Propagation of Modified Bessel-Gaussian Beams in a Misaligned Optical System

doi:10.4236/opj.2012.24039

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

Email: *[email protected]

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

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

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

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

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

x

,

x

GBCDA

BDDzB

















(2d)





21

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

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

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

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

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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