Vector Diffraction Theory for Electromagnetic Waves

$Page 1: Vector Diffraction Theory for Electromagnetic Waves$
A. S. Marathay and J. F. McCalmont Vol. 18, No. 10 /October 2001 /J. Opt. Soc. Am. A 2585

Vector diffraction theory for electromagneticwaves

Arvind S. Marathay

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

John F. McCalmont

U.S. Air Force Research Laboratory/SNJW, 3050 C Street, Hangar 4B, Wright-Patterson Air Force Base, Dayton,Ohio 45433-7300

Received October 27, 2000; revised manuscript received March 16, 2001; accepted March 19, 2001

The scalar Huygens–Fresnel principle is reformulated to take into account the vector nature of light and itsassociated directed electric and magnetic fields. Based on Maxwell’s equations, a vector Huygens secondarysource is developed in terms of the fundamental radiating units of electromagnetism: the electric and mag-netic dipoles. The formulation is in terms of the vector potential from which the fields are derived uniquely.Vector wave propagation and diffraction formulated in this way are entirely consistent with Huygens’s prin-ciple. The theory is applicable to apertures larger than a wavelength situated in dark, perfectly absorbingscreens and for points of observation in the right half-space at distances greater than a wavelength beyond theaperture. Alternatively, a formulation in terms of the fields is also developed; it is referred to as a vectorHuygens–Fresnel theory. The proposed method permits the determination of the diffracted electromagneticfields along with the detected irradiance. © 2001 Optical Society of America

OCIS codes: 260.1960, 260.2110, 260.5430.

1. INTRODUCTIONThe popularity of the Fresnel–Kirchhoff and theRayleigh–Sommerfeld formulations of the scalar diffrac-tion theory of light stems from their applicability andrelative ease of use for instrumental optics. An integraltheorem of Kirchhoff1–3 is the basis for the mathematicalformulation of scalar diffraction. Several authors4,5 havecritically analyzed and refined scalar diffraction theory.Maggi and Rubinowicz6 put Young’s boundary diffractionwave theory on a firm theoretical foundation, which wasfurther developed by Miyamoto and Wolf6 and more re-cently by Marchand and Wolf.7,8

Debye,9 Kottler,10 Stratton and Chu,11 Wolf,12 Ri-chards and Wolf,13 and Tai14 have described the diffrac-tion of electromagnetic waves. More recently, theories ofelectromagnetic wave diffraction under particular physi-cal conditions were compared and evaluated by Karcze-wski and Wolf.15,16 Karczewski and Wolf also investi-gated the degree of polarization of the diffracted fieldwhen the incident light was unpolarized. The formalismof Richards and Wolf was applied to confocal fluorescencemicroscopy by Visser and Wiersma.17 These develop-ments are based on the Stratton–Chu integral,11 which isa vectorial equivalent of the Fresnel–Kirchhoff diffractionintegral with Kirchhoff boundary conditions. Diffractiontheory of electromagnetic waves from apertures in a con-ducting screen is discussed by Bekefi,18 who gives morereferences. Diffraction theory from apertures in conduct-ing screens is also found in several texts19,20; the list ofreferences given here is not exhaustive.

In this paper we propose a modification of Huygens’sscalar secondary sources based on Maxwell’s equations to

0740-3232/2001/102585-09$15.00 ©

account for the vector properties of electromagneticwaves. The propagation and diffraction of the waves isthen described by a method that is consistent with Huy-gens’s principle.

The theoretical work in this paper is based on certainfield equivalence theorems that are due toSchelkunoff.21,22 However, these theorems are notwidely known. They are briefly summarized in the be-ginning of Section 4, where the vector secondary sourcesare defined.

The complete solution of the inhomogeneous waveequation in free space for the vector potential consists of avolume integral over sources contained within a volumeof space and a surface integral of the field and its normalderivative that are present on the surface of that volume.Traditionally in optics, diffraction is formulated by usingonly the surface integral (Kirchhoff formulation), wherethe surface includes the aperture plane. We propose analternative approach. As will be shown, it is possible todescribe vector diffraction phenomena by placement ofappropriate hypothetical vector secondary sources in adiffracting aperture. The sources are employed in the in-homogeneous wave equation to solve for the vector poten-tial at a field point beyond the aperture. With these asthe only sources present, one obtains the vector potentialby performing an all-space volume integration over thesesources; see Jackson.19 The surface integral is zero,since there are no sources at infinity. From the solutionof the vector potential, the fields can be derived uniquely.Alternatively, a direct formulation in terms of the fields isalso developed and is referred to as a vector Huygens–Fresnel theory.

