System and method for single-beam internal reflection tomography

[0002] 1. Field of the Invention

[0003] This invention relates to tomography and, more particularly, to tomography wherein an image of an object is directly reconstructed with sub-wavelength resolution.

[0004] 2. Description of the Background Art

[0005] Total internal reflection microscopy (TIRM) brings to conventional microscopy the added functionality of illumination by evanescent waves. Incorporation of evanescent waves in the illuminating field is an important development for several reasons. First, the exponential decay of such waves along one direction allows for the control of the depth of penetration of the illuminating field. Second, evanescent waves may be used to resonantly excite surface plasmon modes of the sample. Finally, and perhaps most importantly, evanescent waves may be employed to supercede the Rayleigh diffraction limit of order half the wavelength, λ/2.

[0006] TIRM has been in practical use for decades. It has primarily been used as a surface inspection technique, though the sensitivity of the field to distance along the decay axis has been utilized to advantage in applications such as the measurement of distance between two surfaces. Until recently the opportunities for transverse superresolution made possible by the high spatial frequency content of the probe field have been largely over-looked. However, there have been advances in the realization of super-resolved imaging utilizing TIRM. There had been an initial direct imaging approach resulting from the marriage of standing-wave illumination techniques and TIRM achieving transverse resolution of λ/7. A later approach in which the information content of the TIRM experiment for the case of weakly-scattered fields has been determined is reported by Fischer in “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures”, published in Journal of Modern Optics, Vol. 47, No. 8, pages 1359-1374, 2000.

[0007] However, the art does not teach or suggest a technique for the reconstruction of a tomographic image of the object which is accomplished by making use of an analytic solution to an inverse scattering formulation with evanescent waves.

[0008] These shortcomings, as well as other limitations and deficiencies, are obviated in accordance with the present invention via a technique, referred to as total internal reflection tomography (TIRT,) whereby an object is probed with evanesent waves and then waves scattered from the object are processed with a prescribed mathematical algorithm to reconstruct the tomographic image.

[0009] In accordance with a broad method aspect of the present invention, a method for generating a tomographic image of an object includes: (1) probing the object with incident evanescent waves; (2) detecting waves scattered by the object, and (3) reconstructing the tomographic image of the object by executing a prescribed mathematical algorithm with reference to the incident evanescent waves and the scattered waves to generate the tomographic image with subwavelength spatial resolution.

[0010] A broad system aspect of the present invention is commensurate with the broad method aspect.

[0011] The features of the TIRT approach are at least three-fold: (i) the evanescent waves used for illumination encode on the scattered field the subwavelength structure of the scattering object. It is thus possible to obtain subwavelength resolved images of the sample as is done in other near-field techniques such as near-field scanning optical microscopy without the technical difficulties encountered with probe-sample interactions; and (ii) the results of the reconstruction are unambiguous in the sense that the relation between the scattered field and the three-dimensional structure of the sample, as described by the spatial dependence of the susceptibility, is made manifest. This is somewhat analogous to the transition from projection radiography to computed tomography.

[0012] FIG. 1 illustrates a object/scatterer under test showing input evanescent waves generated at a prism face by total internal reflection an whereby the object scatters evanescent modes to the far zone;

[0013] FIGS. 2A and 2B depict direct reconstruction results for an exemplary scatterer using different parameters;

[0014] FIG. 3 is a high-level block diagram of a system for directly reconstructing the tomographic image of the scatterer; and

[0015] FIG. 4 is a flow diagram of the methodology for directly reconstructing the tomographic image of the scatterer.

[0016] In this section we consider the scattering of evanescent waves from weakly-scattering dielectric media. In order to develop the physical ideas in a simple setting, we begin with the case of scalar waves. We then turn to the full vector theory of electromagnetic scattering thereby accounting for the effects of polarization. Note that the vector theory is essential since the scalar approximation to the scattering of electromagnetic waves is invalid when the dielectric susceptibility varies on subwavelength scales.

