Abstract

Pupil plane filtering by radial Walsh filters is a convenient technique for tailoring the axial intensity distribution near the focal plane of a rotationally symmetric imaging system. Radial Walsh filters, derived from radial Walsh functions, form a set of orthogonal phase filters that take on values either 0 or π phase, corresponding to +1 or 1 values of the radial Walsh functions over prespecified annular regions of the circular filter. Order of these filters is given by the number of zero-crossings, or equivalently phase transitions within the domain over which the set is defined. In general, radial Walsh filters are binary phase zone plates, each of them demonstrating distinct focusing characteristics. The set of radial Walsh filters can be classified into distinct groups, where the members of each group possess self-similar structures. Self-similarity can also be observed in the corresponding axial intensity distributions. These observations provide valuable clues in tackling the inverse problem of synthesis of phase filter in accordance with prespecified axial intensity distributions. This paper reports our observations on self-similarity in radial Walsh filters of various orders and corresponding axial intensity distributions.

© 2014 Optical Society of America

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References

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  1. J. L. Walsh, “A closed set of normal orthogonal functions,” Am. J. Math. 45, 5–24 (1923).
    [CrossRef]
  2. H. F. Harmuth, Transmission of Information by Orthogonal Functions (Springer-Verlag, 1972), p. 31.
  3. K. G. Beauchamp, Walsh Functions and Their Applications (Academic, 1985).
  4. M. Fu, G. Wade, J. Ning, and R. Jakobs, “On Walsh filtering method of decoding CPM signals,” IEEE Commun. Lett. 8, 345–347 (2004).
    [CrossRef]
  5. H. C. Andrews, Computer Techniques in Image Processing (Academic, 1970).
  6. L. N. Hazra and A. Banerjee, “Application of Walsh function in generation of optimum apodizers,” J. Opt. 5, 19–26 (1976).
  7. M. De and L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
    [CrossRef]
  8. M. De and L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
    [CrossRef]
  9. L. N. Hazra, “A new class of optimum amplitude filters,” Opt. Commun. 21, 232–236 (1977).
    [CrossRef]
  10. L. N. Hazra and A. Guha, “Farfield diffraction properties of radial Walsh filters,” J. Opt. Soc. Am. A 3, 843–846 (1986).
    [CrossRef]
  11. L. N. Hazra, “Walsh filters in tailoring of resolution in microscopic imaging,” Micron 38, 129–135 (2007).
    [CrossRef]
  12. P. Mukherjee and L. N. Hazra, “Farfield diffraction properties of annular Walsh filters,” Adv. Opt. Technol. 2013, 360450 (2013).
    [CrossRef]
  13. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).
  14. J. A. Monsoriu, W. D. Furlan, and G. Saavedra, “Focusing light with fractal zone plates,” Recent Res. Devel. Optics 5, 1–15 (2005).
  15. G. Saavedra, W. D. Furlan, and A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003).
    [CrossRef]
  16. L. Zunino and M. Garavaglia, “Fraunhofer diffraction by Cantor fractals with variable lacunarity,” J. Mod. Opt. 50, 717–727 (2003).
    [CrossRef]
  17. J. Uozumi, Y. Sakurada, and T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995).
    [CrossRef]
  18. A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
    [CrossRef]
  19. V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21, 2701–2706 (2013).
    [CrossRef]
  20. L. N. Hazra, Y. Han, and C. Delisle, “Sigmatic imaging by zone plates,” J. Opt. Soc. Am. A 10, 69–74 (1993).
    [CrossRef]
  21. L. N. Hazra, Y. Han, and C. Delisle, “Imaging by zone plates: axial stigmatism at a particular order,” J. Opt. Soc. Am. A 11, 2750–2754 (1994).
    [CrossRef]
  22. M. Born and E. Wolf, Principles of Optics (Pergamon Oxford, 1980).
  23. H. H. Hopkins, “Canonical and real space coordinates used in the theory of image formation,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1983), Vol. 9, p. 307.
  24. S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52, 035602 (2013).
    [CrossRef]

2013 (4)

P. Mukherjee and L. N. Hazra, “Farfield diffraction properties of annular Walsh filters,” Adv. Opt. Technol. 2013, 360450 (2013).
[CrossRef]

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52, 035602 (2013).
[CrossRef]

V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21, 2701–2706 (2013).
[CrossRef]

2007 (1)

L. N. Hazra, “Walsh filters in tailoring of resolution in microscopic imaging,” Micron 38, 129–135 (2007).
[CrossRef]

2005 (1)

J. A. Monsoriu, W. D. Furlan, and G. Saavedra, “Focusing light with fractal zone plates,” Recent Res. Devel. Optics 5, 1–15 (2005).

