propagation of laguerre-bessel-gaussian beams

2010 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 Anthony A. Tovar

Propagation of Laguerre–Bessel–Gaussian beams

Anthony A. Tovar

Physics and Engineering Programs, Eastern Oregon University, La Grande, Oregon 97850-2899

Received November 23, 1999; revised manuscript received June 23, 2000; accepted July 13, 2000

New exact solutions to the paraxial wave equation are obtained in the form of a product of Laguerre polyno-mials, Bessel functions, and Gaussian functions. In the limit of large Laguerre–Gaussian beam size, theBessel factor dominates and the solution sets reduce to the modes of closed resonators, hollow metalwaveguides, and dielectric waveguides. In the opposite limit the solutions reduce to Laguerre–Gaussianmodes of open resonators and graded-index waveguides. These solutions are valid for electromagnetic wavestraveling through free space, and they are valid for propagation through circularly symmetric optical systemsrepresentable by ABCD matrices as well. An interesting feature of the new solution set is the existence ofthree mode indices, where only two are required for an orthogonal expansion. As an example, Laguerre–Gaussian beam propagation through an optical system that contains a Bessel-like amplitude filter is discussed.© 2000 Optical Society of America [S0740-3232(00)02111-6]

OCIS codes: 260.0260, 350.5500, 140.3410, 050.1960, 060.2310.

1. INTRODUCTIONIt is well known that the mode profiles of a rectangularmetal waveguide consist of sinusoidal functions. Theamplitude of the sinusoids is approximately zero at themetal waveguide walls, which satisfy the well-definedboundary conditions. These sinusoidal functions can beobtained as eigensolutions to the Maxwell–Heavisideequations. When ends are put on these waveguides thewaveguides become resonators, such as those used formaser oscillators. Since the sinusoidal functions can sat-isfy the boundary conditions of these resonators as well,they also represent the modes of a variety of masers.With the advent of the laser it was found that because ofthe small divergence of coherent laser light, the rectangu-lar waveguide walls were no longer necessary.1 Withoutthe waveguide walls, however, the sinusoidal functionswere no longer appropriate to represent the beam modesof these resonators. Additionally, it was determined thatthe new laser resonators should have slightly sphericalmirrors.

Since the paraxial wave equation derived from theMaxwell–Heaviside equations is similar to the Schro-dinger equation of quantum mechanics, it was deter-mined that Hermite–Gaussian functions are also solu-tions to the paraxial wave equation.2 These Hermite–Gaussian modes have been highly successful incharacterizing the resonant fields of rectangular low-diffraction-loss resonators.3 The Hermite polynomialparts of these original Hermite–Gaussian functions havea real argument. However, these modes are not appro-priate when the resonator contains complex elementssuch as variable-reflectivity mirrors or an amplifier thathas a radial gain profile. Complex argument Hermite–Gaussian functions were subsequently obtained as solu-tions to the paraxial wave equation that are the appropri-ate modes for these lasers. Siegman has shown that theoriginal real-argument modes are simply a special case ofthe complex-argument modes.4 These functions havealso been determined to represent the beam modes of la-

0740-3232/2000/112010-09$15.00 ©

ser amplifiers with radial gain or refractive-index profilesor both. These gain-profiled amplifiers are known ascomplex lenslike media. Thus, whereas sinusoidal func-tions represent the modes of metal waveguides and closedresonators, the Hermite–Gaussian functions representthe modes of complex lenslike media and open resonators.Recently, Hermite–sinusoidal–Gaussian solutions to theparaxial wave equations were obtained.5,6 In the limit oflarge Hermite–Gaussian beam size, these functions re-duce to the sinusoidal modes. In the limit of large sinu-soidal spatial period, these functions reduce to thecomplex-argument Hermite–Gaussian beam modes. Thegenerality of these functions makes it easier to match thefunctions to various boundary conditions.5

A similar historical development has evolved for cylin-drically symmetric systems. Bessel functions representthe beam modes of metal waveguides7 and closed resona-tors, whereas Laguerre–Gaussian function represent thebeam modes’ complex lenslike media and open resonators.There are also Bessel–Gaussian functions, which closelymodel the output beam modes of concentric-circle Grat-ing, surface-emitting lasers.8,9 The existence of rect-angular-symmetry Hermite–sinusoidal–Gaussian func-tion solutions suggests the existence of correspondingLaguerre–Bessel–Gaussian function solutions.