2001 Optical Society of America

2586 J. Opt. Soc. Am. A/Vol. 18, No. 10 /October 2001 A. S. Marathay and J. F. McCalmont

The theory discussed in this paper is for apertureslarger than a wavelength situated in dark (perfectly ab-sorbing) screens and for points of observation in the right-half-space at distances greater than a wavelength beyondthe aperture. Some of the results contained here werepresented at the 1999 Annual Meeting of the Optical So-ciety of America.23,24 For validation, in Section 8 weshow examples of common diffraction patterns producedby the new theory and reproduce the known results.

2. HUYGENS’S CONSTRUCTION FORVECTOR WAVES AND VECTORDIFFRACTIONIn Huygens’s principle, the hypothetical secondarysources are scalar in nature and produce scalar sphericalwaves. For electromagnetic theory, the hypothetical sec-ondary sources are modified to describe vector wavepropagation. The desirable properties of such sourcesare as follows:

1. They are hypothetical point sources,2. They radiate primarily in the forward direction

(with an obliquity factor), and3. They include the vector properties of light.

Electromagnetic point sources are closely approxi-mated by infinitesimal electric and magnetic dipoles. In-dividually, each source radiates into 4p sr in the form of adoughnut-shaped pattern and alone would not be appro-priate as a vector secondary source. However, it turnsout that all three requirements can be fulfilled by aunique secondary source derived through the applicationof the boundary conditions to a set of modified Maxwellequations. As will be shown, the application of theboundary conditions to the set of modified Maxwell equa-tions results in an infinitesimal electric dipole at a rightangle to a corresponding magnetic dipole that has all therequired features of a valid vector secondary source. Weshall refer to this combination as an EM dipole (hypo-thetical and mutually noninteracting).

The secondary waves emanating from these sources arevector waves. Following Huygens’s construction, the en-velope of all these secondary waves will yield the futureshape of the wave front, including the orientation of theelectric E and magnetic H fields.

3. MODIFIED MAXWELL EQUATIONSTo set up the framework of the hypothetical secondarysources, we modify the Maxwell equations by introducinga hypothetical magnetic (monopole) charge density, rm ,and a corresponding magnetic current density, Jm . Sev-eral authors19,25–27 have discussed such modifications.The modified Maxwell equations are

¹ • E 5re

eo, ¹ • H 5

rm

mo, (1)

¹ 3 E 5 2Jm 2 mo

]H

]t, ¹ 3 H 5 Je 1 eo

]E

]t. (2)

The terms rm and Jm obey the conservation of magneticcharge

¹ • Jm 1]rm

]t5 0 (3)

similarly to the terms re and Je for the conservation ofelectric charge, namely,

¹ • Je 1]re

]t5 0. (4)

The subscripts e and m are used to identify the specificsources (electric or magnetic) of the EM quantities cited.Introduction of the fictitious terms rm and Jm also intro-duce symmetry into the Maxwell equations. The fieldsthat appear in Eqs. (1) and (2) consist of the fields thatare due to both electric and magnetic contributions:

E 5 Ee 1 Em , H 5 He 1 Hm . (5)

Evidently, the modified Maxwell equations (1) and (2) canbe separated into two parts, one with only electric sourcesand the other with only magnetic sources. In fact, thetwo parts obey the principle of duality,27–29 which statesthat it is possible to pass from one equation to another bya suitable interchange of dual quantities, namely, theelectric and the magnetic quantities. Equations of a par-ticular gauge (either Lorentz or Coulomb) with only elec-tric sources may be converted to those with magneticsources and vice versa by using the dual replacementsgiven in Table 1. In any particular gauge, we can definethe total vector potential and the total scalar potentialanalogous to Eqs. (5):

A 5 Ae 1 Am , f 5 fe 1 fm . (6)

