1. INTRODUCTIONThe sphere is one of the bodies that have been most ex-tensively investigated in diffraction theory throughouttime. A thorough survey of the early studies of the scat-tering of plane waves by a sphere has been written byLogan.1 The mathematician Clebsch2 seems to be thefirst to have contributed to this field, in 1861, well 57years before Mie3 (1908), after whom this scattering prob-lem got the name Mie scattering.4 This problem is one ofthe few geometries5 where we have exact solutions of thescalar and vectorial wave equations in terms of modal ex-pansions. Asymptotic evaluations of these exact solu-tions are used in geometrical diffraction theory6 to de-scribe diffraction by more general convex shapes. Theexact solutions are also used to evaluate the validity ofapproximation theories.
In this paper we consider the scattering of plane wavesby a sphere in the scalar and vectorial cases and trans-form the modal expansion of the scattered field to an an-gular spectrum representation. Devaney and Wolf 7
were the first to derive an angular spectrum representa-tion for the general electromagnetic (vectorial) radiationproblem, but the angular spectrum representation forvectorial Mie scattering has not been presented earlier.The scalar case has already been treated by several au-thors to solve the problem of diffraction of finite beams byspherical objects8 and to obtain models for confocalmicroscopy.9–12 Lobkis et al.13,14 considered scattering of
elastic waves by a sphere and obtained models for the out-put signal of an acoustic microscope. Recently, the angu-lar spectrum representation of the field scattered by cir-cularly cylindrical objects has been used to evaluate thevalidity of the weak-scattering models in two-dimensionalforward scattering15 and to determine the shift of theshadow boundary in images of circular cylinders.16
By neglecting the inhomogeneous plane waves in theangular spectrum representation, we also obtain the vir-tual, or backpropagated, fields for planes within or behindthe object. These virtual fields are the fields that, to anexternal observer, appear to exist in free space within orbehind the object. Thus these virtual fields are differentfrom the actual fields existing in these regions. The im-age field model is obtained by including aperture limita-tions to the angular spectrum representation. These im-age fields could have been obtained by the morecumbersome approach of first computing the field in theentrance pupil of the imaging system by using the modalexpansion and then propagating it back17–21 to a planewithin or behind the object.
In contrast to the imaging geometry of a confocalmicroscope,9–14 we consider the different imaging situa-tion of an arbitrary orientation of the optical axis relativeto the direction of the incident plane wave. We also de-rive some new physical properties of the scalar imagefield at the center plane of the sphere. For paraxial im-aging situations we show that the sphere can be repre-
sented by a flat circularly symmetric object with sampledfield values equal to the mode coefficients. For losslessspheres (including soft and hard spheres), the equivalentflat object is a pure phase screen. The paraxial image ofthe sphere is then computed by convolving the flat-objectfield by the radial impulse response function of the imag-ing system. Since we consider a telecentric imaging sys-tem, the paraxial image of the sphere is therefore equal tothe low-pass filtered object field. The nonparaxial case isinterpreted as an aberrated version of the paraxial case,the aberration being similar to coma. This aberration isnot an error introduced by the imaging system but a purescattering effect. We finally derive a Rayleigh criterionfor neglecting this aberration.
In this paper the usefulness of the image models forfield computations is illustrated by several examples.Examples are also given of diffraction effects that arepresent only in imaging situations.
In Section 2 the modal expansions and the angularspectrum representations for scalar fields are reviewed.We then derive the angular spectrum representation forthe vectorial Mie-scattering case. Finally, the imagefield models are derived. In Section 3 we derive thephysical properties of scalar image fields and paraxiallyscattered fields. Section 4 contains numerical examplesthat illustrate the main results of this paper. Here wealso compare the image field model with the modal expan-sion. Section 5 contains a brief summary of the paperalong with the final conclusions.
2. THEORYWe here review the modal expansion of acoustic (scalar)and electromagnetic (vectorial) fields diffracted by aspherical object. These modal expansions are finallytransformed into angular spectrum representations. Byconsidering a telecentric imaging system, we obtainsimple image field models with the use of these represen-tations.
A. Review of the Modal Expansion and the AngularSpectrum Representation of the Scalar ScatteredFieldWe consider the three-dimensional case illustrated in Fig.1 with a complex monochromatic field at angular fre-
Fig. 1. Scattering geometry for scalar waves.
quency v: V(x, y, z, t) 5 U(x, y, z)exp(2iv t). The in-cident field is scattered by a sphere with radius r 5 a andindex of refraction n relative to that of the background.The incident field is a plane wave traveling in thedirection ki 5 kei 5 k(cos f0 sin u0 ex 1 sin f0 sin u0 ey1 cos u0 ez) (Refs. 5 and 22–24):
Ui~r! 5 exp~ikir! 5 exp~ikr cos g0!, (1)
where g0 is the angle between the vectors ki and r5 rer [i.e., eier 5 cos g0 5 cos u cos u01 sin u sin u03 cos(f 2 f0)] (Ref. 25). The incident field can be devel-oped in scalar spherical wave functions as follows:
Ui 5 (n50
`
in~2n 1 1 !Pn~cos g0!jn~kr !, (2)
where Pn(x) are the Legendre polynomials and jn(x) arethe spherical Bessel functions of the first kind and ordern. The scattered field outside the sphere (r > a) can bedeveloped as
Us~r! 5 (n50
`
in~2n 1 1 !anPn~cos g0!hn~1 !~kr !, (3)
where the modal coefficients an are determined by theboundary conditions at the sphere’s surface and hn
(1)(x)are the spherical Hankel functions of the first kind andorder n. In Eqs. (2) and (3), we can use the additiontheorem for Legendre polynomials22,23,25,26 to get
Pn~cos g0! 5 Pn@cos u cos u0
1 sin u sin u0 cos~f 2 f0!#
5 (m50
n
em~n 2 m !!~n 1 m !!
Pnm~cos u0!
3 Pnm~cos u!cos@m~f 2 f0!#, (4)
where en is 1 for n 5 0 and 2 for n . 0 and Pnm(x) are the
associated Legendre functions. We can express thespherical Bessel and Hankel functions in terms of cylin-drical Bessel and Hankel functions of fractional order 27:
jn~z ! 5 Ap/2zJn11/2~z !, (5)
hn~1 !~z ! 5 Ap/2zHn 1 1/2
~1 ! ~z !. (6)
Since the total field and its radial derivative have to becontinuous across the spherical boundary r 5 a, the ex-pansion coefficients an are4,5,22,24
an 5 2njn8 ~kan !jn~ka ! 2 jn~kan !jn8 ~ka !