Since Laguerre–Gaussian functions and Hermite–Gaussian functions both represent complete orthogonalsets, it follows that one must be able to write one set interms of the other. In this paper it is shown thatLaguerre–Bessel–Gaussian solutions are a superpositionof off-axis Hermite–Gaussian functions. As a first stepin the derivation, we review the Hermite–Gaussian beamtheory in Section 2. The new Laguerre–Bessel–Gaussian solutions to the wave equation are obtained inSection 3. In Section 4, beam expansions are considered,and an example of these expansions is discussed. In par-ticular, the propagation of a Laguerre–Gaussian beamthrough an optical system that contains a Bessel-like am-plitude filter is examined.

2000 Optical Society of America

Anthony A. Tovar Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 2011

2. REVIEW OF HERMITE–GAUSSIANBEAM MATRIX THEORYA. Hermite–Gaussian Beams in Complex LenslikeMediaAs may be shown by a rigorous density matrix derivation,media that have gain (or loss) can be represented with acomplex propagation constant, k. If this is done, theMaxwell–Heaviside equations may be combined to formthe following Helmholtz equation:

¹2E8 1 k2E8 5 22¹S ¹k

k• E8D . (1)

The primes are a reminder that the harmonic time depen-dence exp(ivt) has been factored out of the real electricfield. The right-hand side of Eq. (1) is identically zero forhomogeneous media and is nearly zero in media that haveslowly varying inhomogeneities, such as those consideredhere.

A wide variety of applications involve only paraxial fo-cusing, and in these situations laser beams have a phasevariation that is similar to that of plane waves. Hence aplane wave traveling predominantly in the positive z di-rection is factored out:

E8~x, y, z ! 5 c~x, y, z !expF2iE0

z

k0~z8!dz8G H ix

iyJ . (2)

Following Kogelnik,10,11 the terms in braces are meant tosuggest a superposition of unit vector components of theform aix 1 biy , where a and b are complex constants. If,for example, a 5 1 and b 5 2i, Eqs. (1) and (2) wouldrepresent a circularly polarized wave. This notationwhere terms written inside braces represent a superposi-tion of functions inside the braces, is used throughout thispaper. The complex propagation constant k0(z)5 2pn0(z)/l 1 ig0(z)/2, where n0(z) is the refractive in-dex of the medium, l is the free-space wavelength of theelectromagnetic field, and g0(z) is the intensity (or power)gain coefficient.

First we consider beam propagation in complex lenslikemedia such as graded-index waveguides and laser ampli-fiers with radial gain profiles. In these media the com-plex propagation constant often takes the form

k~x, y, z ! 5 k0~z ! 2 k2x~z !~x2 1 y2!/2, (3)

which can be expressed for weakly profiled media as

k2~x, y, z ! > k0~z !@k0~z ! 2 k2x~z !~x2 1 y2!#. (4)

With the substitution into Eq. (2) of the complex propaga-tion constant given by expression (4), Eq. (1) becomes

]2c

]x2 1]2c

]y2 2 2ik0~z !]c

]z2 @k0~z !k2x~z !~x2 1 y2!#c 5 0,

(5)

where the second derivative with respect to z of the slowlyvarying envelope has been eliminated. Equation (5) issometimes referred to as the paraxial Helmholtz equa-tion.

The basic goal is to obtain new solutions to Eq. (5).The procedure uses a superposition of known Hermite–Gaussian solutions. Therefore it is useful to review the

solutions to Eq. (5), which are in the form of a product ofHermite functions and Gaussian functions. In terms ofcomplex beam matrices, the Hermite–Gaussian solutionsare

cmn~x, y, z ! 5 HmF 21/2x

W~z !GHnF 21/2y

W~z !G

3 expX2iF k0~z !

2q~z !~x2 1 y2! 1 Sx~z !x

1 Sy~z !y 1 P~z !GC, (6)

where the output parameters of the beam are related tothe corresponding input parameters of the beam by thefollowing transformations:

1

q~z !5

C~z ! 1 D~z !/q~0 !

A~z ! 1 B~z !/q~0 !, (7)

Sx~z ! 5Sx~0 !

A~z ! 1 B~z !/q~0 !, (8)

Sy~z ! 5Sy~0 !

A~z ! 1 B~z !/q~0 !, (9)

W2~z ! 5 W2~0 !@A~z ! 1 B~z !/q~0 !#2

1 4iB~z !@A~z ! 1 B~z !/q~0 !#/k0~0 !, (10)

P~z ! 5 P~0 ! 2 i ln@A~z ! 1 B~z !/q~0 !#

2i

2 X~m 1 n !lnF1 14i

k0~0 !W2~0 !

3B~z !

A~z ! 1 B~z !/q~0 !GC

31

2k0~0 !X@Sx

2~0 ! 1 Sy2~0 !#B~z !