4. VECTOR SECONDARY SOURCESA brief review of Schelkunoff ’s induction theorem andequivalence theorem is adapted from Jordan andBalmain.27 In EM field theory it is often convenient tospecify electric and magnetic current densities as thesources of the fields. The induction theorem deals withtwo homogeneous regions: m1 , e1 , s1 and m2 , e2 , s2 ,where region 1 is enclosed by a closed surface S and con-tains the sources that produce the field incident at theboundary. The incident field sets up reflected and trans-mitted fields. The induction theorem indicates that thediscontinuities in the fields E and H across surface S canbe produced by (induced) current densities on the surfaceS given by

Je 5 n 3 ~Htt 2 Ht

r! 5 n 3 Hti, (7)

Jm 5 2n 3 ~Ett 2 Et

r! 5 2n 3 Eti, (8)

where n is the unit normal in the direction of the trans-mitted wave, the subscript t indicates tangential compo-

Table 1. Dual Replacements of Electric andMagnetic Sources

Type of Source Equivalent for Conversion

Electric Ee He Je re eo mo Ae

Magnetic Hm 2Em Jm rm mo eo Am


nents, and the appropriate superscripts label incident, re-flected, and transmitted fields.

The equivalence theorem follows directly from the in-duction theorem. However, in this case the medium inregion 2 is the same as the medium in region 1, that is,the material constants are the same in both regions, andare separated by the closed surface S. Thus there is noreflected wave. The required hypothetical surface cur-rent densities are

Je 5 n 3 Htt 5 n 3 Ht

i, (9)

Jm 5 2n 3 Ett 5 2n 3 Et

i. (10)

These equivalent hypothetical current densities on thesurface S combine to form the EM dipole secondarysources. They produce fields outside the closed surface Sthat are identical to the fields produced by the originalsources while at the same time producing no fields insideS. These theorems prove to be powerful tools in solvingEM problems that involve radiation from apertures.

The equivalence theorem is employed in formulatingthe vector diffraction problem. The surface of the entirewave front radiated by the original EM sources is takenas the surface S, and the induced electric and magneticcurrent densities combine to form the vector secondaryEM dipoles on this surface. When the wave front is inci-dent on the open area of the diffracting aperture, the com-bined radiation of the EM dipoles exposed by the diffract-ing aperture generates the diffracted field.

The secondary EM dipole strengths are specified interms of the electric and magnetic current densities ofEqs. (9) and (10) through the boundary conditions appliedto the incident wave front itself:

n • E 5rse

eo, n • H 5

rsm

mo, (11)

n 3 E 5 2Jsm , n 3 H 5 Jse , (12)

where n is the outward unit normal to the primary wavefront diverging from its source. The extra subscript s onrse , rsm , Jse , and Jsm , indicates surface values on thewave front. Only the fields on the wave front appear inEqs. (11) and (12). The fields inside S are set to zero, sono backward propagation toward the source region exists.

Although the secondary sources that we seek are hypo-thetical, the incident primary wave front is real. We sus-pect that its electric and magnetic fields are transverse tothe direction of propagation. As such, n • E 5 n • H5 0, implying that rse 5 rsm 5 0. In turn, the surfacecurrent densities, namely, Jse and Jsm , are solenoidal,[see Eqs. (3) and (4)], indicating thereby that they are alsotransverse to the outward unit normal of the primarywave front. We make this explicit by using an extra sub-script, ', as in

n 3 E 5 2J'sm , n 3 H 5 J'se . (13)

It is helpful to note for the special case of a plane wavefront with propagation vector k 5 (v/c) s, where s is theunit vector in the direction of propagation, that the unitnormal is n 5 s. Or, for a spherical wave front, n is inthe direction of the radius vector, n 5 r.

It is hypothesized that the EM dipoles are uniformlydistributed over the primary wave front. The magnitudeand direction of the constituent electric and magnetic di-poles are chosen to be consistent with Eqs. (12) and (13).

