Abstract

There has been considerable interest recently in the generation of azimuthal phase functions associated with photon orbital angular momentum (OAM) for high-dimensional quantum key distribution. The generation of secure quantum keys requires not only this pure phase basis but also additional bases comprised of orthonormal superposition states formed from the pure states. These bases are also known as mutually unbiased bases (MUBs) and include quantum states whose wave functions are modulated in both phase and amplitude. Although modulo 2π optical path control with high-resolution spatial light modulators (SLMs) is well suited to creating the azimuthal phases associated with the pure states, it does not introduce the amplitude modulation associated with the MUB superposition states. Using computer-generated holography (CGH) with the Leith–Upatnieks approach to hologram recording, however, both phase and amplitude modulation can be achieved. We present a description of the OAM states of a three-dimensional MUB system and analyze the construction of these states via CGH with a phase-modulating SLM. The effects of phase holography artifacts on quantum-state generation are quantified and a prescription for avoiding these artifacts by preconditioning the hologram function is presented. Practical effects associated with spatially isolating the first-order diffracted field are also quantified, and a demonstration utilizing a liquid-crystal SLM is presented.

© 2008 Optical Society of America

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References

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  4. T. C. Poon, T. Yatagai, and W. Juptner, "Digital holography-coherent optics of the 21st century: introduction" and collected papers, Appl. Opt. 45, special issue on Digital Holography, 821-983 (2006).
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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. K. Bauchert, S. Serati, and A. Furman, "Advances in liquid crystal spatial light modulators," Proc. SPIE 4734, 35-43 (2002).
    [CrossRef]
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    [CrossRef]
  26. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover Publications, 2000), Chap. 21.8.
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    [CrossRef] [PubMed]

2006 (2)

A. Gehner, M. Wildenhain, H. Neumann, J. Knobbe, and O. Komenda, "MEMS analog light processing--an enabling technology for adaptive optical phase control," Proc. SPIE 6113, 61130K 1-15 (2006).

T. C. Poon, T. Yatagai, and W. Juptner, "Digital holography-coherent optics of the 21st century: introduction" and collected papers, Appl. Opt. 45, special issue on Digital Holography, 821-983 (2006).
[CrossRef]

2005 (2)

M. T. Gruneisen, R. C. Dymale, J. R. Rotge, L. F. DeSandre, and D. L. Lubin, "Wavelength-dependent characteristics of a telescope system with diffractive wavefront control," Opt. Eng. 44, 068002 (2005).
[CrossRef]

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

2004 (1)

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

2003 (1)

X. Wang, B. Wang, J. Pouch, F. Miranda, M. Fisch, J. E. Anderson, V. Sergan, and P. Bos, "Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer," Proc. SPIE 5162, 139-146 (2003).
[CrossRef]

2002 (2)

N. J. Cerf, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

K. Bauchert, S. Serati, and A. Furman, "Advances in liquid crystal spatial light modulators," Proc. SPIE 4734, 35-43 (2002).
[CrossRef]

2001 (1)

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, "Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model," Opt. Eng. 40, 2558-2564 (2001).
[CrossRef]

1999 (1)

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, "High efficiency electrically-addressable phase-only spatial light modulator," Opt. Rev. 6, 339-344 (1999).
[CrossRef]

1989 (2)

W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements," Ann. Phys. (N. Y.) 191, 363-381 (1989).
[CrossRef]

D. A. Buralli, G. M. Morris, and J. R. Rogers, "Optical performance of holographic kinoforms," Appl. Opt. 28, 976-983 (1989).
[CrossRef] [PubMed]

1987 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

1964 (1)

1963 (1)

1962 (1)

1960 (1)

J. Schwinger, "Unitary operator bases," Proc. Natl. Acad. Sci. U.S.A. 46, 570-579 (1960).
[CrossRef] [PubMed]

Ann. Phys. (1)

W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements," Ann. Phys. (N. Y.) 191, 363-381 (1989).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Eng. (3)

