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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1301–1307
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Reflection and transmission of twisted light at phase conjugating interfaces

Anita Thakur and Jamal Berakdar  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1301-1307 (2012)
http://dx.doi.org/10.1364/OE.20.001301


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Abstract

We study the transmission and the reflection of light beams carrying orbital angular momentum through a dielectric multilayer structure containing phase-conjugating interfaces. We show analytically and demonstrate numerically that the phase conjugation at the interfaces results in a characteristic angular and radial pattern of the reflected beam, a fact that can be exploited for the detection and the characterization of phase conjugation in composite optical materials.

© 2011 OSA

1. Introduction

Recently, the behavior of LG beams at dielectric interfaces has been the subject of several studies showing that the beam-interface interactions are dependent on the incident angles of the LG beams. In the case of normal incidence, the azimuthal index of the reflected or the transmitted LG beam is increased and decreased by the cross-polarization coupling in the beam component of the incident beam [11

11. W. Nasalki, Optical Beams at Dielectric Interfaces-Fundamentals (Institute of Fundamental Technological Research Polish Academy of Sciences, Warszawa, 2007).

13

13. W. Nasalski, “Polarization versus spatial characteristics of optical beams at a planar isotropic interface,” Phys. Rev.E 74, 056613 (2006). [CrossRef]

]. In this contribution, we have study the propagation of twisted light in a multi layer dielectric medium containing interfaces that act as phase-conjugating mirrors (PCM) [14

14. R. A. Fischered., Optical Phase Conjugation (Academic Press, Inc.,1983).

17

17. M. Woerdman, C. Alpmann, and C. Denz, “Self-pumped phase conjugation of light beams carrying orbital angular momentum,” Opt. Express 17, 22791–22799 (2009). [CrossRef]

]. The problem for a single PCM has been addressed previously [14

14. R. A. Fischered., Optical Phase Conjugation (Academic Press, Inc.,1983).

17

17. M. Woerdman, C. Alpmann, and C. Denz, “Self-pumped phase conjugation of light beams carrying orbital angular momentum,” Opt. Express 17, 22791–22799 (2009). [CrossRef]

]. Here it is shown that for a structure containing several PCMs interferences lead to a characteristic angular and radial pattern of the reflected beam. This pattern can in turn serve as an indicator for the phase conjugation of a composite optical materials.

2. Theoretical formulation

Fig. 1 Schematic representation of the propagation of LG beam in a multi layer dielectric structure. The interfaces with phase conjugation (pcm) are indicated.

The well known time reversal property of phase conjugating mirror plays a key role in determining the behavior of the scattered beams electric fields. As detailed in Refs. [16

16. A. Y. Okulov, “Angular momentum of photons and phase conjugation,” J. Phys. B: At. Mol. Opt. Phys. 41, 101001 (2008). [CrossRef]

17

17. M. Woerdman, C. Alpmann, and C. Denz, “Self-pumped phase conjugation of light beams carrying orbital angular momentum,” Opt. Express 17, 22791–22799 (2009). [CrossRef]

]) the orbital angular momentum changes sign upon a wavefront reversal at pcm, i.e. changes sign (this goes with excitations in the pcm material such that the total angular momentum balance is guaranteed [16

16. A. Y. Okulov, “Angular momentum of photons and phase conjugation,” J. Phys. B: At. Mol. Opt. Phys. 41, 101001 (2008). [CrossRef]

,17

17. M. Woerdman, C. Alpmann, and C. Denz, “Self-pumped phase conjugation of light beams carrying orbital angular momentum,” Opt. Express 17, 22791–22799 (2009). [CrossRef]

,30

30. V. G. Fedoseyev, “Transformation of the orbital angular momentum at the reflection and transmission of a light beam on a plane interface,” J. Phys A: Math. Theor. 41, 505202 (2008). [CrossRef]

,31

31. V. G. Fedoseyev, “Reflection of the light beam carrying orbital angular momentum from a lossy medium,” Phys. Lett. A 372, 2527–2533 (2008). [CrossRef]