2004 (1)

M. Fu, G. Wade, J. Ning, and R. Jakobs, “On Walsh filtering method of decoding CPM signals,” IEEE Commun. Lett. 8, 345–347 (2004).
[CrossRef]

2003 (2)

L. Zunino and M. Garavaglia, “Fraunhofer diffraction by Cantor fractals with variable lacunarity,” J. Mod. Opt. 50, 717–727 (2003).
[CrossRef]

G. Saavedra, W. D. Furlan, and A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003).
[CrossRef]

1995 (1)

J. Uozumi, Y. Sakurada, and T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995).
[CrossRef]

1994 (1)

1993 (1)

1986 (1)

1977 (3)

M. De and L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[CrossRef]

M. De and L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

L. N. Hazra, “A new class of optimum amplitude filters,” Opt. Commun. 21, 232–236 (1977).
[CrossRef]

1976 (1)

L. N. Hazra and A. Banerjee, “Application of Walsh function in generation of optimum apodizers,” J. Opt. 5, 19–26 (1976).

1923 (1)

J. L. Walsh, “A closed set of normal orthogonal functions,” Am. J. Math. 45, 5–24 (1923).
[CrossRef]

Andrews, H. C.

H. C. Andrews, Computer Techniques in Image Processing (Academic, 1970).

Asakura, T.

J. Uozumi, Y. Sakurada, and T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995).
[CrossRef]

Banerjee, A.

L. N. Hazra and A. Banerjee, “Application of Walsh function in generation of optimum apodizers,” J. Opt. 5, 19–26 (1976).

Beauchamp, K. G.

K. G. Beauchamp, Walsh Functions and Their Applications (Academic, 1985).

Bhattacharya, K.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52, 035602 (2013).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Oxford, 1980).

Calatayud, A.

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21, 2701–2706 (2013).
[CrossRef]

De, M.

M. De and L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[CrossRef]

M. De and L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

Delisle, C.

Ferrando, V.

V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21, 2701–2706 (2013).
[CrossRef]

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

Fu, M.

M. Fu, G. Wade, J. Ning, and R. Jakobs, “On Walsh filtering method of decoding CPM signals,” IEEE Commun. Lett. 8, 345–347 (2004).
[CrossRef]

Furlan, W. D.

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21, 2701–2706 (2013).
[CrossRef]

J. A. Monsoriu, W. D. Furlan, and G. Saavedra, “Focusing light with fractal zone plates,” Recent Res. Devel. Optics 5, 1–15 (2005).

G. Saavedra, W. D. Furlan, and A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003).
[CrossRef]

Garavaglia, M.

L. Zunino and M. Garavaglia, “Fraunhofer diffraction by Cantor fractals with variable lacunarity,” J. Mod. Opt. 50, 717–727 (2003).
[CrossRef]

Giménez, F.

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21, 2701–2706 (2013).
[CrossRef]

Guha, A.

Han, Y.

Harmuth, H. F.

H. F. Harmuth, Transmission of Information by Orthogonal Functions (Springer-Verlag, 1972), p. 31.

Hazra, L. N.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52, 035602 (2013).
[CrossRef]

P. Mukherjee and L. N. Hazra, “Farfield diffraction properties of annular Walsh filters,” Adv. Opt. Technol. 2013, 360450 (2013).
[CrossRef]

L. N. Hazra, “Walsh filters in tailoring of resolution in microscopic imaging,” Micron 38, 129–135 (2007).
[CrossRef]

L. N. Hazra, Y. Han, and C. Delisle, “Imaging by zone plates: axial stigmatism at a particular order,” J. Opt. Soc. Am. A 11, 2750–2754 (1994).
[CrossRef]

L. N. Hazra, Y. Han, and C. Delisle, “Sigmatic imaging by zone plates,” J. Opt. Soc. Am. A 10, 69–74 (1993).
[CrossRef]

L. N. Hazra and A. Guha, “Farfield diffraction properties of radial Walsh filters,” J. Opt. Soc. Am. A 3, 843–846 (1986).
[CrossRef]

L. N. Hazra, “A new class of optimum amplitude filters,” Opt. Commun. 21, 232–236 (1977).
[CrossRef]

M. De and L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

M. De and L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[CrossRef]

L. N. Hazra and A. Banerjee, “Application of Walsh function in generation of optimum apodizers,” J. Opt. 5, 19–26 (1976).

