalvi and J. Hebling "Z-scan study of the thermo-optical effect," Appl. Phys. B 78, 775-780 (2004)
[CrossRef]

2002

2000

R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis,  46, 586-592 (2000).

1999

1997

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

1995

P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995).
[CrossRef]

M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, "Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime)," Optik 100, 49-56 (1995).

1994

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of λ/10 wave-front distortion," Opt. Lett. 19, 317-319 (1994).
[CrossRef] [PubMed]

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, "Reflection Z-scan technique for measurements of optical properties surfaces," Appl. Phys. Lett. 65, 1067 (1994).
[CrossRef]

L. C. Oliveira and S. C. Zilio, "Single beam time-resolved Z-scan measurements of slow absorbers," Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

1993

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

1991

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurement of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991).
[CrossRef]

1990

M. Sheik-Bahae, A. A: Said, T. Wei, D. Hagan, E. W. Van Stryland, and E. W. Van Stryland, "Sensitive measurement of Optical Nonlinearities using a single Beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1989

1982

1973

1966

1965

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

Bermúdez-Cruz, R. S.

R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis,  46, 586-592 (2000).

Bloch, G.

P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995).
[CrossRef]

Chapple, P. B.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Cuppo, F. L. S.

de Araujo, C. B.

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, "Reflection Z-scan technique for measurements of optical properties surfaces," Appl. Phys. Lett. 65, 1067 (1994).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurement of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

Ding, J. P.

Gomes, A. S. L.

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, "Reflection Z-scan technique for measurements of optical properties surfaces," Appl. Phys. Lett. 65, 1067 (1994).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurement of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

Gómez, S. L.

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

Gu, B.

Hagan, D.

M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991).
[CrossRef]

Hagan, D. J.

He, J. L.

Hebling, J.

L. Pálfalvi and J. Hebling "Z-scan study of the thermo-optical effect," Appl. Phys. B 78, 775-780 (2004)
[CrossRef]

Hermann, J. A.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Hu, C.

Iturbe Castillo, M. D.

M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, "Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime)," Optik 100, 49-56 (1995).

Jia, T.

Kim, S. G.

Knight, L. V

Kogelnik, H.

Kwak, C. H.

Lee, Y. L.

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

Li, T.

Ma, H.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurement of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

Márquez Aguilar, P. A.

P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995).
[CrossRef]

McDuff, R. G.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Mckay, T. J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

Neto, A. M. F.

Oliveira, L. C.

L. C. Oliveira and S. C. Zilio, "Single beam time-resolved Z-scan measurements of slow absorbers," Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

Pálfalvi, L.

L. Pálfalvi and J. Hebling "Z-scan study of the thermo-optical effect," Appl. Phys. B 78, 775-780 (2004)
[CrossRef]

Palffy-Muhoray, P.

Peng, X. C.

Petrov, D. V.

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, "Reflection Z-scan technique for measurements of optical properties surfaces," Appl. Phys. Lett. 65, 1067 (1994).
[CrossRef]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991).
[CrossRef]

M. Sheik Bahae, A. A. Said, and E. W. Van Stryland, "High sensitivity single beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
[CrossRef]

Sánchez Mondragón, J. J.

P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995).
[CrossRef]

Sánchez-Mondragón, J. J.

M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, "Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime)," Optik 100, 49-56 (1995).

Sheik Bahae, M.

Sheik-Bahae, M.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, "Eclipsing Z-scan measurement of λ/10 wave-front distortion," Opt. Lett. 19, 317-319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A: Said, T. Wei, D. Hagan, E. W. Van Stryland, and E. W. Van Stryland, "Sensitive measurement of Optical Nonlinearities using a single Beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Sheldon, S. J.

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991).
[CrossRef]

Staromlynska, J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Stepanov, S.

P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995).
[CrossRef]

Stepanov, S. I.

M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, "Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime)," Optik 100, 49-56 (1995).

Takur, M.

R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis,  46, 586-592 (2000).

