Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction

H. Hamam; J. L. de Bougrenet de la Tocnaye

doi:10.1364/JOSAA.12.001920

Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction

H. Hamam and J. L. de Bougrenet de la Tocnaye

Journal of the Optical Society of America A
Vol. 12,
Issue 9,
pp. 1920-1931
(1995)
•https://doi.org/10.1364/JOSAA.12.001920

Get Citation
Copy Citation Text
H. Hamam and J. L. de Bougrenet de la Tocnaye, "Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction," J. Opt. Soc. Am. A 12, 1920-1931 (1995)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (536 KB)
Set citation alerts
Save article

Related Topics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: November 1, 1994
- Revised Manuscript: February 2, 1995
- Manuscript Accepted: April 12, 1995
- Published: September 1, 1995

Not Accessible

Your account may give you access

Abstract

The efficient Fresnel-transform algorithm (EFTA) is a computational tool that facilitates diffracted field analysis. The computational improvement that is achieved with the EFTA relative to the conventional double fast-Fourier-transform (FFT) algorithm results from the properties of fractional Fresnel diffraction. Some programming simplicities are shown to appear at numerous locations along the propagation axis. The number of those locations is determined by the fractional order. Some computational aspects of the method are presented and compared with those of the FFT algorithm.

Full Article | PDF Article

More Like This

Binary logic based purely on Fresnel diffraction

H. Hamam and J. L. de Bougrenet de la Tocnaye
Appl. Opt. 34(26) 5901-5906 (1995)

Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms

Bryan M. Hennelly and John T. Sheridan
J. Opt. Soc. Am. A 22(5) 917-927 (2005)

Discrete fractional Fourier transform as a fast algorithm for evaluating the diffraction pattern of pulsed radiation

Magdy Tawfik Hanna, Amr Mohamed Shaarawi, Nabila Philip Attalla Seif, and Waleed Abd El Maguid Ahmed
J. Opt. Soc. Am. A 28(8) 1610-1619 (2011)

References

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Figures (6)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Tables (2)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Equations (89)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Metrics

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

	p (a/q)

Fractional order	1 (4)	1 (8)	3 (8)	1 (16)	3 (16)	1 (32)	3 (32)	5 (32)	7 (32)
0	$\frac{1}{\sqrt{2}} exp (i \frac{π}{4})$	$\frac{1}{2} exp (i \frac{3 π}{4})$	$\frac{1}{2} exp (- i \frac{3 π}{4})$	$\frac{1}{\sqrt{8}} exp (- i \frac{π}{4})$	$\frac{1}{\sqrt{8}} exp (i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{3 π}{4})$	$\frac{1}{4} exp (i \frac{3 π}{4})$	$\frac{1}{4} exp (i \frac{π}{4})$
1	$\frac{1}{\sqrt{2}} exp (- i \frac{π}{4})$	1/2	1/2	$\frac{1}{\sqrt{8}} exp (i \frac{7 π}{8})$	$\frac{1}{\sqrt{8}} exp (i \frac{5 π}{8})$	$\frac{1}{4} exp (i \frac{13 π}{16})$	$\frac{1}{4} exp (i \frac{15 π}{16})$	$\frac{1}{4} exp (i \frac{9 π}{16})$	$\frac{1}{4} exp (i \frac{11 π}{16})$
2		$\frac{1}{2} exp (- i \frac{π}{4})$	$\frac{1}{2} exp (i \frac{π}{4})$	$\frac{1}{\sqrt{8}} exp (i \frac{π}{4})$	$\frac{1}{\sqrt{8}} exp (- i \frac{π}{4})$	1/4	1/4	1/4	1/4
3		1/2	1/2	$\frac{1}{\sqrt{8}} exp (- i \frac{π}{8})$	$\frac{1}{\sqrt{8}} exp (- i \frac{3 π}{8})$	$\frac{1}{4} exp (- i \frac{11 π}{16})$	$\frac{1}{4} exp (i \frac{7 π}{16})$	$\frac{1}{4} exp (- i \frac{15 π}{16})$	$\frac{1}{4} exp (i \frac{3 π}{16})$
4						$\frac{1}{4} exp (i \frac{3 π}{4})$	$\frac{1}{4} exp (i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{3 π}{4})$
5						$\frac{1}{4} exp (i \frac{5 π}{16})$	$\frac{1}{4} exp (- i \frac{9 π}{16})$	$\frac{1}{4} exp (i \frac{π}{16})$	$\frac{1}{4} exp (- i \frac{13 π}{16})$
6						1/4	1/4	1/4	1/4
7						$\frac{1}{4} exp (- i \frac{3 π}{16})$	$\frac{1}{4} exp (- i \frac{π}{16})$	$\frac{1}{4} exp (- i \frac{7 π}{16})$	$\frac{1}{4} exp (- i \frac{5 π}{16})$

