Decentered elliptical Gaussian beam

Yangjian Cai and Qiang Lin

A new kind of laser beam, called a decentered elliptical Gaussian beam �DEGB�, is defined by a tensormethod. The propagation formula for a DEGB passing through an axially nonsymmetrical paraxialoptical system is derived through vector integration. The derived formula can be reduced to the formulafor a fundamental elliptical Gaussian beam and a decentered Gaussian beam under certain conditions.As an example application of the derived formula, the propagation characteristics of a DEGB in free spaceare calculated and discussed. As another example we study the properties of a generalized laser beamarray constructed by use of a DEGB as the fundamental mode. © 2002 Optical Society of America

OCIS codes: 140.3300, 140.3290.

1. Introduction

As early as 1973, Casperson put forward the idea ofoff-axis Gaussian beams.1 Recently, decenteredbeams have attracted increasing attention because oftheir unique characteristics.2–7 Al-Rahshed andSaleh studied the propagation properties of a decen-tered Gaussian beam and its transmission throughoptical systems.2 Palma developed the concept ofray parameters for characterizing decentered Gauss-ian beams and a model of decentered Gaussian beambundles for characterizing Bessel–Gaussian beams.3Coherent and incoherent combinations of off-axisGaussian beams and off-axis Hermite–Gaussianbeams were studied by Lu and Ma.4–6 The diffrac-tion properties of a scalar, off-axis Gaussian beamwere studied by Cronin et al.7 But all the investi-gations cited above were restricted to axially symmet-ric beams or to beams with separable variables x andy.

Previously tensor methods were used to treat prop-agation and transmission of generalized ellipticalGaussian beams.8,9 In this paper we introduce amore general three-dimensional decentered ellipticalGaussian beam �DEGB�, whose variables cannot beseparated. The formula for propagation of a DEGBthrough an axially nonsymmetric optical system isderived by a generalized Collins integral.8,9 The

Y. Cai and Q. Lin �qlin@mail.hz.zj.cn� are with the Institute ofOptics and the State Key Laboratory of Modern Optical Instru-mentation, Zhejiang University, Hangzhou 310028, China.

Received 4 September 2001; revised manuscript received 22April 2002.

0003-6935�02�214336-05$15.00�0© 2002 Optical Society of America

4336 APPLIED OPTICS � Vol. 41, No. 21 � 20 July 2002

propagation of a DEGB in free space is calculated anddiscussed by use of a derived propagation formula.The application of a DEGB in a laser beam array isalso studied.

2. Definition of a Decentered Elliptical Gaussian Beamand Its Propagation through Paraxial Optical Systems

The generalized elliptical Gaussian beam can be ex-pressed in tensor form as follows8:

E�r1� � exp��ik2

r1TQ1

�1r1� , (1)

where k � 2��� is the wave number, � is the wave-length, and r1 is a position vector given by r1

T � �x1y1�. Q1

�1 is complex curvature tensor for the gen-eralized elliptical Gaussian beam given by8

Q1�1 � �q1xx

�1 q1xy�1

q1xy�1 q1yy

�1� . (2)

The decentered elliptical Gaussian beam at z � 0 canbe defined by use of the tensor method in the follow-ing form:

E�r1� � exp��ik2

�r1 � r0�TQ1

�1�r1 � r0�� , (3)

where r0 is a complex vector called the decenteredparameter and can be expressed by r0

T � �xd � ix0yd � iy0�. When the complex curvature tensor takesthe following form:

Q1�1 � �q1

�1 00 0� , (4)

Eq. �3� is reduced to

E� x1, 0� � exp��ik

2q1�� x1 � xd � ix0�

2� . (5)

Equation �5� is the well-known one-dimensional de-centered Gaussian beam.2–4

Within the framework of the paraxial approxima-tion, the propagation of any beam through a nonsym-metrical optical system can be treated by the generalCollins formula, which can be written in tensor formas follows8,9:

E2�r2� � �in1

��det�B�1�2 exp��ikl0�

� E1�r1�exp��ikl1�dr1, (6)

where l0 is an eikonal along the propagation axis. l1is an eikonal given by

l1 �12 �r1

r2�T� n1B�1A �n1B�1

n2�C � DB�1A� n2DB�1��r1

r2� , (7)

where n1 and n2 are the refractive indices of the inputand the output spaces, respectively. For simplicitywe assume that n1 � n2 � 1 what follows. r1 and r2are position vectors in the input and the outputplanes, respectively. A, B, C, and D are the subma-trices of the optical system, defined by

