SE ESTUDIA LA CINTURA CINTURA DE UN HAZ GAUSSIANO EN TERMINO DE LA TRANSFORMADA FRACCIONAL

1. Waist-to-waist transformation of Gaussian beams using
the fractional Fourier transform
C. O. Torres∗
, L. Mattos∗
, C. Jiménez†
, J. Castillo†
and Y. Torres‡
∗
Laboratorio de Optica e Informática, Universidad Popular del Cesar. Valledupar-Colombia.
†
Grupo de Fı́sica, Universidad de la Guajira. Riohacha-Colombia.
‡
Grupo de Optica y Tratamiento de Señales, Universidad Industrial de Santander. Bucaramanga-Colombia.
Abstract. Because of the importance of Gaussian beams in optics, particularly in the area of lasers, we devote the
present paper to a study of the propagation of these beams. The Gaussian beam, also called a Gaussian spherical
wave field, is basically a spherical wave field whose modulus in a plane transverse to the propagation direction varies
in a Gaussian fashion. The fractional Fourier transforms (FRFT) were originally introduced in 1980 by Namias as a
technique for solving theoretical physical problems. Since then lot of works have been done on its properties, optical
implementations and applications; the fractional Fourier transform became an important tool in optics. It is of
singular importance in the laser physics the study of the propagation and transformation of Gaussian beams. In this
paper the fractional Fourier transform is applied to described the beam waist-to-waist transformation of Gaussian
beams between input and output reference surfaces; for example, the waist-to-waist transformation of a Gaussian
beam passing through a thin lens is necessary for beam focusing or modematching. Using the Collins formula and
the fractional Fourier transform we can obtain the clear physical picture and simple calculation to study the prop-
agation as well as the transformation of Gaussian beam, scaled variables and scaled field amplitudes are defined by
complying with mathematical consistency; this relation provide a convenient way for analyzing and calculating the
beam waist-to-waist transformation of Gaussian beams in the ABCD optical system.
Keywords: Waist-to-waist transformation; Gaussian beams; Fractional Fourier transform.
INTRODUCTION
The study of the ray transfer matrix is particularly useful to simplificate the analysis of the optical situations;
in this case many optical configurations can be treated with the Collins formula [1,2], this formula gives the
relationship between input and output amplitude complex of a light field in paraxial approach. The fractional
Fourier transform may offer the description of the diffraction in the free space [3,4]. We show the way is
resulting from the combination of the ray transfer matrix and the fractional Fourier transform results a new
approach suitable for the study of Huygens principle, laser resonators and simple obtaining of the fractional
Fourier transform in optical structures.
In the corresponding literature they are works where the possibility of using the property of composition of
the fractional Fourier transformed to describe the presence of the Huygens principle; however it has not been
proven in an explicit way as it is suggested of the articles that approach the topic, in this article it is shown as
using this property to show in a direct way the relationship among the composition property and the Huygens
principle, besides illustrating this situation with an example. In the same way the topic optical resonators it
has been developed in multiple articles where diverse strategies are used to get the laser resonators stability
condition, but none shows as the illumination with spherical wave, free space propagation and observation
over spherical surface, besides being a consequence of the Huygens principle, it can be interpreted as the result
of the application of the self consistency field theory, facilitating the obtaining of mathematical relationship
among the two spherical surfaces of where the resonators stability condition is deduced, and then the beam
parameters are calculated. In the case of the optic systems and propagation, with the help of illumination
with spherical wave in the input plane a general condition is obtained for the optic systems that he allows
to deduce a general law to establish a relationship of fractional transform in terms of the elements of the
163
CREDIT LINE (BELOW) TO BE INSERTED ON THE FIRST PAGE OF EACH PAPER
EXCEPT THE PAPER ON PP. 507 - 512
CP992, RIAO/OPTILAS 2007, edited by N. U. Wetter and J. Frejlich
© 2008 American Institute of Physics 978-0-7354-0511-0/08/$23.00
CREDIT LINE (BELOW) TO BE INSERTED ONLY ON THE FIRST PAGE OF
THE PAPER ON PP. 507 - 512