[0017] 1.1 Scalar Case

[0018] Begin by considering an experiment in which a monochromatic scalar field is incident on a dielectric medium with susceptibility 72(r). The field incident on the sample will be taken to be an evanescent wave which is generated by total internal reflection at the interface of two half-spaces. One half-space, taken to be z≧0, will have the vacuum index of refraction 1 while the z<0 half-space will have an index of refraction n. The scalar field U(r) obeys the reduced wave equation

[0019] ∇²U(r)+k₀²n²(z)U(r)=−4πk₀²η(r)U(r),  (1)

[0020] where k₀is the free-space wavenumber, n(z) is the z-dependent index of refraction as described above, and the support of η(r) is contained in the z≧0 half-space. We represent the total field as the sum

[0021] U=U_i+U_s,  (2)

[0022] where U_iand U_srepresent the incident and scattered fields, respectively. The incident field obeys the homogeneous equation

[0023] ∇²U_i(r)+k₀²n²(z)U_i(r)=0.  (3)

[0024] The scattered field obeys the equation

[0025] ∇²U_s(r)+k₀²n²(z)U_s(r)=4πk₀²η(r)U(r).  (4)

[0026] Equation (4) may be recast as the integral equation

[0027] U_s(r)=k₀²∫d³r′G(r,r′)U(r′)η(r′),  (5)

[0028] where G(r,r′) is the Green's function.

[0029] The Green's function G(r,r′) satisfies the equation

[0030] ∇²G(r,r′)+n²(z)k₀²G(r,r′)=−4πδ(r−r′)  (6)

[0031] and obeys the boundary conditions

[0032] G(r,r′)|_z=0=G(r,r′)|_z=0,  (7)

[0033] {circumflex over (z)}·∇G(r,r′)|_z=0{circumflex over (z)}·∇G(r,r′)|_z=0.  (8)

[0034] It may be seen that G(r,r′) admits the plane-wave decomposition $\begin{array}{cc}G\left(r,r^{\prime }\right)=\frac{i}{2\pi }\int d^2{\mathrm{qk}}_z^{-1}\left(q\right)\left\{1+R_1\left(q\right)\exp \left[2{\mathrm{ik}}_z\left(q\right)z^{\prime }\right]\right\}\exp \left[\mathrm{ik}\left(q\right)·\left(r-r^{\prime }\right)\right],& \left(9\right)\end{array}$

[0035] where R₁(q) is the reflection coefficient given by $\begin{array}{cc}{R}_1=\frac{k_z\left(q\right)-k_z^{\prime }\left(q\right)}{k_z\left(q\right)+k_z^{\prime }\left(q\right)},\mathrm{with}& \left(10\right)\\ k_z\left(q\right)=\sqrt{k_0^2-q^2}& \left(11\right)\\ k_z^{\prime }\left(q\right)=\sqrt{n^2{k}_0^2-q^2},& \left(12\right)\end{array}$

[0036] and k(q)=(q, k_z(q)). The plane wave modes appearing in Equation (7) are labeled by the transverse part of the wavevector q. The modes for which |q|≦k₀correspond to propagating waves while the modes with |q|>k₀correspond to evanescent waves.

[0037] Consider now the situation in which the incident field is an evanescent planewave of the form

[0038] U_i(r)=exp[ik(q)·r],  (13)

[0039] where q labels the transverse part of the incident wavevector. With reference to FIG. 1, if incoming wave 101 traverses prism 105 with index of refraction n, then k₀≦|q|≦nk₀. Thus k_zis imaginary with the choice of sign dictated by the physical requirement that the field decay exponentially with increasing values of z. In the first Born approximation we replace the field by the incident field in the right hand side of equation (4) and obtain the following expression for the scattered field, which occurs because object 100 frustrates total internal reflection and scatters evanescent modes to the homogenous mode that propagates to the far field:

[0040] U_s(r)=k₀²∫d³r′G(r,r′)U_i(r)η(r′).  (14)

[0041] In the far zone of the scatterer, the leading term in the asymptotic expansion of the Green's function is given by the expression $\begin{array}{cc}G\left(r,r^{\prime }\right)\sim \left\{1+R_1\left(q\right)\exp \left[2{\mathrm{ik}}_z\left(q\right)z^{\prime }\right]\right\}\frac{\exp \left({\mathrm{ik}}_{0}r\right)}{r}\exp \left[-\mathrm{ik}\left(q\right)·r^{\prime }\right],& \left(15\right)\end{array}$

[0042] where k(q) is parallel to r and |q|≦k₀. Using this result and equation (14) we find that the scattered field behaves as an outgoing homogeneous wave of the form $\begin{array}{cc}{U}_s\left(r\right)\sim \frac{\exp \left({\mathrm{ik}}_{0}r\right)}{r}A\left(q_1,q_2\right),& \left(16\right)\end{array}$