In the study of radiation from an elementary electricdipole, as described in standard textbooks, the currentdensity with harmonic time dependence is

Je~r, t ! 5 Je~r!exp~2ivt !, (14)

where v is the circular frequency and

Je~r! 5 2ivped ~r!. (15)

pe stands for the electric dipole moment. The Dirac deltafunction d (r) [ d (x)d (y)d (z) is three dimensional anddescribes the spatial location of the dipole at the origin.An expression similar to Eq. (14) can also be written forthe magnetic dipole:

Jm~r! 5 2ivpmd ~r!. (16)

To convert the volume current densities of Eqs. (15) and(16) into surface densities requires a proportionality con-stant with units of length. The choice of this constant islargely arbitrary. A natural parameter for the presentproblem is the wavelength l. Thus for the respectivesurface current densities we choose

n 3 H 5 1J'se 5 2i2pcp'ed ~r 2 rs!, (17)

n 3 E 5 2J'sm 5 i2pcp'md ~r 2 rs!, (18)

where the relationship 2pc 5 lv has been used. FromEqs. (17) and (18) the magnitude and direction of the re-spective electric and magnetic dipoles can be identified as

p'e~rs! 5 1i

2pcn~rs! 3 H~rs!, (19)

p'm~rs! 5 2i

2pcn~rs! 3 E~rs!, (20)

where rs denotes points on the surface of the primarywave front. Observe that the ratio of the magnitudes ofthe electric and magnetic dipoles equals the admittance offree space:

p'e

p'm5

uHu

uEu5 Aeo

mo. (21)

Equations (19) and (20) define the hypothetical EM di-pole at rs on the primary wave front. The directions ofp'e in Eq. (19) and p'm in Eq. (20) are mutually perpen-dicular, and these dipoles are hypothetical and noninter-acting. Both are tangential to the primary wave frontand describe the underlying structure of the EM dipole;see Fig. 1. The EM dipole so defined is an indivisibleunit and is the hypothetical vector Huygens secondarypoint source mentioned in Section 2. In Section 5 westudy the scalar and vector potentials of this vector sec-ondary source. A diffraction theory of vector waves asmentioned at the beginning of this paper is then formu-lated in terms of these potentials.


5. SCALAR AND VECTOR POTENTIALS OFTHE VECTOR SECONDARY SOURCESWe continue to assume time-harmonic dependence of thesources and fields, as first described in Eq. (14). For thetransverse current densities found in Section 4, it is moreappropriate to use the Coulomb gauge to obtain the EMpotentials. As discussed in standard texts,19,30 radiatingEM waves are influenced only by the transverse part J' ofthe current density, the source of the vector potential.For the electric current density,

¹2Ae~r! 1 k2Ae~r! 5 2moJ'e~r!. (22)

In this gauge, the Coulomb condition

¹ • Ae~r! 5 0 (23)

implies that the vector potential itself is also entirelytransverse, Ae 5 A'e , although generally it is not madeexplicit. Similarly, for the magnetic current density,

¹2Am~r! 1 k2Am~r! 5 2eoJ'm~r!. (24)

Corresponding to Eq. (23), the Coulomb condition formagnetic sources is

¹ • Am~r! 5 0. (25)

As before, this vector potential is entirely transverse:Am 5 A'm .

In Section 4 we found that the electric and magneticcharge densities are zero. Therefore the correspondingscalar potentials are either constant or zero and the lon-gitudinal components of the respective current densitiesare zero. The associated scalar potentials influence onlythe longitudinal components of the fields, and these com-ponents are zero. The radiated electromagnetic fields ofthe vector secondary source are derived from the (trans-verse) vector potentials discussed above, and these fieldsare given by the prescriptions

E~r! 5 Ee~r! 1 Em~r! 5 ivAe~r! 21

eo¹ 3 Am~r!,

(26)

H~r! 5 He~r! 1 Hm~r! 5 ivAm~r! 11

mo¹ 3 Ae~r!.

(27)

The total vector potential is

Fig. 1. Orientation of the EM dipole at rs on the primary wavefront (solid circle). A source of a monochromatic spherical waveis located at S0 ; point of observation P is shown with positionvector r. Difference vector Rs 5 r 2 rs makes an angle us withn(rs), the unit normal to the wave front. p'e , p'm , electric andmagnetic dipole moments, respectively.

A~r! 5 Ae~r! 1 Am~r!, (28)

which fulfills the inhomogeneous Helmholtz equation

¹2A~r! 1 k2A~r! 5 2moJ'e~r! 2 eoJ'm~r!. (29)

A formal solution of Eq. (29) has the form

A~r! 51

4pEEE

allspace@moJ'e~r8!