njn8 ~kan !hn~1 !~ka ! 2 jn~kan !hn
~1 !8~ka !, (7)
where jn8 (z) 5 (d/dz)jn(z) and hn(1)8(z) 5 (d/dz)hn
(1)(z).If the medium within the sphere is lossless [i.e., Im(n)5 0], the expansion coefficients can be written in theform
is the scattering phase. For soft spheres the correspond-ing scattering phases are5,22,24,28
cn 5 22 arg@hn~1 !~ka !#, (10)
and for hard spheres the scattering phases are5,22,24,28
cn 5 22 arg@hn~1 !8~ka !#. (11)
The modal expansion for the scattered field outside thesphere in Eq. (3) can be transformed into an angular spec-trum representation by using the integral representationfor the scalar spherical eigenfunctions7,22,23,29:
Pn~cos u!hn~1 !~kr ! 5
12pin E
0
2p
dbE0
p/22i`
da
3 sin a
3 exp~ikr!Pn~cos a!, (12)
where kr 5 kreker 5 kr@cos u cos a 1 sin u sin a cos(b2 f)]. Applying the summation formula for Legendrepolynomials in Eq. (12),22,23,25,26 we obtain the integralrepresentation:
Pn~cos g0!hn~1 !~kr ! 5
12pin E
0
2p
dbE0
p/22i`
da
3 sin a exp~ikr!Pn~cos ga0!,
(13)
where cos ga0 5 ekei 5 cos a cos u0 1 sin a sin u0 cos(f02 b). Substituting this integral representation into themodal expansion, we obtain
Us~r! 51
2p E0
2p
dbE0
p/22i`
da sin a exp~ikr!S~cos ga0!,
(14)
which, by comparison with Eq. (7), yields the angularspectrum representation for the scattered field corre-sponding to an incident wave in the ei direction. In Eq.(14) the function S(cos ga0) is defined by
S~cos ga0! 5 (n50
`
~2n 1 1 !anPn~cos ga0!. (15)
We can identify S(cos ga0) 5 S(cos a cos u01 sin a sin u0 cos(f0 2 b)) 5 S(eiek) as the angularspectrum of the scattered waves. For a P @0,p/2# wehave homogeneous plane waves, and for a P @p/22 i0,p/2 2 i`) we have inhomogeneous plane waves.We see that the symmetry axis of S(eiek) 5 S(1) is thedirection of the incident plane wave, ek 5 ei .
In the far zone r → `, the spherical Hankel function inEq. (3) can be expressed by its asymptotic expansion27:
hn~1 !~z ! ;
1z
~2i !n11exp~iz !, z → `. (16)
Thus the scattered field in Eq. (3) can be expressed as
Us~r! ;exp~ikr !
ikrS~cos g0!, kr → `. (17)
Here S(cos g0) is the far-field scattering function.5,24,30
Since g0 is a real angle, the far-field scattering function isdetermined by the homogeneous part of the angular spec-trum of the scattered field.
B. Modal Expansion of the Vectorial Scattered FieldWe consider the three-dimensional case in Fig. 2 and as-sume a monochromatic field at angular frequency v,where the electric-field vector is E (x, y, z, t)5 Re@E(x, y, z)exp(2iv t)# and the magnetic-field vectoris H(x, y, z, t) 5 Re@H(x, y, z)exp(2iv t)#. The inci-dent field is a plane wave traveling in the z direction,with the electric-field vector parallel with the x axis andthe magnetic-field vector parallel with the y axis:
Ei~x, y, z ! 5 ex exp~ikz !, (18)
Hi~x, y, z ! 5 ey exp~ikz !. (19)
The incident field can be expanded in vectorial wavefunctions22,23 to yield
Ei~x, y, z ! 5 (n50
`
in2n 1 1
n~n 1 1 !@mo1n
~1 ! 2 ine1n~1 ! #, (20)
Hi~x, y, z ! 5 2(n50
`
in2n 1 1
n~n 1 1 !@me1n
~1 ! 1 ino1n~1 ! #.
(21)
Defining the angular functions p(cos u) and t(cos u) tobe24
pn~cos u! 5Pn
1~cos u!
sin u, (22)
tn~cos u! 5d
duPn
1~cos u!, (23)
we can write the odd vectorial wave functions22,23 in Eqs.(20) and (21) (with subindex o) as
mo1n~1 ! 5 pn~cos u!jn~kr !cos f eu 2 jn~kr !tn~cos u!
The even vectorial wave functions (with subindex e) aredefined as
me1n~1 ! 5 2pn~cos u!jn~kr !sin f eu 2 jn~kr !tn~cos u!
3 cos f ef , (26)
ne1n~1 ! 5
n~n 1 1 !
krjn~kr !Pn
1~cos u!cos f er
11kr
@krjn~kr !#8tn~cos u!cos f eu
21kr
@krjn~kr !#8pn~cos u!sin f ef . (27)
In Eqs. (24)–(27) the primes denote differentiation withrespect to the argument kr, and the vectors er , eu , andef are unit vectors pointing in, respectively, the r, u, andf directions. The incident field is scattered by a spherewith radius r 5 a and index of refraction n relative tothat of the background. The scattered field outside thesphere (r > a) can be developed in vectorial wave func-tions as follows:
Es~r! 5 (n50
`
in2n 1 1
n~n 1 1 !@anmo1n
~3 ! 2 ibnne1n~3 ! #, (28)
Hs~r! 5 2(n50
`
in2n 1 1
n~n 1 1 !@bnme1n
~3 ! 1 ianno1n~3 ! #,
(29)
where the vectorial wave functions ms1n(3) and ns1n
(3) (s5 e, o) are found by replacing jn(kr) in Eqs. (24)–(27)with hn
(1)(kr).The boundary conditions22 at r 5 a yield the expansion
coefficients an and bn (Refs. 4, 5, 22, and 24):
an 5 2jn~kan !@kajn~ka !#8 2 @kanjn~kan !#8jn~ka !
If the medium inside the sphere is lossless [i.e., Im(n)5 0], the expansion coefficients can be written in theforms
an 5 212 @1 1 exp~icn!#, (32)
bn 5 212 @1 1 exp~ifn!#, (33)
where
cn 5 22 arg( jn~kan !@kahn~1 !~ka !#8
2 @kanjn~kan !#8hn~1 !~ka !), (34)
fn 5 22 arg(@kanjn~kan !#8hn~1 !~ka !
2 n2jn~kan !@kahn~1 !~ka !#8) (35)
are the scattering phases. For perfectly reflectingspheres the corresponding scattering phases are22
cn 5 22 arg@hn~1 !~ka !#, (36)
fn 5 22 arg(@kahn~1 !~ka !#8). (37)
C. Angular Spectrum Representation of VectorialScattered FieldsIn Subsection 2.B we showed that the scattered field out-side the sphere r > a can be expanded in vectorial wavefunctions as follows22,23:
Es~r! 5 (n51
`
in2n 1 1
n~n 1 1 !@anmo1n
~3 ! ~r! 2 ibnne1n~3 ! ~r!#,
(38)
Hs~r! 5 2(n51
`
in2n 1 1
n~n 1 1 !@bnme1n
~3 ! ~r! 1 ianno1n~3 ! ~r!#,
(39)
where ms1n(3) and ns1n
(3) (s 5 e, o) represent the outgoingvectorial wave functions. The angular spectrum repre-sentations of the vectorial wave functions in Eqs. (38) and(39) are given by the following integralrepresentations7,22,23:
ms1n~3 ! ~r!
