Incremental Diffraction Coefficients for Planar Surfaces

A bstruct-Exact expressions for incremental diffraction coefficients at arbitrary angles of incidence and scattering are derived directly in terms of the corresponding two-dimensional, cylindrical diffraction coefficients. Specifically, if one can supply an expression for the conventional diffraction coefficients of a two-dimensional planar scatterer, one can immediately find the incremental diffraction coefficients through direct substitution. No integration, differentiation, or specific knowledge of the current is required. The derivation is limited to perfectly conducting scatterers that consist of planar surfaces, such as the wedge, the slit in an infinite plane, the strip, parallel or skewed planes, polygonal cylinders, or any combination thereof; and requires a known expression (whether exact or approximate) for the two-dimensional diffraction coefficients produced by the current on each different plane. Special attention is given to defining unambiguously all real angles and their analytic continuation into imaginary values when required by the incremental diffraction coefficients. The validity of the general expressions is confirmed by showing that the physical theory of diffraction, geometrical theory of diffraction, and physical optics incremental diffraction coefficients obtained by direct substitution into the general expressions agree with the results of Mitzner, Michaeli, and Knott, respectively, in the case of the infinite wedge. In addition, it is shown that the two-dimensional diffraction coefficients are recovered when the general expressions for the incremental diffraction coefficients are integrated over an infinite straight line. Finally, our general method is used to obtain for the first time the incremental diffraction coefficients for the infinitely long, narrow strip and slit.


