Self-similarity in radial Walsh filters and axial intensity distributions in the far-field diffraction pattern

P. Mukherjee; L. N. Hazra

doi:10.1364/JOSAA.31.000379

Journal of the Optical Society of America A
Vol. 31,
Issue 2,
pp. 379-387
(2014)
•https://doi.org/10.1364/JOSAA.31.000379

Self-similarity in radial Walsh filters and axial intensity distributions in the far-field diffraction pattern

P. Mukherjee and L. N. Hazra

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P. Mukherjee and L. N. Hazra, "Self-similarity in radial Walsh filters and axial intensity distributions in the far-field diffraction pattern," J. Opt. Soc. Am. A 31, 379-387 (2014)

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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.

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Group	Subgroup	First Few Order Members of the Subgroup					$b$	$c$	$Z (σ)$
I	—	1	3	7	15	31	0	$- 1$	$2^{σ} - 1$
I	—	(0)	(0)	(0)	(0)	(0)	0	$- 1$	$2^{σ} - 1$
II	A	2	4	8	16	32	0	0	$2^{σ}$
	A	(1)	(3)	(7)	(15)	(31)	0	0	$2^{σ}$
	B	6	12	24	48	96	1	$2^{σ}$	$2^{σ + 1} + 2^{σ}$
	B	(1)	(3)	(7)	(15)	(31)	1	$2^{σ}$	$2^{σ + 1} + 2^{σ}$
	C	14	28	56	112	224	2	$2^{σ + 1} + 2^{σ}$	$2^{σ + 2} + (2^{σ + 1} + 2^{σ})$
	C	(1)	(3)	(7)	(15)	(31)	2	$2^{σ + 1} + 2^{σ}$	$2^{σ + 2} + (2^{σ + 1} + 2^{σ})$
	D	30	60	88	176	352	3	$2^{σ + 2} + (2^{σ + 1} + 2^{σ})$	$2^{σ + 2} + [2^{σ + 2} + (2^{σ + 1} + 2^{σ})]$
	D	(1)	(3)	(7)	(15)	(31)	3	$2^{σ + 2} + (2^{σ + 1} + 2^{σ})$	$2^{σ + 2} + [2^{σ + 2} + (2^{σ + 1} + 2^{σ})]$
	…	…	…	…	…	…	…	…	…
III	AA	5	11	23	47	95	1	$2^{σ} - 1$	$2^{σ + 1} + (2^{σ} - 1)$
	AA	(2)	(4)	(8)	(16)	(32)	1	$2^{σ} - 1$	$2^{σ + 1} + (2^{σ} - 1)$
	AB	13	27	55	111	223	2	$2^{σ + 1} + (2^{σ} - 1)$	$2^{σ + 2} + [2^{σ + 1} + (2^{σ} - 1)]$
	AB	(2)	(4)	(8)	(16)	(32)	2	$2^{σ + 1} + (2^{σ} - 1)$	$2^{σ + 2} + [2^{σ + 1} + (2^{σ} - 1)]$
	AC	29	59	119	239	479	3	$2^{σ + 2} + [2^{σ + 1} + (2^{σ} - 1)]$	$2^{σ + 3} + [2^{σ + 2} + 2^{σ + 1} + 2^{σ} - 1]$
	AC	(2)	(4)	(8)	(16)	(32)	3	$2^{σ + 2} + [2^{σ + 1} + (2^{σ} - 1)]$	$2^{σ + 3} + [2^{σ + 2} + 2^{σ + 1} + 2^{σ} - 1]$
	…	…	…	…	…	…	…	…	…
	BA	9	19	39	79	159	2	$2^{σ} - 1$	$2^{σ + 2} + (2^{σ} - 1)$
	BA	(6)	(12)	(24)	(48)	(96)	2	$2^{σ} - 1$	$2^{σ + 2} + (2^{σ} - 1)$
	BB	25	51	103	207	415	3	$2^{σ + 2} + (2^{σ} - 1)$	$2^{σ + 3} + [2^{σ + 2} + (2^{σ} - 1)]$
	BB	(6)	(12)	(24)	(48)	(96)	3	$2^{σ + 2} + (2^{σ} - 1)$	$2^{σ + 3} + [2^{σ + 2} + (2^{σ} - 1)]$
	…	…	…	…	…	…	…	…	…
	CA	17	35	71	143	287	3	$2^{σ} - 1$	$2^{σ + 3} + (2^{σ} - 1)$
	CA	(14)	(28)	(56)	(112)	(224)	3	$2^{σ} - 1$	$2^{σ + 3} + (2^{σ} - 1)$
	…	…	…	…	…	…	…	…	…
IV	AAA	10	20	40	80	160	2	$2^{σ}$	$2^{σ + 2} + 2^{σ}$
	AAA	(5)	(11)	(23)	(47)	(95)	2	$2^{σ}$	$2^{σ + 2} + 2^{σ}$
	AAB	26	52	104	208	416	3	$2^{σ + 2} + 2^{σ}$	$2^{σ + 3} + (2^{σ + 2} + 2^{σ})$
	AAB	(5)	(11)	(23)	(47)	(95)	3	$2^{σ + 2} + 2^{σ}$	$2^{σ + 3} + (2^{σ + 2} + 2^{σ})$
	…	…	…	…	…	…	…	…	…
	ABA	18	36	72	144	288	3	$2^{σ}$	$2^{σ + 3} + 2^{σ}$
	ABA	(13)	(27)	(55)	(111)	(223)	3	$2^{σ}$	$2^{σ + 3} + 2^{σ}$
	…	…	…	…	…	…	…	…	…
	BAA	22	44	88	176	352	3	$2^{σ + 2} - 2^{σ}$	$2^{σ + 3} + (2^{σ + 2} - 2^{σ})$
	BAA	(9)	(19)	(39)	(79)	(159)	3	$2^{σ + 2} - 2^{σ}$	$2^{σ + 3} + (2^{σ + 2} - 2^{σ})$
	…	…	…	…	…	…	…	…	…
V	AAAA	21	43	87	175	351	3	$2^{σ + 2} - (2^{σ} + 1)$	$2^{σ + 3} + [2^{σ + 2} - (2^{σ} + 1)]$
	AAAA	(10)	(20)	(40)	(80)	(160)	3	$2^{σ + 2} - (2^{σ} + 1)$	$2^{σ + 3} + [2^{σ + 2} - (2^{σ} + 1)]$
	…	…	…	…	…	…	…	…	….

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Journal of the Optical Society of America A