51
2pin E0
2p
dbE0
p/22i`
exp~ikr!C1ns ~a, b!sin a da,
(40)
ns1n~3 ! ~r!
5i
2pin E0
2p
dbE0
p/22i`
exp~ikr!B1ns ~a, b!sin a da,
(41)
where Cmns (a, b) and Bmn
s (a, b) are the vectorialangle functions23 and the index s 5 e,o. For m 5 1 theeven vectorial angle functions (index e) are defined as23
C1ne ~u, f! 5 2pn~cos u!sin f eu 2 tn~cos u!cos f ef ,
(42)
B1ne ~u, f! 5 tn~cos u!cos f eu 2 pn~cos u!sin f ef ,
(43)
and the odd vectorial angle functions (index o) are
C1no ~u, f! 5 pn~cos u!cos f eu 2 tn~cos u!sin f ef ,
(44)
B1no ~u, f! 5 tn~cos u!sin f eu 1 pn~cos u!cos f ef ,
(45)
where we have used the angular functions defined in Eqs.(22) and (23). Expressing the wave functions in Eqs. (38)and (39) by the integral representations in Eqs. (40) and(41), we obtain
For the scattered electric field in Eq. (46), the expressionwithin the braces can be written as
(n51
` 2n 1 1
n~n 1 1 !@anC1n
o ~a, b! 1 bnB1ne ~a, b!#
5 S1~cos a!cos b ea 2 S2~cos a!sin b eb , (48)
where
S1~cos a! 5 (n50
` 2n 1 1
n~n 1 1 !@anpn~cos a! 1 bntn~cos a!#,
(49)
S2~cos a! 5 (n50
` 2n 1 1
n~n 1 1 !@antn~cos a! 1 bnpn~cos a!#.
(50)
Similarly, for the scattered magnetic field in Eq. (47), theexpression within the braces can be written as
(n51
` 2n 1 1
n~n 1 1 !@bnC1n
e ~a, b! 2 anB1no ~a, b!#
5 2S2~cos a!sin b ea 2 S1~cos a!cos b eb . (51)
Thus we finally obtain the angular spectrum representa-tions for the scattered vectorial field:
Es~r! 51
2p E0
2p
dbE0
p/22i`
exp~ikr!@S1~cos a!cos b ea
2 S2~cos a!sin b eb#sin a da, (52)
Hs~r! 51
2p E0
2p
dbE0
p/22i`
exp~ikr!@S2~cos a!sin b ea
1 S1~cos a!cos b eb#sin a da. (53)
We can now identify S1(cos a) and S2(cos a) as the angu-lar spectra of the scattered vector fields. For a P @p/22 i0,p/2 2 i`) we have inhomogeneous plane waves,and for a P @0,p/2# we have homogeneous plane waves.We also see that the symmetry axis of S1(cos a)5 S1(ezek) 5 S1(1) and S2(cos a) 5 S2(ezek) 5 S2(1) isthe direction of the incident plane wave, ek 5 ez .
For r → ` the asymptotic expansion of the sphericalHankel function in relation (16) yields
@zhn~1 !~z !#8 ; ~2i !n exp~iz !, z → `. (54)
An expression of the scattered field in the far-field regioncan then be obtained by substituting relations (16) and(54) into Eqs. (38) and (39):
Es~r! ;exp~ikr !
ikr@S1~cos u!cos f eu 2 S2~cos u!
3 sin f ef#, kr → `, (55)
Hs~r! ;exp~ikr !
ikr@S2~cos u!sin f eu 1 S1~cos u!
3 cos f ef#, kr → `, (56)
with S1(cos u) and S2(cos u) defined by Eqs. (49) and (50),respectively. In relations (55) and (56), we have ne-glected the radial er terms because they decrease as(kr)22 for kr → `. Here the functions S1(cos u) andS2(cos u) are the far-field scattering functions. Since u isa real angle, the far-field scattering functions are deter-mined by the homogeneous plane-wave part of the angu-lar spectrum.
The angular spectrum representation for the vectorialscattered field in Eqs. (52) and (53) is valid in the half-space z > 0 and corresponds to an incident plane wave inthe z direction. For arbitrary incidence the angular spec-trum representation of the vectorial scattered field is stillvalid in the half-space z > 0. However, now the incidentfield is propagating in the z8 direction, with the electric-field vector parallel with the x8 axis and the magnetic-field vector parallel with the y8 axis:
Ei 5 ex8 exp~ikz8!, (57)
Hi 5 ey8 exp~ikz8!. (58)
Now the z8 axis is the new symmetry axis for the angularspectra of the vectorial scattered fields [i.e., S1(cos a8)5 S1(ez8ek) 5 S1(1) and S2(cos a8) 5 S2(ez8ek) 5 S2(1)].The z and z8 axes define a scattering plane. For TM po-larization Ei is in the scattering plane, and the(x8, y8, z8) coordinates are found by rotating the (x, y, z)coordinate system about the y axis:
ex8 5 cos u0 ex 2 sin u0 ez ,
ey8 5 ey ,
ez8 5 sin u0 ex 1 cos u0 ez , (59)
where u0 is the angle between the z8 and z axes as illus-trated in Fig. 3. The relation between the sphericalcoordinates25 of both coordinate systems can be written as
a8 5 arccos~cos u0 cos a 1 sin u0 sin a cos b!,
b8 5 arcsinS sin b sin a
sin a8 D , (60)
which can be easily evaluated numerically.For TE polarization, with Ei perpendicular to the scat-
tering plane, we obtain the (x8, y8, z8) coordinates by ro-tating the (x, y, z) coordinate system about the x axis:
where u0 is the angle between the z8 and z axes as illus-trated in Fig. 4. The transformation between the primedand unprimed spherical coordinates25 is now
a8 5 arccos~cos u0 cos a 1 sin u0 sin a sin b!,
b8 5 arccosS cos b sin a
sin a8 D , (62)
which also can be evaluated numerically. Finally, theangular spectrum representation for the scattered fieldwith an incident field in an arbitrary direction becomes
Es~r! 51
2p E0
2p
dbE0
p/22i`
exp~ikr!
3 @S1~cos a8!cos b8 ea8
2 S2~cos a8!sin b8 eb8#a8 5 a8~a, b, u0!,
b8 5 b8~a, b, u0!
3 sin a da, (63)
Hs~r! 51
2p E0
2p
dbE0
p/22i`
exp~ikr!
3 @S2~cos a8!sin b8 ea8
1 S1~cos a8!cos b8 ea8#a8 5 a8~a, b, u0!,
b8 5 b8~a, b, u0!
