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Vector-beamsolutionsofMaxwell’swaveequation

January1,1996/Vol.21,No.1/OPTICS LETTERS9 Vector-beam solutions of Maxwell’s wave equation

Dennis G.Hall

The Institute of Optics,University of Rochester,Rochester,New York14627

Received September12,1995

The Hermite–Gauss and Laguerre–Gauss modes are well-known beam solutions of the scalar Helmholtz

equation in the paraxial limit.As such,they describe linearly polarized fields or single Cartesian components

of vector fields.The vector wave equation admits,in the paraxial limit,of a family of localized Bessel–Gauss

beam solutions that can describe the entire transverse electric field.Two recently reported solutions are

members of this family of vector Bessel–Gauss beam modes.?1996Optical Society of America

That electromagnetic waves can propagate through

space in the form of highly directional,transversely

localized beams is an important and useful property of

radiation.Most commonly,the spatial characteristics

of beams in free space are described by solution of

the scalar Helmholtz equation in the paraxial limit

to produce two familiar families of solutions:in rec-

tangular coordinates(x,y,z)the Hermite–Gauss

modes,and in cylindrical coordinates(r,f,z)the

Laguerre–Gauss modes.1The much-used azimu-

thally symmetric Gaussian-beam solution appears as

the lowest-order member of each family.Whereas the

scalar Helmholtz equation can be used to describe

the propagation of either a linearly polarized electro-

magnetic wave or a single Cartesian component of

an arbitrary vector field,it is the vector Helmholtz

equation that describes the propagation of the electric

field as a whole.This Letter reports that,in the

paraxial limit appropriate for beams,Maxwell’s vector

wave equation admits an interesting family of propa-

gating vector-beam solutions that can be expected to

be particularly important when the polarization of the

propagating field is of major concern.

To establish notation used later in this Letter,con-

sider first the scalar case.The elementary Gaussian

beam,the most familiar example,arises as a solution

of the scalar Helmholtz equation?=21k2?E?0in the

paraxial limit.In cylindrical coordinates,with z the

propagation axis and with the electric field E taken

to be of the form E?f?r,f,z?exp?i?kz2v t??,where

k and v are the propagation constant and the(circu-

lar)frequency?k?v?c?,respectively,the paraxial ap-

proximation appropriate for beams neglects≠2f?≠z2so

that f?r,f,z?is a solution of the scalar paraxial wave

equation:

1 r

≠

≠r

≠f

≠r

≠2f

≠f2

12ik

≠f

≠z

?0.(1)

The elementary Gaussian solution is independent of f and can be written as

f?r,z??

w?z?

exp?2i F?z??exp

r2?w02

11iz?L

,(2)

where L?kw02?2is the Rayleigh range,w?z??w0?11?z?L?2?1/2,and F?z??tan21?z?L?;w0is the waist pa-rameter.There also exist higher-order solutions to Eq.(1),as mentioned above.The Gaussian envelope effectively localizes the field along directions trans-verse to the direction of propagation.

A description of the propagation of vector beams begins with the vector wave equation for the electric field E:

=3=3E2k2E?0,(3) where an exp?2i v t?time dependence is assumed and k?v?c.Before treating the case of arbitrary transverse polarization,it is instructive to examine a specific case to motivate what follows.Consider a cir-cularly symmetric,azimuthally polarized(f-polarized) field,for which E takes the form

E?r,z???f F?r,z?exp?i?kz2v t??.(4) Several lasers have been reported to emit beams of light with this polarization.2–5After inserting Eq.(4) into Eq.(3)and making the paraxial approximation, one obtains the single,scalar equation

≠

≠r

≠F

≠r

12ik

≠F

≠z

?0,(5)

which for azimuthal symmetry differs from Eq.(1)by the middle term in Eq.(5).One can show by direct substitution that Eq.(5)has the solution

F?r,z??AJ1

μbr

11iz?L

f?r,z?Q?z?,(6)

where A is a constant,b is a constant scale parameter, f?r,z?is the elementary Gaussian defined in Eq.(2), L is the Rayleigh range defined below Eq.(2),J1is the Bessel function of the first kind of order one,and

Q?z??exp

∑

i b2z??2k?

