Notes on plane wave expansions

8/10/2019 Notes on plane wave expansions

1/6

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Shono and Inuzuka - Representation of a diffracted wave field by the band-limited angular spectrum, J.Opt.Soc.Am 68, n.11, 1978

- in Weyl's expression inhomogeneous waves are usually neglected [3]- validity of this approximation estimated analitically [4] and numerically [5]- alternative expression given by Wittaker as superposition of only homogeneousplane waves [6]- contribution of inhomogeneous waves = contribution of homogeneous waves propagating back ward from a source [7]- Field in space is given by superimposing plane waves with an angular spectrumof the emitter [8, pag.61]- Amplitude distribution of the PWs is equal to the angular spectrum at infinitedistance and approx equal in the far field- angular spectrum contributing to observation point is limited in narrow regiondepending on the distance from the object- significant contributions to the integral about the angular spectrum arise from the vicinity of stationary phase points [9]- all other contributions are neglected due to distructive interference- determine spectral bandwidth- spherical wave exactly represented with the superposition of a part of homogeneous plane waves- approximation to circular-plane-wave expansion [10]- obtain spectral bandwidth as a function of distance from the object and apertu

re size- start from Weyl's expansion, theta integrated over a complex domain C- divide C into C1 (real axis) and C2 (imaginary axis): C1 gives homogeneous waves, C2 gives inhomogeneous waves- Fresnel zone construction- contribution of homogeneous PWs is Uh = U1/2-1/(ikr)- contribution of inhomogeneous PWs is Ui = 1/(ikr)- field is U = U1/2 = half the contribution due to the first zone -> minimum number of PWs without phase jumps- if r >> lambda -> theta1 approx sqrt(lambda/r)- only PWs of which the poles are involved in the solid angle subtended at the source by the first Fresnel'2 zone make an effective contribution

- 1st representation of spherical wave: PW having infinitely extended wave fronts but propagating into a restricted spatial angle- 2nd representation of spherical wave: PW having suitably extended spatial wavefronts and propagating to all directions -> CPW- CPW is invariant under rotation of coordinates != from conventional angular spectrum and Weyl's expression- CPW is a good approximation if r is larger than a few wavelengths

[3] G.C.Sherman, Diffracted wave field expressible by plane-wave expansion containing only homogeneous waves, J.Opt.Soc.Am., 59, 697-711, 1969[4] G.C.Sherman and J.J.Stamnes and A.J.Devaney and E.Lalor, Contribution of theinhomogeneous waves in angular-spectrum representations, Opt.Commun., 8, 271-274, 1973

[5] W.H.Carter, Band-limited angular-spectrum approximation to a spherical scalar wave field, J.Opt.Soc.Am., 65, 1054-1058, 1975[6] E.T.Whittaker, On the partial differential equations of mathematical physics, Math.Ann. 57, 333-355, 1902[7] G.C.Sherman and A.J.Devaney and L.Mandel, Plane Wave expansions of the optical field, Opt.Commun. 6, 115-118, 1972[8] J.V.Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968[9] H.M.Nussenzweig, Diffraction theory in the k-representation, An.Acad.Bras.Cienc. 31, 515-521, 1959[10] Y.Shono, Reconstructed images from volume holograms in the Fraunhofer appro


2/6

ximations: Analysis by a new spherical-wave expansion, J.Opt.Soc.Am. 66, 564-574, 1976

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Lalor - Conditions of validity of the Angular Spectrum of Plane Waves, J.Opt.Soc.Am. 58, n.9, 1968

- Specify conditions on the field which guarantee the existence of angular spectrum representation

- L2 = space of square integrable function, i.e. function which are integrable if we consider their absolute value squared- Plancheral theorem: if f(x,y) is in L2, there exist its Fourier transform f(p,q) in L2