[0043] where q₁and q₂are the transverse parts of the incident and outgoing wavevectors, respectively. Here A(q₁,q₂), which is the scattering amplitude associated with the scattering of evanescent plane waves with transverse wavevector q₁into homogeneous plane waves with transverse wavevector q₂, is related to the susceptibility of the scatting object by the expression

[0044] A(q₁,q₂)=k₀²∫d³r{1+R₁(q₂)exp[2ik_z(q₂)z]}exp{i[k(q₁)−k(q₂)]·r}η(r).  (17)

[0045] We will take up the inversion of this integral equation in Section 2.1.

[0046] 1.2 Vector Case

[0047] We now turn to the vector theory of electromagnetic scattering. As before, we consider an evanescent wave which is incident on a dielectric medium with susceptibility η(r) and assume that the evanescent wave is produced at the interface of two half-spaces. We consider only nonmagnetic materials and accordingly will limit our attention to the electric field E which satisfies the reduced wave equation

[0048] ∇×∇×E(r)−k₀²n²(z)E(r)=4πk₀²η(r)E(r),  (18)

[0049] where k₀is the free space wave number and n(z) is the background index of refraction as described in Section 1.1. We decompose the field into a sum of two parts,

[0050] E=Eⁱ+E^s.  (19)

[0051] The incident field Eⁱobeys the homogeneous equation

[0052] ∇×∇×Eⁱ(r)−k₀²n²(z)Eⁱ(r)=0.  (20)

[0053] The scattered field E^sobeys

[0054] ∇×∇×E^s(r)−k₀²n²(z)E^s(r)=4πk₀²η(r)E(r).  (21)

[0055] Equation (21) may be reformulated as the integral equation

[0056] E_α^s(r)=k₀²∫d³r′G_αβ(r,r′)E_{β (}r′)η(r),  (22)

[0057] where G_αβ(r,r′) is the Green's tensor for the half-space and the summation convention over repeated indices applies and will throughout.

[0058] The Green's tensor G_αβ(r,r′) satisfies the equation

[0059] ∇×∇×G(r,r′)−k₀²n²(z)G(r,r′)=4πδ(r−r′)I  (23)

[0060] where I is the unit tensor. The Green's tensor must also satisfy the boundary conditions

[0061] {circumflex over (z)}×G(r,r′)|_z=0={circumflex over (z)}×G(r,r′)|_z=0,  (24)

[0062] {circumflex over (z)}×∇×G(r,r′)|_z=0={circumflex over (z)}×∇×G(r,r′)|_z=0.  (25)

[0063] Making use of a planewave decomposition, it may be found that $\begin{array}{cc}{G}_{\mathrm{\alpha \beta }}\left(r,r^{\prime }\right)=\frac{i}{2\pi }\int \frac{d^{2}q}{k_z\left(q\right)}{S}_{\alpha \gamma }^{-1}\left(q\right)g_{\gamma \delta }\left(q,z\right)S_{\mathrm{\delta \beta }}\left(q\right)\exp \left[\mathrm{ik}\left(q\right)·\left(r-r^{\prime }\right)\right],& \left(26\right)\end{array}$

[0064] where it is assumed that z>z′>0. Here the matrix S(q) rotates k(q) into the xz plane, or more explicitly $\begin{array}{cc}S\left(q\right)={\left|q\right|}^{-1}\left(\begin{array}{ccc}{q}_x& q_y& 0\\ -q_y& q_x& 0\\ 0& 0& \left|q\right|\end{array}\right)& \left(27\right)\\ g_{\mathrm{xx}}={\left(\frac{k_z\left(q\right)}{k_0}\right)}^2\left\{1+R_2\left(q\right)\exp \left[2{\mathrm{ik}}_z\left(q\right)z^{\prime }\right]\right\},& \left(28\right)\\ g_{\mathrm{yy}}=1+R_1\left(q\right)\exp \left[2{\mathrm{ik}}_z\left(q\right)z^{\prime }\right],& \left(29\right)\\ g_{\mathrm{zz}}={\left(\frac{\left|q\right|}{k_0}\right)}^2\left\{1-R_2\left(q\right)\exp \left[2{\mathrm{ik}}_z\left(q\right)z^{\prime }\right]\right\},& \left(30\right)\\ g_{\mathrm{zx}}=\frac{-\left|q\right|k_z\left(q\right)}{k_0^2}\left\{1+R_2\left(q\right)\exp \left[2{\mathrm{ik}}_z\left(q\right)z^{\prime }\right]\right\},& \left(31\right)\\ g_{\mathrm{xz}}=\frac{-\left|q\right|k_z\left(q\right)}{k_0^2}\left\{1-R_2\left(q\right)\exp \left[2{\mathrm{ik}}_z\left(q\right)z^{\prime }\right]\right\},& \left(32\right)\end{array}$

