Abstract

We study the lateral Goos–Hanchen and the transverse Imbert–Fedorov shift produced during the reflection of Hermite–Gauss beams Hm0 or H0m at a plane interface. The vector angular spectrum method for a light beam in terms of a two-form angular spectrum consisting of the two orthogonal polarized components was used. We have carried out a detailed numerical calculation of these shifts at different angles of incidence, over the whole range of incidence without making the usual approximations. The shift variation as a function of refractive index and order of the Hermite–Gauss beam is studied. We also compare the shift variations with the orientation of the lobes of the Hermite–Gauss beam. We observed that the shifts are nearly equal for the two cases Hm0 (lobe oriented in the plane of incidence) and H0m (lobe oriented perpendicular to plane of incidence). These are the first quantitative estimates of the shifts for Hermite–Gauss beams as per our knowledge and are relevant for all cases of slab geometry.

© 2012 Optical Society of America

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  1. F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 1, 333–346 (1947).
  2. F. I. Fedorov, “K teorii polnogo otrazheniya,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).
  3. C. Imbert, “Calculation and experimental proof of the transverse shift induced by TIR of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
    [CrossRef]
  4. K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 2, 87–102 (1948).
  5. R. H. Renard, “Total reflection: a new evaluation of the Goos-Hanchen shift,” J. Opt. Soc. Am. 54, 1190–1197 (1964).
    [CrossRef]
  6. C. A. Risset and J. M. Vigoureux, “An elementary presentation of the Goos-Hanchen shift,” Opt. Commun. 91, 155–157 (1992).
    [CrossRef]
  7. M. McGuirk and C. K. Carniglia, “An angular spectrum representation approach to the Goos-Hanchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  8. B. R. Horowitz, and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  9. H. K. V. Lotsch, “Reflection and refraction of a beam of light at a plane interface,” J. Opt. Soc. Am. 58, 551–561 (1968).
    [CrossRef]
  10. C. C. Chan and T. Tamir, “Beam phenomena at and near critical incidence upon a dielectric interface,” J. Opt. Soc. Am. A 4, 655–663 (1987).
    [CrossRef]
  11. T. Tamir and H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]
  12. J. J. Cowan and B. Anicin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. 67, 1307–1314 (1977).
    [CrossRef]
  13. F. Pillon, H. Gilles, and S. Girard, “Experimental observation of the Imbert-Fedorov transverse displacement after a single total reflection,” Appl. Opt. 43, 1863–1869 (2004).
    [CrossRef]
  14. R. Briers, O. Leroy, and G. Shkerdin, “Bounded beam interaction with thin inclusions. characterization by phase differences at Rayleigh angle incidence,” J. Acoust. Soc. Am. 108, 1622–1630 (2000).
    [CrossRef]
  15. B. M. Jost, Abdul-Azeez R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).
    [CrossRef]
  16. V. K. Ignatovich, “Neueron reflection from condensed matter, the Goos-Hanchen effect and coherence,” Phys. Lett. A 322, 36–46 (2004).
    [CrossRef]
  17. J. Jose, F. B. Segerink, J. P. Korterik, and H. L. Offerhaus, “Near-field observation of spatial phase shifts associated with Goos-Hanchen and surface plasmon resonance effects,” Opt. Express 16, 1958–1964 (2008).
    [CrossRef]
  18. X. Yin, and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
    [CrossRef]
  19. H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96, 243903 (2006).
    [CrossRef]
  20. C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos-Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
    [CrossRef]
  21. M. Onoda, S. Murakami, and N. Nagosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
    [CrossRef]
  22. K. Yu Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
    [CrossRef]
  23. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
    [CrossRef]
  24. C.-F. Li, “Unified theory for Goos-Hanchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
    [CrossRef]
  25. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hanchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
    [CrossRef]
  26. A. Aiello and J. P. Woerdman, “The reflection of a Maxwell-Gaussian beam by planar interface,” arXiv:0710.1643v2 (2007).
  27. D. Golla and S. D. Gupta, “Goos-Hanchen shift for higher order Hermite-Gauss beams,” Pramana J. Phys. 76, 603–612 (2011).
    [CrossRef]
  28. H. M. Lai, F. C. Cheng, and W. K. Tang, “Goos-Hanchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3, 550–557 (1986).
    [CrossRef]
  29. O. Costa de Beauregard, “Translational inertial spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
    [CrossRef]
  30. M. Born and E. Wolf, Principles of Optics (Pergamon, 1987), p. 47.