q

Number of Samples	1	2	4	8	16	32	64	128	256	512	1024	2048	4096
					s = 4, A = 0		s = 8, A = 0		s = 16, A = 4		s = 32, A = 32		s = 64, A = 176

(C/N)	0	0	0	1	2	4	2	4	4.5	8.5	10	18	21.5

	p (a/q)

Fractional order	1 (4)	1 (8)	3 (8)	1 (16)	3 (16)	1 (32)	3 (32)	5 (32)	7 (32)
0	$\frac{1}{\sqrt{2}} exp (i \frac{π}{4})$	$\frac{1}{2} exp (i \frac{3 π}{4})$	$\frac{1}{2} exp (- i \frac{3 π}{4})$	$\frac{1}{\sqrt{8}} exp (- i \frac{π}{4})$	$\frac{1}{\sqrt{8}} exp (i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{3 π}{4})$	$\frac{1}{4} exp (i \frac{3 π}{4})$	$\frac{1}{4} exp (i \frac{π}{4})$
1	$\frac{1}{\sqrt{2}} exp (- i \frac{π}{4})$	1/2	1/2	$\frac{1}{\sqrt{8}} exp (i \frac{7 π}{8})$	$\frac{1}{\sqrt{8}} exp (i \frac{5 π}{8})$	$\frac{1}{4} exp (i \frac{13 π}{16})$	$\frac{1}{4} exp (i \frac{15 π}{16})$	$\frac{1}{4} exp (i \frac{9 π}{16})$	$\frac{1}{4} exp (i \frac{11 π}{16})$
2		$\frac{1}{2} exp (- i \frac{π}{4})$	$\frac{1}{2} exp (i \frac{π}{4})$	$\frac{1}{\sqrt{8}} exp (i \frac{π}{4})$	$\frac{1}{\sqrt{8}} exp (- i \frac{π}{4})$	1/4	1/4	1/4	1/4
3		1/2	1/2	$\frac{1}{\sqrt{8}} exp (- i \frac{π}{8})$	$\frac{1}{\sqrt{8}} exp (- i \frac{3 π}{8})$	$\frac{1}{4} exp (- i \frac{11 π}{16})$	$\frac{1}{4} exp (i \frac{7 π}{16})$	$\frac{1}{4} exp (- i \frac{15 π}{16})$	$\frac{1}{4} exp (i \frac{3 π}{16})$
4						$\frac{1}{4} exp (i \frac{3 π}{4})$	$\frac{1}{4} exp (i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{π}{4})$	$\frac{1}{4} exp (- i \frac{3 π}{4})$
5						$\frac{1}{4} exp (i \frac{5 π}{16})$	$\frac{1}{4} exp (- i \frac{9 π}{16})$	$\frac{1}{4} exp (i \frac{π}{16})$	$\frac{1}{4} exp (- i \frac{13 π}{16})$
6						1/4	1/4	1/4	1/4
7						$\frac{1}{4} exp (- i \frac{3 π}{16})$	$\frac{1}{4} exp (- i \frac{π}{16})$	$\frac{1}{4} exp (- i \frac{7 π}{16})$	$\frac{1}{4} exp (- i \frac{5 π}{16})$

q

Number of Samples	1	2	4	8	16	32	64	128	256	512	1024	2048	4096
					s = 4, A = 0		s = 8, A = 0		s = 16, A = 4		s = 32, A = 32		s = 64, A = 176

(C/N)	0	0	0	1	2	4	2	4	4.5	8.5	10	18	21.5

Abstract

References

Cited By

Figures (6)

Tables (2)

Equations (89)

Metrics

Login or Create Account