I. INTRODUCTION
ONSIDER an electromagnetic wave in free space incident C upon a perfectly electrically conducting scatterer.To a first approximation the surface current induced on the scatterer will be the physical optics (PO) current, i.e., 28 x H, on the illuminated side of the scatterer and zero on the shadow side.To obtain the scattered fields radiated by the PO current, the PO current (multiplied by the free space Green's function) can usually be integrated numerically, sometimes analytically, and often asymptotically for certain regions of observation.In particular, a variety of computer programs exist for calculating the far fields of reflector antennas by integrating the PO current induced on the reflector by a given feed illumination [I].
Of course, the scattered fields obtained from the PO current will not be exact unless the PO current equals the total surface current.Thus the accuracy of the computed scattered fields can be improved if one can estimate the contribution to the scattered fields of the difference between the total and PO currents.Even for electrically large scatterers this difference current or "nonuniform current"' to use the terminology of Ufimtsev [21, [31 can strongly affect the scattered fields.For example, the far fields of the nonuniform current near the rims of reflectors can appreciably change the further-out sidelobes of the copolarized fields, and all the lobes of the crosspolarized fields [4], [5].In general, the inclusion of the fields radiated by the nonuniform currents is especially important for the accurate determination of cross-polarized fields, sidelobe fields, and fields near nulls.
If one does not know the exact solution to the scattering problem, one must approximate the nonuniform current.Ufimtsev [2], [3], in developing his "physical theory of diffraction" (PTD), assumes that the nonuniform current at a given point on a general electrically large scatterer is approximately equal to the nonuniform current of a corresponding canonical scatterer that conforms to the shape of the general scatterer in the locality of the given point.(The incident field for the canonical scatterer is also chosen as the incident field in the locality of the given point on the general scatterer.)For example, the nonuniform current near the rim of an illuminated thin-metal reflector would be approximated locally by the nonuniform current near the edge of a correspondingly illuminated, perfectly conducting half-plane.Similarly, the nonuniform current near the slits between panels that may form the reflector surface would be approximated by the nonuniform current near an infinite straight slit in an infinite perfectly conducting plane.
Typically, the predominant nonuniform currents vary rapidly over a transverse distance less than a wavelength.Moreover, closed-form expressions for the nonuniform currents of canonical problems are not generally available.Thus, unlike the PO current, it is usually impractical to numerically integrate the nonuniform currents on a general scatterer to obtain their radiated fields.
Fortunately, the far fields radiated by the nonuniform currents of a number of two-dimensional canonical scatterers can be expressed in closed form, even though the nonuniform currents themselves usually cannot.Specifically, the total diffracted far fields are known in closed form for a number of canonical scatterers illuminated by a plane wave.In addition, the PO current for these scatterers under plane-wave illumination can usually be integrated to obtain closed-form expressions for the PO diffracted far fields.Thus the far fields radiated by the nonuniform currents of these two-dimensional canonical problems can be found in closed form, simply by 1271.
subtracting the PO far fields from the total far fields since the nonuniform current is defined as the difference between the total and PO currents.Of course, this technique for finding the fields radiated by the nonuniform currents of canonical scatterers is precisely the one that Ufimtsev used to obtain closed-form expressions for the nonuniform or PTD diffraction coefficients for the perfectly conducting wedge [2], [ 3 ] .Because the total diffracted fields define the coefficients of the geometrical theory of diffraction (GTD) [6], [7], one is also correct in saying that Ufimtsev obtained the PTD diffraction coefficients by subtracting the PO diffraction coefficients from the GTD diffraction coefficients.* Through an asymptotic analysis of the diffraction integral, Van Kampen [9], [IO] shows that the dominant high-frequency diffracted fields of a general three-dimensional scatterer emanate (or appear to emanate) from "critical points" on the scatterer where the ray path length becomes stationary or discontinuous (including discontinuities in derivatives of the path length).Van Kampen shows further that the critical points can be divided into critical points of the first, second, or third kind depending on whether they are 1) stationary points on the surface of integration (specular reflection), 2) stationary points along a curve (edge, shadow boundary, etc.) bounding the surface of integration, or 3 ) discontinuities (such as corners and tips) in the slope (or higher order derivatives) of a bounding curve.
When the frequency is high enough that several Fresnel zones are contained in a neighborhood of a critical point, and the amplitude of the incident field does not vary appreciably over this neighborhood, the diffracted field of this critical point is well approximated by the first term of the highfrequency asymptotic expansion [9].Under these conditions the three-dimensional diffracted fields from a critical point of the second kind lie on the local cone of diffraction and can be expressed directly in terms of the diffraction coefficients of the two-dimensional canonical scatterer that conforms locally to the general scatterer at that point [2], [ 3 ] , [6], [7].This powerful result, which combines the assumption that at high * For two-dimensional canonical problems of infinite cross section, like the perfectly conducting wedge, the far fields of the total and PO current for an incident plane wave approach infinity at the shadow and reflection boundaries.Since the PO current is removed from the total current in the determination of the PTD diffraction coefficients, the PTD diffraction coefficients for such canonical problems have the advantage over the GTD coefficients of remaining finite at shadow and reflection boundaries, except for grazing incidence along an infinite face.The GTD coefficients, when applied to general scatterers, have the advantage over the PTD coefficients of not requiring a separate evaluation of the PO fields away from the shadow and reflection boundaries.The GTD singularities at the shadow and reflection boundaries are removed in "uniform geometrical theories of diffraction" that consider the source or field points at a finite distance from the scatterer.However, the shadow and reflection singularities remain a difficulty for GTD under scattering conditions most commonly encountered in practice, namely large distances to the source and field points.Also, in the particular case of a plane H-wave illuminating a wedge, both the GTD and PTD diffraction coefficients are discontinuous across the faces of the wedge, as a function of either the incident angle or the scattering angle.In applying the wedge diffraction coefficients to general scatterers, these face-angle discontinuities can be removed but only by ad hoc truncation of the current tails or by considering multiple interactions, if possible, between different parts of the scatterer [8].
frequencies canonical currents approximate the local currents on the general scatterer, with the derivation of a generalized Fermat's principle, is the keystone and main reason for the success of both the physical and geometrical theories of diffraction.The approximation of local scattering by canonical scattering and the generalization of Fermat's principle are sometimes stated as the two fundamental postulates of the geometrical theory of diffraction [ 1 11.However, only the assumption that high-frequency scattering is a local phenomenon that can be approximated by canonical scattering need be postulated, because, as discussed above, the generalization of Fermat's principle is derivable from an asymptotic treatment of the diffraction integral [9], For critical points of the second kind that do not have several neighboring Fresnel zones (for example, observation points near caustics of the scattered field) or that have an incident field with an appreciable variation in amplitude along the curve near the critical point, and for critical points of the third kind (i.e., points of abrupt change in smooth curves), the diffracted fields, in general, are not well approximated by the conventional plane-wave diffraction coefficients of the corresponding two-dimensional canonical problems.To remedy this deficiency, Mitzner introduced ''incremental length diffraction coefficients" [ 171, which, when multiplied by the incident field, could be integrated along a bounding curve of the scatterer to obtain the diffracted fields of the nonuniform current for arbitrary angles of incidence and scattering.
The PTD incremental diffraction coefficients as defined by Mitzner give the far fields (in any direction) radiated by the nonuniform current on a differential length of an infinite cylinder of arbitrary cross section (two-dimensional canonical problem) illuminated by a plane wave at an arbitrary angle of incidence.Under the assumption that high-frequency diffraction is a local phenomenon, the fields radiated by each differential length of the nonuniform current on a general scatterer will be the same as the fields radiated by the differential length of the corresponding two-dimensional canonical scatterer.Thus, the scattered fields radiated by a general scatterer can be determined by integrating the appropriate incremental diffraction coefficients over the bounding curves of the scatterer.For example, a computer program that computes the fields of a reflector antenna by integrating the PO current can be improved to include the contribution from the nonuniform current by integrating the incremental diffrac-' As early as 1902 Schwarzschild [I21 presented the idea of using the canonical half-plane solution to obtain an improved solution to diffraction by a slit.However, the first paper to combine the idea of using canonical solutions with a stationary phase evaluation that included edges (generalized Fermat's principle) is apparently the classic 1912 paper by MacDonald [13], [14].In 1950, Braunbek [IS] developed further these two basic ideas of the modern theories of diffraction and applied them in detail to the circular aperture and disc.Both Ufimtsev [2] and Keller et al. [6] reference Braunbek [IS], who formulated a physical theory of diffraction by adding nonuniform "correction" fields to the uniform Kirchhoff fields in the aperture of a plane screen or on a disk.Although we concentrate on the Ufimtsev current formulation in the present paper, it is emphasized that similar results could be derived as well for a Braunbek field formulation.Such a formulation involving nonuniform correction near fields instead of nonuniform currents would apply conveniently to improving the accuracy of the radiated fields computed from a geometrical-optics aperture field integration [I], [ 161.
tion coefficients (multiplied by the feed illumination) around the edge of the reflector and along the slits between the panels that form the reflector.Separate, ad hoc analyses to obtain the fields in caustic (focal) regions, main beam and near-in sidelobe regions, or from corners (neglecting the higher order distortion of the current near corners) become unnecessary.Once the computer program is fed the geometry of the scatterer and the incident field, the fields everywhere can be computed straightforwardly by two algorithms-one that integrates the PO current and one that integrates the incremental diffraction coefficients.Moreover, tangential slope and higher tangential derivative diffraction coefficients are computed implicitly when integrating the incremental diffraction coefficients multiplied by the incident field.(In the remainder of this paper we will sometimes loosely refer to "integration of incremental diffraction coefficients" without explicit mention that the coefficients are multiplied by the incident field.) Of course, for the curves that have only stationary points illuminated by incident fields with slowly varying amplitudes over several Fresnel zones, the integration should yield approximately the same values of the fields as those obtained without integration directly from the conventional two-dimensional coefficients.However, as mentioned above, integration of the incremental diffraction coefficients has the advantage of determining accurately the fields of the non-uniform current of a general scatterer for many geometries, observation angles, and incident fields for which the first-order diffraction fields are a poor approximation; in addition, tangential slope or higher tangential derivative diffraction coefficients are not required when the variation in amplitude of the incident field becomes appreciable along the direction tangent to the curve.Also, the extra computer time required to integrate the incremental diffraction coefficients along bounding curves is generally small compared to the computer time required to do a two-dimensional integration of the PO current.
As with the conventional two-dimensional diffraction coefficients, incremental diffraction coefficients can be defined for the total current of cylinders to yield GTD incremental diffraction coefficients [ 181, or for the PO current to yield PO incremental coefficients.The PTD (Mitzner) incremental diffraction coefficients can be obtained by subtracting the PO incremental diffraction coefficients from the GTD incremental diffraction coefficients [ 191.The comparative advantages mentioned in footnote 2 for PTD and GTD two-dimensional diffraction coefficients hold also for the incremental diffraction coefficients.
The implicit use of the concept of incremental diffraction coefficients was made in work that preceded that of Mitzner [17], notably that of Ufimtsev [2], [3], Braunbek [15], , and the "equivalent (edge) current" work of Ryan and Peters [23].However, this previous work required the fields of the incremental diffraction coefficients only near focal regions where slight, ad hoc modifications to the scattered fields along the diffraction cones were adequate.Knott and Senior [24] review the use of GTD "equivalent currents" before 1974, as well as extend the technique of Ryan and Peters to obtain approximate GTD incremental diffraction coefficients for arbitrary angles of incidence and scattering.In a 1985 publication [33], Knott reviews further the subject of equivalent currents and incremental length diffraction coefficients.
To the authors' knowledge, exact expressions for the incremental diffraction coefficients have been found previously only for the perfectly conducting wedge.Mitzner [I71 determined the PTD incremental diffraction coefficients of the wedge by matching the required nonuniform current integrals to similar integrals that occur for the two-dimensional wedge diffraction coefficients.Ten years later, Michaeli [ 181 determined the GTD incremental diffraction coefficients of the wedge by integrating the total wedge current in closed form.
The main objective of this paper is to complement the work of Mitzner [17] and Michaeli [18] by providing a more general, convenient method for determining incremental diffraction coefficients.Specifically, we derive exact expressions for the incremental diffraction coefficients at arbitrary angles of incidence and scattering directly in terms of the corresponding two-dimensional diffraction coefficients.The derivation is limited to perfectly conducting scatterers that consist of planar surfaces, such as the wedge, the slit in an infinite plane, the strip, parallel or skewed planes, polygonal cylinders, or any combination thereof.The derivation also requires a closed-form expression, whether exact or approximate, for the two-dimensional diffraction coefficients produced by the current on each different plane.In other words, if one can supply a closed-form expression for the conventional diffraction coefficients of a two-dimensional planar scatterer.one can immediately find the incremental diffraction coefficients through direct substitution.No integration, differentiation or specific knowledge of the current is required.(Also, the restriction to "closed-form'' expressions can be removed for the many problems that require only real angles of the diffraction coefficients.) We show that the PTD, GTD, and PO incremental diffraction coefficients obtained by direct substitution into the general expressions agree with the results of Mitzner [17], Michaeli [18], [27], and Knott [19],[33], respectively, in the case of the infinite wedge.In addition, it is shown that the twodimensional diffraction coefficients are recovered when the general expressions for the incremental diffraction coefficients are integrated over an infinite straight line.Finally, we use our general method to obtain for the first time the incremental diffraction coefficients for the infinitely long, narrow strip and slit.
The present p a p x is a condensed version of an RADC report [37] that should be consulted if greater detail is desired on the derivations in Sections IV and V.In sequels to this report [38] the incremental diffraction coefficients for the halfplane and for the narrow slit are integrated around the rims of reflectors and along the slits between panels of the reflectors to obtain the far fields produced by the nonuniform currents.These computed far fields of the nonuniform current are added to the far fields computed from the PO current to improve the accuracy of the far fields especially, as mentioned above, in cross polarization, in the further-out sidelobes, and near nulls.