- consider scalar wavefield U(x,y,z) which satisfies 3 conditionsa) is solution to Helmholtz equation when z>0b) boundary values U(x,y,0) = f(x,y)

i) f(x,y) is sectionally continuous in the xy planeii) f(x,y) is continuous and has continuous derivatives outside

a circle of radius R0 s.t. abs(f)0 there exist a constant C s.t. U and U_R satisfyabs(U)0 there exist a constant D(theta) s.t. abs(U_R - ikU) < D/R^2- if conditions a,b,c are satisfied, U is expressed by Rayleigh integral formulation (pressure->pressure)- terms in the integrand are in L2

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Y. Hsu and K.T.Wong and L.Yeh - Mismatch of Near-Field Bearing-Range Spatal Geometry in Source-Localization by a Uniform Linear Array, IEEE Trans. on Antennas and Propagation, vol.59, n.10, 2011

- many near-field source-localization algorithms simplify the exact spatial geometry to speed up the signal processing involved- e.g. Fresnel approximation is adopted -> 2nd order Taylor-series approximation- approximation introduces a systemic error in the algorithm's modeling -> systematic non-random bias added to random estimation errors- propose explicit formulas of the degrading effects in 3d source localization

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%L.J.Ziomek - Three Necessary Conditions for the Validity of the Fresnel Phase Approximation for the Near-Field Beam Pattern of an Aperture - IEEE J. of OceanicEngineering, vol.18, n.1, 1993

- Fresnel diffraction integral is defined as near-field directivity function of

an array- perform 2nd order binomial expansion of the range term in the phase factor offree field Green's function- derive region of validity for the Fresnel approximation

- binomial expansion of the range term- use first expansion term to approximate the amplitude term- use first 2 terms to approximate the phase- first necessary condition: 72


3/6

- far field criterion: r>pi R^2/lambda>1.356R- near field criterion: 1.356 R < r < pi R^2/lambda (third necessary condition)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%A.J.Devaney and G.C.Sherman - Plane-Wave Reprsentations for Scalar Wave Fields -SIAM Review vol.15, n.4, Oct. 1973

- fields that satisfy the homogeneous wave equation are represented by integralexpansions of monochromatic plane waves all propagating with same speed

i) each expansion term is a solution to wave equation -> can be dealt with individually as physical entity

ii) simple form (in contrast to spherical, cylindrical functions), differing each other only by change in temporal frequency and direction of propagation

iii) solution for 1 PW leads to solution for all PWsiv) in certain regions of the space-time domain, PW expansion can be inv

erted to determine PW amplitudes for knowledge of the field- PW representations related to 4d Fourier transform -> superposition of function exp(i(vec(k) vec(r) -omega t))

i) 4D-FT is a four-fold integral over kx, ky, kz, omega -> exp functionis not in general a solution to homogeneous wave equation with fixed wave speed

ii) PW expansions are three-fold integrals with kx, ky, kz, omega constrained by a dispersion relation -> exp function is a solution to hom. wave eq. with fixed wave speed

iii) 4D-T can be used to represent arbitrary L2 function, while PW expansions can represent a more restricted class of functions- Whittaker: field that satisfies hom. wave eq. with speed c in space-time region D can be represented in D as 3-fold integral

i) PW of the form F(k,alpha,beta)exp(ik(xsin(alpha)cos(beta)+ysin(alpha)sin(beta)+zcos(alpha)-ct)), omega in (-inf,inf), alpha in [0,pi], beta in [-pi,pi]

ii) valid if field can be expanded in 4D Traylor series convergent in Diii) does not provide method to determine PW amplitudesiv) highly restrictive condition, e.g. monocromatic spherical wave in re

gion D which does not contain the center of the wave can't be expanded for all times in Whittaker form

v) contains only homogeneous PWs travelling in all directions

- Weyl: spherical wave can be expanded in PW form over a different domain of integrationi) alpha integration over a complex contour: portion of the real axis th

en toward -i infii) exp(ik(xsin(alpha)cos(beta)+ysin(alpha)sin(beta)+zcos(alpha)-ct)) st

ill satisfies the hom. wave eq. when alpha is complex, but it is no longer constant on plane surfaces

iii) phase and amplitude are constant on different planes -> inhomogeneous plane waves

iv) spherical wave expanded as superposition of homogeneous and inhomogeneous plane waves -> angular spectrum

v) contains homogeneous PWs traveling only into one half-space

- Employ 4D-FT to obtain 2 3D integral expansionsa) PW amplitude varies with timeb) PW amplitude varies with position