].) This property has to be imposed as an additional requirement on the beam when traversing the structure. To keep the notation simple we can incorporate this condition on by the ansatz
E0i=E¯0ieiϕ=Cplw0(2rw0)exp(r2w02)Lp(2r2w02)ei(k0n0z+ϕ),(z0)
(2)
E0r=E¯0reiϕ=r0Cpw0(2rw0)exp(r2w02)Lp(2r2w02)ei(k0n0z+ϕ),(z0)
(3)
E1t=E¯1teiϕ=t1Cplw0(2rw0)exp(r2w02)Lp(2r2w02)ei(k0n1z+ϕ),(0zd1)
(4)
E1r=E¯1reiϕ=r1Cplw0(2rw0)exp(r2w02)Lp(2r2w02)ei(k0n1z+ϕ),(0zd1)
(5)
E2t=E¯2teiϕ=t2Cplw0(2rw0)exp(r2w02)Lp(2r2w02)ei[k0n2(zd1)+ϕ],(zd1).
(6)

Explicitly, the reflection and the transmission coefficients are
r0=(n0A+n1A+n0An1A+),
(11)
r1=1+r01e2iα1N,
(12)
t1=r1e2iα1N,
(13)
t2=n1n2[t1eiα1r1eiα1].
(14)
where
A+=1+e2iα1N,A=1e2iα1N,N=(n2+n1n2n1).
After substituting for the reflection and the transmission coefficients in the Eqs.(2)(6), we can obtain the electromagnetic fields that describe the propagation of the LG beam through the system depicted in Fig. 1.

3. Results

Fig. 2 For the structure depicted in Fig. 1 we show the calculated total radial (r) intensity (in CGS system) of the LG laser beam in the medium 0 for = 1, p = 0 (a), and for = 10, p = 2 (b). The material parameters and laser properties are chosen as: ϕ = 30°, n0 = 1(air), n1 = 1.77 (Al2O3), n2 = 1.457 (SiO2), d1 = 20 μm, w0=1 μm, λ = 632.9 nm.
Fig. 3 The same as in Fig. 2 for = 1, p = 0 but here we show the angular (ϕ) distribution of the LG beam intensity (in CGS system) for a different thickness d1 of the medium 1. The blue solid curve is for d1 = 11πλ/2 and the dashed curve is for d1 = 4πλ. The radial distance r is fixed to be w0/2.

4. Conclusion

We studied the propagation of a light beam carrying orbital angular momentum (OAM) in a dielectric multi layer structure with phase conjugating properties. Analytical expressions for the reflection and the transmission of the fields at individual layers are provided. We demonstrated that the scattering of OAM beams from phase conjugating refractive inhomogeneities leads to characteristic interferences that depend on the depth profile. This can be tested by, e.g. varying the light wave length.

Acknowledgments

We thanks G.F Quinteiro for discussions on twisted light and Andrey Moskalenko, Koray Köksal, and Yaroslav Pavlyukh for fruitful discussions and support. The work is supported by the International Max-Planck School for the Science and Technology of Nanostructures.

References and links

1.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol, 2003). [CrossRef]

2.

G. F. Quinteiro and J. Berakdar, “Electric currents induced by twisted light in Quantum Rings,” Opt. Express 17, 20465–20475 (2009). [CrossRef] [PubMed]

3.

G. F. Quinteiro and P. I. Tamborenea, “Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics,” Phys. Rev. B 82, 125207 (2010). [CrossRef]

4.

G. F. Quinteiro, P. I. Tamborenea, and J. Berakdar, “Orbital and spin dynamics of intraband electrons in quantum rings driven by twisted light,” Opt. Express 19, 26733–26741 (2011). [CrossRef]

5.

A. Thakur and J. Berakdar, “Self-focusing and defocusing of twisted light in non-linear media,” Opt. Express 18, 27691–27696 (2010). [CrossRef]

6.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

7.

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

8.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. 3, 305–310 (2007). [CrossRef]

9.

M. Padgett, J. Courtial, and L. Allen, “Lights orbital angular momentum,” Phys. Today 57, 35–40 (2004). [CrossRef]

10.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

11.

W. Nasalki, Optical Beams at Dielectric Interfaces-Fundamentals (Institute of Fundamental Technological Research Polish Academy of Sciences, Warszawa, 2007).

12.

W. Szabelak and W. Nasalski, “Transmission of Elegant Laguerre-Gaussian beams at a dielectric interface numerical simulations,” Bulletin of the Polish Academy of Sciences: Technical Sciences , 57, 181–188 (2009). [CrossRef]

13.

W. Nasalski, “Polarization versus spatial characteristics of optical beams at a planar isotropic interface,” Phys. Rev.E 74, 056613 (2006). [CrossRef]

14.