3 sin a da, (64)
which is valid for z > 0. The integrals in Eqs. (63) and(64) are evaluated by first expressing the primed spheri-cal unit vectors in terms of primed Cartesian unitvectors25 (see Fig. 5):
ea8 5 cos a8 cos b8 ex8 1 cos a8 sin b8 ey8
2 sin a8 ez8 , (65)
eb8 5 2sin b8 ex8 1 cos b8 ey8 . (66)
For TM polarization the primed Cartesian unit vectorsare expressed in terms of the unprimed unit vectors byusing Eqs. (59), and the primed spherical coordinates arecomputed in terms of unprimed spherical coordinates byusing Eqs. (60). For TE polarization Eqs. (61) are used
Fig. 3. Coordinate transformations for TM polarization.
for the transformation of the Cartesian unit vectors, andEqs. (62) are used for the transformation of the sphericalcoordinates.
D. Homogeneous Plane Waves and Image FieldsIn most of the practical imaging and scattering situations,we can neglect the inhomogeneous plane waves, and thenthe general integral representation of the scalar scatteredfield in Eq. (14) reduces to
Us~r! 51
2p E0
2p
dbE0
p/2
S~cos ga0!exp~ikr!sin a da,
(67)
where cos ga0 5 cos a cos u0 1 sin a sin u0 cos(f0 2 b).The image field formed by an external circularly symmet-ric imaging system with the z axis as optical axis is foundby introducing a filter function H(a):
Us~r!
51
2p E0
2p
dbE0
p/2
S~cos ga0!exp~ikr!H~a!sin a da.
(68)If the entrance pupil of the imaging system is in the farzone, i.e., if the imaging system is telecentric, the filterfunction is simply the aperture transmission function.For a hard aperture of half-angle ua , it becomes particu-larly simple:
H~a! 5 H 1, a < ua
0, a . ua, (69)
which yields the image field
Fig. 4. Coordinate transformations for TE polarization.
The incident plane wave may also contribute to the totalimage field. For the telecentric case the image of a planewave is also a plane wave provided that it passes theaperture.8,17,31 With an incident plane wave Ui(r)propagating in the direction u0 with respect to the opticalaxis of the imaging system (i.e., the z axis), the total im-age field formed by a telecentric imaging system is
U~r! 5 H Ui~r! 1 Us~r!, u0 < ua
Us~r!, u0 . ua. (71)
We now immediately see that for the telecentric imagingcase the image of a sphere has contributions only from theincident field for forward scattering with u0 < ua . In theother cases, including the special case of backscattering(i.e., for u0 5 p) and Ui(r) normally incident to the opti-cal axis of the imaging system (i.e., for u0 5 p/2), thereare contributions only from the scattered field to the im-age field. Moreover, Eq. (70) shows that for a given im-aging geometry the angular spectrum of the scatteredfield, S(cos g a 0), contributes to the image field for onlyone of the following possible ranges of ga0 :
uga0u P H @0,u0 1 ua#, u0 < ua
@u0 2 ua ,u0 1 ua#, ua , u0 , p 2 ua
@u0 2 ua ,p#, p 2 ua , u0 < p
.
(72)
Thus, to evaluate the integral in Eq. (70) numerically, wecan first compute a table of the angular spectrum of thescattered field within the limits of ga0 indicated in expres-sion (72). For all the possible values of the integrationvariables a and b in Eq. (70), the integrand can then becomputed by interpolation from this table. In this waythe number of calculations of the angular spectrum(which involves a summation over modal orders) can beminimized.
The vectorial scattered fields in Eqs. (63) and (64) canbe written in terms of their vectorial components, whichfor the Cartesian case are
Es~r! 5 Ex~r!ex 1 Ey~r!ey 1 Ez~r!ez , (73)
Hs~r! 5 Hx~r!ex 1 Hy~r!ey 1 Hz~r!ez , (74)
where Ex(r) 5 Es(r)ex , etc., are scalar functions given byexpressions of the same form as those of Eqs. (69)–(71).The ranges for ga0 in expression (72) also apply for theangular argument a8 to the vectorial scattering functionsS1(cos a8) and S2(cos a8) in Subsection 2.C. For an axi-ally symmetric imaging geometry (with u0 5 0 or p), theb integral in Eq. (70) can be evaluated analytically interms of cylindrical Bessel functions for both the scalarand vectorial cases.
3. SCALAR WAVESIn this section we consider the scalar case in Eq. (14) withan incident field propagating in the z direction. The
scattered field backpropagated to z 5 0 is discussed inparticular, and we derive the physical properties of thisfield.
A. Homogeneous Plane WavesIf we neglect the inhomogeneous plane waves in Eq. (14)as shown in Eq. (67), the backpropagated scattered fieldat z 5 0 is
Us~h, 0! 51
2p E0
2p
dbE0
p/2
S~cos a!
3 exp@ikh sin a
3 cos~f 2 b!#sin a da. (75)
The b integral in Eq. (75) can be evaluated analytically byusing the formula25
E0
2p
db exp@ikh sin a cos~f 2 b!# 5 2pJ0~kh sin a!,
(76)
where J0(z) is the cylindrical Bessel function of zero or-der. The scattered field at z 5 0 can then be written inthe form
Us~h, 0! 5 E0
p/2
S~cos a!J0~kh sin a!sin a da.
(77)
Since the integral limits are finite and the integrand isnonsingular, we conclude that when we neglect the inho-mogeneous plane waves, the backpropagated scatteredfield Us(h, 0) is nonsingular.
B. Paraxial ImagingIn this subsection we show that in the paraxial approxi-mation the sphere can be represented by a flat circularlysymmetric object. This result is obtained by comparingthe paraxial images of the sphere with that of a flat ob-ject.
With the use of Eq. (70), the image field at z 5 0 is
Us~h, 0! 51
2p E0
2p
dbE0
ua
S~cos a!