FIELDS RADIATED BY CURRENTS ON TWO-DIMENSIONAL SCATTERERS
Fig. 1 shows the surface current sheet on a planar facet of a two-dimensional, perfectly electrically conducting scatterer in free space.The scatterer extends uniformly to infinity in the +z-direction and the planar facet lies in the xz-plane.The scatterer is illuminated by a plane wave with propagation vector k , whose direction is denoted by the spherical angles eo and 40.The magnitude of k equals k = 2a/X = d c , where X is the free-space wavelength, o is the angular frequency of the suppressed exp ( -i d ) time dependence (o > 0), and c is the speed of light in free space.
Maxwell's equations and the boundary condition of zero tangential electric field on the perfectly conducting scatterer show that for plane-wave illumination the surface current X ( r ' ) at any point r' on the current sheet can be factored as That is, the 2'-dependence can be written explicitly in terms of the elevation angle Bo of the incident plane wave, and thus the remaining current factor K(x' ) depends on the transverse coordinate x' , but not on the longitudinal coordinate z ' .Although the phase of the 2'-dependence in (1) is zero at z' = 0, (1) does not imply that the phase of the incident plane wave is zero at 2' = 0 because an arbitrary constant phase factor can be implicitly contained in K ( x ' ) .Of course, K ( x ' ) also depends implicitly on the angles (Bo, $o) defining the direction of propagation of the incident plane wave, and is proportional to the complex amplitude of the incident plane wave.Also, the surface current described by (1) need not refer to the total current; in particular it could refer to the PO current or the nonuniform current (total minus PO current).

