Optics and Photonics Journal, 2012, 2, 318-325
http://dx.doi.org/10.4236/opj.2012.24039 Published Online December 2012 (http://www.SciRP.org/journal/opj)
Propagation of Modified Bessel-Gaussian Beams in a
Misaligned Optical System
Lahcen Ez-Zariy1, Hamid Nebdi1, El Hassane Bentefour2, Abdelmajid Belafhal1*
1Laboratoire de Physique Nucléaire, Atomique et Moléculaire Département de Physique, Faculté des Sciences,
Université Chouaïb Doukkali, El Jadida, Morocco
2LigthOnKnowledge Foundation, Chicago, USA
Received August 15, 2012; revised September 15, 2012; accepted September 30, 2012
ABSTRACT
The formalism of generalized diffraction integral for paraxial misaligned optical systems is used to investigate the
propagation of the Modified Bessel-Gaussian (MBG) beam through a misaligned thin lens. The properties of the
propagation of MBG beam traveling through this misaligned ABCD optical system are discussed. A special case of
misaligned circular thin lens is illustrated analytically and numerically. The shape of the MBG beam at the exit of the
misaligned optical system is unchanged; however the center of the beam is shifted from the propagation axis in corre-
lated manner with the design parameters of the optical system.
Keywords: Generalized Diffraction Integral; Modified Bessel-Gaussian Beams; Propagation; Misaligned ABCD
Optical Systems; Circular Thin Lens
1. Introduction
Modified optical beams, such as zero central intensity
beams and vortex beams, gained increasing importance
in the recent years [1,2]. Among their applications are the
guiding and manipulating of particle beam such as in
atoms interferometry, atoms trapping and cooling, and in
wireless Tera-bite-speed information transport. Particular
attention is given to the propagation of Bessel and Modi-
fied Bessel beams, which their apodization by a Gaussian
transmittance leads to the so called, in literature, Modi-
fied Bessel-Gaussian beams (referred in the follows as
MBG beams).
Several studies have examined the propagation of
MBG beams through a turbulent atmosphere [3,4], and
through aligned optical systems [5-11]. These studies and
others [12-21] intend to comprehend practical matters
such as the impact of imperfections in the optical systems
on the propagation of the MBG beams. Such imperfec-
tions, which translate into misalignment of the optical
systems, may be due in one hand to the design and
manufacturing mishaps and in other hand due to external
perturbations such as thermally induced deformations
and or the accumulation of dust and fingerprints, etc. In
this perspective, the propagation of nondiffracting and
pseudo-nondiffracting Laser beams through misaligned
optical systems is of great interest.
Recently, Zhao et al. [17] have used the generalized
diffraction integral to derive a generalized formula for
high-order Bessel-Gaussian beams through a misaligned
first order ABCD optical system. In 2009, Chafiq et al.
[19] have applied the generalized diffraction integral to
develop an analytical formalism for generalized Mathieu-
Gauss beams passing through an aperture misaligned
optical system. Moreover, in a recent work, Belafhal et al.
[20] have studied the propagation of truncated Bessel
modulated Gaussian (QBG) beams traveling any mis-
aligned ABCD optical system using the generalized dif-
fraction integral formalism.
To our knowledge, the propagation of MBG beams
through misaligned optical systems has not been studied
yet. In this paper, we use the generalized diffraction in-
tegral introduced by Wang and Ronchi [22], and all our
previous work, in this field, to develop the analytical
solution of the propagation of MBG beams traveling
through any misaligned optical system. For illustration,
we apply our solution to the particular case of the propa-
gation of MBG beams through a misaligned lens. More
interestingly, we demonstrate that our analytical result is
a generated formalism which allows for retrieving the
solution of the propagation of pure Bessel, Gaussian and
of Bessel-Gaussian beams through a misaligned optical
system [17] as particular cases.
The paper is organized as follows: in Section 2, we
calculate the analytical equation of MBG beams passing
*Corresponding author.
C
opyright © 2012 SciRes. OPJ

L. EZ-ZARIY ET AL. 319
through any misaligned optical system ABCD. Then,
from this equation, we derive the solutions for the special
cases of the Bessel, Gaussian and Bessel-Gaussian beams
propagating through the same optical system. In Section
3, we use the main expression of Section 2 to perform
detailed analysis of the propagation of MBG through a
misaligned thin lens analytically and numerically.
2. Propagation of MBG Beams through
Misaligned Paraxial Optical System
MBG beams are solutions of the 3D Helmholtz equation
in cylindrical coordinates. These beams are the general-
ized Modified Bessel beams modulated by Gaussian
transmittance. The field distribution of MBG beams in
the source plane situated at z = 0 is composed by the
product of a Gaussian function part and a summation
over the modified Bessel functions of different ampli-
tudes An. This field is expressed by [11]