Thorne, J. M.

Torres Quintero, R.

R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis,  46, 586-592 (2000).

Van Stryland, E. W.

Wang, H. T.

Whinnery, J. R.

C. Hu and J. R. Whinnery, "New thermo optical measurement method and comparison with other methods," Appl. Opt. 12, 72-79 (1973).
[CrossRef] [PubMed]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

Xia, T.

Zambrano-Valencia, L.

R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis,  46, 586-592 (2000).

Zhao, W.

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

Zilio, S. C.

L. C. Oliveira and S. C. Zilio, "Single beam time-resolved Z-scan measurements of slow absorbers," Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

Appl. Opt.

Appl. Phys. B

L. Pálfalvi and J. Hebling "Z-scan study of the thermo-optical effect," Appl. Phys. B 78, 775-780 (2004)
[CrossRef]

Appl. Phys. Lett.

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurement of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

D. V. Petrov, A. S. L. Gomes, and C. B. de Araujo, "Reflection Z-scan technique for measurements of optical properties surfaces," Appl. Phys. Lett. 65, 1067 (1994).
[CrossRef]

L. C. Oliveira and S. C. Zilio, "Single beam time-resolved Z-scan measurements of slow absorbers," Appl. Phys. Lett. 65, 2121-2123 (1994).
[CrossRef]

IEEE J. Quantum Electron.

M. Sheik-Bahae, A. A: Said, T. Wei, D. Hagan, E. W. Van Stryland, and E. W. Van Stryland, "Sensitive measurement of Optical Nonlinearities using a single Beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

J. Appl. Phys.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-transient effects in lasers with inserted liquid samples" J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

J. Nonlinear Opt. Phys. Mater.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, "Single-beam z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and G. Bloch, "Z-scan experiments with cubic photorefractive crystal Bi12Ti20," Opt. Commun. 118,165-174 (1995).
[CrossRef]

Opt. Eng.

M. Sheik-Bahae, A. A. Said, D. Hagan, M. J. Soileau, E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. 38, 1228-1235 (1991).
[CrossRef]

Opt. Lett.

Optik

M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. I. Stepanov, "Peculiarities of Z-scan technique in liquids with nonlinearity (steady regime)," Optik 100, 49-56 (1995).

Rev. Mex. Fis

R. Torres Quintero, L. Zambrano-Valencia, R. S. Bermúdez-Cruz, and M. Takur, "Z-scan like results produced by linear optical approximation of a nonlinear material," Rev. Mex. Fis,  46, 586-592 (2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1.
Fig. 1.

Z-scan technique scheme.

Fig. 2.
Fig. 2.

Z-scan curves for different values of m: 4(solid); 3 (dot); 2 (dashdot) and 1 (dashed). With the following values of the constant: a4= 1.6×1012; a3=3.8×109; a2=1×107; a1=3.8×104. w0 = 20μm and λ=457 nm.

Fig. 3.
Fig. 3.

(a). Z-scan curves for m=4 and different beam waists: w0 (solid); 1.5w0 (dashdot) and 2w0 (dot) a4=3.2×1011; w0 = 20 μm and λ=457 nm. b) ΔT p-ν as function of the beam waist.

Fig. 4.
Fig. 4.

Z-scan curves for m=2 and different beam waists: w0 (solid); 1.5w0 (dashdot); 2w0 (dot). a2 = 2×106; w0 = 20μm. ΔT p-ν is independent of the beam waist. b) ΔT p-ν as function of the beam waist.

Fig. 5.
Fig. 5.

Z-scan curves for m=3 and different beam waists: w0 (solid); 1.5w0 (dashdot); 2w0 (dot). a3 = 7.4×108; w0 = 20μm. b) ΔT p-ν as function of the beam waist.

Fig. 6.
Fig. 6.

Z-scan curves for m=1 and different beam waist: w0 (solid); 1.5w0 (dashdot) and 2w0 (dash). b) ΔT p-ν as function of the beam waist.