1 eoJ'm~r8!#exp~ikur 2 r8u!

ur 2 r8ud ~3 !r8, (30)

provided that no boundary surfaces are present.19 Recallthat, with the time-harmonic assumption, we have ex-pressed

A~r, t ! 5 A~r!exp~2ivt !, (31)

and Eq. (30) is consistent with the requirements of cau-sality.

6. ELECTROMAGNETIC DIPOLE FIELDSThe EM dipole fields are found through the vector poten-tials whose sources are the transverse current densities ofEq. (29). Having defined the EM dipole by Eqs. (19) and(20), we find the corresponding current densities:

J'e~Rs8! 5 2ivp'ed ~Rs8!, (32)

J'm~Rs8! 5 2ivp'md ~Rs8!, (33)

with the difference vector Rs8 [ r8 2 rs .It is assumed that no other sources are present except

the ones given in Eqs. (32) and (33). Hence, with noboundary surfaces, the all-space evaluation of Eq. (30)gives the usual form of the vector potential for the electricand magnetic dipoles:

AEM~r, rs! 5 Ae 1 Am 5 2iv

4p~mop'e 1 eop'm!

3exp~ikRs!

Rs, (34)

where Rs is the magnitude of Rs . The vector potential isgiven as a function of field point r that is due to the EMdipole located at rs on the primary wave front. Calcula-tion of the fields in Eqs. (26) and (27) requires a curl op-eration with respect to r on the vector potentials, which iscarried out for constants p'e and p'm . For the radiatedfield (the r22 terms are neglected) we get

EEM~r, rs! 5k2

4peo

exp~ikRs!

RsF ~Rs 3 p'e! 3 Rs

2 Aeo

mo~Rs 3 p'm!G , (35)


HEM~r, rs! 5k2

4pmo

exp~ikRs!

RsF ~Rs 3 p'm! 3 Rs

1 Amo

eo~Rs 3 p'e!G , (36)

where the dipole moments p'e and p'm are those definedrespectively in Eqs. (19) and (20). Rs 5 r 2 rs ; Rs is itsmagnitude and Rs is the corresponding unit vector. TheEM dipole fields have the form of an expanding sphericalwave from the point rs . The fields EEM , HEM and theunit vector Rs from a right-hand triad of mutually per-pendicular vectors.

The time-averaged Poynting vector is

^SEM& 5k4

8p2eomo

up'euup'mu

Rs2 S 1 1 cos us

2 D 2

Rs (37)

and points in the direction of the difference vector Rs .p'e , p'm , and Rs are the respective magnitudes of the as-sociated vector quantities, and us is the angle made by Rswith the normal n(rs) to the wave front at the point rs ;see Fig. 1. Observe that the angular dependence in Eq.(37) is the same as the obliquity factor that appears in theFresnel–Kirchhoff diffraction theory of scalar waves; see,for example, Ref. 1. The electric and the magnetic time-averaged energy densities also have the same angular de-pendence.

Figure 2 shows the magnitude of the time-averagedPoynting vector in a polar plot. In this figure p'e is per-pendicular to the plane of the figure and is located at theorigin. p'm is perpendicular to p'e , oriented in the ver-tical direction, also located at the origin, and lies in theplane of the page. Maximum radiation is in the forward,u 5 0, direction; no radiation is in the backward, u 5 p,

Fig. 2. Mapping of the EM dipole’s normalized irradiance on aspherical wave front in the far zone (represented by the outer-most ring) as a function of the angle between the direction of ob-servation and n, the outward unit normal to the primary wavefront. (Here n lies along the u 5 0 direction.) The electric di-pole is located at the origin, and its axis is perpendicular to thepage. The magnetic dipole is also located at the origin, orientedin the vertical direction, and lies in the plane of the page. ThePoynting vector points radially outward. The electric field vec-tor is tangent to the spherical wave front and is perpendicular tothe plane of the page. The magnetic field vector is also tangentto the wave front but lies in the plane of the page. The normal-ized magnitude of the Poynting vector is proportional to thelength of the chord from the origin to the cardioid along the ra-dius to the point of tangency. The plot is rotationally symmetricabout n.

direction; and there are small fractions in the directionsp/2 , uuu , p. The figure is rotationally symmetricabout the u 5 0 axis.