A~z ! 1 B~z !/q~0 !C. (11)

The complex beam parameter, q(z), was introduced byKogelnik when he first obtained Eq. (7).10 That equation(formerly called the ABCD law) is now known as theKogelnik transformation. For an important early reviewof Gaussian beam theory, see Ref. 11. A given optical el-ement is represented by an axial complex propagationconstant k0(z) and a matrix

Melement 5 FA~z ! B~z !

C~z ! D~z !G . (12)

The input Hermite–Gaussian beam is characterized by acomplex phase parameter P(0), a complex beam param-eter q(0), two complex displacement parameters Sx(0)and Sy(0), and a complex spot size W(0). The outputbeam has the corresponding parameters P(z), q(z),Sx(z), Sy(z), and W(z). The propagation of more gen-eral types of Hermite–Gaussian beams through complexoptical systems that are misaligned and astigmatic havebeen reported,12 but only this level of generality is neededhere.


B. Hermite–Gaussian Beams in Complex OpticalSystemsIn optical systems theory it is common to designate refer-ence planes numerically. If the input and output planesare represented by subscripts 1 and 2, respectively, thenEqs. (7)–(12) can be rewritten as

cmn,2~x, y !

5 HmS 21/2x

W2DHnS 21/2y

W2D

3 expF2iS k02

2q2~x2 1 y2! 1 Sx2x 1 Sy2y 1 P2D G ,

(13)

1

q25

C 1 D/q1

A 1 B/q1, (14)

Sx2 5Sx1

A 1 B/q1, (15)

Sy2 5Sy1

A 1 B/q1, (16)

W22 5 W1

2~A 1 B/q1!2 1 4iB~A 1 B/q1!/k01 , (17)

P2 5 P1 2 i ln~A 1 B/q1!

1i

2 F ~m 1 n !lnS 1 14i

k01W12

B

A 1 B/q1D G

21

2k01F ~Sx1

2 1 Sy12 !B

A 1 B/q1G . (18)

Equation (14) can be written as a ratio of two equa-tions, and these two equations can be put into matrixform:

F uu/q G

25 FA B

C DG F uu/q G

1. (19)

The matrix in Eq. (19) is referred to as the beam matrixfor a particular optical element. The beam matrix for anadjacent optical element between planes 2 and 3 will begoverned by

F uu/q G

35 FA2 B2

C2 D2G F u

u/q G2. (20)

For a two-element optical system, Eqs. (19) and (20) maybe combined to yield

F uu/q G

35 FA2 B2

C2 D2GFA B

C DG F uu/q G . (21)

It can be readily seen that the system matrix is

Msystem 5 FA2 B2

C2 D2GFA B

C DG . (22)

It follows by induction that, for a system of n elements,the system matrix is

Msystem 5 MnMn21¯M3M2M1 . (23)

Thus the system matrix is the product of the matrices ofthe individual optical elements multiplied in the reverseof the order in which they are encountered by the laserbeam.

Like the Gaussian portion of the beam, the plane-waveportion may also be written in matrix form:

E0,28 5 AJ E0,18 , (24)

where

AJ 5 expF2iE0

z

k0~z8!dz8G . (25)

Equation (24), which follows directly from Eq. (2), can beconsidered the 1 3 1 special case of a 2 3 2 Jones ma-trix.

3. LAGUERRE–BESSEL–GAUSSIAN BEAMSOLUTIONIn this section, it is shown how rectangularly polarizedLaguerre–Bessel–Gaussian beams may be obtained as so-lutions to Eq. (5). There are three primary methods ofobtaining solutions to Eq. (5). In the first method, vari-ous substitutions are made that reduce Helmholtz equa-tion (5) to a set of ordinary differential equations, whichare subsequently solved. The second commonly used ap-proach is to convert Eq. (5) into a diffraction integral andlook for eigensolutions from it. The third method in-volves adding up different known solutions by means ofsuperposition. The third method is used here. In par-ticular, it is shown that Laguerre–Bessel–Gaussian solu-tions to the Helmholtz equation [Eq. (5)] may be obtainedas a linear combination of the well-known Hermite–Gaussian solutions.

A. SuperpositionWhile the TEMm,n mode is given in Eq. (13), the corre-sponding TEM2r22s1l,2p22r12s solution at plane 2 would be

c2r22s1l,2p22r12s,2~x, y !

5 H2r22s1lS 21/2x

W2DH2p22r12sS 21/2y

W2D

3 expX2iF k02

2q2~x2 1 y2! 1 Sx2x 1 Sy2y 1 P2GC.