Fig. 7.
Fig. 7.

Z-scan curves for m=4 and different values of F0/z0: 5 (point), 1 (dash), 0.5 (dashdot), 0.16 (minus sign) and 0.11 (solid).

Fig. 8.
Fig. 8.

Z-scan curves for m=4, a4= 1.8×109, and different beam waist: w0 (solid);1.5w0 (dashdot) and 2w0 (dash).

Fig. 9.
Fig. 9.

Z-scan curves for m=2, and different values of F0/z0: 2.5 (dot), 1.6 (dash), 1 (dashdot), 0.72 (minus-sign), 0.56 (solid).

Fig. 10.
Fig. 10.

Z-scan curves for m=2, a2 = 4.9×104 and different beam waists: w0 (solid); 1.5w0 (dashdot) and 2w0 (dash).

Fig. 11.
Fig. 11.

Z-scan curves for m=1 and different values of F01 /z0 : 2.6 (solid), 3.1 (minus sign), 3.7 (dash-dot), 4.6 (dash) and 6.2 (point).

Fig. 12.
Fig. 12.

ΔT p-ν as function of F0 /z0 and different values of m: 2(solid), 3(dashdot) and 4(dot).

Fig. 13.
Fig. 13.

ΔZ p-ν as function of F0 /z0 for different values of m: 2 (solid), 3 (dashdot) and 4 (dot).

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

T = F 2 z 0 2 + ( F z ) 2 .
n = n 0 + n 2 I ,
I ( r , z ) = 2 P π w 2 ( z ) exp [ 2 r 2 w 2 ( z ) ] ,
w ( z ) = w 0 ( 1 + ( z z 0 ) 2 ) 1 2 ,
I ( r , z ) 2 P π w 2 ( z ) [ 1 2 r 2 w 2 ( z ) ] ,
n n 0 + 2 n 2 P π w 2 4 n 2 P π w 4 r 2 ,
[ A B C D ] = [ cos   γd sin γd n 0 γ n 0 γ   sin   γd cos    γd ]
γ 2 = n 2 n 0 , n 0 = n 0 + 2 n 2 P π w 2 and n 2 = 8 n 2 P π w 4 ,
F kerr = 1 8 n 2 dP w 4 ,
F ther = πκ P abs ( n T T ) w 2 ,
F = a m w m ( z ) ,
T = 1 + 2 z F .
T = 1 + 2 z F 0 m ( 1 + ( z z 0 ) 2 ) m 2 ,
Δ z p ν = z peak z valley = 2 m 1 z 0 .
Δ T p ν = T peak T valley = 2 Δ z p ν F 0 m [ ( m 1 m ) m ] 1 2 = 2 k a m w 0 m 2 [ ( m 1 ) m 1 m m ] 1 2 ,
T = 1 + 2 z F 0 m ,
T = 1 + 2 z 0 m F 0 m z m 1 ,
Δz p ν = 2 3 z 0 , Δ T p ν = 3 3 8 k a 4 w 0 2 ,
T = 1 + 2 z F 04 for z z 0 , and     T = 1 + 2 z 0 4 F 04 z 3 for z z 0 ,
Δz p ν = 2 z 0 , Δ T p ν = k a 2 ,
T = 1 + 2 z F 02 for z z 0 , and     T = 1 + 2 z 0 2 F 02 z for z z 0 ,
Δz p ν = 2 z 0 , Δ T p ν = 4 3 3 k a 3 w 0
T = 1 + 2 z F 03 for z z 0 , and     T = 1 + 2 z 0 3 F 03 z 2 , for z z 0 ,
Δ T = 2 w 0 k a 1 ,
T = 1 1 4 x ( 1 + x 2 ) 2 ( z 0 2 F 0 m ) + 4 ( 1 + x 2 ) 3 ( z 0 2 F 0 m ) 2 ,
ΔΦ 0 = z 0 2 F 04 .
ΔΦ 0 m = z 0 2 F 0 m .

Metrics