11iz?L

∏

.(7)

This f-polarized Bessel–Gauss beam solution was de-scribed in the recent paper by Jordan and Hall.6Gori et al.7had shown earlier that Eq.(1)admits of a solu-tion of the same general character as Eq.(6)but with J1replaced by J0.A f-polarized field of necessity contains an on-axis(r?0)null,which J1provides. The J0Bessel–Gauss solution to Eq.(1)describes a

0146-9592/96/010009-03$6.00/0?1996Optical Society of America

10OPTICS LETTERS/Vol.21,No.1/January1,1996 linearly polarized beam and reduces to the elementary Gaussian in Eq.(2)for b?0.Representative plots of the fields for several values of z for both the J0and J1 Bessel–Gauss modes appear in Ref.6.

The f-polarized field given by Eqs.(4)and(6)is reminiscent of the electric-field distribution for the TE01mode in the core region of the step-index opti-cal fiber(TE is transverse electric).This mode car-ries the same polarization and is described by the J1 Bessel function but differs in detail from Eqs.(4)and (6)because transverse localization is achieved in the optical fiber by total internal ref lection at the interface between the fiber’s core and cladding regions.The re-lationship between the fiber TE01mode and the solu-tion in Eqs.(4)and(6)can be stated simply.When light propagating in the TE01mode reaches the end of a length of fiber,it excites the mode of free space defined

by Eqs.(4)and(6).In free space,for both polariza-tion states discussed so far,transverse localization is achieved by a multiplicative Gaussian.It is natural then to inquire whether Gaussian localization can be used in a similar way to transform the set of bound vec-tor modes of the optical fiber into a family of free-space solutions of the vector wave equation,in the paraxial limit.This turns out indeed to be the case,and the solutions given here provide a natural description of the free-space modes excited by light exiting a fiber,al-though the solutions are of more general interest,too.

The new solutions emerge in a straightforward man-ner as the vector amplitude F?r,f,z?is introduced according to

E?F?r,f,z?exp?i?kz2v t??,(8) similar to Eq.(4).Next,invoke the paraxial approxi-mation to eliminate≠2F?≠z2in Eq.(3).Then,taking the view of that Gaussian localization is fundamental

to propagation in free space,write F as the product of two factors:

F?′?r,f,z?f?r,z?,(9) where f?r,z?is the Gaussian defined in Eq.(2)and

′?r,f,z???′r,′f?is transverse,containing only r and f components.Inserting Eq.(8)into the paraxial form of Eq.(3)produces two coupled differential equa-tions for the components of′?r,f,z?:

1 r

≠

≠r

≠′r

≠r

4r?w02

11iz?L

≠′r

≠r

≠2′r

≠f2

′r2

≠′f

≠f

12ik

≠′r

≠z

?0,(10)

1 r

≠

≠r

≠′f

≠r

4r?w02

11iz?L

≠′f

≠r

≠2′f

≠f2

′f1

≠′r

≠f

12ik

≠′f

≠z

?0.(11)

Equations(10)and(11)can be solved in closed form, with the results that

′r,m?r,f,z??Q?z??a m J m21?u?

1b m J m11?u??

64cos?m f?

sin?m f?

75,(12)′f,m?r,f,z??Q?z??a m J m21?u?

2b m J m11?u??

642sin?m f?

cos?m f?

75,(13) where

11iz?L

,(14)

b is again a constant,m is the azimuthal mode integer, a m and b m are constants,and Q?z?is as defined in Eq.(7).

If Q?z??f?r,z??1and u?k r r,Eqs.(8),(9),(12), and(13)are simply the transverse components of the electric-field distributions of the modes(in the core)of the familiar optical fiber.8In the core region of the fiber,such solutions are known to satisfy the vector wave equation but do not themselves exhibit proper behavior for large r,something that multiplication by Q?z?f?r,z?accomplishes in free space.With u as defined in Eq.(14),Eqs.(8),(9),(12),and(13) constitute a family of transverse-electric-field solutions to Eq.(3)in the paraxial limit.

One technical point deserves https://www.wendangku.net/doc/60231306.html,x et al.9 pointed out that the paraxial approximation does not produce a field with zero divergence,as required by Maxwell’s equations.This is readily apparent from the elementary Gaussian beam in Eq.(2).They showed that the fields produced by making the parax-ial approximation are,in fact,the lowest-order terms in an expansion of the solutions of Maxwell’s equations in terms of certain transverse and longitudinal scaling distances.9Failure to produce a truly divergence-free electric field is therefore understood to be an intimate part of the paraxial approximation.The electric field in Eqs.(8),(9),(12),and(13)also has a nonzero divergence because it depends on the paraxial approxi-mation.However,one can show that the divergence is minimized by the choice a m?b m in Eqs.(12)and (13),which retains in the divergence only the term analogous to that for the elementary linearly polarized Gaussian beam.Whereas the above fields satisfy the vector paraxial wave equation for all a m and b m, the best vector-beam solutions are those for which a m?b m:

′r,m?r,f,z??a m Q?z??J m21?u?1J m11?u??