3 exp@ikh sin a cos~f 2 b!#sin a da,
(78)
where h 5 Ax2 1 y2. By using Eq. (76) for the b inte-gral in Eq. (78), we can then write the image field at z5 0 in the form
Us~h, 0! 5 E0
ua
S~cos a!J0~kh sin a!sin a da. (79)
Substituting Eq. (15) for the angular spectrum into Eq.(79), we can write the image field at z 5 0 in the form
The Legendre polynomial in Eq. (81) has the following ex-act integral representation26:
Pn~cos a! 5A2
pE
0
a cos@~n 112 !t#
~cos t 2 cos a!1/2 dt
5A2a
pE
0
p/2 cos t
@cos~a sin t! 2 cos a#1/2
3 cos@~n 112 !a sin t#dt. (82)
In the paraxial approximation we have sin a ' a andcos a ' 1 2
12a 2. Thus, by substituting cos(a sin t) ' 1
212 a2 sin2 t into Eq. (82) and identifying the resulting
integral to be equal to Eq. (76), we obtain the paraxial ap-proximation to the Legendre polynomial26:
Pn~cos a! ' J0(~n 112 !a). (83)
Note that relation (83) is valid for any order n (since wehave not made any assumptions regarding the value ofn), even though the well-known asymptoticrepresentation26,32 of Pn(cos a), which is valid for n → `and a → 0, also is equal to relation (83). Substituting re-lation (83) into Eq. (81), we obtain the paraxial approxi-mation of the function Dn(h):
Relation (86) can be further simplified for khn8 , kh @ 1.By using Debye’s asymptotic representations for the cy-lindrical Bessel functions,27
J0~b ! ; A 2pb
cosS b 2p
4 D , b @ 1, (88)
J1~a ! ; A 2pa
sinS a 2p
4 D , a @ 1, (89)
we obtain
D~hn8 , h! ;kua
pAhn8
h
sin@kua~hn8 2 h!#
kua~hn8 2 h!,
khn8 , kh @ 1. (90)
To interpret the expressions in Eqs. (80) and (81), we con-sider the general case of a flat circularly symmetric objectfield U0(x, y) 5 U0(Ax2 1 y2) at z 5 0. The corre-sponding image field formed by the external imaging sys-tem with aperture half-angle ua is then
UI~x, y ! 5 E2`
` E2`
`
U0~Ax82 1 y82!
3 h~x 2 x8, y 2 y8!dx8dy8, (91)
where
h~x, y ! 51
~2p!2 E2`
` E2`
`
H~kx , ky!
3 exp@i~kxx 1 kyy !#dkxdky (92)
is the amplitude point spread function of the imagingsystem.17,31 In Eq. (92) the aperture function is
H(kx , ky) 5 1 for Akx2 1 ky
2 < k sin ua and 0 otherwise[compare with Eq. (69)]. With spherical coordinates h5 Ax2 1 y2, kx 5 k sin a cos b, and ky 5 k sin a sin b,the point spread function can be written as
h~h! 5k2
~2p!2 E0
2p
dbE0
ua
da sin a
3 exp@ikh sin a cos~f 2 b!#
5k2
2pE
0
ua
da~sin a!J0~kh sin a!, (93)
where we have used Eq. (76) for the b integral. In theparaxial approximation we have sin a ; a, and the corre-sponding point spread function is
h~h! ;k2
2p E0
ua
J0~kha!a da. (94)
The convolution integral in Eq. (91) can be expressed interms of spherical coordinates to yield
UI~h! 5 E0
`
dh8 U0~h8!h~h8, h!, (95)
where h(h8, h) is the radial impulse response function ofthe imaging system:
h~h8, h! 5 h8E0
2p
df8 h(~h2 1 h82 2 2h8h cos f8!1/2).
(96)
The radial impulse response function can be found ana-lytically by using the summation formula for Besselfunctions26:
where R 5 (r2 1 r2 2 2rr cos f)1/2. Equation (96) nowreduces to the expression
h~h8, h! 5 k2h8E0
ua
J0~kh8a!J0~kha!a da (98)
by using the orthogonality properties of the trigonometricfunctions.25,26 Comparing relation (84) and Eq. (98), weimmediately see that h(hn8 , h) 5 D(hn8 , h), where hn85 Dh8(n 1
12 ) and Dh8 5 1/k. Thus, in the paraxial
approximation, Dn(h) is interpreted as the radial impulseresponse function of the imaging system. The discreteversion of Eq. (95) is found by using the discrete variablehn8 :
UI~h! 5 (0
` 1
kU0~hn8 !h~hn8 , h!. (99)
Comparing Eqs. (80) and (99), we see that UI(h)5 Us(h, 0) for U0(hn8 ) 5 2an . Thus, in a focusedparaxial image of the center plane of a sphere, the modecoefficients are equal to the effective object field sampledat discrete positions. For a pure phase object, the totalobject field is exp@iC(h)#, where C(h) is a real phase func-tion. The scattered part of this field can be expressed asfollows:
U0~h! 5 exp@iC~h!# 2 1. (100)
For lossless spheres (including hard and soft spheres), weexpress an in the form of Eq. (8), and by comparing Eqs.(80) and (99), we now see that UI(h) 5 Us(h, 0) for2exp@iC(hn8)# 5 exp(icn), where cn is the scatteringphase. Thus all lossless spheres (including hard and softspheres) can be represented by a pure phase screen.
In summary, we have now shown that in the paraxialapproximation, a sphere can be represented by a flat cir-cularly symmetric object. In particular, for losslessspheres (including hard and soft spheres) this flat objectis a pure phase screen. The flat object is defined by anobject field, and we have shown that sampled values ofthis object field are equal to the expansion coefficients.We have also interpreted the integral coefficient Dn(h) tobe the radial point spread function of a telecentric imag-ing system with the symmetry Dn(h) 5 D(hn8 , h)5 D(h, hn8 ) in the paraxial approximation. The imagefield can finally be obtained either by convolving the ob-ject field with the radial impulse response function or bylow-pass filtering the object field.
C. Paraxial Scattering of Scalar WavesIf a sphere’s scattering function S(cos a) is narrowlypeaked in the forward direction and decreases rapidly to 0before the angular argument a approaches ua , we have aparaxial scattering situation. This happens if the sphereis both weakly scattering (i.e., n ' 1) and large comparedwith the wavelength (i.e., a @ l). The discussion of theparaxial scattering case is similar to that of the paraxialimaging case in Subsection 3.B, and the radial impulseresponse function D(hn8 , h) in relation (86) is obtained.
However, since the scattering function is narrowly peakedin the forward direction, we may formally choose ua5 Np, where N is an integer integer @ 1, as the upperintegration limit in Eq. (79). This operation is equivalentto zero padding in signal processing. Thus the paraxiallyscattered field at z 5 0 can be written in the same formas that of Eq. (80):
Us~h, 0! 5 (n50
`1k
2anDn~h!. (101)
Formally, in the limit ua @ 1, relation (86) reduces to re-lation (90), and we obtain the following radial impulse re-sponse function for paraxial scattering:
D~hn8 , h! ; kAhn8
h
sin@k~hn8 2 h!Np#
k~hn8 2 h!Np, N @ 1.
(102)
Since N is an integer, we have D(hn8 , h) 5 k for hn8 5 hand 0 for hn8 Þ h. Thus the paraxially scattered field atz 5 0 consists of the mode coefficients at discrete posi-tions hn 5 Dh(n 1
12 ) with sampling distance Dh
5 1/k:
Us~hn , 0! 5 2an . (103)
Note that the effective sampling distance is l/2p and lessthan the Nyquist limit for homogeneous plane waves:DhNyq 5 l/A2 [i.e., DhNyq 5 (DxNyq
2 1 DyNyq2 )1/2 with
DxNyq 5 DyNyq 5 l/2 (Refs. 4, 17, and 31)].For a paraxial scattering sphere, all scattering occurs
into the half-space z . 0, and by backpropagating thewhole scattered field in the half-space z . 0 back to z5 0, we obtain the paraxial scattered field in Eq. (103).However, for the paraxial image of the center plane of asphere in Eq. (80) we backpropagate only the part of thescattered field that enters the imaging aperture.