2. ray transfer matrix, that which opens the possibility of other assemblies only with the consideration of the
elements of the ray transfer matrix system and the application of the mathematical condition found in this
paper.
THE FRACTIONAL FOURIER TRANSFORM AND THE GENERALIZED
COLLINS FORMULA
The well-known Collins formula can be rewritten as:
UP (u; v) =
−i
¸B
exp
Ã
i¼D
¡
u2
+ v2
¢
¸B
!
Z
R2
Z
R2
exp
Ã
i¼A
¡
»2
+ ´2
¢
¸B
!
exp
µ
−2i¼ (u:» + v:´)
¸B
¶
UA (»; ´) d»d´ (1)
From this expression a conventional standard Fourier transform (® = 1)[5] multiplied by a factor phase;
between the output field complex amplitude UP (u; v) in the plane P and the input field complex amplitude
UA (»; ´), in the plane A can be obtained if and only if A = 0.
Furthermore illuminating with spherical wave of radius R1
A and introducing appropriate input and output
scale parameters; by making use of the fractional Fourier transform and comparing with (1) the following
expression is obtained:
UP (u; v) =
2¼sen®
i¸ (B)
exp
⎡
⎣
⎛
⎝D
B
−
cos2
®
A
³
1 − B
R1
´
B
⎞
⎠
i¼
¡
u2
+ v2
¢
¸
⎤
⎦
h
exp
h
−i
³¼
4
−
®
2
´ii2
F®
[UA (»; ´)] (2)
An interesting fact is that exist a fractional Fourier transform F(®)
relation of order ® between the output field
complex amplitude UP (u; v) and the input field complex amplitude UA (»; ´). The phase factor is associated
in paraxial approach with a divergent wave; therefore the field complex amplitude over the output UP (u; v)
is over spherical surface with radius R2 and proportional to the fractional Fourier transform of order ® of
the input field complex amplitude UA (»; ´)[6,7]. Where:
R2 =
−AB
³
1 − B
R1
´
AD
³
1 − B
R1
´
− cos2®
(3)
This relationship establishes the general obtainment condition of a fractional transform between emitter and
spherical detectors when is known the ray transfer matrix of the optical system and their physical meaning
it corresponds to the presence of the Huygen’s principle. for the particular case of emitter flat R1 → ∞, the
fractional Fourier transform is over a spherical surface with raidus R2 = −AB
AD−cos2® .
In the general form; when it is wanted to obtain receptor flat, the fractional Fourier transform of orfer ® can
be obtained illuminating with spherical emitter of radius R1 calculated of:
AD
µ
1 −
B
R1
¶
= cos2
® (4)
In the particular situation to obtain a fractional Fourier transform [8] over the plane surface its necesary
that B different to zero, ® different to zero and:
164

3. AD = cos2
® (5)
That which means that if one knows the elements of the ray transfer matrix, one can obtain the condition
of the optic system for which the fractional Fourier transform is reached.
In summary the fractional Fourier transfom provides a compact and powerful formulation for wave propa-
gation in an optic system whose ray transfer matrix it is known.
THE TRANSFER MATRIX FOR SCALED FRACTIONAL FOURIER
TRANSFORM IN THE WAIST-TO-WAIST TRANSFORMATION OF A
GAUSSIAN BEAM
Assume an incident optical field is:
UA (»; ´) = exp
"
−
¡
»2
+ ´2
¢
w2
01
#
(6)
Figure 1. Transformation of a Gaussian beam passing through a thin lens.
The whole transfer matrix is:
Ã
1 − d2
f d1 + d2 − d1d2
f
− 1
f 1 − d1
f
!
UP (u; v) =
2¼sen®
i¸
³
d1 + d2 − d1d2
f
´
h
exp
h
−i
³¼
4
−
®
2
´ii2
F®
"
exp
"
−
¡
»2
+ ´2
¢
w2
01
##
(7)
The former analysis leads to a classical result, the resonator stability condition:
µ
1 −
d2
f
¶ µ
1 −
d1
f
¶
= cos2
® (8)
By Equation (8) we have:
d1 = f
⎡
⎣1 −
cos2
®
³
1 − d2
f
´
⎤
⎦ (9)
165

