Abstract

Light’s orbital angular momentum (OAM) is a conserved quantity in cylindrically symmetric media. However, it is easily destroyed by free-space turbulence or fiber bends, because anisotropic perturbations impart angular momentum. We observe the conservation of OAM even in the presence of strong bend perturbations, with fibers featuring air cores that appropriately sculpt the modal density of states. Analogous to the enhanced stability of spinning tops with increasing angular velocity, these states’ lifetimes increase with OAM magnitude. Consequently, contrary to conventional wisdom that ground states of systems are the most stable, OAM longevity in air-core fiber increases with mode order. Aided by conservation of this fundamental quantity, we demonstrate fiber propagation of 12 distinct higher order OAM modes, of which eight remain low loss and >98 % pure from near-degenerate coupling after kilometer-length propagation. The first realization of long-lived higher order OAM states, thus far posited to exist only in vacuum, is a necessary condition for achieving the promise of higher dimensional classical and quantum communications over practical distances.

© 2015 Optical Society of America

Full Article  |  PDF Article

Corrections

P. Gregg, P. Kristensen, and S. Ramachandran, "Conservation of orbital angular momentum in air-core optical fibers: erratum," Optica 4, 1115-1116 (2017)
https://www.osapublishing.org/optica/abstract.cfm?uri=optica-4-9-1115
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References

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2014 (2)

C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, L. A. Rusch, Opt. Express 22, 26117 (2014).
[Crossref]

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

2013 (3)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, M. J. Padgett, Science 341, 537 (2013).
[Crossref]

S. Ramachandran, P. Kristensen, Nanophotonics 2, 455 (2013).

2012 (5)

N. Bozinovic, S. Golowich, P. Kristensen, S. Ramachandran, Opt. Lett. 37, 2451 (2012).
[Crossref]

C. N. Alexeyev, J. Opt. 14, 085702 (2012).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, P. St. J. Russell, Science 337, 446 (2012).
[Crossref]

B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, R. W. Boyd, Opt. Lett. 37, 3735 (2012).
[Crossref]

2011 (1)

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

2010 (1)

2009 (2)

2006 (1)

P. Z. Dashti, F. Alhasen, H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006).
[Crossref]

2005 (1)

2003 (3)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, Nature 421, 925 (2003).
[Crossref]

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[Crossref]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, IEEE Photon. Technol. Lett. 15, 1171 (2003).
[Crossref]

2000 (1)

C. N. Alexeyev, M. S. Soskin, A. V. Volyar, Semiconductor Phys. Quantum Electron. Optoelectron. 3, 501 (2000).

1994 (1)

S. J. Van Enk, G. Nienhuis, Europhys. Lett. 25, 497 (1994).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
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1987 (1)

1986 (1)

A. Bjarklev, J. Lightwave Technol. 4, 341 (1986).
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Ahmed, M.

J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Alexeyev, C. N.

C. N. Alexeyev, J. Opt. 14, 085702 (2012).
[Crossref]

C. N. Alexeyev, M. S. Soskin, A. V. Volyar, Semiconductor Phys. Quantum Electron. Optoelectron. 3, 501 (2000).

Alhasen, F.

P. Z. Dashti, F. Alhasen, H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Andrews, D. L.

D. L. Andrews, Structured Light and Its Applications (Academic, 2008).

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, Nature 421, 925 (2003).
[Crossref]

Auksorius, E.

L. Yan, E. Auksorius, N. Bozinovic, G. J. Tearney, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu3N.2.

Awaji, Y.

Y. Awaji, N. Wada, Y. Toda, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2012), paper JTu2K.3

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, M. J. Padgett, Science 341, 537 (2013).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Biancalana, F.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, P. St. J. Russell, Science 337, 446 (2012).
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A. Bjarklev, J. Lightwave Technol. 4, 341 (1986).
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Blake, J. N.

Boyd, R. W.

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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
[Crossref]

N. Bozinovic, S. Golowich, P. Kristensen, S. Ramachandran, Opt. Lett. 37, 2451 (2012).
[Crossref]

L. Yan, E. Auksorius, N. Bozinovic, G. J. Tearney, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu3N.2.

Brunet, C.

Choi, S.

Conti, C.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, P. St. J. Russell, Science 337, 446 (2012).
[Crossref]

D’Ambrosio, V.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Dashti, P. Z.

P. Z. Dashti, F. Alhasen, H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006).
[Crossref]

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Engan, H. E.