2011 (1)

D. Golla and S. D. Gupta, “Goos-Hanchen shift for higher order Hermite-Gauss beams,” Pramana J. Phys. 76, 603–612 (2011).
[CrossRef]

2009 (1)

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos-Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

2008 (3)

2007 (1)

C.-F. Li, “Unified theory for Goos-Hanchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[CrossRef]

2006 (3)

K. Yu Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

X. Yin, and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef]

2004 (3)

F. Pillon, H. Gilles, and S. Girard, “Experimental observation of the Imbert-Fedorov transverse displacement after a single total reflection,” Appl. Opt. 43, 1863–1869 (2004).
[CrossRef]

V. K. Ignatovich, “Neueron reflection from condensed matter, the Goos-Hanchen effect and coherence,” Phys. Lett. A 322, 36–46 (2004).
[CrossRef]

M. Onoda, S. Murakami, and N. Nagosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef]

2000 (1)

R. Briers, O. Leroy, and G. Shkerdin, “Bounded beam interaction with thin inclusions. characterization by phase differences at Rayleigh angle incidence,” J. Acoust. Soc. Am. 108, 1622–1630 (2000).
[CrossRef]

1998 (1)

B. M. Jost, Abdul-Azeez R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).
[CrossRef]

1992 (1)

C. A. Risset and J. M. Vigoureux, “An elementary presentation of the Goos-Hanchen shift,” Opt. Commun. 91, 155–157 (1992).
[CrossRef]

1987 (1)

1986 (1)

1977 (2)

1972 (1)

C. Imbert, “Calculation and experimental proof of the transverse shift induced by TIR of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

1971 (2)

1968 (1)

1965 (1)

O. Costa de Beauregard, “Translational inertial spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
[CrossRef]

1964 (1)

1955 (1)

F. I. Fedorov, “K teorii polnogo otrazheniya,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).

1948 (1)

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 2, 87–102 (1948).

1947 (1)

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 1, 333–346 (1947).

Aiello, A.

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hanchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[CrossRef]

A. Aiello and J. P. Woerdman, “The reflection of a Maxwell-Gaussian beam by planar interface,” arXiv:0710.1643v2 (2007).

Akhmerov, A. R.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos-Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Al-Rashed, Abdul-Azeez R.

B. M. Jost, Abdul-Azeez R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).
[CrossRef]

Anicin, B.

Artmann, K.

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 2, 87–102 (1948).

Beenakker, C. W. J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos-Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Bertoni, H. L.

Bliokh, K. Yu

K. Yu Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

Bliokh, Y. P.

K. Yu Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987), p. 47.

Briers, R.

R. Briers, O. Leroy, and G. Shkerdin, “Bounded beam interaction with thin inclusions. characterization by phase differences at Rayleigh angle incidence,” J. Acoust. Soc. Am. 108, 1622–1630 (2000).
[CrossRef]

Carniglia, C. K.

Chan, C. C.

Cheng, F. C.

Costa de Beauregard, O.

O. Costa de Beauregard, “Translational inertial spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
[CrossRef]

Cowan, J. J.

Fedorov, F. I.

F. I. Fedorov, “K teorii polnogo otrazheniya,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).

Gilles, H.

Girard, S.

Golla, D.

D. Golla and S. D. Gupta, “Goos-Hanchen shift for higher order Hermite-Gauss beams,” Pramana J. Phys. 76, 603–612 (2011).
[CrossRef]

Goos, F.

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 1, 333–346 (1947).

Gupta, S. D.

D. Golla and S. D. Gupta, “Goos-Hanchen shift for higher order Hermite-Gauss beams,” Pramana J. Phys. 76, 603–612 (2011).
[CrossRef]

Hanchen, H.

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 1, 333–346 (1947).

Hentschel, M.

H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef]

Hesselink, L.

X. Yin, and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

Horowitz, B. R.

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Ignatovich, V. K.

V. K. Ignatovich, “Neueron reflection from condensed matter, the Goos-Hanchen effect and coherence,” Phys. Lett. A 322, 36–46 (2004).
[CrossRef]

Imbert, C.

C. Imbert, “Calculation and experimental proof of the transverse shift induced by TIR of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Jose, J.

Jost, B. M.

B. M. Jost, Abdul-Azeez R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).
[CrossRef]

Korterik, J. P.

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Lai, H. M.

Leroy, O.

