Abstract

We present a ray transfer matrix for a spiral phase plate. Using this matrix we determine the stability of an optical resonator made of two spiral phase plates and trace stable ray orbits in the resonator. Our results should be relevant to laser physics, optical micromanipulation, quantum information, and optomechanics.

© 2013 Optical Society of America

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References

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  1. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, 2003).
  2. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
    [CrossRef]
  3. Y. S. Rumala and A. E. Leanhardt, “Multiple beam interference in a spiral phase plate,” J. Opt. Soc. Am. B 30, 615–621 (2013).
    [CrossRef]
  4. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  5. A. Mawardi, S. Hild, A. Widera, and D. Meschede, “ABCD-treatment of a propagating doughnut beam generated by a spiral phase plate,” Opt. Express 19, 21205–21210 (2011).
    [CrossRef]
  6. W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796–1798 (2004).
    [CrossRef]
  7. M. Bhattacharya and P. Meystre, “Using a Laguerre–Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99, 153603 (2007).
    [CrossRef]
  8. O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
    [CrossRef]
  9. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
    [CrossRef]
  10. G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
    [CrossRef]
  11. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science, 1986).
  13. P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).
  14. Y. Rumala, “A new etalon geometry: the spiral phase plate etalon,” Ph.D. Dissertation (University of Michigan, 2012).
  15. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
    [CrossRef]
  16. A. V. Kudryashov and A. H. Paxton, Laser Resonators III (SPIE, 2000).
  17. T. M. Jeong, D.-K. Ko, and J. Lee, “Generalized ray-transfer matrix for an optical element having an arbitrary wavefront aberration,” Opt. Lett. 30, 3009–3011 (2005).
    [CrossRef]
  18. J. C. Gutiérrrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A 20, 2113–2122 (2003).
    [CrossRef]
  19. J. C. Minano, “Application of the conservation of Etendue theorem for 2-D subdomains of the phase space in nonimaging concentrators,” Appl. Opt. 23, 2021–2025 (1984).
    [CrossRef]

2013 (1)

2011 (1)

2010 (1)

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[CrossRef]

2008 (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
[CrossRef]

2007 (1)

M. Bhattacharya and P. Meystre, “Using a Laguerre–Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99, 153603 (2007).
[CrossRef]

2005 (3)

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

T. M. Jeong, D.-K. Ko, and J. Lee, “Generalized ray-transfer matrix for an optical element having an arbitrary wavefront aberration,” Opt. Lett. 30, 3009–3011 (2005).
[CrossRef]

G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
[CrossRef]

2004 (1)

2003 (1)

2001 (1)

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

1996 (1)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1984 (1)

’t Hooft, G. W.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Aiello, A.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Allen, L.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

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

Barnett, S. M.

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

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
[CrossRef]

Bhattacharya, M.

M. Bhattacharya and P. Meystre, “Using a Laguerre–Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99, 153603 (2007).
[CrossRef]

Chávez-Cerda, S.

Cheong, W. C.

Cirac, J. I.

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Davidson, N.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

Eliel, E. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Foo, G.

Friesem, A. A.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
[CrossRef]

Gutiérrrez-Vega, J. C.

Hasman, E.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Hild, S.

Jeong, T. M.

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
[CrossRef]

Juan, M. L.

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[CrossRef]

Ko, D.-K.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Kudryashov, A. V.

A. V. Kudryashov and A. H. Paxton, Laser Resonators III (SPIE, 2000).

Leanhardt, A. E.

Lee, J.

Lee, W. M.

Ma, X.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
[CrossRef]

Mawardi, A.

Meschede, D.

Meystre, P.

M. Bhattacharya and P. Meystre, “Using a Laguerre–Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99, 153603 (2007).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

Minano, J. C.

Oemrawsingh, S. S. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Padgett, M. J.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

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

Palacios, D. M.

Paxton, A. H.

A. V. Kudryashov and A. H. Paxton, Laser Resonators III (SPIE, 2000).

Quidant, R.

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
[CrossRef]

Robertson, D. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Rodríguez-Masegosa, R.

Romero-Isart, O.

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[CrossRef]

Rumala, Y.

Y. Rumala, “A new etalon geometry: the spiral phase plate etalon,” Ph.D. Dissertation (University of Michigan, 2012).

Rumala, Y. S.

Siegman, A. E.

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

Smith, G. M.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Swartzlander, G. A.

Turnbull, G. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

Voigt, D.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Widera, A.

Woerdman, J. P.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Yuan, X.-C.

Appl. Opt. (1)

J. Microsc. (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

New J. Phys. (1)

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” New J. Phys. 12, 033015 (2010).
[CrossRef]

Opt. Commun. (2)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre–Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

M. Bhattacharya and P. Meystre, “Using a Laguerre–Gaussian beam to trap and cool the rotational motion of a mirror,” Phys. Rev. Lett. 99, 153603 (2007).
[CrossRef]

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Prog. Opt. (1)

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Other (5)

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

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

P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

Y. Rumala, “A new etalon geometry: the spiral phase plate etalon,” Ph.D. Dissertation (University of Michigan, 2012).

A. V. Kudryashov and A. H. Paxton, Laser Resonators III (SPIE, 2000).

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Figures (3)

Fig. 1.
Fig. 1.

Spiral phase plate discussed in this work. An incident ray is shown being reflected by the plate.

Fig. 2.
Fig. 2.

Spiral resonator with two identical plates separated by a distance L. The steps of the plates have been aligned.

Fig. 3.
Fig. 3.

Stable ray orbits that can occupy the spiral plate resonator of Fig. 2. The ray orbits are drawn to scale and ray trajectories are indicated using arrowheads. The plates P1 and P2 are not shown for clarity. Solid arrows denote twisted rays traveling from P1 to P2 and dotted arrows represent the annulus of rays traveling from P2 to P1.

Equations (21)

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tanα(r)=s2πr.
Mc(s,R)=(Mr00Mt),
Mr=(101r1W(r,ϕ)r|r=r11),
Mt=(101ϕ1W(r,ϕ)rϕ|r=r11),
W(r,ϕ)=r2R+s(1ϕ2π).
Mc(s,R)=(10002R100001000sπr1ϕ11).
x=rcosϕ,y=rsinϕ,
r=x2+y2,ϕ=tan1(yx),
drdz=xdxdz+ydydzx2+y2,dϕdz=xdydzydxdzx2+y2.
M(s,R)=(10002R1sπr1200010sπr1202R1),
σ·MT·σ·M=I,
σ=(0100100000010010).
x2up2vx2vp2u+y2uq2vy2vq2u={1,u,v=x1,p1ory1,p1,0,u,v=x1,y1orx1,q1ory1,p1orp1,q1.
M(L)=(1L000100001L0001).
MT=M(s,R)·M(L)·M(s,R)·M(L),
P(λ)=|MTλI|=0,
λ±=e±iθ,
cosθ=14LR+2(LR)2+12(sLπr2)2.
1cosθ1.
(1LR)2+(sL2πr2)20,
L2R1+(sR2πr2)2.

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