The paraxially scattered field at z 5 0 can be regardedas a flat circularly symmetric object field, in analogy withthe results in Subsection 3.B. For example, a weaklyscattering lossless sphere can be modeled by a phasescreen at z 5 0 [i.e., Eq. (100)] by using the straight-rayapproximation of geometrical optics.15 Assuming that n' 1, so that all rays remain parallel throughout thesphere, we obtain the phase function C(h) 5 2k(n2 1)Aa2 2 h2 for h < a and 0 otherwise.
D. Stationary-Phase Approximation and NonparaxialImagingIn this subsection we derive a simple geometrical modelfor a defocused image of a sphere at planes z @ l. Thisexample yields a simple physical explanation of the differ-ence between the paraxial and nonparaxial images of thecenter plane of a sphere.
From Eqs. (14) and (70), we see that the defocused im-age field at z 5 r cos u can be written as
By substituting the exact integral representation for theLegendre polynomial in Eq. (82) into Eq. (105) and by in-troducing the variable hn8 5 (1/k)(n 1
12 ), we write Dn(r)
as D(hn8 , r) with the following integral representation:
D~hn8 , r! 5 k2hn8E0
ua
daE0
2p
dbE0
p/2
dt f~a, t!
3 $exp@iC1~a, b, t!#
1 exp@iC2~a, b, t!#%. (106)
In Eq. (106) the amplitude term is
f~a, t! 5a sin a
2A2p2
cos t
@cos~a sin t! 2 cos a#1/2 , (107)
and the phase functions are
C6~a, b, t! 5 kr@cos a cos u
1 sin a sin u cos~b 2 f!#
6 khn8a sin t. (108)
We define the angle sin g 5 hn8/r and evaluate the integralin Eq. (106) asymptotically by the method of stationaryphase.33 The interior stationary-phase points are foundby solving ]C1 /]a 5 ]C1 /]b 5 ]C1 /]t 5 0, whichyields (as1 , bs1 , ts1) 5 (u 2 g, f 1 p, p/2) 5 (u1 , f1 p, p/2) and (as2 , bs2 , ts2) 5 (u 1 g, f, p/2)5 (u2 , f, p/2) provided that u1 ,u2 < ua . The phasefunction can now be approximated by its Taylor expan-sion about each of the stationary-phase points, whichyields
C1~a, b, t! ; C1~as , bs , ts!
11
2 S ]2C1
]a2 D a5as ,b5bs ,t5ts
~a 2 as!2
11
2 S ]2C1
]b2 D a5as ,b5bs ,t5ts
~b 2 bs!2
11
2 S ]2C1
]t 2 D a5as ,b5bs ,t5ts
~t 2 ts!2, (109)
where we have retained only the nonzero terms andwhere s 5 s1, s2. The amplitude term is approximatedby its value at the stationary points, which yields
f~as , ts! ;~as sin as!
1/2
2p2 , (110)
with s 5 s1, s2. Letting the integration limits of the aand b integrals in Eq. (106) formally extend to 6` (i.e.,neglecting the edge contribution for a 5 ua), we canevaluate these integrals by using the formula27,33
E2`
`
expS iB2
b2Ddb 5 A2p
uBuexpF i
p
4sgn~B !G ,
(111)
where sgn (B) 5 1 for B . 0 and 21 for B , 0. The t in-tegral in Eq. (106) can be formally written as
E2`
ts
expF iC2
~t 2 ts!2Gdt 5
12 E
2`
`
expS iC2
t 2Ddt,
(112)
which can also be evaluated by using Eq. (111). Alto-gether, we finally obtain
D~hn8 , r! ; kAhn8
h
expS ikr8 2 ip
4 DA2pkr8
3 @2i exp~ikhn8u2! 1 exp~ikhn8u1!#,
(113)
where r8 5 Ar2 1 hn82 and h 5 r sin u 5 Ax2 1 y2.
The expression in relation (113) can also be found byusing an alternative approach. Equation (105) is themodal expansion of the scattered field outside the sphere,with
Dn~r! 5 k~n 112 !inhn
~1 !~kr !Pn~cos u!, (114)
where hn(1)(kr) 5 Ap/2krHn11/2
(1) (kr) and Hn11/2(1) (kr) is the
cylindrical Hankel function of fractional order27 n 112 .
In the paraxial approximation we use relation (83) for theLegendre polynomials and obtain
Dn~r! 5 kS n 112 D expS i
p
2n DA p
2krHn11/2
~1 ! ~kr !
3 J0XS n 112 DaC. (115)
In Eq. (115) the cylindrical Bessel functions can be repre-sented by the asymptotic representation of Debye in rela-tions (88) and (89). The corresponding asymptotic repre-sentation for the cylindrical Hankel function is26
Hn~1 !~z ! 5 S 2
pAz2 2 n2D 1/2
expF iSAz2 2 n2
2 n arccosn
z2
p
4 D G , z @ n. (116)
By using that n 5 khn8 212 and that arccos(hn8/r) 5 p/2
2 g, we finally obtain
D~hn8 , r! ; kAhn8
ru
expS ikr8 2 ip
4 DA2pkr8
@2i exp~ikhn8u2!
1 exp~ikhn8u1!#, r @ hn8 . (117)
We now immediately see that this expression is theparaxial version of relation (113) with h 5 r sin u ; ru.Thus the stationary-phase solution in relation (113)equals the modal contribution in relation (117) with theDebye asymptotic representation of the Hankel and Leg-
endre functions (also called the approximation bytangents24). Referring to Fig. 6, we see that D(hn8 ,r) de-scribes two rays in, respectively, the u1 and u2 directions(the so-called significant directions24). The rays are tan-gent to a spherical caustic of radius hn8 5 (1/k)(n 1
12 )
centered at the origin. The focused image at z 5 0 of themode coefficient an is then found by projecting these raysback to z 5 0. Since these backprojected rays are closeto the caustic surface of radius hn8 , a nonparaxial imageof an can be interpreted as an aberrated version of theparaxial case, the aberrations being caused by the caus-tic. This aberration is a pure scattering effect and not anaperture diffraction effect.
E. Image Planes Close to Caustic RegionsIn Subsection 3.D the nonparaxial image of the centerplane of a sphere was interpreted to be an aberrated ver-sion of the paraxial case, the aberration being a pure scat-tering effect. In this subsection we derive an expressionfor this aberration and a criterion for neglecting it.