R. Briers, O. Leroy, and G. Shkerdin, “Bounded beam interaction with thin inclusions. characterization by phase differences at Rayleigh angle incidence,” J. Acoust. Soc. Am. 108, 1622–1630 (2000).
[CrossRef]

Li, C.-F.

C.-F. Li, “Unified theory for Goos-Hanchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[CrossRef]

Lotsch, H. K. V.

McGuirk, M.

Murakami, S.

M. Onoda, S. Murakami, and N. Nagosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef]

Nagosa, N.

M. Onoda, S. Murakami, and N. Nagosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef]

Offerhaus, H. L.

Onoda, M.

M. Onoda, S. Murakami, and N. Nagosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef]

Pillon, F.

Renard, R. H.

Risset, C. A.

C. A. Risset and J. M. Vigoureux, “An elementary presentation of the Goos-Hanchen shift,” Opt. Commun. 91, 155–157 (1992).
[CrossRef]

Saleh, B. E. A.

B. M. Jost, Abdul-Azeez R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).
[CrossRef]

Schomerus, H.

H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef]

Segerink, F. B.

Sepkhanov, R. A.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos-Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Shkerdin, G.

R. Briers, O. Leroy, and G. Shkerdin, “Bounded beam interaction with thin inclusions. characterization by phase differences at Rayleigh angle incidence,” J. Acoust. Soc. Am. 108, 1622–1630 (2000).
[CrossRef]

Tamir, T.

Tang, W. K.

Tworzydlo, J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos-Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

Vigoureux, J. M.

C. A. Risset and J. M. Vigoureux, “An elementary presentation of the Goos-Hanchen shift,” Opt. Commun. 91, 155–157 (1992).
[CrossRef]

Woerdman, J. P.

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hanchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[CrossRef]

A. Aiello and J. P. Woerdman, “The reflection of a Maxwell-Gaussian beam by planar interface,” arXiv:0710.1643v2 (2007).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987), p. 47.

Yin, X.

X. Yin, and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

Ann. Phys. (2)

F. Goos and H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. 1, 333–346 (1947).

K. Artmann, “Calculation of the lateral shift of totally reflected beams,” Ann. Phys. 2, 87–102 (1948).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

X. Yin, and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[CrossRef]

Dokl. Akad. Nauk SSSR (1)

F. I. Fedorov, “K teorii polnogo otrazheniya,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).

J. Acoust. Soc. Am. (1)

R. Briers, O. Leroy, and G. Shkerdin, “Bounded beam interaction with thin inclusions. characterization by phase differences at Rayleigh angle incidence,” J. Acoust. Soc. Am. 108, 1622–1630 (2000).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. A. Risset and J. M. Vigoureux, “An elementary presentation of the Goos-Hanchen shift,” Opt. Commun. 91, 155–157 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

V. K. Ignatovich, “Neueron reflection from condensed matter, the Goos-Hanchen effect and coherence,” Phys. Lett. A 322, 36–46 (2004).
[CrossRef]

Phys. Rev. (1)

O. Costa de Beauregard, “Translational inertial spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
[CrossRef]

Phys. Rev. A (1)

C.-F. Li, “Unified theory for Goos-Hanchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[CrossRef]

Phys. Rev. D (1)

C. Imbert, “Calculation and experimental proof of the transverse shift induced by TIR of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Phys. Rev. Lett. (5)

B. M. Jost, Abdul-Azeez R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81, 2233–2235 (1998).
[CrossRef]

H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos-Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef]

M. Onoda, S. Murakami, and N. Nagosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef]

K. Yu Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

Pramana J. Phys. (1)

D. Golla and S. D. Gupta, “Goos-Hanchen shift for higher order Hermite-Gauss beams,” Pramana J. Phys. 76, 603–612 (2011).
[CrossRef]

Science (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987), p. 47.

A. Aiello and J. P. Woerdman, “The reflection of a Maxwell-Gaussian beam by planar interface,” arXiv:0710.1643v2 (2007).

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Figures (9)

Fig. 1.
Fig. 1.

Variation of the GH shift for s-polarized H10 (black curve) and H01 (red curve) mode Hermite–Gauss beams with θ at μ=1.5. The two curves coincide.

Fig. 2.
Fig. 2.

Variation of the GH shift for s-polarized H10 (black curve) and H01 (red curve) mode Hermite–Gauss beams with the refractive index μ at θ=42°. The two curves coincide.