64cos?m f?

sin?m f?

75,

(15)′f,m?r,f,z??a m Q?z?

3?J m21?u?2J m11?u??

642sin?m f?

cos?m f?

75.(16)

January1,1996/Vol.21,No.1/OPTICS LETTERS11 The two Bessel–Gauss modes already known and

mentioned above can be found as members of the

infinite set of solutions just presented.For m?

0,′r,0vanishes,leaving the azimuthally polarized,f-

independent field

m?0:E?r,z??2?f2a0J1μbr

11iz?L

3f?r,z?Q?z?exp?i?kz2v t??,(17) the same as that discussed in Eqs.(4)–(7),as the

lowest-order vector-beam solution.The J0Bessel–

Gauss beam reported by Gori et al.7can be found as

part of the m?1solution.To see this,recall that their f-independent solution emerges from the scalar

Helmholtz equation in the paraxial limit[Eq.(1)],

which means that it is valid for a linearly polarized field.Taking the direction of that linear polarization

along x,and noting that the Cartesian component′x

can be built up according to′x?′r cos f2′f sin f, one finds that choosing the upper trigonometric func-

tions and m?1in Eqs.(15)and(16)gives

m?1:E x?r,f,z??a1?J0?u?1J2?u?cos?2f??

3f?r,z?Q?z?exp?i?kz2v t??,(18)

where u is given by Eq.(14).The first term in Eq.(18)is precisely that given by Gori et al.7;the second term is also a solution of Eq.(1).In fact,one can show that the solution written down explicitly by Gori et al.7is indeed the lowest-order member of the set of scalar solutions of Eq.(1)given by

E n?r,f,z??A n J n?u?f?r,z?Q?z?

3cos?n f?exp?i?kz2v t??,(19)

where n is the azimuthal mode integer,A n is a constant,and all other quantities are as defined above. One can similarly find the other members of the set in Eq.(19)by using the vector modes defined by Eqs.(8),(9),(15),and(16)by specializing to the case of linear polarization.One can find the familiar linearly polarized,symmetric Gaussian,Eq.(2),by taking the limit b?0in the first term of Eq.(18).

The vector-beam solutions that are the subject of this Letter have so far been discussed in terms of the electric field E,but it is clear that they apply also to the magnetic field H because both H and E satisfy Eq.(3).This means,for example,that there exists a magnetic-field analog of the azimuthally polarized solution in Eq.(17):m?0:H?r,z??2?f2b0J1

μbr

11iz?L

3f?r,z?Q?z?exp?i?kz2v t??,(20) a free-space solution with the same general character as the TM01mode of the optical fiber(b0is a con-stant).The azimuthal E and H fields of the TE01 and TM01modes(TM is transverse magnetic)are ac-companied by H and E fields,respectively,that are purely radial with no dependence on the azimuthal angle f.Modes with this polarization state are well known in waveguides8but are somewhat less famil-iar for free space.These m?0solutions exhibit com-plete(vector)rotational symmetry about the direction of propagation,which is not the case for the linearly po-larized Gaussian beam,for which only the magnitude of the field exhibits rotational symmetry.

As discussed in Ref.6,in which only the azimuthal polarization was discussed,there is some connection between the solutions presented here and the J0Bessel beam discussed by Durnin10and Durnin et al.11Those described here are in no sense diffractionless,however, as they have finite size,are square integrable,and do indeed diverge as they propagate,as illustrated ex-plicitly in Ref.6for the azimuthal polarization.It de-serves mention that Bouchal and Olivik recently solved the(nonparaxial)vector wave equation to obtain a fam-ily of vector Bessel beams similar to the scalar solutions described in Refs.10and11,solutions described there as nondiffractive.12

The author gratefully acknowledges the support of the U.S.Army Research Office and the National Science Foundation.

References

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250.

https://www.wendangku.net/doc/60231306.html,x,W.H.Louisell,and W.B.McKnight,Phys.Rev.

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