The image field at z 5 0 can also be represented by theintegral in Eq. (106) with the same phase and amplitudefunctions but for u 5 p/2. However, the asymptotic rep-resentation of Dn(r) in relation (113) breaks down if theinterior phase points (as1 , bs1 , ts1) and (as2 , bs2 , ts2)coalesce at an inflection point, where both the first andsecond derivatives of the phase function vanish [i.e., ifas1 → as2 → 0, the parabolic terms in relation (109) van-ish]. Thus we here approximate the phase functionswith the corresponding Taylor expansions about the criti-cal points, where we include the cubic terms. The criticalpoints are the stationary-phase points for the b and t in-tegrals and the inflection point for the a integral.33
Thus, by solving ]2C1 /]a2 5 ]C1 /]b 5 ]C1 /]t 5 0, weobtain the critical point (as1 , bs1 , ts1) 5 (01, f1 p, p/2), and by solving ]2C2 /]a2 5 ]C2 /]b5 ]C2 /]t 5 0, we obtain the critical point(as2 , bs2 , ts2) 5 (01, f, p/2). The phase function canthen be approximated by
C1~a, b, t! ; S ]C1
]aD a501,
b5bs1 ,t5ts1
a 11
6 S ]3C1
]a3 D a501,b5bs1 ,t5ts1
a3
11
2 S ]3C1
]b2]aD a501,
b5bs1 ,t5ts1
a~b 2 bs1!2
11
2 S ]3C1
]t 2]aD a501,
b5bs1 ,t5ts1
a~t 2 ts1!2, (118)
Fig. 6. Modal contribution in the approximation by tangents.
where we have retained the nonzero terms only. A simi-lar expression can be found for C2 by replacing the indi-ces s1 with s2 in relation (118). The amplitude term isapproximated by the Taylor expansion of relation (110)with respect to a:
f~a, ts! ;~a sin a!1/2
2p2 51
2p2 @a 1 O~a3!#, (119)
with s 5 s1, s2. The b and t integrals can now be for-mally solved by using, respectively, Eqs. (111) and (112),and we obtain
D~hn8 , h! ;k
2pAhn8
h E2ua
ua
3 expH iFk~hn8 2 h!a 1kh
6a3G J da,
hn8 ; h. (120)
Letting the integration limits in relation (120) formally goto 6` (i.e., we neglect the edge contributions from a5 6ua), we finally obtain
D~hn8 , h! ; kAhn8
h S 2kh D 1/3
3 AiF S 2kh D 1/3
k~hn8 2 h!G ,hn8 ; h, (121)
where
Ai~z ! 51
2p E2`
`
expF iS za 113
a3D Gda (122)
is the Airy function.27 The Airy solution in relation (121)is a caustic solution,33 as expected from the geometricalinterpretation of D(hn8 , r) in Subsection 3.D. Thus, innonparaxial imaging, the cubic phase term in relation(120) causes aberrations that are similar to coma.4,31
Since we have neglected the edge term in the a integralfor a 5 ua , this aberration is a pure scattering effect.Returning to relation (120), we can neglect the cubic termif the following Rayleigh criterion holds (i.e., the cubicterm is less than p/2 (Refs. 4 and 31)]:
kh , 3p/ua3. (123)
In this case we obtain
D~hn8 , h! ;k
2pAhn8
h E2ua
ua
exp@ik~hn8 2 h!a#da
5kua
pAhn8
h
sin@kua~hn8 2 h!#
kua~hn8 2 h!, (124)
which equals the expression for the paraxial radial im-pulse response function in relation (90). For h 5 a5 the sphere radius, the Rayleigh criterion in relation(123) is a criterion for using the paraxial results. We im-mediately see that, in a paraxial imaging situation, thesize of the aperture ua limits the size of the sphere a to beimaged (and vice versa). For example, for ua 5 16° (i.e.,
f-number 5 0.57) the paraxial results are limited tospheres of radius a , 2250l ('1.4 mm for l 5 628 nm).
4. NUMERICAL EXAMPLESIn this section we illustrate the usefulness of the angularspectrum representation of the scattered field for field cal-culations. In all the following examples, the incidentplane wave propagates along the z axis. In the scalarcases the images are the squared modulus of the total sca-lar field,4,5,22,24 uU(r)u2 5 uexp(ikz) 1 Us(r)u2. In the vec-torial case the images are the normalized z component ofthe Poynting vector,4,5,22,24 Sz 5 (1/2N)@(Ei 1 Es) 3 (Hi
1 Hs)#z , with the normalization constant N 512 (Ei
3 Hi)z .Figure 7 shows the image at z 5 0 of a perfectly reflect-
ing sphere of radius a 5 2l, where the incident electricvector is parallel with the x axis. The dashed circle is theposition of the sphere’s geometrical shadow boundary.Figure 7 illustrates that the image of a perfectly reflectingsphere is elliptical because of polarization effects. In Fig.8 the solid and dashed curves are the fields along, respec-tively, the x- and y axes in Fig. 7. The position of theshadow boundary found for cylinders16 is also plotted asstep functions in Fig. 8. We see that the position of theshadow boundary for spheres agrees with the position ofthe shadow boundary for cylinders: The field along the xaxis (solid curve) corresponds to the TM-polarization casefor cylinders, and the field along the y axis (dashed curve)agrees with the TE-polarization case for cylinders.
Figure 9 shows the virtual focal region of an acousti-cally transparent sphere with index of refraction n5 0.9 relative to that of the background and radius a5 10l. The boundary of the sphere is indicated with adashed curve. Here the sphere acts like a concave lenswith a virtual focus on the negative z axis. The positionof the virtual focal plane as predicted by paraxialoptics4,31 is plotted with a dashed line in Fig. 9.
Fig. 7. Image at z 5 0 of a perfectly reflecting sphere of radiusa 5 2l. The dashed circle is the geometrical shadow boundaryof the sphere. The incident field is linearly polarized with theelectric-field vector parallel with the x axis, and the imaging ap-erture half-angle ua ' 60°. Because of polarization effects theimage of the sphere is elliptical.
In Fig. 10 the usefulness of the paraxial imaging re-sults in Subsection 3.B for numerical field computationsis illustrated. Figure 10(a) shows a cross section of thephase function of a flat circularly symmetric object, thephase function being equal to the scattering phase cn fora soft sphere with ka 5 10. The dashed curve in Fig.10(b) is the paraxial image of this object for an aperturehalf-angle ua 5 20°, and the solid curve is the corre-sponding exact image of the sphere. Both curves in Fig.10(b) yield the same results. Two two-dimensional fastFourier transform algorithms have been used to low-passfilter the object function U0(hn8 ) 5 exp(icn) in Fig. 10(a)and thereby yield the paraxial image in Fig. 10(b).
The main results in Subsections 3.C and 3.D are illus-trated in Fig. 11. Here the radial impulse response func-tion of the imaging system D64(h) is plotted as a function
Fig. 8. Image fields along the x and y axes in Fig. 7 plottedwith, respectively, a dashed curve and a solid curve. The geo-metrical shadow boundary is at h 5 a. The steps are the posi-tions of the shadow boundary found for cylinders. The imagefield along the x and y axes corresponds to, respectively, the TM-and TE-polarization cases for cylinders.