A . Fields Produced by an Incremental Current Sheet
Let dH,(r) be the scattered magnetic field at the point r radiated by the incremental current sheet of length dz' .
(Initially, the axis of the incremental current sheet is chosen perpendicular to the z-axis; however, the analysis is generalized in Section 111-C to allow the axis of the incremental current sheet to make an arbitrary angle with the z-axis.)Through Maxwell's equations we can express dH,(r) in terms of the current by taking the curl of the vector potential: To obtain the magnetic far field from ( 2 ) , take the limit as r -+ 00, bring the limit under the integral, and expand the scalar

B. The Cylindrical Fields of the Current Sheet
The cylindrical fields of the entire current sheet in Fig. 1 can be found by integrating (2) from z' = -03 to 03.In the Appendix we show that the I' integral can be expressed in closed form as where H t ) is the Hankel function of the first kind and p is the transverse part of r, i.e., p = r -2%.Substitution of (4) into (2) integrated over z' gives the magnetic field H,(r) of the entire current sheet as To obtain the magnetic far field from (5) by taking the limit as At first sight, this interchange may not appear valid for scatterers with currents that extend to infinity in the transverse plane (such as the infinite wedge).However, the interchange can also be applied rigorously to these infinite scatterers by assuming the currents have a small exponential decay as x ' + i m [26].p -+ CO, bring the limit under the integral (see footnote 4), and replace the Hankel function by its large-argument asymptotic form, to get e;kiO.rei7r/4 ikio Hs(r) "m (8xkp sin O0)1/2 The unit vector i o in (6) is defined as the unit vector i evaluated at 8 = a -8,.i.e., io = 6 sin 80 -4 cos Bo.Thus we see that (6) represents a cylindrical wave propagating along the generators of the diffraction cone, i.e., the scattered rays that make an angle 8 with the longitudinal ( z ) axis equal to x For a number of canonical scatterers, the cylindrical far field given by ( 6) is known in closed form for the total current, PO current, and thus the nonuniform current.Consequently, the cross product of io and the current integral in (6) can be written in terms of these convenient, closed-form canonical expressions.Moreover, in the next section we show that the cross product of i and the current integral in (3) can be rewritten in terms of the cross product of i o and the current integral in (6), so that the incremental far fields (dH,) can be written directly in terms of the canonical expressions for the cylindrical far fields (H,). e,,.

INCREMENTAL FAR FIELDS IN TERMS OF CYLINDRICAL FAR FIELDS
As a preliminary to expressing the incremental far fields in terms of the canonical cylindrical fields, let us rewrite (3) and (6) in the following form: where the vectors A and A. are the current integrals in (3) and (6), respectively, i.e., ( 9 4 The constants C and CO are defined by inspection of ( 7) and ( 8); z' = 0.) Since the current sheet is in the xz-plane, K,, is zero and ( 7) and ( 8) can be written in terms of the x-and z- The unit vectors 6 and 6 are retained in (1 1) and ( 12) because far fields conveniently polarize into 8and $-spherical components.The unit vector 6; in (12) denotes the unit vector 8 evaluated at the diffraction-cone angle 8 = x -BO.
We are assuming a known expression for the 8and 4components of the far fields H,(r) of the canonical current sheet.In other words, we are given H,# and HSQ such that We could, of course, just as well be given the far electric field E,(r), which is related to H,(r) by the cylindrical far-field formula in which Z , is the impedance of free space.Equating the right sides of ( 12) and (1 3a) determines the rectangular components of A,, in terms of the given 8and +-components of H,: A,,= --1 (Hd cot O0 cot $+H,/sin 8,).
(14b) C O Thus, the problem of finding the incremental far fields in terms of the canonical far fields reduces to finding A , and A , in (1 1) in terms of Aox and Aoz, because Aox and Aoz are known through (14).

A . Determination of the Rectangular Components of A in Terms of A,,
Inspection of (9b) reveals that the rectangular components of A. are functions of + only through the cos $ in the exponential of the integrand, because the rectangular components of K(x') do not depend on $ or 8. Therefore, the rectangular components of A in (9a) can be obtained merely by replacing $ with cos-' (sin 8 cos $/sin 8,) in A,, of (9b).Specifically, we have from (9a) and (9b) A(8, 4) = A0 ( $+COS-' ( s i n s y n c ; : 4)) (15) (A similar substitution was used by Mitzner in [17, sec. 3.2.11for the special case of the wedge.)Furthermore, the rectangular components of A can be related to the given components of the canonical cylindrical far fields by substituting Ao, and Aoz of ( 14) into (15) to get The far electric field dE, of the incremental current is found from dH, through the spherical far-field relationship