� r2

r2�� � �A B

C D�� r1

r1�� . (8)

Substituting Eq. �3� as E1�r1� into Eq. �6�, after atedious but straightforward vector integral operationwe obtained the following expression for the outputDEGB:

E�r2� � �det�A � BQ1�1�1�2 exp��ikl0�

� exp��ik2

r2TQ2

�1r2�exp��ik2

r0T�Q1

� A�1B��1r0�exp�ikr0T�AQ1 � B��1r2, (9)

where

Q2�1 � �C � DQ1

�1��A � BQ1�1��1. (10)

In the derivation of Eq. �9� the following relationswere used8:

�B�1A�T � B�1A, ��B�1�T � �C � DB�1A�,

�DB�1�T � DB�1. (11)

Equation �10� is the well-known tensor ABCD law forthe propagation of the generalized elliptical Gaussianbeam.

When decentered parameter r0T � �0 0�, Eq. �9� is

reduced to the following form:

E2�r2� � �det�A � BQ1�1�1�2

� exp��ikl0�exp��ik2

r2TQ2

�1r2� . (12)

We can find that Eq. �12� is the same as the propa-gation formula for the generalized elliptical Gaussianbeam through the asymmetrical paraxial optical sys-tem in Refs. 8 and 9.

For the one-dimensional beam Eq. �9� can easily bereduced to the following form:

E� x2� � �a � b�q1��1�2 exp��

ik2q2

� x2 � ax0�2

� ikcx0 x2 �ik2

acx02�exp��ikl0�, (13)

where

q2 �aq1 � bcq1 � d

; (14)

a, b, c, and d are the matrix elements of the opticalsystem, defined by

� x2

x2�� � �a b

c d�� x1

x1�� . (15)

It can easily be verified that Eq. �13� is the same asthe propagation formula for the one-dimensional de-centered Gaussian beam given in Refs. 2–4.

3. Propagation of a Decentered Elliptical GaussianBeam in Free Space

In this section we study the propagation properties ofa DEGB in free space, using the derived propagationformula for a DEGB. The submatrices for free spaceof distance z read as

A � �1 00 1� , B � �z 0

0 z� ,

C � �0 00 0� , D � �1 0

0 1� . (16)

Substituting Eqs. �16� into Eqs. �9� and �10�, we canobtain the three-dimensional relative intensity dis-tribution of a DEGB at various propagation dis-tances, as shown in Fig. 1. The parameters used inthe calculation are r0

T � �2 � i 2 � i�, � � 632.8 nm,w0x � 1 mm, w0y � 1.5 mm, w0xy � 2 mm, and

Q1�1 � � �0.201i �0.0503i

�0.0503i �0.0895i��m��1. (17)

The propagation distances are normalized to the Ray-leigh distance in the x direction, zx � �w0x

2��.From Fig. 1�a� we can easily learn that the peak

position of the intensity distribution at z � 0 is lo-cated at rd � �xd, yd�. The calculated results showthat the intensity distribution of the DEGB at z � 0

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is closely related to the imaginary part of the decen-tered parameter but not to the real part of the decen-tered parameter. From Figs. 1�b�–1�d� we can seethat the beam spot of the DEGB rotates with propa-gation distance z. The ratio of the long axis to the

short axis of the intensity distribution of the DEGBchanges with propagation distance z. At a certainpropagation distance the intensity distribution of theDEGB assumes a circular shape, as shown in Fig.2�b�. We can also find that the position of peak in-tensity of the DEGB will move during propagationwhen the imaginary part of the decentered parame-ter is not zero. In the far field, the intensity distri-bution evolves to an elliptical shape again, but thelong axis and the short axis have changed their po-sitions from the initial positions. With a further in-crease of the propagation distance, the beam spotrotates slowly, but the position of the peak intensitymoves continuously.

4. Application of a Decentered Elliptical GaussianBeam in a Laser Beam Array

In this section we discuss the applications of thenewly introduced DEGB in characterizing laser beamarrays. In recent years, laser beam arrays were de-veloped to permit slab lasers to be used in high-powersystems, and a number of linear, rectangular, andradial arrays have been developed and studied.10–14

There are two ways to combine beams, i.e., by phase-locked and by non-phase-locked methods. The prop-agation properties of such beam arrays weredescribed in Refs. 13 and 14, but the descriptionswere limited to properties of beam arrays constructedwith a combination of decentered circular Gaussianbeams or elliptical Gaussian beams with separablevariables x and y.