(26)

Beam transformation equations (14)–(17) are unchangedbut since the phase parameter Eq. (18) contains mode-order subscripts, the corresponding phase parametertransformation for the TEM2r22s1l,2p22r12s mode wouldbe

P2 5 P1 2 i ln~A 1 B/q1!

1i

2 F ~2p 1 l !lnS 1 14i

k01W12

B

A 1 B/q1D G

3 21

2k01F ~Sx1

2 1 Sy12 !B

A 1 B/q1G . (27)


Since Eq. (5) is linear, we may choose an arbitrary linearcombination of these beam modes. In particular, we con-sider the following superposition:

ctotal~x, y ! 5~21 !p

22p1lp! (r50

p

(s50

l/2

~21 !sS pr D S l

2s D3 c2r22s1l,2p22r12s,2~x, y !. (28)

This superposition is reminiscent of the following math-ematical identity13:

S 21/2r

W D l

Lpl S 2r2

W2 D cos~lf!

5~21 !p

22p1lp! (r50

p

(s50

l/2

~21 !sS pr D S l

2s D

3 H2r22s1lS 21/2x

W DH2p22r12sS 21/2y

W D . (29)

This identity, combined with the fact that beam param-eter q2 in Eq. (14), displacement parameters Sx and Sy inEqs. (15) and (16), and phase parameter P in Eq. (27) areindependent of r and s and therefore come out of the sum-mation of Eq. (28), yields

ctotal 5 S 21/2r

W D l

LPl S 2r2

W2 D cos~lf!expF2iS k02r2

2q21 P2D G

3 exp@2i~Sx2x 1 Sy2y !#. (30)

A similar summation could be performed to produce theidentical result with a corresponding sin(lf) term.13 Su-perposition can be used on these sin(lf) and cos(lf) solu-tions to garner exp(ilf) and exp(2ilf) terms, and the re-sult could be written as

ctotal 5 S 21/2r

W D l

Lpl S 2r2

W2 D expF2iS k02r2

2q21 P2D GJ~r, f!,

(31)

where

J~r, f! [ exp(2i@Sx2r cos~ f! 1 Sy2r sin~ f!#)

3 H exp@2i~l 2 q !f#exp~2iqf!

exp@i~l 2 q !f#exp~iqf! J , (32)

where we have converted to polar coordinates and intro-duced an integer constant q. As in Eq. (2), the bracketsdenote a superposition of the two terms inside them.Note that Eq. (32) is independent of the constant q, whichshould not be confused with the beam parameter.

It may be noted that a superposition of ctotal is also asolution of linear differential equation (5). It follows thata superposition (or linear combination) of J(r, f) func-tions when substituted into Eq. (31) for J(r, f) will alsobe a solution to Eq. (5). An example of such a linear com-bination is

J8~r, f! [1

4pH exp(2i@~l 2 q !f#exp(2i@q~ f0 1 f! 1 Sx2r cos~ f! 1 Sy2r sin~ f!#)

exp(i@~l 2 q !f#exp(2i@q~ f0 2 f! 1 Sx2r cos~ f! 1 Sy2r sin~ f!#) J , (33)

which is obtained by multiplying J(r, f) byexp(2iqf0)/4p, where f0 is another introduced constant.If Sx1 5 Sy1 5 0, then, from Eqs. (15) and (16), Sx25 Sy2 5 0, and, if J8(r, f) is independent of r, then Eq.(31) reduces to the well-known Laguerre–Gaussian beamsolution. On the other hand, if we rename Sx2 and Sy2 inaccordance with

Sx2 [ V2 sin f0 , Sy2 [ 6V2 cos f0 , (34)

however, J8(r, f) becomes

J8~r, f! 51

4pH exp@2i~l 2 q !f#exp(2i@q~ f0 1 f! 1 V2 r sin~ f0 1 f!#)

exp@i~l 2 q !f#exp(2i@q~ f0 2 f! 1 V2 r sin~ f0 2 f!#) J , (35)

where the top and bottom signs of Eqs. (34) were used toobtain the top and bottom components, respectively, ofEq. (35). We can slightly modify the procedure leadingfrom Eq. (32) to Eq. (35) to achieve an alternative result.If we instead multiply Eq. (32) by exp(iqf0)/4p, and re-name Sx2 and Sy2 in accordance with

Sx2 [ 2V2 sin f0 , Sy2 [ 7V2 cos f0 , (36)

then J8(r, f) will become

J8~r, f! [1

4pH exp(2i~l 2 q !f]exp(i@q~ f0 2 f! 1 V2r sin~ f0 2 f!#)

exp~i~l 2 q !f#exp(i@q~ f0 1 f! 1 V2r sin~ f0 1 f!#) J . (37)