B. Evaluation of Incremental Far Fields for I t I > I .
The far fields radiated by an incremental current sheet can be found from ( 17) and ( 18) provided one knows the far fields of the corresponding two-dimensional canonical problem.Specifically, one must evaluate H, o and H,, for 4 equal to cos-1 t.
Although cos-' t is a multiple-valued function, when it is substituted for 4 in (9b) the unique value t is recovered.That is, Ao, and Aoz, and thus Hd/sin (cos-' t) and H,+ in (17) remain independent of which value is chosen for cos-' t , as long as the same value is used throughout.
For I t I < 1, + is a real angle and Hd and H,+ can be evaluated from either numerical or closed-form expressions.
However, for I t 1 > 1, 4 is a complex angle and thus Hss and Hs+ must be evaluated outside their usual domain of scattering in real, physical space.This means that if one merely has numerical values for HSo and H,, over the usual domain of 0 < + < 2n, one would not be able to determine their values directly for5 I t ( > 1. Fortunately, (9b) shows that Aox and Aoz, and thus Hdlsin (cos -' t) and H,, , are analytic functions of t for continuous6 currents K(x') that have finite extent in the x'-direction [26].
Therefore, if one has closed-form analytic expressions of Hd/ sin (cos-' t) and H,, for these finite problems solved originally for the domain 1 t 1 < 1, the same expressions can be used directly in the domain I t I > 1 as well.For a current sheet that extends to infinity along the x'-axis, such as the current on the face of the perfectly conducting wedge, the integral of (9b) is also an analytic function o f t on the real axis, provided a small negative or positive imaginary part is added to t that produces an exponentially decaying integrand for an x' integration that extends to plus or minus infinity, respectively [26].Thus, for current sheets of infinite transverse extent the closed-form expressions of Hdlsin (cos-I t ) and Hs+ in the domain I t I < 1 can also be evaluated by direct substitution o f t (with a small loss approaching zero, if needed, to ensure analyticity over the entire real axis) in the domain 1 t 1 > 1. (The theorems on analyticity of integra!s with finite limits [26], assure that the singularities of l/sin (cos-' t) in ( 17) at t = k 1 will be canceled by zeros of Hd at t = k 1.
These singularities may remain, however, for infinite current sheets, if a small imaginary part is not added to t.) In summary, if closed-form expressions for the cylindrical far fields, H, o and H,,, of the two-dimensional canonical scatterer are available over the usual domain of I t I < 1, these closed-form expressions can be used directly and unambiguously for all t to find the incremental far fields from ( 17) and ( 18).Of course, for normal incidence (8, = a/2), 1 tI never exceeds one.

C. Singularities in the Incremental Far Fields
Singularities in the incremental far fields given by ( 17) and (18) will occur only at angles (0, 4) where Hd(+ -+ cos-' t)/ sin (cos-' t ) or Hs4(+ -+ cos-' t) are singular.For canonical scatterers of finite transverse dimensions, integrability of the current in (9b) demands that these two functions be finite for all t , and thus the incremental fields of finite canonical scatterers will have no singularities for all (0, 4).
For infinite sheets of current like the total currents on the faces of the wedge, the cylindrical far-field functions HJo/sin + and H,+ have singularities at isolated values of azimuth angle + for which the current integral in (9b) becomes infinite.
Denoting these singular angles by +,,, we see from ( 17) that Of course, if the current K(x') were known, one could conceivably integrate (9a) numerically or possibly analytically to obtain the A , and A, required to determine the incremental far fields immediately from (1 1).In fact, as mentioned in the Introduction, Michaeli [18] was able to perform the current integration in (9a) analytically for the faces of the perfectly conducting wedge to obtain the GTD incremental fields of the wedge.In the present paper we want to avoid such involved current integrations by working directly with the cylindrical far fields of the corresponding two-dimensional canonical scatterers.
For scatterers with sharp edges, the current may be singular at the edge.However, since these singularities are integrable they can be removed from the integration by moving the limit of the integration just inside the edge, and thus the theorems of [26] still apply.
the corresponding incremental far fields will, in general, have singularities for all 8 and 4 that satisfy f = cos &, i.e., sin 8 cos 4 = sin Bo cos 4,.Equation ( 19) defines a cone of far-field singularities in the incremental fields about the x-axis, i.e., about the transverse coordinate axis of the current sheet.(This cone of singularities is not to be confused with the far-field cone of diffraction for the cylindrical fields about the z-axis, i.e., the longitudinal axis of the planar current sheet.)The half-angle CY, of the cone of singularities measured from the positive x-axis is given by C Y , , = C O S -~ (cos 4,, sin eo). (20) For normal incidence eo equals a / 2 and CY, equals 4,.
Geometrically, the cone of the far-field singularities of the incremental fields is generated by rotating about the x-axis the ray that lies in the direction of the singularity of the cylindrical far-field function (Hd/sin 4 or Hs4).This generating ray, of course, lies on the cone of diffraction of the cylindrical far fields as well.
The total current on the illuminated face of a wedge has two associated cones of singularities given by ( 19).One is generated by the singularity in the diffracted fields at the shadow or reflection boundary (4, = a k 40), i.e., the singularity produced by integration of the PO current in (9b).
The second is generated by the singularity of Hdlsin 4 (or Hs4 for a TM plane wave grazing at $0 = a) in the plane of the face (4, = 0, 2a), i.e., the singularity produced by integration of the nonuniform current in (9b) excited by an incident TE plane wave (or TM plane wave grazing at 4o = a).Both diffracted components, H,, and Hd/sin 4, produced by the PO current remain finite at a face angle of the wedge (except for grazing incidence at 40 = a), whereas these same components produced by the nonuniform current remain finite at the shadow and reflection boundaries (except for grazing incidence at 4o = a).Therefore in calculating the far fields of three-dimensional scattering bodies locally approximated by infinite two-dimensional canonical scatterers, like the wedge, with cylindrical far fields containing singularities at the shadow and reflection boundaries, it is especially advantageous to use the incremental far fields of the nonuniform (PTD) currents of the canonical scatterer, because the far fields of the nonuniform currents of these infinite scatterers do not contain singularities at the shadow and reflection boundaries, unlike the far fields of the total (GTD) and PO currents of these infinite scatterers.Moreover, in the PTD formulation there are no singularities in the PO far fields if they are obtained by integrating numerically the PO current on the finite three-dimensional scatterer.
The cone of singularities defined by ( 19) that may occur at q5,, = 0, 2 a for canonical scatterers of infinite transverse extent can be collapsed to a single line and sometimes ' A notable exception can occur when 00 = r / 2 , Qn = 0 so that ( 19) is satisfied only by the single far-field point 0 = 7r/2, Q = 0 where the sin Q and (cos 0 + sin2 0 cos Oo/sin2 0,) factors in the numerators of (17) cancel the sin (cos-' t ) factors in the denominators of ( 17).This exception is a particular example of the generalization (discussed later in Section 111-C) to current increments lying along the direction of the grazing diffracted ray.eliminated by choosing the axis of the increment of current sheet skewed rather than perpendicular to the z-axis.In particular, Michaeli [27] has recently shown that the cone of singularities generated by the diffracted ray grazing the face of a wedge (4, = 0, 2a) can be eliminated (except when the direction of incidence is also grazing at 40 = a) by choosing the axis of the incremental length in the direction of the grazing diffracted ray, i.e., in the direction of propagation of the nonuniform current far from the edge.Butorin and Uftimtsev [39] derived the same result for scalar (acoustic) diffraction from the wedge.
The present analysis shows that choosing the positive x ' integration axis of the incremental current sheet to make an arbitrary angle $ with the positive z-axis (rather than 90", as shown in Fig. 1) multiplies the current in (3) by exp ( -ikx' cos Bo cot $) and the exponential in the integrand of (3) by exp ( -ikx' cos 8 cot $); thereby changing the integral on the right side of (3) to Thus our general analysis and final results ( 17) and ( 18) for the incremental far fields remain the same when the axis of the incremental current sheet is skewed, except that f is set equal to t!b, sin 8 cos 4 + c o t $ (cos 8+cos 80) sin 80 instead of to sin 8 cos 4/sin Bo.Equation ( 22) reduces to Michaeli's corresponding result [27] for the faces of the infinite wedge when $ is set equal to a -eo, the angle of the grazing diffracted ray for a face along the positive x-axis.In that case, as mentioned above, the original singularity of the incremental diffracted fields at $,, = 0, 2 a is eliminated (except for grazing incidence at $, , = a) by substituting f+ for t; note, t$ 5 1 for $ = ado.For a face extending to infinity along the negative x-axis, the angle of grazing is $ = eo, and the singularity at t = -1 is eliminated; note, f$ 2 -1 for $ = Oo.
In summary, the differential increment of current on the face of a wedge forms a semi-infinite line source along the x- axis of Fig. 1, and thus radiates with a symmetric vector potential about the x-axis.The total current of the line source can be divided into the PO current and the nonuniform current.The PO current of the line source radiates a far field that becomes singular in a cone of directions that includes the shadow and reflection directions of the corresponding twodimensional wedge face.The nonuniform current of the line source radiates a far field that becomes singular in a cone of directions that includes the diffracted ray along the corresponding two-dimensional wedge face.Skewing the axis of the differential increment of current to form a line source in the direction of the grazing diffracted ray rather than the x-axis collapses the cone of singularities of the nonuniform current to a single line in the direction of the grazing diffracted ray.
Skewing the axis of the current strip cannot eliminate the GTD and PO cone of singularities associated with the shadow and reflection boundaries (& = a t &) of the face of a wedge because for nongrazing incidence the shadow and reflection boundaries of a face of a wedge do not lie along that face.Also, the directions of propagation of the PO and nonuniform currents on the face of a wedge are different except for normal (0, = a/2) incidence or grazing incidence at +, , = a.Of course, the singularities in the PO far fields at the shadow and reflection boundaries of the infinite wedge can be eliminated by giving the faces of the wedge finite radii of curvature [28].However, the resulting incremental diffraction coefficients apply then only to three-dimensional scatterers with curved rather than flat surfaces.