4. Inserting the scaled variables » ⇒ − u
f
∙
1− cos2 ®
¡
1−
d2
f
¢
¸
¸
y ´ ⇒ − v
f
∙
1− cos2 ®
¡
1−
d2
f
¢
¸
¸
into Equation (9), one obtains:
using the beam Gaussian beam inavariance to the fractional Fourier transform:
UP (u; v) =
2¼sen®
i¸
³
d1 + d2 − d1d2
f
´
h
exp
h
−i
³¼
4
−
®
2
´ii2
⎡
⎢
⎢
⎢
⎣
exp
⎡
⎢
⎢
⎢
⎣
−
¡
u2
+ v2
¢
w2
01f2
∙
1 − cos2 ®
¡
1−
d2
f
¢
¸2
¸2
⎤
⎥
⎥
⎥
⎦
⎤
⎥
⎥
⎥
⎦
(10)
The Equation (10) can be written as:
UP (u; v) =
2¼sen®
i¸
³
d1 + d2 − d1d2
f
´
h
exp
h
−i
³¼
4
−
®
2
´ii2
"
exp
"
−
¡
u2
+ v2
¢
w2
02
##
(11)
The resulting output waist radii of Gaussian beams at the output plane can be expressed by:
w2
02 = f2
¸2
⎡
⎣1 −
cos2
®
³
1 − d2
f
´
⎤
⎦
2
w2
01 (12)
Now let us consider some particular cases:
When ® = ¼
2 ; d1 = d2 = f; Then:
UP (u; v) =
2¼
i¸f
∙
exp
µ
−
u2
+ v2
w2
02
¶¸
(13)
Where:
w02 = f¸w01 (14)
When ® = 0; d1 = d2 = 0 Then:
UP (u; v) = exp
"
−
¡
u2
+ v2
¢
w2
02
#
(15)
Where:
w02 = w01 (16)
When ® = ¼; 1
d1
+ 1
d2
= 1
f parity operator, then:
UP (u; v) = exp
"
−
¡
u2
+ v2
¢
w2
02
#
(17)
The resulting output waist radii of Gaussian beams can be expressed by:
166

5. w02 =
⎡
⎣−
¸d2
³
1 − d2
f
´
⎤
⎦w01 (18)
We know, a thin lens is extensively used on the transformation of Gaussian beams for focusing or mode
matching. The above results give us a direct relation between the beam waist radius and the parameters of
lens (focusing length, and position). The above analyses [9,10] have shown that:
1. A beam waist in the front focal plane always produces a beam waist in the back focal plane.
2. When d1 = d2 = 0; we have shown that the width of the waist produced by the lens, the right minimum
beam width is obtained.
3. When d1 = d2 = f; the right maximum beam width is obtained.
4. When d1 = d2 6= 0 and d1 = d2 6= f; always produces inverted beam width.
5. When d1 = d2 and f
2 ≤ d1 < f; always produces a image that increases of size when d1 tends a focal
distance.
6. When d1 = d2 and 0 < d1 < f
2 ; always produces a image that decreases of size when d1 approaches the
position of the lens.
Examinations of equation (18) shows that beam waists are not imaged into beam waists in the usual
sense, as the initial waist is moved toward the lens, the second waist may also move toward the lens rather
than away from it as we might expect from normal image considerations.
CONCLUSIONS
In this paper we have analyzed the beam waist-to-waist transformation of Gaussian beam between input
and output reference planes by the scaled fractional Fourier transform. In other words,the waist-to-waist
transformation of the Gaussian beams is just a scaled fractional Fourier transform.
REFERENCES
1. D. Zhao, “Multi-element resonators and scaled fractional Fourier transforms,” Opt. Comunn. 168, 85-88 (1999).
2. J. Hua, L. Liu and G. Li, ”Scaled fractional Fourier transform and its optical implementation” Appl. Opt. 36,
8490-8492 (1997).
3. C.O. Torres, Y. Torres and P. Pellat—Finet, Electronic Edition. Academia Colombiana de Ciencias Exactas,
Físicas y Naturales, paper 090,Santafé de Bogotá, October, (1998).
4. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform”, Opt. Let., 19, pp 1388- 1390,
1994.
5. W. Goodman, “Introduction to Fourier optics,” McGraw-Hill, New York, Chap. 3, p. 34, (1996).
6. X. Wang and J. Zhou, “Scaled fractional Fourier transform and optical system,” Opt. Comunn. 147, 341-348
(1998).
7. W A. W. Lohmann, Z. Zalevsky, R. G. Dorsh and D. Mendlovic, “Experimental considerations and scaling
property of the fractional Fourier transform,” Opt. Comunn. 146, 55-61 (1998).
8. A. W. Lohmann, “A fake lens for fractional Fourier experiments,” Opt. Com. 115, 437-443 (1995).
9. J. D. Gaskill, “Linear system, Fourier transforms; and optics,” John wiley sons, New York, Chap. 10, p. 420,
(1980).
10. Fan Ge, Shaomin Wang, Daomu Zhao ”The representation of waist-to-waist transformation of Gaussian beams
by the scaled fractional Fourier transform”. , Optics and laser technology. Article in press.
167