M. T. Gruneisen, R. C. Dymale, J. R. Rotge, L. F. DeSandre, and D. L. Lubin, "Wavelength-dependent characteristics of a telescope system with diffractive wavefront control," Opt. Eng. 44, 068002 (2005).
[CrossRef]

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, "Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model," Opt. Eng. 40, 2558-2564 (2001).
[CrossRef]

Opt. Rev. (1)

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, "High efficiency electrically-addressable phase-only spatial light modulator," Opt. Rev. 6, 339-344 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

N. J. Cerf, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. Schwinger, "Unitary operator bases," Proc. Natl. Acad. Sci. U.S.A. 46, 570-579 (1960).
[CrossRef] [PubMed]

Proc. SPIE (3)

K. Bauchert, S. Serati, and A. Furman, "Advances in liquid crystal spatial light modulators," Proc. SPIE 4734, 35-43 (2002).
[CrossRef]

A. Gehner, M. Wildenhain, H. Neumann, J. Knobbe, and O. Komenda, "MEMS analog light processing--an enabling technology for adaptive optical phase control," Proc. SPIE 6113, 61130K 1-15 (2006).

X. Wang, B. Wang, J. Pouch, F. Miranda, M. Fisch, J. E. Anderson, V. Sergan, and P. Bos, "Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer," Proc. SPIE 5162, 139-146 (2003).
[CrossRef]

Other (10)

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover Publications, 2000), Chap. 21.8.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1966).

http://www.holoeye.com/phase_only_modulator_heo1080p.html.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, 1997), Chap. 1.4, pp. 38-45.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, 1984), pp. 175-179.
[PubMed]

G. J. Swanson, "Binary optics technology: the theory and design of multi-level diffractive optical elements," MIT Lincoln Laboratory Tech. Rep. 854 (MIT, 1989), pp. 1-47.

P. Hariharan, Optical Holography--Principles, Techniques and Applications (Cambridge U. Press, 1991).

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Figures (6)

Fig. 1
Fig. 1

Calculated amplitudes and phases comparing the theoretical complex field with that generated by CGH with phase modulation holography for various values of the phase-scaling parameter σ .

Fig. 2
Fig. 2

Calculated inner-product-based probability quantifying the fidelity with which the complex field is generated via CGH versus the phase-scaling parameter σ for each of the mutually unbiased bases in three-dimensional Hilbert space.

Fig. 3
Fig. 3

Numerically calculated amplitudes and phases resulting from holographic generation of the complex field. The calculation includes the effects of precompensating phase holography artifacts and of spatially isolating the relevant diffracted order. Several values of the reference wave tilt are shown.

Fig. 4
Fig. 4

Calculated inner-product-based probability associated with the numerical calculation illustrated in Fig. 3.

Fig. 5
Fig. 5

Schematic of the experimental setup showing locations of the spatial light modulator (SLM), aperture stop (AS), beam splitter (BS), reference mirror (RM), objective lens (L1), spatial filter (SF), field lens (L2), and CCD camera.

Fig. 6
Fig. 6

Demonstration of | c 3 state generation utilizing CGH and LCOS SLM technology comparing calculated irradiance and interferograms with those generated experimentally.

Tables (1)

Tables Icon

Table 1 Dirac Notation Description of Mutually Unbiased Bases in Three-Dimensional Hilbert Space

Equations (40)