Essiambre, R.-J.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Ghalmi, S.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, IEEE Photon. Technol. Lett. 15, 1171 (2003).
[Crossref]

Golowich, S.

N. Bozinovic, S. Golowich, P. Kristensen, S. Ramachandran, Opt. Lett. 37, 2451 (2012).
[Crossref]

P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2.

Gregg, P.

P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2.

P. Gregg, P. Kristensen, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM2N.2.

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Jennewein, T.

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[Crossref]

Kang, M. S.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, P. St. J. Russell, Science 337, 446 (2012).
[Crossref]

Karimi, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

Kim, B. Y.

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, Nature 421, 925 (2003).
[Crossref]

Kramer, G.

Kristensen, P.

S. Ramachandran, P. Kristensen, Nanophotonics 2, 455 (2013).

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
[Crossref]

N. Bozinovic, S. Golowich, P. Kristensen, S. Ramachandran, Opt. Lett. 37, 2451 (2012).
[Crossref]

S. Ramachandran, P. Kristensen, M. F. Yan, Opt. Lett. 34, 2525 (2009).
[Crossref]

P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2.

P. Gregg, P. Kristensen, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM2N.2.

LaRochelle, S.

Lavery, M. P. J.

Lee, H. P.

P. Z. Dashti, F. Alhasen, H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006).
[Crossref]

Lee, H. W.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, P. St. J. Russell, Science 337, 446 (2012).
[Crossref]

Lee, J. W.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Malik, M.

Marrucci, L.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

Messaddeq, Y.

Mirhosseini, M.

Miyamoto, K.

Nagali, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

Nicholson, J. W.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, IEEE Photon. Technol. Lett. 15, 1171 (2003).
[Crossref]

Nienhuis, G.

S. J. Van Enk, G. Nienhuis, Europhys. Lett. 25, 497 (1994).
[Crossref]

O’Sullivan, M. N.

Oh, K.

Okida, M.

Olsen, J.

P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2.

Omatsu, T.

Padgett, M.

Padgett, M. J.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, M. J. Padgett, Science 341, 537 (2013).
[Crossref]

Pan, J.-W.

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[Crossref]

Piccirillo, B.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
[Crossref]

S. Ramachandran, P. Kristensen, Nanophotonics 2, 455 (2013).

N. Bozinovic, S. Golowich, P. Kristensen, S. Ramachandran, Opt. Lett. 37, 2451 (2012).
[Crossref]

S. Ramachandran, P. Kristensen, M. F. Yan, Opt. Lett. 34, 2525 (2009).
[Crossref]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, IEEE Photon. Technol. Lett. 15, 1171 (2003).
[Crossref]

P. Gregg, P. Kristensen, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM2N.2.

P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2.

L. Yan, E. Auksorius, N. Bozinovic, G. J. Tearney, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu3N.2.

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Robertson, D. J.

Rodenburg, B.

Rusch, L. A.

Russell, P. St. J.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, P. St. J. Russell, Science 337, 446 (2012).
[Crossref]

Santamato, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

Sciarrino, F.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

Shaw, H. J.

Slussarenko, S.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011).
[Crossref]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Soskin, M. S.

C. N. Alexeyev, M. S. Soskin, A. V. Volyar, Semiconductor Phys. Quantum Electron. Optoelectron. 3, 501 (2000).

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, M. J. Padgett, Science 341, 537 (2013).
[Crossref]

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, Nature 421, 925 (2003).
[Crossref]

Sponselli, A.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Steinvurzel, P.

P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2.

Tanaka, Y.

Tearney, G. J.

L. Yan, E. Auksorius, N. Bozinovic, G. J. Tearney, S. Ramachandran, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu3N.2.

Toda, Y.

Y. Awaji, N. Wada, Y. Toda, in CLEO, OSA Technical Digest (online) (Optical Society of America, 2012), paper JTu2K.3

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Vahala, K. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, Nature 421, 925 (2003).
[Crossref]

Vaity, P.

Vallone, G.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

Van Enk, S. J.

S. J. Van Enk, G. Nienhuis, Europhys. Lett. 25, 497 (1994).
[Crossref]

Vaziri, A.