[0065] all other elements of g being zero, R₁(q) is defined in equation (10), and R₂(q) is given by $\begin{array}{cc}{R}_2\left(q\right)=\frac{k_z^{\prime }\left(q\right)-{\mathrm{nk}}_z\left(q\right)}{k_z^{\prime }\left(q\right)+{\mathrm{nk}}_z\left(q\right)}.& \left(33\right)\end{array}$

[0066] We take the incident field to be an evanescent plane wave with polarization E⁽⁰⁾

[0067] E_αⁱ(r)=E_α⁽⁰⁾exp [ik(q)·r],  (34)

[0068] where k(q) is the incident wavevector and k₀≦|q|≦nk₀. Within the accuracy of the first Born approximation, we find that the scattered field is given by the expression

[0069] E_α^s(r)=k₀²∫d³r′G_αβ(r,r′)E_β⁽⁰⁾exp[ik(q)·r′]η(r′).  (35)

[0070] In the far zone of the scatterer the Green's tensor assumes the asymptotic form $\begin{array}{cc}{G}_{\mathrm{\alpha \beta }}\left(r,r^{\prime }\right)\sim S_{\alpha \gamma }^{†}\left(q\right)g_{\gamma \delta }\left(q,z\right)S_{\mathrm{\delta \beta }}\left(q\right)\frac{\exp \left({\mathrm{ik}}_{0}r\right)}{r}\exp \left[-\mathrm{ik}\left(q\right)·r^{\prime }\right]& \left(36\right)\end{array}$

[0071] where k(q) lies in the direction of r. Thus we find that the sccattered field becomes $\begin{array}{cc}{E}_{\alpha }^s\left(r\right)\sim A_{\mathrm{\alpha \beta }}\left(q_1,q_2\right)E_{\beta }^{\left(0\right)}\frac{\exp \left({\mathrm{ik}}_{0}r\right)}{r},& \left(37\right)\end{array}$

[0072] where q₁and q₂are the transverse parts of the of the incident and outgoing wavevectors, respectively. A_αβ(q₁,q₂) denotes the tensor scattering amplitude which is related to the susceptibility by

[0073] A_αβ(q₁,q₂)=∫d³rw_αβ(q₂,z)exp{i[k(q₁)−k(q₂)]·r}η(r),  (38)

[0074] where

[0075] w_αβ(q,z)=k₀²S_αγ^†(q)g_γδ(q,z)S_δβ(q).  (39)

[0076] Inversion of the above integral equation will be discussed in Section 2.2.

[0077] The inverse problem consists of reconstructing the susceptibility from measurements of the scattering amplitude. To this end we will construct the pseudoinverse solution of the integral equations (17) and (38). We begin by briefly reviewing the singular value decomposition (SVD) of linear operators on Hilbert spaces. Let A denote a linear operator with kernel A(x,y) which maps the Hilbert space H₁into the Hilbert space H₂. By the SVD of A we mean a representation of the form $\begin{array}{cc}A\left(x,y\right)=\sum _n{\sigma }_n{g}_n\left(x\right)f_n^{*}\left(y\right),& \left(40\right)\end{array}$

[0078] where σ_nis the singular value associated with the singular functions f_nand g_n. The {f_n} and {g_n} are orthonormal bases of H₁and H₂, respectively and are eigenfunctions with eigenvalues σ_n²of the positive self-adjoint operators A*A and AA*:

[0079] A*Af_n=σ_n²f_n  (41)

[0080] AA*g_n=σ_n²g_n.  (42)

[0081] In addition, the f_nand g_nare related by

[0082] Af_n=σ_ng_n  (43)

[0083] A*g_n=σ_nf_n.  (44)

[0084] The pseudoinverse solution to the equation Af=g is defined to be the minimizer of ∥Af−g∥ with smallest norm. This well-defined element f⁺∈N(A)^⊥ is unique and may be shown to be of the form f⁺=A⁺g, where the pseudoinverse operator A⁺ is given by A⁺=A*(AA*)⁻¹and N(A)¹⁹⁵ is the orthogonal complement of the null space of A. The SVD of A may be used to express A⁺ as $\begin{array}{cc}{A}^{+}\left(x,y\right)=\sum _n\frac{1}{{\sigma }_n}{f}_n\left(x\right)g_n^{*}\left(y\right).& \left(45\right)\end{array}$

[0085] We now apply the foregoing to the inverse problem for scalar waves.