- Assumptions:i) Source bounded to a region of space time: t in [0,T], r>=Rii) Cauchy conditions: field V(r,t) = 0 at t=0; time derivative V_t(r,t)

= 0 at t=0iii) Field continuous and with continuous first partial derivatives

- Under assumptions i,ii,iii the wave eq has unique solution obtained via Greenfunction techniques -> retarded Green function (2.4)


4/6

- V = int_(over source spatial region) int_(when source is active) source function * retarded Green function (2.5)- 4D-FT (better: Laplace transform) of retarded Green function: int_(line distant eps above real omega axis) int_(wavenumbers) function with pole in k=omega/c (2.6)

- function with pole in k=omega/c is equal to 4D-FT of source function *function with pole in k=omega/c (2.7)

- 4D-FT of source function computed over the region in the space-time domain where the source is active (2.8)- a) omega integration

- lead to expansion of V in terms of homogeneous PWs with time-dependentamplitude (2.12)

- in general does not satisfy hom. wave eq. even in regions of space where the source is inactive

- expansion can be found employing Duhamel's principle [19, par. 11.2]- transform from Cartesian to spherical polar coordinates of the integra

tion variables- V = real part of int_(positive real frequencies) int_(azimuth)

int_(elevation) source function * exp (2.13)- frequency and directions of propagation of the PWs are the var

iables of integration- b) integration over one Cartesian component

- leads to expansion in terms of monocromatic PWs with variable amplitudes

- in general does not satisfy hom. wave eq. because PW amplitude dependson z (spatially dependent)- phase of PWs is real only over part of the domain of integration

i) homogeneous waves: purely oscillatory, constant modulusii= inhomogeneous waves: propagate in directions perpendicular t

o z axis and have modulus which depends exponentially on z- transform to spherical polar coordinates of the integration variables

- V = real part of int_(homogeneous waves) + real part of int_(inhomogeneous waves) (2.24)

- frequency and direction of propagation are variables of integration

- Expansion a) reduces to Whittaker valid over all space for times after the sou

rce has ceased to radiate; b) reduces to Weyl valid for all times in half spacesthat do not include the source- PW amplitudes are determined in terms of the source- Relationship between the 2 expansions in regions where both are valid:

- if the source has ceased to radiate and considering only the half-space where there are no sources, inhomog. waves in Weyl are equal to a superposition of homog. waves in Whittakera) source has ceased to radiate

- V = real part of int_(wavenumbers) 4D-FT of source function * exp / k(3.2)

- PW amplitude independent on observation point -> Whittaker expansion valid for all points in space and for times t >= T

- transform to spherical coordinates to get integration variables as fre

quency and directions of propagation -> (3.3)b) half-space with no sources- V = real part of int_(real omega) \int_(kx, ky) exp * mod 4D-FT of sou

rce function / gamma (3.5)- PW amplitude independent of observation point -> Weyl expansio

n- transform to spherical polar coordinates -> (3.6)

- Whittaker representation includes homog. PWs which are not present in the Weylrepresentation: only PWs propagating into half space z>0 occur in Weyl, Pws propagating in all directions occur in Whittaker


5/6

- Compare (3.3) and (3.6): PW amplitudes are identical, integration contour differs

- in common domain of validity t>=T and abs(z)>R superposition of inhomog. PWs in Weyl is equivalent to superposition of homog. PWs in Whittaker propagating in half-space not containing observation point

- inhomog. PWs decay exponentially with abs(z) -> superposition gives decreasing contribution as abs(z) increases

- if t>=T and abs(z)>=R superposition of homog. PWs towards source givedecreasing contribution to the total field as abs(z) increases

- Extend Whittaker expansion in restricted regions of space during times while source is radiating- Obtain PW amplitudes in terms of source function and time intervals of validity- Fields with which we are dealing are not, in general, analytic -> need not admit Taylor series expansions in domains of interest- Start with retarded Green function representation for the field (2.5)