00
2
00
,,0
expexp i,
N
nn
n
Er
krAn Iar

  


0B
(1)
where n denotes the order of modified Bessel function In,
aB is the width parameter, 2π
k

 is the wave number
with

being the wavelength,


2
1i
2
s
s
F
k
 ,
where s

and
s
F
respectively refer to radial Gaussian
source and focusing parameter,

i1.
In Figure 1, we present the plots of the incident single
and multiple MBG beams intensity at different orders
and with various amplitude coefficients. Figure 1(a)
shows a single modified Bessel-Gaussian beam of order
10 which can be obtained from summation in Equation (1)
by setting n = N = 10. The diagram of the considered
figure shows the single modified Bessel beam of order 10
which is a doughnut beam (this observation holds true for
any single beam of order n > 0). Figure 1(b) displays an
example of a superposition of the modified Bessel-
Gaussian beams for different odd orders, n, and with
various amplitudes, An. From this figure, it appears that
the beam is a dark hollow beam, and the plot presents
two principal lobes surrounded by secondary lobes. In
addition, we note that the considered figure presents a
(a)
(b)
Figure 1. Normalized transverse irradiance distributions of input single and multiple MBG beams at various amplitude s: (a)
; (b) 10 10
n
nA






1357 10.50.21
n
nA
.

L. EZ-ZARIY ET AL. 320
y
O
z
z
Misaligned
MBG beam
Input plane Output plane
x
Input MBG
beam
ε
y
ε
y
ε
x
ε
z
m
RP
2
RP
1
RP
1 m
RP
2 m
A
B
CD



O
2 m
ε
εʹ
Figure 2. Schematic representation of MBG beams propagation trough a misaligned paraxial ABCD optical system.
symmetry in the lobes, because the products of
generate terms defining the azi-
muthal intensity as

00 00
,*,urqu rq



0
cos 2.nq
Let us now consider a misaligned ABCD optical sys-
tem illuminated by a MBG beam as shown in Figure 2.
RP1,2 are the alignment reference planes, RP1m,2m are the
misalignment reference planes,
 denotes the transverse
offset and
 ' is the tilted angle. A, B, C and D are the ma-
trix transfer elements of the aligned optical system.
In a cylindrical coordinates system, the genera-
lized diffraction integral formula between input electric
field and output plane electric field

000
,, 0Er z  
, ) z(,Er

is given by [17]
  

 
2π
00
00
22
00 00
00 000
i
,,expi, ,0
2π
i
exp2 coscos
2
sincossind d
k
Er zkzErz
B
kArrrDr Er
B
FrGrHrr r

0




 

 




(2a)
The misalignment parameters E, F, G, and H charac-
terizing the optical system are given by
 
21 ,
x
EAzB
x



  

(2b)
 
21 ,
y
FAzB
y


 
 

(2c)


21
21
x
,
x
GBCDA
BDDzB


 


 

(2d)


21
21
y
y
HBCDA
BDDzB


  


 

(2e)
where
x

,
x

,
y

and
y


denote the two-dimen-
sional misalignment parameters.
x

and
y

are the
displacement element in x and y direction, respectively.
x

 and
y

 are the tilting angle of the element in x and
y direction, respectively. The parameters
x

and
y

are given by
cos
x




, (2f)
and
sin
y




. (2g)
Here

is the misaligned azimuth angle of the opti-
cal elements in a cylindrical coordinates system. Also
x


and
y


are given by

sinarctancostan' ,
yx


 

 (2h)
and


arctansintan' .
x


 (2i)
Substituting Equation (1) into Equation (2a), we get



  
i2
2
00
0
2π
00
0
0000 000
,,
ii
e expcossin
2π2
i
exp 2
i
exp iexp2cos
2
cossindd.
kz
N
nnB
n
Er z
kk
DrG rHr
BB
k
AAkrIar
B
k
nrr
B
ErFrr r

0


























  
 





 