[0086] 2.1 Scalar Case

[0087] The integral equation (17) may be rewritten in the form

[0088] A(q₁,q₂)=∫d³rK(q₁,q₂;r)η(r),  (46)

[0089] where the scattering operator K(q₁,q₂;r) is given by

[0090] K(q₁,q₂;r)=exp[i(q₁−q₂)·ρ]κ(q₁,q₂;z),  (47)

[0091] with

[0092] κ(q₁,q₂;z)=k₀²{1+R₁(q₂)exp[2ik_z(q₂) z]}exp[i(k_z(q₁)−k_z(q₂))z]χ(q₁,q₂).  (48)

[0093] Here we have assumed that A(q₁,q₂) is measured for (q₁,q₂), in the data set and have introduced a function χ(q₁,q₂) which is unity if (q₁,q₂) ∈ and is zero otherwise. Note that specifies the available orientations of the incoming and outgoing wavevectors while χ(q₁,q₂) enforces the constraints k₀≦|q₁|≦nk₀and |q₂|≦k₀. In general, the wavevectors q₁and q₂vary continuously over all space. If, however, the transverse support of η(r) is contained in the region [−L,L]×[−L,L], then q₁and q₂may be taken to be discrete and restricted to the set Λ={(n_xπ/L,n_{yπ/L): nx},n_y=0,±1, . . . }. Thus η(r) may be expressed as the Fourier series $\begin{array}{cc}\eta \left(\rho ,z\right)=\sum _{q\in \Lambda }{c}_q\left(z\right)\exp \left(\mathrm{iq}·\rho \right),& \left(49\right)\end{array}$

[0094] where c_q(z) are appropriate coefficients.

[0095] In order to obtain the SVD of K(q₁,q₂;r) it will prove useful to introduce the following identity: $\begin{array}{cc}K\left(q_1,q_2;r\right)=\sum _{Q\in \Lambda }\exp \left(\mathrm{iQ}·\rho \right)\delta \left(Q+q_2-q_1\right)\kappa \left(Q+q_2,q_2;z\right),& \left(50\right)\end{array}$

[0096] where δ denotes the kronecker delta. Using this result, we find that the matrix elements of the operator KK* are given by $\begin{array}{cc}{\mathrm{KK}}^{*}\left(q_1,q_2;q_1^{\prime },q_2^{\prime }\right)=\sum _{Q\in \Lambda }M\left(q_2,q_2^{\prime };Q\right)\delta \left(Q+q_2-q_1\right)\delta \left(Q+q_2^{\prime }-q_1^{\prime }\right),& \left(51\right)\end{array}$

[0097] where $\begin{array}{cc}M\left(q_2,q_2^{\prime };Q\right)={\int }_0^{L}dz\kappa \left(Q+q_2,q_2;z\right){\kappa }^{*}\left(Q+q_2^{\prime },q_2^{\prime };z\right),& \left(52\right)\end{array}$

[0098] with L the range of η(r) in the {circumflex over (z)} direction. To find the singular vectors g_QQ′ of K which satisfy

[0099] KK*g_QQ′=σ_QQ′²g_QQ′;  (53)

[0100] we will make the ansatz

[0101] g_QQ′(q₁,q₂)=C_Q′(q₂;Q)δ(Q+q₂−q₁),  (54)

[0102] where Q,Q′∈Λ. Equation (51) now implies that $\begin{array}{cc}\sum _{q^{\prime }\in \Lambda }M\left(q,q^{\prime };Q\right)C_{Q^{\prime }}\left(q^{\prime };Q\right)={\sigma }_{{\mathrm{QQ}}^{\prime }}^2{C}_{Q^{\prime }}\left(q;Q\right).& \left(55\right)\end{array}$