7. VECTOR DIFFRACTION THEORYThe diffraction geometry is shown in Fig. 3(a). It showsa monochromatic point source S0 producing a wave frontW incident onto an open aperture A in an opaque (dark)planar screen S. Point of observation P is shown behindthe screen. The primary source S0 is in the left-half-space, and the point of observation P is restricted to theright-half-space with respect to the screen S and at dis-tances greater than a wavelength beyond the aperture.

Following Huygens, we regard the surface of the pri-mary wave front to be uniformly populated by the (hypo-thetical) vector secondary sources, namely, the EM di-poles; see Eqs. (19) and (20). For the study of diffraction,the EM dipole is to be taken as one indivisible unit. Thefields from such a unit are given by Eqs. (35) and (36).The diffracting aperture A exposes some of the EM di-poles, and the screen blocks the rest.

Now the entire problem of diffraction can be describedby an equivalent geometry. The primary source S0 , thewave front W, and the opaque screen S are no longerpresent; only the exposed EM dipoles and the point of ob-servation P are; see Fig. (3b). Since these exposed EMdipoles are the only sources that are present in theequivalent geometry, we can perform an all-space-unbounded evaluation of Eq. (30) to obtain the vector po-tential A(r) at the point of observation P. As mentionedabove, the surface integral is zero. This procedureamounts to summing the component contributions [Eq.(34)] over all the relevant points rs in the aperture to con-struct the total A(r). Thus

A~r! 5 EEEESD

@Ae~r, rs! 1 Am~r, rs!#d~3 !rs , (38)

where ESD under the volume integral stands for the ex-posed secondary dipoles. Since the exposed secondary di-poles reside only on the surface of the primary wave front,the volume integral effectively reduces to a surface inte-gral. For example, if the primary wave front is a planewave propagating along the 1z axis, then the volume dis-tribution of EM dipoles may be described by the dipoles inthe x, y plane at d (z 2 z0), where z0 is the location of theplane wave along the z axis. For a spherical wave the ex-posed dipoles reside on the surface of the sphere at d (r2 r0), where r0 is the radius of the primary wave front.

The corresponding fields, E(r) and H(r), are found bydifferentiation according to the prescriptions of Eqs. (26)and (27), respectively. These fields permit us to calculatethe time-average Poynting vector and the irradiance atthe field point r.

However, when one is using a digital computer, takingderivatives is not the best way to compute fields. An al-ternative way to compute the fields at r is to perform thesum over EM dipole fields, namely, EEM(r, rs) of Eq. (35)and HEM(r, rs) of Eq. (36) over all the relevant points rs toconstruct the total fields E(r) and H(r) at point of obser-vation P at r. Thus the calculation of the vector diffrac-


Fig. 3. (a) Diffraction geometry. The primary source of a monochromatic spherical wave is located at S0 . The primary wave front isW. A dark screen, S, contains a clear aperture A. The point of observation is located at P. (b) EM dipoles are represented as smallopen circles along the arc of the wave front in the geometry of (a). Only the EM dipoles exposed by the aperture are shown; they con-tribute to the diffracted field at P. Primary source S0 , primary wave front W, the open aperture A, and dark screen S have been re-moved; they are shown as dotted lines.

tion of EM waves is considerably simplified. In the lit-erature a discrete dipole approximation has been usedsuccessfully in describing scattered light.31,32

Although from the computational point of view the pro-cedure described in the preceding paragraph is useful,formally we may wish to express the total fields in termsof the fields of Eqs. (35) and (36). The electric field at thepoint of observation r is

E~r! 5 EEEESD

EEM~r, rs!d~3 !rs

52i

2l2 EEEEPW

XAmo

eo$Rs 3 @H~rs! 3 n~rs!#%

3 Rs 2 $Rs 3 @ n~rs! 3 E~rs!#%C3

exp~ikRs!

Rsd~3 !rs ,

(39)and the magnetic field is

H~r! 5 EEEESD

HEM~r, rs!d~3 !rs

52i

2l2 EEEEPW

XAeo

mo$Rs 3 @ n~rs!

3 E~rs!#% 3 Rs 1 $Rs 3 @H~rs!

3 n~rs!#%C exp~ikRs!