D. Division of Incremental Far Fields into TE and TM Fields
The scattered fields of two-dimensional perfectly conducting cylinders divide conveniently into transverse electric (TE) and transverse magnetic (TM) fields depending on whether the incident plane wave is T E (E, = 0) or TM (H, = 0), respectively [25].Specifically, for the TE case the far field component Hs4 is zero everywhere, and for the TM case Hd is zero everywhere.Thus the incremental magnetic far fields (17) for these TE and TM cylindrical fields reduce to sin2 8 cos eo) $1 The incremental electric far fields are obtained, as usual, from (18): where E,, and E,e, the cylindrical electric far fields, have been substituted from (1 3b).Note especially that the incremental TE magnetic and electric fields have both 0and +-components even though the corresponding TE cylindrical fields have only a &component of magnetic field and a +-component of electric field.For the TM incremental magnetic or electric field, however, the single component of the corresponding TM cylindrical field remains as the only component.Recall that Although the diffracted fields radiated by the total currents of twodimensional perfectly conducting cylinders divide conveniently into TE and TM fields, the fields radiated by the PO currents, and thus the fields radiated by the nonuniform currents, do not divide generally into TE and TM fields, except for normal incidence (6'0 = 90").Consequently, (17) or (18) must be used, in general, rather than (23) or (24) to determine the incremental diffraction coefficients for the PO and nonuniform currents.(For example, see Section IV-C.)E,@ and H,e as defined by ( 13) refer to the 8; components of the cylindrical far fields.
In dyadic notation (24) can be written dE,(r) L r n E,(+-+COS-' t ) D,dz' where the components of the dyadic D, are the elements of the matrix defined by ( 24) that converts the cylindrical electric far field of the canonical scatterer to the corresponding incremental electric far field.If, in addition, we express the cylindrical electric far field as a dyadic D multiplied by the incident field E, (see e. (In ( 25), ( 26), and ( 27) we have included the radial dependence of the far fields in the dyadics.)The dyadic incremental diffraction coefficient D, provides an extremely compact notation ( 27) for the incremental diffraction fields.However, for detailed analytical or computational purposes the explicit vector expressions ( 17), (18), ( 23) or ( 24) prove the more useful.