(3)
For solving the double integral of Equation (3), one
introduce a new parametric angle and a new radial
coordinate


which satisfy the following relations






22
cos 2
cos
cos2 sin2
rE
rErF





 
, (4a)


   
22
sin 2
sin
cos2 sin2
rF
rErF





 
, (4b)
and

L. EZ-ZARIY ET AL. 321
   
22
cos2 sin2rErF   
. (4c)
After a tedious calculations and with help of the below
integral formulae [23],


2π
0
exp icosid
2πiexp i
mm
xm
mJx



 





(5a)
and
  
222
0
1
edexpwithRe0,Re1 ,
242
x
xIxJxx J

 
 
 
 


 






 (5b)
the receiver electric field of a MBG beam passing by any misaligned ABCD optical system, is expressed by


i2
22
ii
,,e expcossin
2i 2
1π
expexpi (),
242 2
kz
N
nn
n
kk
ErzDr GrHr
kB kAB
An J


 

 




 








 



(6)
where
i2
kA
kB   , (7a)
k
B

 , (7b)
and
B
a


. (7c)
Taking in account the expression of
 quation (6)
can be written as

  






i2
22
2
2
2
22
ii
,,e expcossin
2i 2
2cos 2sin
exp
4i16 i
22
2cos 2sin
π
exp i24i
2
kz
B
NB
nn
n
kk
Er zDr GrHr
kB kAB
krE rF
a
kA kA
kBk
BB
ka rErF
An JkA
Bk B














 



  

 

 


 



  

 


.







(8)
This formula is the general analytical equation of out-
put electric field of a MBG beam traveling a misaligned
paraxial ABCD optical system and it’s the main result of
this paper. From Equations (4c) and (8), we can easily
deduce that the beam obtained at the misaligned plane
after the optical system becomes decentered. The posi-
tion of the center of the output beam is shifted from the
center of emitted plane beam by E/2 in x-direction and
by F/2 in y-direction.
The validity of the present work withstands the gener-
alized formalism obtained in the previous investigations
about propagating of pure Bessel beams, pure Gaussian
beams and Bessel-Gaussian beams through a misaligned
optical system [20].
Taking in account the following variables changing
Nn, (9a)

0
in
n
A
E

, (9b)
i
B
Z
a


, (9c)
2
1
kw 

, (9d)
Equation (1) becomes


 
00
2
0
00 0
2
0
,,0
expexp i,
nZ
Er
r
EJ rn
w








(10)
that is the high order Bessel-Gaussian beam which is the
incident electric field expressed by Equation (7) of Ref.
[17]. By substituting Equations (9) in Equation (8) of our
nvestigation, we obtain i

L. EZ-ZARIY ET AL. 322
 


i2 0
2
0
22
2
2
2
22
00
22
2
0
ii
, ,eexpcossinexpi
2
1
2i
2
2cos 2sin
exp
11
4i16 i
22
i2cos 2sin
1
4i
2
Z
kz
BG
Z
n
kE k
ErzDrGrHrn
B
kA
BB
w
krE rF
kA kA
B
BB
ww
kr ErF
JkA
Bw




 













 


 

 


 

 

 


 

.
B










(11)
This result describes the Bessel-Gaussian beam after
passing through a misaligned paraxial optical system that
is the main finding of Zhao et al. [17]. When


0
w,
this equation can be reduced to the propagation equation
through a misaligned paraxial optical ABCD system of a
pure Bessel beam, which is given by





i2
0
2
22
i
,,eexpcossin
2
i
exp2 cos2 sin
24
2cos2sin .
2
kz n
B
Z
Z
n
Ek
Er zDr GrHr
AB
BkrErF
Ak B
JrErF
A




 
















2

(12)
The Gaussian case corresponds to 0

 and 0

n, so the corresponding output electric field is given by



i2 0
2
0
222
2
2
0
ii
,,eexpcossin
2
1
2i
2
exp2 cos2 sin.
1
16i 2
kz
G
kE k
Er zDrGrHr
B
kA
BB
w
krErF
kA
BB
w















 









(13)
Next, we will be interested to the normalized intensity
of the receiver beam which is given by

*
max
,,,, ,
N
IErzErzI   where * denotes the
conjugate and max
I
is the maximal intensity.
3. Numerical Example: Propagation of MBG
Beams through a Misaligned Thin Lens
In order to validate our calculation of the propagation of
MBG beams through a misaligned optical system, we
studied the propagation of the considered beam through a
circular thin lens.
The displacements and angle misalignments of the lens
with respect to the optical axis of the system are respec-
tively: 0
x


and 0
yxx





 (see Figure (3)).
The thin lens is located at z = 0 and the exit plane is
located at z. The ray transfer matrix of the optical system
between the input plane and the exit plane are
1
11
A
Bzf
CD f






z



, (14a)
and from Equations (2), one deduces
2
x
z
E
f

 and (14b) 0.FGH 
where
f
is the focal length of the thin lens.

