Abstract

The analysis of wavefront-coding systems is explored via the joint fractional Fourier signal representation (JFF) of the pupil function. The properties of the JFF of the pupil function are presented and are shown to be revealing with regard to the system response to defocus. Numerical examples that illustrate the properties are given.

© 2009 Optical Society of America

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References

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  1. E. Dowski and W. Cathey, Appl. Phys. Lett. 34, 1859 (1995).
  2. G. E. Johnson, P. Silveira, and E. Dowski, Proc. SPIE 5817, 34 (2005).
    [CrossRef]
  3. Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Lu, Appl. Opt. 45, 8586 (2006).
    [CrossRef] [PubMed]
  4. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  5. L. Durak, A. Ozdemir, and O. Arikan, J. Opt. Soc. Am. A 25, 765 (2008).
    [CrossRef]

2008 (1)

2006 (1)

2005 (1)

G. E. Johnson, P. Silveira, and E. Dowski, Proc. SPIE 5817, 34 (2005).
[CrossRef]

1995 (1)

E. Dowski and W. Cathey, Appl. Phys. Lett. 34, 1859 (1995).

Arikan, O.

Cathey, W.

E. Dowski and W. Cathey, Appl. Phys. Lett. 34, 1859 (1995).

Dowski, E.

G. E. Johnson, P. Silveira, and E. Dowski, Proc. SPIE 5817, 34 (2005).
[CrossRef]

E. Dowski and W. Cathey, Appl. Phys. Lett. 34, 1859 (1995).

Durak, L.

Johnson, G. E.

G. E. Johnson, P. Silveira, and E. Dowski, Proc. SPIE 5817, 34 (2005).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Liu, L.

Lu, W.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Ozdemir, A.

Silveira, P.

G. E. Johnson, P. Silveira, and E. Dowski, Proc. SPIE 5817, 34 (2005).
[CrossRef]

Sun, J.

Yang, Q.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zhu, Y.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. Dowski and W. Cathey, Appl. Phys. Lett. 34, 1859 (1995).

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

Proc. SPIE (1)

G. E. Johnson, P. Silveira, and E. Dowski, Proc. SPIE 5817, 34 (2005).
[CrossRef]

Other (1)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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Figures (3)

Fig. 1
Fig. 1

(a) Grayscale plot of J p ( 1.0 , 1.3 ) ( t , s ) for a cubic phase mask with 2 π α = 5 and (b) the corresponding IFC.

Fig. 2
Fig. 2

(a) Grayscale plot of J p ( 1.0 , 1.3 ) ( t , s ) for a cubic phase mask with 2 π α = 10 . In (b) note that the corresponding IFC fits inside that of Fig. 1b for 2 π α = 5 .

Fig. 3
Fig. 3

D ¯ , as defined in Eq. (17), is plotted versus a 2 for cubic phase masks with 2 π α = 2.5 , 10, 15, and 20.

Equations (18)

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F p a ( x ) = sin ( ϕ ) 1 2 e j ( π sgn ( a ) 4 + ϕ 2 ) e j π ( x 2 cot ( ϕ ) 2 u x csc ( ϕ ) + u 2 cot ( ϕ ) ) p ( u ) d u , 2 < a < 2 ,
p ( u ) = 2 1 2 exp ( j 2 π θ ( u ) ) for u 1 ,
p ( u ) = 0 for u > 1 ,
W p ( u , v ) = p ( u + τ 2 ) p * ( u τ 2 ) exp ( j 2 π v τ ) d τ ,
F p a ( u ) 2 = W p ( u cos ϕ v sin ϕ , u sin ϕ + v cos ϕ ) d v .
W p ( u , v ) d v = F p 0 ( u ) 2 = p ( u ) 2 ,
W p ( u , v ) d u = F p 1 ( v ) 2 = h ( v ) 2 ,
J p a ( t , s ) d s = F p a 1 ( t ) 2 , J p a ( t , s ) d t = F p a 2 ( s ) 2 .
J p a ( t , s ) = csc ϕ 12 W p ( u ( t , s ) , v ( t , s ) ) ,
h ( x , z ) 2 = sin ϕ F p a ( u ) 2 ,
J p a ( t , s ) d s = csc ϕ 1 h ( x , z 1 ) 2 , t = x sin ϕ 1 ,
J p a ( t , s ) d t = csc ϕ 2 h ( x , z 2 ) 2 , s = x sin ϕ 2 ,
t = θ ( γ ) with γ = ( s t sin ϕ 2 ) cos ϕ 2 ,
J p ( 1 , a 2 ) ( t , s ) = 0.5 sec ϕ 2 2 ( 1 γ ) 2 ( 1 γ ) exp ( j Γ ( τ ) ) d τ for γ < 1 ,
Γ ( τ ) = 2 π [ θ ( γ + τ 2 ) θ ( γ τ 2 ) τ t ] .
J p ( 1 , a 2 ) ( t , s ) 2 sec ϕ 2 cos ( Γ ( τ s ) + sgn ( Γ ( τ s ) ) π 4 ) θ ( γ + τ s 2 ) θ ( γ τ s 2 ) 1 2 ,
Γ ( τ s ) = 0 or t = [ θ ( γ + τ s 2 ) + θ ( γ τ s 2 ) ] 2 .
J p ( 1 , a 2 ) ( t , s ) sec ϕ 2 4 π 2 ϕ ( γ ) 1 3 Ai { 4 π 2 ϕ ( γ ) 1 3 β sgn ( ϕ ( γ ) ) } ,

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