IV. CHECKS FOR THE INCREMENTAL FAR FIELD EXPRESSIONS
In this section we present a series of checks on the expressions that have been derived for the incremental far field.In Section IV-A we integrate the incremental magnetic far field (17) from minus to plus infinity, and show that by so doing we recover the cylindrical magnetic far field, (13a).In Section IV-B we specialize our general results to the perfectly conducting infinite wedge illuminated by a plane wave, obtain incremental far fields corresponding to both TM and TE incident illumination, and compare the resulting expressions with those obtained independently by Michaeli [ 181 through integration of the surface currents.Complete agreement is found between our expressions and his.In Section IV-C we use our method to derive expressions for the incremental physical optics far field of a wedge illuminated by a plane wave.The resulting expressions for both TM and TE incident illumination are found to agree with those obtained independently by Knott [19],[33] in his paper comparing the incremental diffraction coefficients of Michaeli [ 181 and  Mitzner [ 171.

A . Integration of the Incremental Far Field to Obtain the Cylindrical Far Field
A basic check on the expression (17) obtained for the incremental far field is that the integral of (17) over z' from minus to plus infinity should yield the cylindrical far field (1 3a).With the plan of evaluating the integral of (1 7) by the method of stationary phase, we write jm -m dH,(r)= jm -C O erkpq(z')f(z') dz' For large kp the integral ( 28) is asymptotically approximated by the stationary phase formula [29] lm ekpq(Z')f(Z') dz' pT_" edfPdZl ) + l p r / 4 -m where zs' is the point of stationary phase defined by q ' ( z s ' ) = 0 and p = sign q" (zs').
Evaluating the functions in (32) at the stationary phase point zS' = p(cot eo + Cot e), yields the desired relation

B. Comparison with Michaeli's Incremental Diffraction Coefficients for the Wedge
As a second check on the expressions we have derived for the incremental far fields of planar surfaces, we obtain the incremental far fields for the particular case of an infinite wedge, and show that these fields agree with the expressions obtained independently by Michaeli [ 181 from integration of the surface currents.The wedge is defined in terms of either a circular cylindrical coordinate system (p, 4, z ) or a spherical coordinate system ( r , 8, 4) by the equations 4 = 0 (face 1) and 4 = N a (face 2), 0 < N < 2, with the edge coincident with the z-axis.The direction vector of the illuminating plane wave forms angles of a + c $ ~ and aeo with the positive xaxis and z-axis, respectively (see Fig. 3).
We begin with a TM incident plane wave with the incident electric field given by E; = -Ei exp [ -ikp sin do cos (4 -40) -ikz cos e,] 8, (34) where El is the complex electric field amplitude of the plane wave.To apply (24b) we require expressions for the far-zone electric field radiated separately by the current on each of the faces of the wedge.(Of course, for the special case of the halfplane, the two faces of the wedge lie in the same plane and thus one would need only the diffracted field of the total current [30] .)Michaeli [32] has derived expressions for the contributions, El, and E2sz, of each of the wedge faces to the total diffracted field when the incident wave vector is normal to the edge of the wedge (00 = a/2).For a TM incident wave where Making the substitution Y = a + i{ in (37) and evaluating the resulting integral by the method of stationary phase yields from which, with the aid of some elementary trigonometric with M = 0 (TM illumination) and dn sin -2 i N I' kZo sin2 eo Z=E--- which is seen to be identical to our expression given by ( 43) with ( 44) and (45).To compare with Michaeli's expressions we have used the correspondences j ft -i, S + + i, ?++ 2, 0' ++ aeo, 0 ft 8, and +' + + +o between Michaeli's notation and ours.
A similar analysis for an infinite wedge illuminated by the T E (H-polarized) incident field given by H i = -Hi exp [ -ikp sin Bo cos (4 -+o) -ikz cos eo] 8, (47) where Hi is the complex magnetic field amplitude of the plane wave, yields which again is identical to Michaeli's expression [18] given by ( 46) with Z and M inserted from his (31).
C. Comparison of the PO Incremental Far Fields for a Wedge with Knot t ' s Expressions Just as total incremental far fields are obtained by our method starting with the total cylindrical far field, one can obtain PO incremental far fields by starting with expressions for the cylindrical PO far field.Such PO incremental far fields are important since in practical computational applications it is often desired to supplement the PO far field (obtained by numerical integration of the PO surface currents) with the far field radiated by the nonuniform currents.This latter far field can then be obtained by integrating the nonuniform current incremental far field which has been found by subtracting the PO incremental far field from the total incremental far field.
In this section we derive PO incremental far fields for the wedge and compare these with expressions given by Knott [19], [33], who shows that the difference between Michaeli's [ 181 and Mitzner's [ 171 incremental far fields are simply the PO incremental far fields.The starting point for obtaining the PO incremental far fields for the wedge is the cylindrical PO far field of each face of the wedge (see Fig. 3 for the wedge geometry).To obtain expressions for the incremental total far field, in Section IV-B we began with the two-dimensional far field corresponding to normal incidence of the illuminating plane wave, from which the cylindrical far field for oblique incidence was obtained by a simple substitution using a method described in [25] and [3 I].The same procedure cannot in general be used to derive the cylindrical PO far field for oblique incidence from the two-dimensional PO far field for normal incidence since, unlike the total field, the PO fields do not satisfy the boundary condition that the tangential electric field vanishes on the surfaces of the scatterer.In particular, the PO fields associated with a TE incident plane wave do not remain transverse for oblique incidence.Hence we will obtain the cylindrical PO far field for oblique incidence directly.
We begin with a TM (E-polarized) incident plane wave with the incident electric field given by (34).The incident magnetic field is then On face 1 of the wedge + = 0 and the unit normal is 9 so that the PO current is given by E,  [33] for the TE incremental PO electric far field for the wedge if, in addition to the correspondences already noted between Knott's notation and ours, we use Hi, + + Hi, and I, + + M .
In concluding this section we want to draw attention to the essential simplicity of our method for obtaining incremental far fields.As seen above with the example of the infinite wedge, all that is needed is an expression for the cylindrical far field of each of the planar surfaces comprising the scattering object.The incremental far field is then obtained by straightforward substitution of the components of the cylindrical far field in the general expressions ( 23) and ( 24) or ( 17) and (1 8).
No current integrations are required.