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. (a) Free-space OAM states are coupled into air-core fiber and conserved despite bends and random fiber shape deformations. Each state possesses total angular momentum, J , which comprises orbital, L , and spin, S , parts, which may be positive or negative (right or left handed). Thus, there are four OAM states for every | L | , each of which can carry information. (b) Microscope image (top) and measured refractive index profile (bottom) for air-core fiber supporting 12 OAM states. The radius of the air core is 3 μm, and the outer radius of the ring region is 8.25 μm. (c) Example of n eff splitting among OAM states. In conventional fibers, states of the same | L | are near-degenerate and freely couple. Via the air-core design, this near–degeneracy is broken such that states with spin and OAM aligned separate from states with spin and OAM anti–aligned. (d) Effective index splitting in a typical air-core fiber: ∼ 10 − 4 is considered sufficient for OAM propagation [13] achieved in this design for | L | = 5 , 6, and 7.

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Fig. 2.
Fig. 2. (a) Setup: light from an external cavity laser (ECL; 1530 nm) or a picosecond pulsed laser (1550 nm) is converted to free-space OAM modes via a spatial light modulator (SLM) followed by a quarter-wave plate, and launched into the air-core fiber. The fiber’s output is imaged through a circular polarization beam splitter (CPBS), separating σ ^ − from σ ^ + . For interference measurements, the reference is tapped from a 50/50 splitter at the input (orange path). For stability measurements, the air-core fiber passes through a polcon. For time-of-flight measurements (blue path), a fast detector and oscilloscope (Rx and Osc) are used. Dashed lines indicate free space, solid lines indicate fiber. Fiber output images are for | L | = 7 . (b) OAM states after 10 m of the air-core fiber, interfered with an expanded Gaussian reference. Text around images indicates launch conditions. 12 states for | L | = 5 , 6, 7 in all SAM/OAM combinations are shown. See Supplement 1, Section 6 for additional experimental details.

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Fig. 3.
Fig. 3. (a) Theoretical prediction of degenerate-state coupling for different OAM orders due to a 2.8 cm radius-of-curvature fiber bend. Coefficients rapidly decrease with increasing OAM content, p . (b) Illustration of power binning measurement for L = 7 , σ ^ + . As the polcon paddles [Fig. 2(a)] are tuned, negligible coupling from σ ^ + to σ ^ − is observed, indicating degenerate-state stability. Legend “pol 1” indicates launched polarization; “pol 2” indicates parasitic polarization. (c) Polcon measurement for SMF, indicating complete degenerate state mode coupling. (d) Experimentally measured average values of degradation factor α for each | L | , plotted against a shifted 1 / | L | trend line (dashed line). Degradation drops with increasing OAM. This concept was tested experimentally on states for which spin-orbit aligned to spin-orbit anti-aligned coupling is suppressed. (e) Schematic indicating the perturbation OAM content necessary to couple degenerate fiber states with opposite values of L .

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Fig. 4.
Fig. 4. (a) Time-of-flight measurements, using setup of Fig. 2(a), for four different OAM states. Traces vertically offset for visual clarity, in order of increasing group delay: L = + 5 σ ^ − , L = + 5 σ ^ + , L = + 6 σ ^ − , and L = + 6 σ ^ + . Inset: fiber output image after 1 km propagation. (b) Close-up of time-domain trace for spin-orbit anti-aligned L = + 5 σ ^ − mode (peak around 519.5 ns is spurious from the detector’s electrical impulse response). (c) Close-up of time-domain trace for spin-orbit aligned L = + 5 σ ^ + . The time difference between the two L = + 5 peaks, 0.75 ns, corresponds well to the theoretical value of 0.7 ns. (d) and (e) show close-ups of L = + 6 σ ^ − and L = + 6 σ ^ + traces, indicating even better parasitic mode suppression. The peaks from (d) and (e) would overlap in conventional step-index fibers due to mixing. In each case, the excited mode is approximately 18–20 dB pure relative to the background. See Supplement 1, Section 7 for additional details.

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Equations (6)

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J = L + S .
〈 P j , k 〉 = k 2 Φ ( k j − k k ) ( ∬ r d r d φ Δ n 2 ( r , φ ) ψ j * ψ k ) 2 .
Δ n 2 ( r , φ ) = ∑ p = − ∞ ∞ a p ( r ) e i p φ ,
〈 ψ 1 | Δ n 2 ( r , φ ) | ψ 2 〉 = ∑ p = − ∞ ∞ 〈 F 1 ( r ) | a p ( r ) | F 2 ( r ) 〉 〈 e i L 1 φ | e i p φ | e i L 2 φ 〉 ,
p − ( L 1 − L 2 ) = 0 .
α = 10   log ( P peak   1 − P ¯ noise P peak   2 − P ¯ noise ) ,

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