Abstract

After a small aperture the spatial information of a complex optical wavefront is lost, but amplitude and phase information is mixed and transferred to the smoothed wave that emerges from the pinhole. This mixing effect is described in the case of a wavefront with a phase step, which is shifted over the input plane of an optical processor with a pinhole as spatial filter in the Fourier plane. We constructed a polarizing interferometer to demonstrate this continuous phase shift and show that it can be used as a variable retardation wave plate similar to a birefringent compensator, but without crystalline wedges.

© 2005 Optical Society of America

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References

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  1. S. Prasat, G. Loos, “Spatial filtering of atmospheric decorrelation from wavefronts for interferometry,” Opt. Commun. 99, 380–392 (1993).
    [CrossRef]
  2. C. Zhang, “Preparation of polarization-entangled mixed states of two photons,” Phys. Rev. A 69, 014304-1-3 (2004).
    [CrossRef]
  3. J. D. Gaskill, Linear Systems, Fourier Transform, and Optics (Wiley-Interscience, 1978).
  4. G. W. Stroke, Coherent Optics and Holography (Academic, 1966), p. 78.
  5. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1964), p. 27.
  6. N. Bhattacharya, H. B. van Linden, van den Heuvell, R. J.C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901-1-4 (2002).
    [CrossRef]

2004 (1)

C. Zhang, “Preparation of polarization-entangled mixed states of two photons,” Phys. Rev. A 69, 014304-1-3 (2004).
[CrossRef]

2002 (1)

N. Bhattacharya, H. B. van Linden, van den Heuvell, R. J.C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901-1-4 (2002).
[CrossRef]

1993 (1)

S. Prasat, G. Loos, “Spatial filtering of atmospheric decorrelation from wavefronts for interferometry,” Opt. Commun. 99, 380–392 (1993).
[CrossRef]

Bhattacharya, N.

N. Bhattacharya, H. B. van Linden, van den Heuvell, R. J.C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901-1-4 (2002).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1964), p. 27.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transform, and Optics (Wiley-Interscience, 1978).

Heuvell, van den

N. Bhattacharya, H. B. van Linden, van den Heuvell, R. J.C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901-1-4 (2002).
[CrossRef]

Loos, G.

S. Prasat, G. Loos, “Spatial filtering of atmospheric decorrelation from wavefronts for interferometry,” Opt. Commun. 99, 380–392 (1993).
[CrossRef]

Prasat, S.

S. Prasat, G. Loos, “Spatial filtering of atmospheric decorrelation from wavefronts for interferometry,” Opt. Commun. 99, 380–392 (1993).
[CrossRef]

Spreeuw, R. J.C.

N. Bhattacharya, H. B. van Linden, van den Heuvell, R. J.C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901-1-4 (2002).
[CrossRef]

Stroke, G. W.

G. W. Stroke, Coherent Optics and Holography (Academic, 1966), p. 78.

van Linden, H. B.

N. Bhattacharya, H. B. van Linden, van den Heuvell, R. J.C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901-1-4 (2002).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1964), p. 27.

Zhang, C.

C. Zhang, “Preparation of polarization-entangled mixed states of two photons,” Phys. Rev. A 69, 014304-1-3 (2004).
[CrossRef]

Opt. Commun. (1)

S. Prasat, G. Loos, “Spatial filtering of atmospheric decorrelation from wavefronts for interferometry,” Opt. Commun. 99, 380–392 (1993).
[CrossRef]

Phys. Rev. A (1)

C. Zhang, “Preparation of polarization-entangled mixed states of two photons,” Phys. Rev. A 69, 014304-1-3 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

N. Bhattacharya, H. B. van Linden, van den Heuvell, R. J.C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901-1-4 (2002).
[CrossRef]

Other (3)

J. D. Gaskill, Linear Systems, Fourier Transform, and Optics (Wiley-Interscience, 1978).

G. W. Stroke, Coherent Optics and Holography (Academic, 1966), p. 78.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1964), p. 27.

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Figures (5)

Fig. 1
Fig. 1

(a) Optical processor with n input beams and a pinhole PH as spatial filter. (b) Input beam with a phase plate φ in the front focal plane.

Fig. 2
Fig. 2

Mixed phase κ of the exit beam of the processor (Fig. 1) as a function of the shift parameter ( Δ 2 a ) and the phase delay φ of the phase plate in the entrance beam.

Fig. 3
Fig. 3

Intensity R 2 of the exit beam of the processor (Fig. 1) as a function of the shift parameter ( Δ 2 a ) and the phase delay φ.

Fig. 4
Fig. 4

Experiment with a mica retardation plate MP in the front focal plane of the optical processor L 1 , L 2 and a pinhole PH in the Fourier plane. The exit state of polarization is analyzed with a Soleil–Babinet compensator SBC and the rotating analyzer P using the ground glass plate GGP and the eye. “A” represents the aperture. The laser is polarized (E-vector) in the paper plane.

Fig. 5
Fig. 5

Rotation angle η of the analyzer depending on the shift Δ of the mica plate and phase retardation γ in fractions of the wave length λ for two different thickness of retardation plate, 14.2 μ m (solid squares) and 28.8 μ m (open circles).

Equations (9)

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A 0 ( x ) = i A i rect [ ( x x i ) H i ] exp ( i φ i ) ,
A g ( x g ) = F { F [ A 0 ( x ) ] δ ( x f ) } = c A 0 * 1 = c + A 0 ( x ) d x = c i A i H i exp ( i φ i ) c r exp ( i ψ ) ,
r = [ ( i I i 1 2 cos φ i ) 2 + ( i I i 1 2 sin φ i ) 2 ] 1 2 ,
ψ = arctan { [ i ( I i ) 1 2 sin φ i ] [ i ( I i ) 1 2 cos φ i ] } ,
A 0 ( x , y ) = A ( x , y ) { rect [ ( x Δ 2 ) ( 2 a Δ ) ] + rect [ ( x + a Δ 2 ) Δ ] exp ( i φ ) } { rect [ y ( 2 a ) ] } .
F { F [ A 0 ( x , y ) ] δ ( x f , y f ) } = F { F [ A 0 ( x , y ) ] } * * 1 = + A 0 ( x , y ) d x d y = 2 a [ A 0 ¯ ( 2 a Δ ) + A φ ¯ Δ exp ( i φ ) ] ,
A g = 1 Δ ( 2 a ) + Δ ( 2 a ) exp ( i φ ) R exp ( i κ ) .
κ = arctan { Δ ( 2 a ) sin φ [ 1 Δ ( 2 a ) ( 1 cos φ ) ] } ,
R 2 = { 1 Δ ( 2 a ) [ 1 cos φ ] } 2 + { Δ ( 2 a ) sin φ } 2 .

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