V. INCREMENTAL FAR FIELDS FOR THE INFINITE STRIP AND SLIT
We now obtain incremental far fields, total and physical optics, for the perfectly conducting infinite strip and the complementary infinite slit.To the best of our knowledge, this is the first time that such expressions have been derived.Our expressions for the total incremental far fields are given for the low frequency approximation (i.e., narrow strip and slit) but the method used to obtain them is fully applicable to any size strip or slit.

A . Incremental Total Far Field for the Strip
Following [34], the strip of width d is defined in terms of Cartesian coordinates (x, y, z ) by y = 0, 1x1 5 d/2, so that the strip lies in the xz-plane with the edges of the strip parallel to the z-axis; in other words, as shown in Fig. 1 but with the center of the strip at x = 0.The direction vector of the incident plane wave forms angles of a + 40 and a -80 with the positive x-axis and z-axis, respectively.
We begin with a TM incident plane wave and the incident electric field given by (34).For 80 = a / 2 the scattered electric These expressions taken from [34] are valid for all values of 6 and +o .For brevity we have included only terms through c4 in the series for PI.Expressions for higher order terms are given in [34] and [35].Next we consider the strip illuminated by a TE (Hpolarized) incident plane wave given by (47).For 6 0 = a12 the scattered magnetic far field is given by [34] where, in the low frequency approximation with To = --1 sin 4o sin 4, 4 complementary slit is the PO diffracted far field of the entire plane which is equal to zero if it is assumed, as is done consistently in this report, that the PO surface current vanishes at infinity.Hence the PO diffracted far field of the slit is the negative of the PO far field of the complementary strip.
The geometry of the strip and incident illumination are defined as in Section V-A.We note that the PO surface currents excited on the strip for either TM or TE illumination are identical in form to those excited on face 1 of the wedge for the given illumination.Hence in applying (6) to obtain the PO magnetic far field for the strip, the strip differs from face 1 of the wedge only in that the effective limits of the K(x') integration are -dl2 to dl2 for the strip, and zero to infinity for the wedge.Thus the PO incremental far fields for the strip can be obtained from those found in Section IV-C for face 1 of the wedge, (56) and (59), simply by multiplying the face 1 incremental wedge fields by the factor sin Bo (cos 9o + cos 9) .(73) 1 VI.SUMMARY After introducing incremental diffraction theory in Section I, we derived in Sections 11-IIIA exact expressions (17), ( 18) for the three-dimensional incremental diffraction coefficients in terms of the conventional two-dimensional diffraction coefficients of perfectly conducting, planar scatterers.Section 111-B explained how the necessary two-dimensional far-field functions are analytically continued into the domain of imaginary values of the azimuthal angle 9.
In Section 111-C we showed that an isolated singularity in the two-dimensional far field transforms to a cone of singularities in the incremental far fields.Moreover, Section 111-C showed that the exact expressions for the incremental far fields can be generalized to allow increments of current that are skewed rather than normal to the axis of the two-dimensional scatterer, merely by replacing 1 in (17), (18) with t$ given in (22).When this generalization is applied to the infinite wedge and the skew angle is chosen along the grazing diffracted ray, the cone of singularities associated with the diffracted ray reduces to a single direction and the corresponding results of [27] and [39]  are obtained.
In Section 111-D the incremental far fields were separated into TE and TM fields (23), ( 24).
In Section IV-A the expression (17) for the three-dimensional incremental far magnetic field was integrated over an infinite straight line to prove that the two-dimensional far magnetic field is recovered.In the remainder of Section IV we further confirmed the validity of the general expressions by showing that PTD, GTD, and PO incremental diffraction coefficients obtained by direct substitution into (17), ( 18) and (23), ( 24) agree with the results of Mitzner [17], Michaeli [ 181,[27], and Knott [ 191,[33], respectively, in the case of the infinite wedge.
In Section V we determined the total and PO incremental diffracted fields of the infinite strip and complementary infinite slit by direct substitution into the general expressions derived in Sections I1 and 111.
As a concluding invitation to further analysis, there remains the question of whether the method of direct substitution derived here for determining three-dimensional incremental far fields from the corresponding two-dimensional far fields can be extended, at least in some cases, to curved surfaces and penetrable scatterers.

APPENDIX
To prove (4) change the integration variable z' to z" = z' -z, so that the integral becomes
Green's function, exp (ikJr -r' J)/lr -r' I, to get eikrdZ f eikz / (cos 0 +cos 0o)ikf r-a, dH,(r) --4a rThe spherical angles 6' and 4 define the direction of scattering, i.e., the direction to the observation point r (see Fig.l), and the caret denotes unit vectors throughout.Although the limits of the x' integrations in (2) and (3) extend from -03 to 03, they reduce to finite values for scatterers of finite transverse dimensions (like the one shown in Fig.1).

Fig. 2 .
Fig. 2. Geometry of integration of the incremental far field d E f"(r ) = d E T,"(r) + d E l,"(r)