As shown later in the text, the displacement parameter substitution [Eqs. (34) and (36)] leads to the same transformationfor V2 . Thus the V2 in Eqs. (35) and (37) are identical. The J8(r, f) in Eqs. (35) and (37) are also independent, and, bysuperposition, may be summed. In particular, we consider the sum and the difference of the top component of Eq. (35)and the bottom component of (37) to obtain


J9~r, f! [1

2pH cos@~l 2 q !f 1 q~ f0 1 f! 1 V2r sin~ f0 1 f!#sin@~l 2 q !f 1 q~ f0 1 f! 1 V2r sin~ f0 1 f!#J . (38)

The bottom component of Eq. (35) and the top componentof Eq. (37) would ultimately yield redundant solutions, sothey are no longer considered. For the moment, we ex-amine the top solution in Eq. (38), which may be rewrit-ten with elementary trigonometric identities as

J9~r, f!utop [1

2p(cos@~l 2 q !f#cos@q~ f0 1 f!

1 V2r sin~ f0 1 f!# 2 sin@~l 2 q !f#

3 sin@q~ f0 1 f! 1 V2 r sin~ f0 1 f!).

(39)

This J9(r, f) is valid for any f0 . By superposition, thetotal field can be represented by a summation (discrete orcontinuous) of fields with different values of f0 . Thecontinuous summation of interest here is similar to a con-ventional diffraction integral that is the result of a con-tinuous summation of spherical waves with differentwave centers:

J-~r, f! [ E2p

p

J9~r, f, f0!df0 . (40)

In this case, Eq. (39) becomes

J-~r, f!utop [1

2p(cos@~l 2 q !f#E

2p

p

cos@q~ f0 1 f!

1 V2r sin~ f0 1 f!#df0

2 sin@~l 2 q !f#E2p

p

sin@q~ f0 1 f!

1 V2r sin~ f0 1 f!#df0), (41)

which may be rewritten as

J9~r, f!utop 51

2p(cos@~l 2 q !f#E

f2p

f1p

cos@qf

1 V2r sin~f !#df 2 sin@~l 2 q !f#

3 Ef2p

f1p

sin@qf 1 V2r sin~f !#df). (42)

Because the integrands are either symmetric or antisym-metric about p and 2p, Eq. (42) becomes

J-~r, f!utop 51

2p(cos@~l 2 q !f#E

2p

p

cos@qf

1 V2r sin~f !#df 2 sin@~l 2 q !f#

3 E2p

p

sin@qf 1 V2r sin~f !#df). (43)

It can be readily seen that the second integral in Eq. (43)is zero, since the integral of any odd function from 2p top vanishes. The first integrand is even, so Eq. (43) re-duces to

J-~r, f!utop [1

pcos@~l 2 q !f#

3 E0

p

cos@qf 1 V2r sin~f !#df. (44)

However, from integral relation14

Jn~a ! 51

pE

0

p

cos@nf 1 a sin~f !#df, (45)

which is valid for any integer n, it follows that Eq. (43)leads to

J-~r !utop 5 cos@~l 2 q !f#Jq~V2r !, (46)

where Jq is the qth- (integer) order Bessel function. Asimilar procedure performed on the bottom solution of Eq.(38) yields a similar result, with a sin term replacing thecos term in Eq. (46). Any of J, J8, J9, or J- may be usedin Eq. (31) for J(r, f). Our primary interest is in thelast case. With J-, the new beam solution is

ctotal 5 S 21/2r

W D l

Lpl S 2r2

W2 D H cos@~l 2 q !f#sin@~l 2 q !f#J

3 Jq~V2r !expF2iS k02r2

2q21 P2D G , (47)

which is a Laguerre–Bessel–Gaussian beam. In thelimit of small V1 and small q, this result reduces to thewell-known complex-argument Laguerre–Gaussian beammodes [which reduce to the real-argument Laguerre–Gaussian beam modes if the input complex spot size, W1 ,is chosen to be real and to have the same value as the in-put spot size, w1 (Ref. 4)]. Likewise, in the limit of smalll and p, Eq. (47) reduces to Bessel–Gaussian modes. Itmay be noted that J0(0) 5 1, whereas JqÞ0(0) 5 0 for in-teger q.