Hopkins, H. H.

H. H. Hopkins, “Canonical and real space coordinates used in the theory of image formation,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1983), Vol. 9, p. 307.

Jakobs, R.

M. Fu, G. Wade, J. Ning, and R. Jakobs, “On Walsh filtering method of decoding CPM signals,” IEEE Commun. Lett. 8, 345–347 (2004).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).

Monsoriu, A.

Monsoriu, J. A.

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

V. Ferrando, A. Calatayud, F. Giménez, W. D. Furlan, and J. A. Monsoriu, “Cantor dust zone plates,” Opt. Express 21, 2701–2706 (2013).
[CrossRef]

J. A. Monsoriu, W. D. Furlan, and G. Saavedra, “Focusing light with fractal zone plates,” Recent Res. Devel. Optics 5, 1–15 (2005).

Mukherjee, P.

P. Mukherjee and L. N. Hazra, “Farfield diffraction properties of annular Walsh filters,” Adv. Opt. Technol. 2013, 360450 (2013).
[CrossRef]

Mukhopadhyay, S.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52, 035602 (2013).
[CrossRef]

Ning, J.

M. Fu, G. Wade, J. Ning, and R. Jakobs, “On Walsh filtering method of decoding CPM signals,” IEEE Commun. Lett. 8, 345–347 (2004).
[CrossRef]

Saavedra, G.

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

J. A. Monsoriu, W. D. Furlan, and G. Saavedra, “Focusing light with fractal zone plates,” Recent Res. Devel. Optics 5, 1–15 (2005).

G. Saavedra, W. D. Furlan, and A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28, 971–973 (2003).
[CrossRef]

Sakurada, Y.

J. Uozumi, Y. Sakurada, and T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995).
[CrossRef]

Sarkar, S.

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52, 035602 (2013).
[CrossRef]

Uozumi, J.

J. Uozumi, Y. Sakurada, and T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995).
[CrossRef]

Wade, G.

M. Fu, G. Wade, J. Ning, and R. Jakobs, “On Walsh filtering method of decoding CPM signals,” IEEE Commun. Lett. 8, 345–347 (2004).
[CrossRef]

Walsh, J. L.

J. L. Walsh, “A closed set of normal orthogonal functions,” Am. J. Math. 45, 5–24 (1923).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Oxford, 1980).

Zunino, L.

L. Zunino and M. Garavaglia, “Fraunhofer diffraction by Cantor fractals with variable lacunarity,” J. Mod. Opt. 50, 717–727 (2003).
[CrossRef]

Adv. Opt. Technol. (1)

P. Mukherjee and L. N. Hazra, “Farfield diffraction properties of annular Walsh filters,” Adv. Opt. Technol. 2013, 360450 (2013).
[CrossRef]

Am. J. Math. (1)

J. L. Walsh, “A closed set of normal orthogonal functions,” Am. J. Math. 45, 5–24 (1923).
[CrossRef]

IEEE Commun. Lett. (1)

M. Fu, G. Wade, J. Ning, and R. Jakobs, “On Walsh filtering method of decoding CPM signals,” IEEE Commun. Lett. 8, 345–347 (2004).
[CrossRef]

J. Mod. Opt. (2)

L. Zunino and M. Garavaglia, “Fraunhofer diffraction by Cantor fractals with variable lacunarity,” J. Mod. Opt. 50, 717–727 (2003).
[CrossRef]

J. Uozumi, Y. Sakurada, and T. Asakura, “Fraunhofer diffraction from apertures bounded by regular fractals,” J. Mod. Opt. 42, 2309–2322 (1995).
[CrossRef]

J. Opt. (1)

L. N. Hazra and A. Banerjee, “Application of Walsh function in generation of optimum apodizers,” J. Opt. 5, 19–26 (1976).