Fig. 9. Virtual focal region for an acoustically transparentsphere of radius a 5 10l and index of refraction n 5 0.9 relativeto that of the background. The sphere acts like a concave lenswith a virtual focus on the negative z axis. The dashed line isthe position of the virtual focal plane found from paraxial optics.
of h648 2 h for various values of the aperture half-angleua . The figure can be interpreted as the image of an off-axis object point at h648 . Figure 11 illustrates that forsmall aperture angles the radial impulse response func-tion is the symmetric paraxial point spread function in re-lation (86). For increasing aperture angles the pointspread function becomes asymmetric because of the aber-rations discussed in Subsections 3.D and 3.E. The Ray-leigh criterion derived in Subsection 3.E is a criterion forneglecting this aberration.
Figure 12 shows the image at z 5 0 for the sphere inFig. 9. The dashed curve is the exact image field, and thesolid curve is the backpropagated field in the geometrical-optics approximation (see Subsection 3.C). Figure 13shows the scattering phase (dashed curve) of the spherein Fig. 9 along with the phase of the backpropagated fieldin the geometrical-optics approximation. Figure 13 illus-trates that the paraxially scattered field at z 5 0 is equalto the mode coefficients at discrete sampling points hn
5 (1/k)(n 112 ), where n is the mode order.
Figure 14 shows the scattered field at z 5 20a for anacoustically soft sphere of radius a 5 10l. The solid
curve and the plusses are the scattered field computedwith, respectively, the angular spectrum representationand the modal expansion. We see that both approachesyield the same results in the Fresnel region of the sphere.
5. SUMMARY AND CONCLUSIONSWe have considered scattering of plane waves by a spherein the scalar and vectorial cases and transformed themodal expansion of the scattered field to an angular spec-trum representation.
The inhomogeneous and homogeneous plane-waveparts of the angular spectrum of the scattered field arefound for an arbitrary direction of the incident field. Thesymmetry axis of the angular spectrum is found to coin-cide with the direction of the incident field. In contrastto the usual interpretations,11,30 we identify the far-fieldscattering function to be given by the homogeneous plane-wave part of the angular spectrum (and not vice versa).
By neglecting the inhomogeneous plane waves in thisrepresentation, we also obtain the virtual, or backpropa-gated, fields for planes within or behind the object.
Fig. 10. Paraxial imaging of a soft sphere with ka 5 10: (a) cross section of the phase function of a flat circularly symmetric object, thephase function being equal to the scattering phase cn for the sphere; (b) the dashed curve is the intensity in a paraxial image withaperture half-angle ua 5 20°, and the solid curve is the corresponding exact image intensity.
Fig. 11. Point image D64(h) of the off-axis point h648 for several values of the aperture half-angle ua .
These virtual fields are the fields that, to an external ob-server, appear to exist in free space within or behind theobject. Thus these virtual fields are different from theactual fields existing in these regions. The image fieldmodel is obtained by including aperture limitations to theangular spectrum representation. The incident planewave contributes to the total image field only as long asthe wave passes the aperture. Otherwise, only the scat-tered field contributes to the total image field. The modelalso shows that the scattered part of the image field hascontributions from only certain angular ranges of the an-gular spectrum. Thus, in image field algorithms, thenumber of calculations of the angular spectrum (which in-volves a summation over mode orders) can be minimized.
For scalar waves we have shown that when the inho-mogeneous plane waves are neglected, the singularity ofthe eigenfunctions at the origin r 5 0 vanishes. Eventhough the inhomogeneous plane waves are neglected, in-formation about creeping waves, multiple reflected waves,multiple diffracted edge waves, totally reflected waves,etc., is still present in the angular spectrum, since it is de-fined by the expansion coefficients.
In the paraxial approximation we have shown that asphere can be represented by a flat circularly symmetricobject. For lossless spheres (including soft and hardspheres), this object is a pure phase screen. Sampledvalues of this object field are directly given by the expan-sion coefficients at positions hn 5 (1/k)(n 1
12 ), n
5 0, 1, 2, ..., where n is the mode order. A paraxial im-age of the center plane of the sphere is then equal to thediscrete convolution between the flat-object field and theradial impulse response function of the imaging system.Since we consider a telecentric imaging system, the imagefield at z 5 0 is equal to the two-dimensional low-pass fil-tered mode coefficients. We have implemented this re-sult and computed a paraxial image of a soft sphere by us-ing two two-dimensional fast Fourier transformalgorithms.
For paraxial scattering (i.e., weakly scattering sphereswith index of refraction n ' 1 and ka @ 1), the mode co-efficients actually equal the backpropagated field at z5 0 at discrete sampled positions hn 5 (1/k)(n 1
12 ), n
Fig. 12. Image field amplitude at z 5 0 of the sphere in Fig. 9.The dashed and solid curves are, respectively, the exact andgeometrical-optics fields.
Fig. 13. Paraxial scattering case where the backpropagatedfield at z 5 0 is directly given by the mode coefficients. Thescattering phase of the sphere in Fig. 9 is plotted with a dashedcurve along with the field phase in the geometrical-optics ap-proximation.
5 0, 1, 2, ..., where n is the mode order. The field issampled at discrete radial positions hn with a samplingdistance Dh 5 l/2p, which is well beyond the Nyquistlimit for homogeneous waves, DhNyq 5 l/A2.
The nonparaxial case is interpreted as an aberratedversion of the paraxial case, the aberration being a purescattering effect. This aberration is similar to coma andintroduces an asymmetry to the radial impulse responsefunction of the imaging system Dn(h) for nonparaxial ap-ertures and object points away from the optical axis. Toillustrate this result, the radial impulse response functionis plotted for several imaging apertures. A Rayleigh cri-terion for neglecting this aberration is also derived. Thiscriterion shows that for a paraxial imaging system thesize of the aperture is limited by the size of the sphere.
We have checked numerically and theoretically thatboth the modal expansion and the angular spectrum rep-resentation yield the same fields in the Fresnel region.
Examples have been shown of field calculations thatcontain diffraction effects that are present only in imag-ing situations: Acoustically transparent spheres with anindex of refraction less than that of the background (n, 1) act like concave lenses with virtual foci behind the
spheres. Images of the center plane of perfectly reflect-ing spheres appear to be elliptical because of polarizationeffects. In the last case the results agree with the resultsfound for perfectly reflecting cylinders.16 Thus a focusedimage of a general three-dimensional convex shape maybe found by locally approximating the shape with a circu-lar cylinder.
ACKNOWLEDGMENTSI thank the Department of Physics at The NorwegianUniversity of Science and Technology for financing myPh.D. studies. I also thank H. M. Pedersen for all thevaluable feedback I got throughout the time I spent onthis problem.
The author can be reached by tel: 47-67-128809; fax:47-67-126910; e-mail: [email protected].
Fig. 14. Scattered field amplitude at z 5 20l of an acousticallysoft sphere with radius a 5 2l. The solid curve and the plussesare the field amplitudes computed with, respectively, the angularspectrum representation and the modal expansion.
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