R. A. Fischered., Optical Phase Conjugation (Academic Press, Inc.,1983).

15.

B. Y. Zeldovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, Berlin, 1985).

16.

A. Y. Okulov, “Angular momentum of photons and phase conjugation,” J. Phys. B: At. Mol. Opt. Phys. 41, 101001 (2008). [CrossRef]

17.

M. Woerdman, C. Alpmann, and C. Denz, “Self-pumped phase conjugation of light beams carrying orbital angular momentum,” Opt. Express 17, 22791–22799 (2009). [CrossRef]

18.

R. K. Jain and R. C. Lind, “Degenerate four-wave mixing in semiconductor-doped glasses,” J. Opt. Soc. Am. 73(5), 647–653 (1983). [CrossRef]

19.

T. Geethakrishnan and P. K. Palanisamy, “Optical phase-conjugation in erioglaucine dye-doped thin films,” Pranama-J. Phys. 66, 473–478 (2008). [CrossRef]

20.

T. Bach, K. Nawata, M. Jazbinšek, T. Omatsu, and P. Günter, “Optical phase conjugation of picosecondpulses at 1.06 mm in Sn2P2S6: Te for wavefront correction in high-power Nd-doped amplifier systems,” Opt. Express 18, 87–95 (2010). [CrossRef] [PubMed]

21.

N. G. Basov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, and A.Y. Okulov, “Laser interferometer with wavefront-reversing mirrors,” Sov. Phys. JETP 52, 847–851 (1980).

22.

F. A. Starikov, Yu. V. Dolgopolov, A. V. Kopalkin, G. G. Kochemasov, S. M. Kulikov, and S. A. Sukharev, “About the correction of laser beams with phase front vortex,” J. Phys. IV France 133, 683–685 (2006). [CrossRef]

23.

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002). [CrossRef]

24.

P. V. Polyanskiǐ and K. V. Fel’de, “Static holographic phase conjugation of vortex beams,” Opt. Spectrosc. 98(6), 913–918 (2005). [CrossRef]

25.

I. G. Marienko, M. S. Soskin, and M. V. Vasnetsov, “Phase conjugation of wavefronts containing phase singularities,” in International Conference on Singular Optics, M. S. Soskin, ed., 3487, 39–41 (Proceedings of SPIE, 1998).

26.

L. C. Dávila Romero, D. L. Andrews, and M. Babikar, “A Quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B: Quantum Semiclass. Opt. 4, S66–S72 (2002). [CrossRef]

27.

A. E. Siegman, Lasers (University Science Books, 1986).

28.

J.D. Jackson, Classical Electrodynamics, 2nd ed. (New York: Wiley, 1962).

29.

E. M. Liftshitz, L. P. Pitaevskii, and V. B. Berestetskii, Quantum Electrodynamics, Landau and Liftshitz Course of Theoretical Physics 4 (Oxford: Butterworth-Heineman) Ch.I section 6,8.

30.

V. G. Fedoseyev, “Transformation of the orbital angular momentum at the reflection and transmission of a light beam on a plane interface,” J. Phys A: Math. Theor. 41, 505202 (2008). [CrossRef]

31.

V. G. Fedoseyev, “Reflection of the light beam carrying orbital angular momentum from a lossy medium,” Phys. Lett. A 372, 2527–2533 (2008). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.5040) Nonlinear optics : Phase conjugation

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 23, 2011
Revised Manuscript: December 15, 2011
Manuscript Accepted: December 19, 2011
Published: January 5, 2012

Citation
Anita Thakur and Jamal Berakdar, "Reflection and transmission of twisted light at phase conjugating interfaces," Opt. Express 20, 1301-1307 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1301