Rsd~3 !rs , (40)

where EPW under the volume integral stands for exposedparts of the primary wave front. Since the fields that ap-pear in the integrands reside only on the surface of theprimary wave front, the volume integrals of Eqs. (39) and(40) effectively reduce to surface integrals over the wave

front as described for the vector potential A(r) in the dis-cussion following Eq. (38). In Eqs. (39) and (40) the EMdipoles were expressed in terms of the fields of the pri-mary wave front; see Eqs. (19) and (20). However, sincethe fields of the wave front obey

H~rs! 5 Aeo

mon~rs! 3 E~rs!, (41)

the diffracted fields may also be expressed as

E~r! 52i

2l2 EEEEPW

H @Rs 3 E~rs!# 3 Rs

2 Amo

eo@Rs 3 H~rs!#J exp~ikRs!

Rsd~3 !rs (42)

52i

2l2 EEEEPW

(E~rs!$1 1 @Rs • n~rs!#%

2 @Rs • E~rs!#[Rs 1 n~rs!])exp~ikRs!

Rsd~3 !rs,

(43)

H~r! 52i

2l2 EEEEPW

H @Rs 3 H~rs!# 3 Rs

1 Aeo

mo@Rs 3 E~rs!#J exp~ikRs!

Rsd~3 !rs (44)

52i

2l2 EEEEPW

$H~rs!@1 1 Rs • n~rs!#

2 @Rs • H~rs!#[Rs 1 n~rs!#%exp~ikRs!

Rsd~3 !rs,

(45)

so the diffracted fields E(r) and H(r) may be calculatedfrom the fields of the incident wave front in the aperture.


As a reminder we note that Rs [ r 2 rs , where Rs is itsmagnitude and Rs is the corresponding unit vector. Thevolume integrals in Eqs. (39) and (40) and in Eqs. (42)–(43) and (43)–(45) effectively reduce to surface integralsover the wave front only, regardless of its shape. A similarset of equations can be found in Ref. 33.

A. Limitations and ApproximationsIn the Kirchhoff and the Rayleigh–Sommerfeld diffrac-tion theories the primary wave front incident on the ap-erture is regarded as undisturbed by the aperture(Kirchhoff-type boundary conditions) for the calculation ofdiffracted field amplitudes in the right half-space. It iswell known that the presence of the aperture does perturbthe incoming field. These considerations also apply tothe procedure described in this paper for calculating theelectric and magnetic fields at r. However, if the exactboundary conditions are known, the proposed theorycould be used to calculate the exact diffracted fields.

The next caution pertains to the size of the aperture.The EM dipoles are described as point dipoles. However,the approach to diffraction described in this paper can bevalid only for aperture dimensions that are comparable toor larger than the wavelength of light.

The usual appearance of the quadratic-phase factor forFresnel diffraction and the Fourier relation for Fraun-hofer diffraction in relation to the EM dipoles is describedby McCalmont.33 This explicit distinction for the twotypes of diffraction is not necessary in the computer cal-culation of the coherent superposition of EM dipole fields.Computer calculation generates the appropriate diffrac-tion pattern regardless of the placement of the point of ob-servation r relative to the dimensions of the aperture.

B. Undisturbed Primary WaveIn the Huygens–Fresnel theory of diffraction the primaryspherical wave front is divided into annular regions called

the Fresnel half-period zones. The contributions of allthe zones with the assumed obliquity factor are coher-ently summed at the point of observation P. The resultso obtained is compared with the original (undisturbed)primary wave front arriving at the same point P. Thiscomparison compels us to conclude that the secondary(scalar) waves oscillate a quarter of a period ahead of theprimary wave and that the amplitudes of the secondarywaves are in the ratio 1:l to the amplitudes of the pri-mary.

The above procedure is not mandatory in our presentcontext, owing to the fundamental theorem, Schelkunoff ’sfield equivalence theorem.21,22 It is described in detail inthe texts by Jordan and Balmain,27 Harrington,28

Johnk,29 and Yamashita.34

We can apply a special case of the field equivalencetheorem to the problem of the diffraction of light by iden-tifying the closed surface C with the full primary wavefront W. The appropriate placement of the charge andcurrent densities on the surface of the primary wave frontis accomplished through the boundary conditions [Eqs.(7)–(9)] of Section 4. The field equivalence theorem as-sures us that the fields outside W, namely, at the point ofobservation r, are the same as the fields produced by theoriginal source. The theorem also states that the fieldsdue to the secondary (vector) sources placed on W add upto zero inside W.