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t ( θ ) = m = sinc ( 1 m ) exp ( i m l θ ) ,
M U B 0 { | a , | b , | c } .
| ψ = α | a + β | b + γ | c ,
α * α + β * β + γ * γ = 1 .
i | j = δ i , j for i , j { a , b , c } .
| i | j | | 2 = 1 3 for i { a , b , c } and j { a 1 , b 1 , c 1 } .
θ | a = exp ( i a θ ) ,
θ | b = exp ( i b θ ) ,
θ | c = exp ( i c θ ) .
| d = α | a + β | b + γ | c ,
θ | d = A ( θ ) exp [ i Φ ( θ ) ] .
A ( θ ) = d | d , Φ ( θ ) = tan 1 [ Im ( θ | d ) Re ( θ | d ) ] .
| c 3 = 1 3 [ | 1 + exp ( i 2 π / 3 ) | 0 + | 1 ] ,
A ( θ ) = 1 3 [ 1 2 cos ( θ ) + 4 cos 2 ( θ ) ] 1 / 2 ,
Φ ( θ ) = tan 1 [ 3 4 cos ( θ ) 1 ] .
E ( ρ , θ ) = a ( ρ , θ ) exp [ i ϕ ( ρ , θ ) ] .
R ( ρ , θ ) = b exp [ i k α ρ sin θ ] ,
I ( ρ , θ ) = a 2 ( ρ , θ ) + a max 2 + 2 a ( ρ , θ ) a max cos [ ϕ ( ρ , θ ) k α ρ sin θ ] .
f c g h ( ρ , θ ) = σ I ( ρ , θ ) I max ,
Δ ϕ S L M ( ρ , θ ) = σ I ( ρ , θ ) I max ,
t ( ρ , θ ) = exp [ i σ I ( ρ , θ ) I max ] .
t ( ρ , θ ) = exp ( i σ 2 ) m = ( i ) m J m [ σ a ( ρ , θ ) a max ] × exp { i m [ ϕ ( ρ , θ ) k α ρ sin θ ] i σ 2 [ a ( ρ , θ ) a max ] 2 } ,
E o u t ( ρ , θ ) = exp ( i σ 2 ) m = ( i ) m J m [ σ a ( ρ , θ ) a max ] × exp [ i ϕ m ( ρ , θ ) ] ,
ϕ m ( ρ , θ ) = m ϕ ( ρ , θ ) + ( m + 1 ) k α ρ sin θ + σ 2 [ a ( ρ , θ ) a max ] 2 .
a 1 ( ρ , θ ) = J 1 [ σ a ( ρ , θ ) a max ] .
η 1 ( ρ , θ ) = J 1 2 [ σ a ( ρ , θ ) a max ] ,
ϕ 1 ( x , y ) = ϕ ( ρ , θ ) + σ 2 [ a ( ρ , θ ) a max ] 2 .
σ 2 [ a ( ρ , θ ) a max ] 2 .
a 1 ( ρ , θ ) = J 1 [ σ a ( ρ , θ ) a max ] = c A ( ρ , θ ) ,
ϕ 1 ( ρ , θ ) = ϕ ( ρ , θ ) + σ 2 [ a ( ρ , θ ) a max ] 2 = Φ ( ρ , θ ) .
A ( θ ) = 1 3 [ 1 2 cos ( θ ) + 4 cos 2 ( θ ) ] 1 / 2 ,
Φ ( θ ) = tan 1 [ 3 4 cos ( θ ) 1 ] .
a 1 ( θ ) = J 1 [ σ A ( θ ) A max ] ,
ϕ 1 ( θ ) = Φ ( θ ) + σ 2 [ A ( θ ) A max ] 2 .
a 1 ( θ ) = J 1 [ σ 3 A max [ 1 2 cos ( θ ) + 4 cos 2 ( θ ) ] 1 / 2 ] ,
ϕ 1 ( θ ) = tan 1 [ 3 4 cos ( θ ) 1 ] + σ 6 A max 2 × [ 1 2 cos ( θ ) + 4 cos 2 ( θ ) ] .
ψ t | ψ h = 0 a 0 2 π ψ t * ( ρ , θ ) ψ h ( ρ , θ ) ρ d ρ d θ | ψ t | | ψ h | .
| ψ | = 0 a 0 2 π ψ * ( ρ , θ ) ψ ( ρ , θ ) ρ d ρ d θ .
P = | ψ t | ψ h | 2 .
ϕ ( ρ , θ ) = Φ ( θ ) σ 2 [ a ( θ ) a max ] 2 + k W ( ρ , θ ) ,

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