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References

  1. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol, 2003). [CrossRef]
  2. G. F. Quinteiro and J. Berakdar, “Electric currents induced by twisted light in Quantum Rings,” Opt. Express17, 20465–20475 (2009). [CrossRef] [PubMed]
  3. G. F. Quinteiro and P. I. Tamborenea, “Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics,” Phys. Rev. B82, 125207 (2010). [CrossRef]
  4. G. F. Quinteiro, P. I. Tamborenea, and J. Berakdar, “Orbital and spin dynamics of intraband electrons in quantum rings driven by twisted light,” Opt. Express19, 26733–26741 (2011). [CrossRef]
  5. A. Thakur and J. Berakdar, “Self-focusing and defocusing of twisted light in non-linear media,” Opt. Express18, 27691–27696 (2010). [CrossRef]
  6. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45, 8185–8189 (1992). [CrossRef] [PubMed]
  7. M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96, 123–132 (1993). [CrossRef]
  8. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys.3, 305–310 (2007). [CrossRef]
  9. M. Padgett, J. Courtial, and L. Allen, “Lights orbital angular momentum,” Phys. Today57, 35–40 (2004). [CrossRef]
  10. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt.39, 291–372 (1999). [CrossRef]
  11. W. Nasalki, Optical Beams at Dielectric Interfaces-Fundamentals (Institute of Fundamental Technological Research Polish Academy of Sciences, Warszawa, 2007).
  12. W. Szabelak and W. Nasalski, “Transmission of Elegant Laguerre-Gaussian beams at a dielectric interface numerical simulations,” Bulletin of the Polish Academy of Sciences: Technical Sciences, 57, 181–188 (2009). [CrossRef]
  13. W. Nasalski, “Polarization versus spatial characteristics of optical beams at a planar isotropic interface,” Phys. Rev.E74, 056613 (2006). [CrossRef]
  14. R. A. Fischered., Optical Phase Conjugation (Academic Press, Inc.,1983).
  15. B. Y. Zeldovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, Berlin, 1985).
  16. A. Y. Okulov, “Angular momentum of photons and phase conjugation,” J. Phys. B: At. Mol. Opt. Phys.41, 101001 (2008). [CrossRef]
  17. M. Woerdman, C. Alpmann, and C. Denz, “Self-pumped phase conjugation of light beams carrying orbital angular momentum,” Opt. Express17, 22791–22799 (2009). [CrossRef]
  18. R. K. Jain and R. C. Lind, “Degenerate four-wave mixing in semiconductor-doped glasses,” J. Opt. Soc. Am.73(5), 647–653 (1983). [CrossRef]
  19. T. Geethakrishnan and P. K. Palanisamy, “Optical phase-conjugation in erioglaucine dye-doped thin films,” Pranama-J. Phys.66, 473–478 (2008). [CrossRef]
  20. T. Bach, K. Nawata, M. Jazbinšek, T. Omatsu, and P. Günter, “Optical phase conjugation of picosecondpulses at 1.06 mm in Sn2P2S6: Te for wavefront correction in high-power Nd-doped amplifier systems,” Opt. Express18, 87–95 (2010). [CrossRef] [PubMed]
  21. N. G. Basov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, and A.Y. Okulov, “Laser interferometer with wavefront-reversing mirrors,” Sov. Phys. JETP52, 847–851 (1980).
  22. F. A. Starikov, Yu. V. Dolgopolov, A. V. Kopalkin, G. G. Kochemasov, S. M. Kulikov, and S. A. Sukharev, “About the correction of laser beams with phase front vortex,” J. Phys. IV France133, 683–685 (2006). [CrossRef]
  23. G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron.26, 131–191 (2002). [CrossRef]
  24. P. V. Polyanskiǐ and K. V. Fel’de, “Static holographic phase conjugation of vortex beams,” Opt. Spectrosc.98(6), 913–918 (2005). [CrossRef]
  25. I. G. Marienko, M. S. Soskin, and M. V. Vasnetsov, “Phase conjugation of wavefronts containing phase singularities,” in International Conference on Singular Optics, M. S. Soskin, ed., 3487, 39–41 (Proceedings of SPIE, 1998).
  26. L. C. Dávila Romero, D. L. Andrews, and M. Babikar, “A Quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B: Quantum Semiclass. Opt.4, S66–S72 (2002). [CrossRef]
  27. A. E. Siegman, Lasers (University Science Books, 1986).
  28. J.D. Jackson, Classical Electrodynamics, 2nd ed. (New York: Wiley, 1962).
  29. E. M. Liftshitz, L. P. Pitaevskii, and V. B. Berestetskii, Quantum Electrodynamics, Landau and Liftshitz Course of Theoretical Physics 4 (Oxford: Butterworth-Heineman) Ch.I section 6,8.
  30. V. G. Fedoseyev, “Transformation of the orbital angular momentum at the reflection and transmission of a light beam on a plane interface,” J. Phys A: Math. Theor.41, 505202 (2008). [CrossRef]
  31. V. G. Fedoseyev, “Reflection of the light beam carrying orbital angular momentum from a lossy medium,” Phys. Lett. A372, 2527–2533 (2008). [CrossRef]

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