[0103] Thus C_Q′(q₂;Q) is an eigenvector of M(Q) labeled by Q′ with eigenvalue σ_QQ′². Since M(Q) is self-adjoint, the C_Q′(q₂;Q) may be taken to orthonormal. Next, the f_QQ′ may be found from K*g_QQ′=σ_QQ′f_QQ′ and are given by $\begin{array}{cc}{f}_{{\mathrm{QQ}}^{\prime }}\left(r\right)=\frac{1}{{\sigma }_{{\mathrm{QQ}}^{\prime }}}\sum _{q\in \Lambda }\exp \left(-\mathrm{iQ}·\rho \right){\kappa }^{*}\left(Q+q,q;z\right)C_{Q^{\prime }}^{*}\left(q;Q\right).& \left(56\right)\end{array}$

[0104] It follows that the SVD of K(q₁,q₂;r) is given by the expression $\begin{array}{cc}K\left(q_1,q_2;r\right)=\sum _{Q,Q^{\prime }}{\sigma }_{{\mathrm{QQ}}^{\prime }}{f}_{{\mathrm{QQ}}^{\prime }}^{*}\left(r\right)g_{{\mathrm{QQ}}^{\prime }}\left(q_1,q_2\right).& \left(57\right)\end{array}$

[0105] The SVD of equation (57) may now be used to obtain the pseudoinverse solution to the integral equation (46): $\begin{array}{cc}{\eta }^{+}\left(r\right)=\sum _{q_1,q_2}{K}^{+}\left(r;q_1,q_2\right)A\left(q_1,q_2\right),& \left(58\right)\end{array}$

[0106] where K⁺(r;q₁,q₂) is the pseudoinverse of K(q₁,q₂;r). Using the result of equation (45), the pseudoinverse K⁺ may be seen to be given by $\begin{array}{cc}{K}^{+}\left(r;q_1,q_2\right)=\sum _{Q,Q^{\prime }}\frac{1}{{\sigma }_{{\mathrm{QQ}}^{\prime }}}{f}_{{\mathrm{QQ}}^{\prime }}\left(r\right)g_{{\mathrm{QQ}}^{\prime }}^{*}\left(q_1,q_2\right).& \left(59\right)\end{array}$

[0107] Substituting equations (54) and (56) into equation (59) and using the spectral decomposition $\begin{array}{cc}\sum _{Q^{\prime }}\frac{1}{{\sigma }_{{\mathrm{QQ}}^{\prime }}^2}{C}_{Q^{\prime }}\left(q;Q\right)C_{Q^{\prime }}^{*}\left(q^{\prime };Q\right)=M^{-1}\left(q,q^{\prime };Q\right),& \left(60\right)\end{array}$

[0108] where M⁻¹(q,q′;Q) is the qq′ matrix element of M⁻¹(Q) we obtain $\begin{array}{cc}{\eta }^{+}\left(r\right)=\sum _{q_1,q_2,q_2^{\prime }}\sum _Q\exp \left(-iQ·\rho \right)\delta \left(Q+q_2-q_1\right)M^{-1}\left(q_2,q_2^{\prime };Q\right)\times {\kappa }^{*}\left(Q+q_2^{\prime },q_2^{\prime };𝓏\right)A\left(q_1,q_2\right),& \left(61\right)\end{array}$

[0109] which is the inversion formula for scalar TIRT.

[0110] 2.2 Vector Case

[0111] The integral equation (38) may be rewritten in the form

[0112] A_αβ(q₁,q₂)=∫d³rK_αβ(q₁,q₂;r)η(r),  (62)

[0113] where the scattering operator K_αβ(q₁,q₂;r) is given by

[0114] K_αβ(q₁,q₂;r)=exp[i(q₁−q₂)·ρ]κ_αβ(q₁,q₂;z),  (63)

[0115] and

[0116] κ_αβ(q₁,q₂;z)=w_αβ(q₂;z)exp[i(k_z(q₁)−k_z(q₂))z]χ(q₁,q₂).  (64)

[0117] As in the scalar case we have assumed that A(q₁, q₂) is measured for (q₁, q₂), in the data set and have introduced the appropriate blocking function χ(q₁, q₂). The vector integral equation (62) differs from its scalar counterpart equation (46) only by a factor associated with the polarization. Evidently, by measuring a fixed component of the scattered field for a particular incident direction we see that the scalar inversion formula (61) may be used to reconstruct η(r).