Fig. 3.
Fig. 3.

Variation of the GH shift for s-polarized Hm0 (black circles) and H0n (red squares) mode Hermite–Gauss beams with the orders m and n, respectively, at μ=1.5 and θ=42°. The two plots coincide.

Fig. 4.
Fig. 4.

Variation of the GH shift for p-polarized H10 (black curve) and H01 (red curve) mode Hermite–Gauss beams with θ at μ=1.5. The two curves coincide.

Fig. 5.
Fig. 5.

Variation of the GH shift for p-polarized H10 (black curve) and H01 (red curve) mode Hermite–Gauss beams with refractive index μ at θ=42°. The two curves coincide.

Fig. 6.
Fig. 6.

Variation of the GH shift for p-polarized Hm0 (black circles) and H0m (red squares) mode Hermite–Gauss beams with the orders m and n, respectively, at μ=1.5 and θ=42°. The two plots coincide.

Fig. 7.
Fig. 7.

Variation of the IF shift for left circularly polarized H10 (black curve) and H01 (red curve) mode Hermite–Gauss beams with θ at μ=1.5. The two curves coincide.

Fig. 8.
Fig. 8.

Variation of the IF shift for left circularly polarized H10 (black curve) and H01 (red curve) mode Hermite–Gauss beams with refractive index μ at θ=42°. The two curves coincide.

Fig. 9.
Fig. 9.

Variation of the IF shift for left circularly polarized Hm0 (black circles) and H0n (red squars) mode Hermite–Gauss beams with the orders m and n, respectively, at μ=1.5 and θ=42°. The two plots coincide.

Equations (21)

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

E⃗(x,y,z)=12πA⃗(ky,kz)ei(kxx+kyy+kzzωt)dkydkz.
[yz]=1A⃗A⃗dkydkz[A⃗iA⃗kyA⃗iA⃗kz]dkydkz,
A⃗A⃗dkydkz=A˜A˜dkydkz=1.
A˜i=LiA=[li1li2]A,whereLi=UCiand|li1|2+|li2|2=1.
A⃗A⃗ky=A˜A˜kykxkyk(ky2+kz2)(As*ApAp*As),A⃗A⃗kz=A˜A˜kz+kxkyk(ky2+kz2)(As*ApAp*As).
U=12[1111]andLi=12[11].
As*ApAp*As=0,
y=1A˜A˜dkydkziA˜A˜kydkydkz.
A˜r=LrA=[lr1lr2]A,whereLr=UCr=URcCi=RlUCi=RlLi.
Rl=[Rs00Rp],withRs=|Rs|expiϕsandRp=|Rp|expiϕp
A˜r=[Rsli1Rpli2]A.
DGH=1|li1|2+|li2|2(|li1|2ϕsky+|li2|2ϕpky)A2dkydkz.
A(ky,kz)=ωyωz2m+nπm!n!Hm(ωy2(kyky0))Hn(ωzkz2)exp(ωy22(kyky0)2)exp(ωz2kz22).
Rs=μcosθ1cosθ2μcosθ1+cosθ2,Rp=cosθ1μcosθ2cosθ1+μcosθ2,
DGHs=12ωyωz2m+nπm!n!2μkxk2(μ21)(k2kx2)kx2kyexp(ωy2(kyky0)2)exp(ωz2kz2)[Hm(ωy2(kyky0))]2[Hn(ωzkz2)]2dkydkz,
DGHp=12ωyωz2m+nπm!n!2μkx((k2kx2)μ2kx2)(μ21)(k2kx2)kx2kyexp(ωy2(kyky0)2)exp(ωz2kz2)[Hm(ωy2(kyky0))]2[Hn(ωzkz2)]2dkydkz.
z=1AAdkydkzi(AAkz+kxkyk(ky2+kz2)(As*ApAp*As))dkydkz.
z=1AAdkydkzikxkyk(ky2+kz2)(As*ApAp*As)dkydkz.
U=12[11ii],andLi=12[1i];henceCi=[10].
Ar˜=RlUCiA=12[Rs(Ci1+Ci2)Rp(iCi1+iCi2)]A.
DIF=(kxkyk(k2kx2))ωyωz2m+nπm!n!|Rs||Rp||Rs|2+|Rs|2[Hm(ωy2(kyky0))]2[Hn(ωzkz2)]2cos(ϕpϕs)exp(ωy2(kyky0)2)exp(ωz2kz2)dkydkz.

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