Equation (47) is a new solution to differential equation(5). If each of the factors in Eq. (44) except the Besselfunction were substituted into Eq. (5), then the resultingequation would be Bessel’s equation. But Bessel’s equa-tion has Bessel functions of the first kind, Jq , and Besselfunctions of the second kind, Yq , as its solutions. It fol-lows that an additional solution to Eq. (5) is Eq. (43), withJq replaced by Yq . Similarly, if V2 were replaced withiV2 and the relation

Iq~x ! 5 ~21 !qJq~ix ! (48)

were used, it would also follow that yet another solutionto Eq. (5) would be Eq. (44) with Jq replaced by the modi-fied Bessel function Iq . There is also an equation corre-


sponding to Eq. (45) for Kq , and it follows that the com-plex amplitude of the electric field may be written as

E28~r, f! 5 E0,28 expF2iS k02r2

2q21 P2D G S 21/2r

W2D l

3 Lpl S 2r2

W22 D 5

Jq~V2r !

Iq~V2r !

Yq~V2r !

Kq~V2r !6

3 5cos@~l 2 q !f#

sin@~l 2 q !f#

exp@i~l 2 q !f#

exp@2i~l 2 q !f#

6 H ix

iyJ . (49)

Equation (49) represents the Laguerre–Bessel–Gaussiansolution in its most general form. Because of the super-position braces in Eq. (46), there are a total of 32 differentpossible linearly polarized solution forms. Since any twolinearly polarized solutions can be used to achieve an ar-bitrary elliptical polarization, Eq. (47) may also bethought of as 16 different elliptically polarization solu-tions.

Equation (49) represents a wide variation in beam pro-files, and it is useful to consider a special case as an ex-ample. In particular, the l 5 2, p 5 3, q 5 0 mode isconsidered. For specificity, the complex spot size is cho-sen to be the same as the conventional beam spot size, asin the case of real-argument Laguerre–Gaussian beams.Furthermore, the radius of the phase curvature is chosento be infinite, such as when a Gaussian beam is at itswaist. With these constraints, the beam intensity is

I2~r, f!

I0,25 @2r82 exp~2r82!L3

2~2r82!J0~ar8!cos~2f!#2,

(50)

where r8 [ r/w is the scaled radial coordinate and a[ Vw is the inverse width of the Bessel factor. Equa-

tion (50) is plotted in Fig. 1 for a 5 0, 0.5, 1, 2, 10. In thefirst case the beam becomes a pure Laguerre–Gaussianmode. In the last case the beam is largely a Bessel beam.In the intermediate cases the width of the Bessel factor isapproximately one half of the Laguerre–Gaussian beamwidth, then it is the same, and then it is double.

B. New Beam TransformationsThe transformations for q2 , W2 , and E0.28 are given byEqs. (14), (16), and (24), respectively. The transforma-tions for V2 and P2 are discussed here. We can obtainthe beam transformation for our newly defined parameterV by substituting Eqs. (34) into Eqs. (15) and (16):

V2 sin~ f0! 5V1 sin~ f0!

A 1 B/q1, (51)

6V2 cos~ f0! 56V1 cos~ f0!

A 1 B/q1. (52)

Both of these equations reduce to

V2 5V1

A 1 B/q1. (53)

Similarly, the phase parameter [Eq. (27)] becomes

P2 5 P1 2 i ln~A 1 B/q1!

1i

2 F ~2p 1 l !lnS 1 14i

k01W12

B

A 1 B/q1D G

71

2k01S V1

2B

A 1 B/q1D . (54)

In this Eq. (54), the top sign applies when the Jq and YqBessel functions in Eq. (49) are used and the bottom signis to be used when the Iq and Kq modified Bessel func-tions are used.

4. BEAM EXPANSIONSIn the above analysis new Laguerre–Bessel–Gaussianbeams were found as solutions to the paraxial wave equa-tion. For the remainder of this study, we focus on someof the practical applications of the Laguerre–Bessel–Gaussian beam solutions. Because the new Bessel indexq appears in the polarization braces, is not as obviouswith the Bessel functions as with the more standard La-guerre polynomials how these solutions can be used as abasis for the expansion of an arbitrary electromagneticbeam field, and such expansions are the subject of thisdiscussion.

In this section we consider a Laguerre–Gaussian beamof the form

E18~r, f! 5 E0,18 expF2iS k01r2

2q11 P1D G

3 S 21/2r

W1D l

Lpl S 2r2

W12 D H cos~lf!

sin~lf!J (55)

striking a thin transmission filter, such as an optical grat-ing, whose transmission function is given by T(r). Onemight be inclined to expand T(r) in a series of Besselfunctions and multiply Eq. (55) by each of the Bessel func-tions and propagate them individually. Though the re-sulting functions would be Laguerre–Bessel–Gaussianfunctions, this procedure would not work because thesefunctions would not be in the form of Eq. (49). In par-ticular, the azimuthal variation in the braces in Eq. (55)would not match that in Eq. (49).