Here we construct a beam array by using theDEGB given above. Assume that the beam arrayconsists of N equal elements, which are located sym-metrically on a ring with radius �xd

2 � yd2. Then

Fig. 1. Three-dimensional relative intensity distributions ofDEGBs on planes of various propagation distances: �a� z � 0, �b�z � 1.6zx, �c� z � 4zx, �d� z � 8zx.

Fig. 2. Three-dimensional relative intensity distributions and corresponding contour graphs of the phase-locked beam array on planesof several propagation distances: �a� z � 0, �b� z � 2zx, �c� z � 10zx, �d� z � 20zx.

4338 APPLIED OPTICS � Vol. 41, No. 21 � 20 July 2002

the field distribution of the beam array for the phase-locked method at z � 0 is given by

E�r1, 0� � n�0

N�1

En�r1n, 0�, (18)

where En�r1n, 0� is the nth element of the laser beamarray, expressed by

En�r1n, 0� � exp��ik2

�r1n � r0�TQ1

�1�r1n � r0�� ,

(19)

with

r1n � � cos � sin ��sin � cos ��r1 � �x1 cos � � y1 sin �

y1 cos � � x1 sin �� ,

(20)

where � � n�0, n � 0, 1, 2, . . . N � 1, and �0 � 2��N.Applying Eq. �9�, we can easily get the field distribu-tion of En�r2n� after its propagation through a parax-ial system, which is expressed as

En�r2n� � �det�A � BQ1�1��1�2

� exp��ikl0�exp��ik2

r2nTQ2

�1r2n�� exp��

ik2

r0T�Q1 � A�1B��1r0�

� exp�ikr0T�AQ1 � B��1r2n, (21)

where

r2n � �x2 cos � � y2 sin �y2 cos � � x2 sin �� . (22)

The corresponding intensity distribution of the la-ser beam array for the phase-locked method reads as

I � E*�r2� E�r2�. (23)

For the non-phase-locked method we have

I � n�0

N�1

In � n�0

N�1

En*�r2n� En�r2n�. (24)

Using Eqs. �19�–�24�, we can easily study the prop-agation properties of the newly constructed beam ar-ray through the paraxial optical system. Therelative intensity distribution and corresponding con-tour graphs of the phase-locked and non-phase-locked beam arrays at several propagation distancesin free space are depicted in Figs. 2 and 3, respec-tively. The parameters used in the calculation arer0

T � �5 5� and N � 5. Q1�1 is given in Eq. �17�.

The propagation distances are normalized to the Ray-leigh distance in the x direction, zx � �w0x

2��.From Figs. 2 and 3 we can find that there are

essential differences between propagation of phase-locked and non-phase-locked beam arrays. In thenear field the beam arrays have similar intensitydistributions, because the elements that constitutethe beam arrays do not overlap. With the increaseof the propagation distance, the difference betweenphase-locked and non-phase-locked arrays becomesobvious. In the far field, the intensity distribution ofthe phase-locked beam array becomes complicated,but the intensity distribution of the non-phase-lockedbeam array evolves in a comparatively simple way.

5. Conclusions

In conclusion, we have introduced a new kind of laserbeam, called a decentered elliptical Gaussian beam

Fig. 3. Three-dimensional relative intensity distributions and corresponding contour graphs of the non-phase-locked beam array onplanes of several propagation distances: �a� z � 0, �b� z � 2zx, �c� z � 10zx, �d� z � 20zx.

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�DEGB�, by using a tensor method. The propaga-tion formula of a DEGB passing through a nonsym-metrical paraxial optical system was derived throughvector integration. The propagation characteristicsof the DEGB through free space were calculated anddiscussed by use of the derived formula. The nu-merical results have shown that the propagationproperty of a DEGB is closely related to the decen-tered parameter. The beam spots of a DEGB rotatewith propagation distance z. We also constructed ageneralized laser beam array by using DEGBs asfundamental modes and investigated the propaga-tion properties of both phase-locked and non-phase-locked beam arrays.

This research was supported by the National Nat-ural Science Foundation of China �grant 60078003�and the Huo Ying Dong Education Foundation ofChina �grant 71009�.

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