8. ELEMENTARY EXAMPLES OF THECALCULATION OF VECTOR DIFFRACTIONIn support of the validity of the approach to diffractionproposed in this paper, a comparison with published re-sults and expected diffraction patterns under commonly

Fig. 4. Axial distribution of the normalized field amplitude beyond circular apertures of diameters D calculated by the vector Huygens–Fresnel model as a function of distance z from the aperture. (a) D 5 1l, (b) D 5 3l, (c) D 5 6l, (d) D 5 10l. The incident field isa plane wave of unit amplitude falling normally onto the plane of the aperture.


Fig. 5. Irradiance profiles along the horizontal y axis on an observation screen at distances (a) 5l, (b) 100l, (c) 500l, (d) 15,000l that aredue to diffraction by a rectangular slit and produced by the vector Huygens–Fresnel model. Slit width, W 5 20l. The first zero of theFraunhofer pattern is at 2.81°. The incident field is a plane wave of unit amplitude falling normally onto the plane of the aperture andpolarized in the vertical x direction. The irradiance profile is given as a function of wavelengths from the origin of the observation planealong the horizontal y axis.

known circumstances is in order. An additional benefitof the proposed method is that, along with the detected ir-radiance, the EM fields are also obtained in the diffrac-tion pattern. The diffraction patterns are computed bythe method of summation of fields due to the EM dipolesas prescribed in Eqs. (42)–(43) and Eqs. (44)–(45).

As illustrations, we consider a circular aperture of di-ameter D/l and a slit aperture of width w/l. Each ap-erture in turn is placed in a planar dark screen in the x, yplane with the origin at the midpoint of the aperture.The time-averaged Poynting vector ^S& is in the directionof the unit vector r in the radial direction. The irradi-ance I in the plane of observation is computed by the defi-nition I 5 u^S& • nu. For a plane of observation normalto the z axis, n 5 k. This particular definition of irradi-ance is necessitated by the fact that, if the dimensions ofthe apertures are comparable to the wavelength of the in-cident radiation, the diffraction patterns consequentlyhave a very large angular spread. However, when theangles involved are small enough that tan u ' sin u ' u,the scalar product can be ignored and we may set I5 u^S&u, as is usually defined; see Sec. 8.4 of Born andWolf.1

First consider a circular aperture centered about the zaxis. Figure 4 shows the absolute value of the diffractedfield amplitude as a function of the distance z/l along theaxis. The figure shows four cases in which the diameterD/l of the aperture is given. These curves compare fa-vorably with the ones given by Marchand and Wolf8 andWolf35; the comparison is qualitative, since no numericalvalues are available.

Next consider a rectangular one-dimensional apertureof width w/l 5 20. Figure 5 shows several plots of theirradiance at several distances z/l from the rectangular

aperture. It shows a progression of diffraction patternsfrom the Fresnel region through the Fraunhofer region, inagreement with known theory.

9. CONCLUSIONSVector diffraction of electromagnetic waves is consider-ably simplified by being described in terms of vector Huy-gens secondary sources, namely, the EM dipoles definedin terms of the fields of the primary wave front. Diffrac-tion of EM waves for apertures larger than the wave-length situated in dark screens with points of observationrestricted to the right-half-space beyond the aperture andat distances greater than a wavelength is formulated interms of the vector potential. Also, it is formulated interms of the fields and is referred to as the vectorHuygens–Fresnel theory. An additional benefit of theproposed method is that, along with the detected irradi-ance, the EM fields are also obtained in the diffractionpattern. Thus vector diffraction can be looked upon as asimple summation of EM dipole fields.

The EM dipole is a conceptual device with which thediffraction of EM waves may be studied, but the key con-cept is the pair of hypothetical current densities derivedfrom Schelkunoff ’s theorem.

ACKNOWLEDGMENTThe authors gratefully acknowledge the efforts of JimWilson of Tucson, Arizona, in the development of the com-puter simulation code used to validate the theory pre-sented in this paper and in the production of Figs. 4 and5.

J. F. McCalmont’s e-mail address [email protected].


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Vector Diffraction Theory for Electromagnetic Waves