To see how an arbitrary field can be expanded in termsof the appropriate Laguerre–Bessel–Gaussian functions,we note that, whereas the above procedure would notwork for a Bessel function with an arbitrary index, it


Fig. 1. l 5 2, p 5 3, q 5 0 Laguerre–Bessel–Gaussian modeprofiles where the ratio of the width of the Laguerre–Gaussianportion of the beam to the J0 Bessel portion of the beam is (a) 0, (b)0.5, (c) 1, (d) 2, (e) 10. The first plot is a pure Laguerre–Gaussianmode, and the last plot is primarily a Bessel–Gaussian beam.


would work if the transmission function were a J0 Besselfunction. In this case q 5 0, and the resultant fieldwould be

E28~r, f! 5 E0,28 expF2iS k02r2

2q21 P2D G S 21/2r

W2D l

3 Lpl S 2r2

W22 D H cos~lf!

sin~lf!J J0~V2r !, (56)

which is of the form of Eq. (49). Thus, if T(r) could beexpanded in terms of a J0 Bessel function, the procedurewould be straightforward. However, this expansion iswell known. In particular, if

T~r ! 5 (k

AkJ0~lkr !, (57)

then the expansion coefficients would be15

Ak 52

J12~lk!

E0

1

rT~r !J0~lkr !dr, (58)

where lk is the kth root of J0 .With this background information on Bessel series, we

can now indicate a procedure for expanding an arbitraryfield distribution in a series of Laguerre–Bessel–Gaussian beams. The first step is to match the spatialpolarization of the desired beam with an appropriate azi-muthal variation of a Laguerre–Bessel–Gaussian beamto be used in the expansion. Second, select a Gaussianfactor to represent an approximate fit to the beam beingrepresented. These factors should then be divided intothe given beam, and the quotient should be expanded in aJ0 Bessel series. It is necessary only that the expansioninterval be large compared with the beam and its Gauss-ian approximation. Each term in such an expansion canbe propagated analytically by using the formulas of Sec-tion 3.

It is, of course, also possible to use an expansion inter-val that is much larger than the beam diameter. Thiswould, however, require the inclusion of a larger numberof terms in the expansion. With an infinite expansion in-terval, the series representation evolves into a continuouscomplex Bessel transform. Such a transform is never re-quired in this method, though, because the finite Gauss-ian beam factor width renders any larger expansion re-gion unnecessary. Quite complicated beam profilesshould be representable with only a few terms in the ex-pansion.

To illustrate some of the concepts discussed above, wewill briefly sketch a specific example. Consider thepropagation of a Laguerre–Gaussian beam through thehypothetical optical system shown in Fig. 2. In this sys-tem the lenses have a focal length of f, and the distancebetween each of the lenses and the transmission filter isd. The transmission filter in this case is a thin elementthat has the amplitude transfer characteristic

T~r ! 5 T0uJ0~ar !u. (59)

For this purpose we wish to represent the filter character-

istic in a Bessel series, and thus Eq. (56) may be used;and the expansion coefficients will be

Ak 52T0

J12~lk!

E0

1

ruJ0~ar !uJ0~lkr !dr. (60)

Similar methods would also be applicable with phase,rather than amplitude, filters.

The first step in analyzing the transfer of a Laguerre–Gaussian beam through the system shown in Fig. 2 is topropagate the beam from the input plane to the transmis-sion filter. In this example the beam matrix for this pur-pose is

Mbefore filter 5 F1 d

0 1 GF 1 0

21 /f 1G 5 F1 2 d/f d

21 /f 1 G . (61)

The propagation methods for this region are already wellknown, and the initial Laguerre–Gaussian beam will stillbe in the Laguerre–Gaussian beam form. Next, for theresulting beam to propagate through the transmission fil-ter, the beam at the filter must be multiplied by T(r) asgiven in Eq. (57) with coefficients in Eq. (60). The resultin this case will be a set of beams, each of which is aLaguerre–Bessel–Gaussian in the form of Eq. (56). ThusEq. (56) together with the propagation formulas pre-sented in Sections 2 and 3 allows for the further propaga-tion of the filtered beam. For the case shown in this ex-ample, the propagation to the output plane is governed bythe matrix

Mafter filter 5 F 1 0

21 /f 1GF1 d

0 1 G 5 F 1 d

21 /f 1 2 d/fG .(62)

At the output plane one can find the resultant field byadding up the individual complex Laguerre–Bessel–Gaussian beam components. This general procedurewould be applicable for any system containing a filter forwhich the transmission characteristic can be representedby a Bessel series. The method could be extended to sys-tems with multiple filters, and in this case each compo-nent resulting from the expansion at one filter would it-self need to be expanded on transmission through thenext filter.