- V = int_(source spatial region) retarded source function / distance (4.1)

- a source confined in r (4.3)

- source function rho bounded in space-time produces same effectof alternative source bounded only in time- V = int_(space-time region where source is active) source function * r

etarded Green's function - int_(space-time region where source is active) sourcefunction * advanced Green function -> (4.8)

- 4D-FT of advanced Green function -> (4.9) -> (4.10) V = int_(contour C) int_(wavenumbers) source function in [0,t0] / poles * exp

- (4.11) V = real part of int_(wavenumbers) source function in [0,t0] atomega=ck * exp / k, hold at all observation points that satisfy (4.7)

- transform to spherical polar coordinates in integration variables -> (4.12)

- (4.11) and (4.12) can be used to represent the field at any observation point outside the spatial region occupied by the source, valid at all space-ti

me points that satisfy (4.7)- Field admits a PW espansion of Whittaker type in some region about every space-time point that lies outside the space-time region occupied by the source

- in restricted space-time domains some PW do not contribute to the field -> explicit expressions to determine which PW can be ignored- inhomogeneous PW in Weyl expansion and homog. PWs propagating towards the source in Whittaker do not contribute to the field in certain unbounded regions of space-time (as abs(z) increases and as t increases)

[19] P.M.Morse and H.Feshback, Methods of Mathematical Physics, vols I and II, McGraw-Hill, New York, 1953

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%M.Nieto-Vesperinas - Incoming and Outgoing Components of Source-Free Wavefields,Opt.Comm. 67, n.6, 1988

- source free fields are expressed by Whittaker representation- Whittaker representation can be generalized for outgoing radiating sources andscattered fields [3,4] (Weyl representation)- equating Whittaker with Weyl in common domain of validity we obtain that homog. PWs in Whittaker propagating towards the source contribute equally to inhomog.


6/6

PWs in Weyl- focus on the contribution of incoming fields (bandlimited to homog. components), extend angular spectrum representation in complex alpha-plane for incoming fields- a source free-field represented by Whittaker expansion can be considered as standing field = superposition of incoming and outcoming each with Weyl representation

- Source free field Usf satisfies Helmholtz eq. (1)- represented everywhere in terms of angular spectrum by means of Whitta

ker expansion (2)- Outgoing field Ud generated by source distribution rho satisfies inhomogeneousHelmholtz eq. (5)

- expressed by integral (6)- Outgoing spherical wave expressed in Weyl form (7)

- polar angle alpha is compex and takes values on the contours D+ and D-- Weyl representation for the outgoing field generated by rho -> (8), wh

ose angular spectrum is given by (9)- PW amplitude defined by (9) is valid everywhere in the complex alpha-p

lane- values of Ud are obtained in R+ or in R- by taking boundary va

lues of A along the contours D+ or D-- Incoming field satisfy (5) but are subjectied to boundary condition of behaving as a convergent spherical wave at inf -> rho given by (4) represents a sink an

d not a source- Uc given by (10)- Weyl representation of incoming spherical wave in (11) proven in Appen

dix, different contours of integration C+ and C-- angular spectrum for incoming fields due to sink rho is given in (12),

A is given in (9)- In both cases, A is the 3D-FT of the distribution rho

- if rho is real, Uc = hermitian(Ud)- inhomogeneous components contribution of contours C+ and C- are the sa

me of thos of D+ and D-

- Given rho, one obtains A from (9) and from A one can construct- Usf through Whittaker representation (2) everywhere

- Ud choosing contours D through Weyl representation (8) in R+- Uc choosing contours C thorugh Weyl representation (12) in R-- Inspecting the alpha-contours of integration in (2), (8), (12) we have Usf=(1/2i)(Ud-Uc) (13)

- the integration along the complex alpha-part of the contours cancels- A source free field is the standing field obtained by the difference between an outgoing field and an incoming field

- each of these fields are generated from the corresponding boundary values of the same angular spectrum- Contribution of the homogeneous waves of Whittaker propagating into z

Documents

Notes on plane wave expansions