J. Opt. Soc. Am. A (3)

Micron (1)

L. N. Hazra, “Walsh filters in tailoring of resolution in microscopic imaging,” Micron 38, 129–135 (2007).
[CrossRef]

Opt. Acta (2)

M. De and L. N. Hazra, “Walsh functions in problems of optical imagery,” Opt. Acta 24, 221–234 (1977).
[CrossRef]

M. De and L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

Opt. Commun. (2)

L. N. Hazra, “A new class of optimum amplitude filters,” Opt. Commun. 21, 232–236 (1977).
[CrossRef]

A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu, “Fractal square zone plates,” Opt. Commun. 286, 42–45 (2013).
[CrossRef]

Opt. Eng. (1)

S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Opt. Eng. 52, 035602 (2013).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Recent Res. Devel. Optics (1)

J. A. Monsoriu, W. D. Furlan, and G. Saavedra, “Focusing light with fractal zone plates,” Recent Res. Devel. Optics 5, 1–15 (2005).

Other (6)

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1982).

M. Born and E. Wolf, Principles of Optics (Pergamon Oxford, 1980).

H. H. Hopkins, “Canonical and real space coordinates used in the theory of image formation,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1983), Vol. 9, p. 307.

H. C. Andrews, Computer Techniques in Image Processing (Academic, 1970).

H. F. Harmuth, Transmission of Information by Orthogonal Functions (Springer-Verlag, 1972), p. 31.

K. G. Beauchamp, Walsh Functions and Their Applications (Academic, 1985).

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Figures (9)

Fig. 1.
Fig. 1.

Radial Walsh functions φk(r), k=0,1,,3.

Fig. 2.
Fig. 2.

Radial Walsh functions φk(t), k=0,1,,3.

Fig. 3.
Fig. 3.

Derivation of two-valued radial Walsh functions φk(t) over the interval (0, 1) for k=0,1,31. Each two-valued Walsh function is represented by a row; elements of a row, specified by either “+” (+1) or “” (1), give values of the function over the subintervals. The number on the right of a row gives the order of the Walsh function, and the number on the left gives the order of the Walsh function from which it is derived by means of the alternating process.

Fig. 4.
Fig. 4.

Genealogical chart of self-similar groups and subgroups in radial Walsh functions.

Fig. 5.
Fig. 5.

Exit pupil and image plane in the image space of an axially symmetric imaging system.

Fig. 6.
Fig. 6.

Axial intensity distribution curves for Group I, orders 1, 3, 7, 15, and 31.

Fig. 7.
Fig. 7.

(a) Axial intensity distribution curves for Group IIA orders 2, 4, 8, and 16. (b) Axial intensity distribution curves for Group IIB, orders 6, 12, 24. (c) Axial intensity distribution curves for Group IIC, orders 14 and 28.

Fig. 8.
Fig. 8.

(a) Axial intensity distribution curves for Group IIIAA, orders 5, 11, 23. (b) Axial intensity distribution curves for Group IIIAB, orders 13 and 27. (c) Axial intensity distribution curves for Group IIIBA, orders 9 and 19.

Fig. 9.
Fig. 9.

Axial intensity distribution curves for Group IVAAA, orders 10 and 20.

Tables (1)

Tables Icon

Table 1. Composition of Self-Similar Groups and Subgroupsa

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

k=i=0γ1ki2i,
φk(r)=i=0γ1sgn[cos(2iπtki)]=φk(t),
sgn(x)=+1,x>0=0,x=0=1,x<0.
01φk(r)φ£(r)rdr=12δk£,
δk£={0,k£1,k=£.
01φk(t)φ£(t)dt=δk£.
Z(σ)=2σ+b+c,σ=1,2,3,,
F(p)=01f(r)J0(pr)rdr,
p=2πλ(nsinα)χ,
f(r)=1over the aperture=0elsewhere.
FN(p)=F(p)F(0)=201f(r)J0(pr)rdr.
f(r)=exp[ikQ(r)].
Q(r)=m=1MψmBm(r).
Bm(r)=1;rm1rrm=0;otherwise,
rm=[mM]12andrm1=[m1M]12.
W(r)=W20r2.
kW20=aπ,
a=1nλ(nsinα)2Δζ,
rj=[jJ]12andrj1=[j1J]12.
[FN(0)]Δζ=2m=1Mexp[ikψm]j=[(m1)T+1]mTexp[ikWj]rj1rjrdr,
Wj=rj1rjW(r)rdrrj1rjrdr=2j12MTW20
rj1rjrdr=Rj=[rj22rj122]=[12J].
[FN(0)]Δζ=1Jm=1Mexp[ikψm]j=[(m1)T+1]mTexp[ikWj].
IN(Δζ)=1J2m=1Mu=1Mj=[(m1)T+1]mTv=[(u1)T+1]uTcos[{kψmkψu}+{kWjkWv}].

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