It may be noted here that the propagation of beamsthat have been transmitted through filters represented by

Fig. 2. Example of an optical system containing a Bessel-liketransmission filter, as discussed in Section 4. In this system thefilter is centered between two identical lenses of focal length fand two lengths of free space, each with distance d.


their series expansions is well known in studies of spatialmodulation.16 In earlier treatments, however, the propa-gation of the filtered beams was based on diffraction inte-gral methods. With the procedure reported here, inputLaguerre–Gaussian beams are expanded by the trans-mission filter into a set of components that can be propa-gated analytically. If the number of components neededfor this expansion is not large, this procedure is an effi-cient alternative to the brute-force diffraction calcula-tions.

5. SUMMARYBeams of electromagnetic energy whose amplitude atsome input plane may be represented mathematically bya product of Laguerre polynomials, Gaussian functions,and Bessel functions given by Eq. (49) retain their math-ematical form at the output plane of any optical systemrepresentable by ABCD beam matrices. However, theparameters of the beam change according to Eqs. (14),(17), (53), and (54). One of the interesting properties ofthese beams is that they represent an overcomplete set.Thus an arbitrarily shaped beam may be written as a se-ries of these modes in more than one way. For example,one may perform a Laguerre–Gaussian expansion, as isconventional, or one could write an input beam as a seriesof Bessel functions as discussed in Section 4. This over-completeness allows one to represent a given beam profileby a small number of modes, providing an important al-ternative to brute-force diffraction calculations.

ACKNOWLEDGMENTThe author gratefully acknowledges Lee Casperson forseveral fruitful discussions.

The author’s e-mail address is [email protected].

REFERENCES1. R. H. Dicke, ‘‘Molecular amplification and generation sys-

tems and methods,’’ U.S. patent 2,851,652 (September 9,1958).

2. G. D. Boyd and J. P. Gordon, ‘‘Confocal multimode resona-tor for millimeter through optical wavelength masers,’’ BellSyst. Tech. J. 40, 489–508 (1961).

3. G. D. Boyd and H. Kogelnik, ‘‘Generalized confocal resona-tor theory,’’ Bell Syst. Tech. J. 41, 1347–1369 (1962).

4. A. E. Siegman, Lasers (University Science, Mill Valley, Ca-lif., 1986), 798–804.

5. L. W. Casperson and A. A. Tovar, ‘‘Hermite–sinusoidal–Gaussian beam in complex optical systems,’’ J. Opt. Soc.Am. A 15, 954–961 (1998).

6. L. W. Casperson, D. G. Hall, and A. A. Tovar, ‘‘Sinusoidal–Gaussian beams in complex optical systems,’’ J. Opt. Soc.Am. A 14, 3341–3348 (1997).

7. J. W. Strutt (Lord Rayleigh), ‘‘On the passage of electricwaves through tubes, or the vibrations of dielectric cylin-ders,’’ Phil. Mag. Suppl. 43, 125–132 (1897).

8. R. H. Jordan and D. G. Hall, ‘‘Free-space azimuthalparaxial wave equation: the azimuthal Bessel–Gaussbeam,’’ Opt. Lett. 19, 427–429 (1994).

9. A. A. Tovar and G. H. Clark, ‘‘Concentric-circle-grating,surface-emitting laser beam propagation in complex opticalsystems,’’ J. Opt. Soc. Am. A 14, 3333–3340 (1997).

10. H. Kogelnik, ‘‘Imaging of optical modes—resonators withinternal lenses,’’ Bell Syst. Tech. J. 44, 455–494 (1965).

11. H. Kogelnik and T. Li, ‘‘Laser beams and resonators,’’ Appl.Opt. 5, 1550–1567 (1966).

12. A. A. Tovar and L. W. Casperson, ‘‘Generalized beam ma-trices. II. Mode selection in lasers and periodic mis-aligned complex optical systems,’’ J. Opt. Soc. Am. A 13,90–96 (1997).

13. I. Kimel and L. R. Elias, ‘‘Relations between Hermite andLaguerre Gaussian Modes,’’ IEEE J. Quantum Electron. 29,2562–2567 (1993).

14. M. Abramowitz and I. A. Stegun, Handbook of Mathemati-cal Functions (Dover, New York, 1970), Eq. (9.1.21).

15. See, for example, M. R. Spiegel, Schaum’s Outline Series:Mathematical Handbook (McGraw-Hill, New York, 1991),Chap. 24.

16. L. W. Casperson, ‘‘Spatial modulation of Gaussian laserbeams,’’ Opt. Quantum Electron. 10, 483–493 (1978).

propagation of laguerre-bessel-gaussian beams

Documents