L. EZ-ZARIY ET AL. 323

z
x
y

z
x
y
(a) (b)
Figure 3. Misaligned thin lens: (a) displacement εx in
x-direction, (b) displacement εy in y-direction.
In this case, Equation (4c) becomes

22
2
x
E   y. From this, one can deduce that
the beam center in exit plane is deviated from the optical
axis only in x-direction by 2x Ezf

.
For another use of a defined lens, we choose 0
y


and 0
xxx



 . In this case, we have
2y
z
F
f

 and 0.EGH

 (14c)
Similarly in this case, one obtains

2
22xyF
  and the center beam will be
shifted only in y-direction by 2y
F
zf

.
To confirm our theoretical finding numerically, we il-
lustrate in Figures 4 and 5 the analytical results related
to the propagation of MBG beam through the considered
misaligned thin lens. In the numerical calculations, the
parameters of the beam and of the optical system chosen
are: 0.6328 μm


, the width parameter aB is fixed
at 1 24.810 mm
B
a


1cm
, the radial Gaussian size is
s
a

, and we choose that the focusing parameter


s. For all figures, we present the normalized in-
tensity distributions of the single and multiple MBG
beams of various orders. At z = 0, before propagation
through the misaligned lens, the center of the beam is
located on the optical axis. However, in the case of dis-
placement of the optical system in x-direction by
F
x1mm


(see Figure 3(a)) and taking
0
yxy





, from the curves of Figures 4(a) and
5(a), it can be seen that the center of the existing beam
effectively is decentred in x-direction. The corresponding
displacements for the propagation distances z = 0.25; 0.5;
1 and 1.5 m are respectively: 0.625;1.25
2
E and 2.5.
Z = 250 mm (E = 1.25) z = 500 mm (E = 2.5) z = 1000 mm (E = 5)
(a)
z = 250 mm (F = 1.25) z = 500 mm (F = 2.5) z = 1000 mm (F = 5)
(b)
Figure 4. Contour maps of normalized three-dimensional intensity distributions of the output single MBG beam of order n =
10 through a misaligned thin lens at various z. Displacement of optical system by: (a) εx = 1 mm in x-direction, (b) εy = 1 mm
in y-direction.

L. EZ-ZARIY ET AL. 324
Z = 250 mm (E = 1.25) z = 500 mm (E = 2.5) z = 1000 mm (E = 5)
(a)
z = 250 mm (F = 1.2.5) z = 500 mm (F = 2.5) z = 1000 mm (F = 5)
(b)
Figure 5. Contour maps of normalized three-dimensional intensity distributions of the output multiple MBG beams at odd
orders






135710.50. 1
n
nA through a misaligned thin lens at various z. Displacement of optical system by: (a) εx
= 1 mm in x-direction, (b) εy = 1 mm in y-direction.
In the same way, for the case of a displacement of op-
tical system in y-direction by a value of 1mm
y


0
(see
Figure 3(b)) and if we take xxy

 

 , the Fig-
ures 4(b) and 5(b) show that the normalized intensity
center at receiver plane is shifted in y-direction. The new
center is located at (0; F/2; z). The deviation angle de-
pends on the misalignment parameters and the propaga-
tion distance z.
The MBG beams passing through a misaligned optical
system has the same properties of the other beams in the
same situation.
4. Conclusion
Based on the generalized diffraction integral formalism,
a convenient analytical solution to the propagation of the
MBG beam throughany misaligned optical system is de-
rived. The solutions for the propagation of pure Bessel,
Gaussian and Bessel-Gaussian beams, traveling through
the same misaligned optical system can be deduced as
special cases using our result. For the purpose of illustra-
tion, the propagation properties of the MBG beam through
a misaligned thin lens were addressed as numerical ex-
ample. The analysis of the result showed that the MBG
beam becomes decentered after passing through a misa-
ligned thin lens and the centeroid of the intensity distri-
bution is displaced in dependence of the optical system
misalignment parameters.
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