[0118] We now consider the SVD for the vector case. This is a simple generalization of the scalar case. Following the previous development we find that the SVD of K_αβ(q₁,q₂;r) is of the form $\begin{array}{cc}{K}_{\alpha \beta }\left(q_1,q_2;r\right)=\sum _{Q,Q^{\prime }}{\sigma }_{{\mathrm{QQ}}^{\prime }}{f}_{{\mathrm{QQ}}^{\prime }}^{*}\left(r\right)g_{{\mathrm{QQ}}^{\prime }}^{\alpha \beta }\left(q_1,q_2\right).& \left(65\right)\end{array}$

[0119] Here the singular functions are given by

[0120] g_QQ′^αβ(q₁,q₂)=C_Q′^αβ(q₂;Q)δ(Q+q₂−q₁),  (66)

[0121] $\begin{array}{cc}{f}_{{\mathrm{QQ}}^{\prime }}\left(r\right)=\frac{1}{{\sigma }_{{\mathrm{QQ}}^{\prime }}}\sum _{q\in \Lambda }\exp \left(-iQ·\rho \right){\kappa }^{*}\left(Q+q,q;z\right)C_{Q^{\prime }}^{*}\left(q;Q\right).& \left(67\right)\end{array}$

[0122] The C_Q′^αβ(q₂; Q) are eigenfuntions of M_αβ^α′β′(q₂, q′₂; Q) with eigenvalues=σ_QQ′² $\begin{array}{cc}\sum _{q^{\prime }\in \Lambda }{M}_{\alpha \beta }^{{\alpha }^{\prime }{\beta }^{\prime }}\left(q,q^{\prime };Q\right)C_{Q^{\prime }}^{{\alpha }^{\prime }{\beta }^{\prime }}\left(q^{\prime };Q\right)={\sigma }_{{\mathrm{QQ}}^{\prime }}^2{C}_{Q^{\prime }}^{\alpha \beta }\left(q;Q\right),& \left(68\right)\end{array}$

[0123] where $\begin{array}{cc}{M}_{\alpha \beta }^{{\alpha }^{\prime }{\beta }^{\prime }}\left(q_2,q_2^{\prime };Q\right)={\int }_0^{L}dz{\kappa }_{\alpha \beta }\left(Q+q_2,q_2;z\right){\kappa }_{{\alpha }^{\prime }{\beta }^{\prime }}^{*}\left(Q+q_2^{\prime },q_2^{\prime };z\right).& \left(69\right)\end{array}$

[0124] The pseudoinverse solution to the integral equation (62) is

[0125] given by $\begin{array}{cc}{\eta }^{+}\left(r\right)=\sum _{q_1,q_2}{K}_{\alpha \beta }^{+}\left(r;q_1,q_2\right)A_{\alpha \beta }\left(q_1,q_2\right),& \left(70\right)\end{array}$

[0126] where $\begin{array}{cc}{K}_{\alpha \beta }^{+}\left(r;q_1,q_2\right)=\sum _{Q,Q^{\prime }}\frac{1}{{\sigma }_{{\mathrm{QQ}}^{\prime }}}{f}_{{\mathrm{QQ}}^{\prime }}\left(r\right){g_{{\mathrm{QQ}}^{\prime }}^{\mathrm{\alpha \beta }}}^{*}\left(q_1,q_2\right).& \left(71\right)\end{array}$

[0127] More explicitly, we have $\begin{array}{cc}{\eta }^{+}\left(r\right)=\sum _{q_1,q_2,q_2^{\prime }}\sum _Q\exp \left(-iQ·\rho \right)\delta \left(Q+q_2-q_1\right){\left(M^{-1}\left(Q\right)\right)}_{\alpha \beta }^{{\alpha }^{\prime }{\beta }^{\prime }}\left(q_2,q_2^{\prime }\right)\times {\kappa }_{{\alpha }^{\prime }{\beta }^{\prime }}^{*}\left(Q+q_2^{\prime },q_2^{\prime };z\right)A_{\alpha \beta }\left(q_1,q_2\right),& \left(72\right)\end{array}$

[0128] which is the inversion formula for vector TIRT.

[0129] In order to avoid numerical instabilty and set the resolution of the reconstructed image to be comensurate with the available data, the SVD inversion formulas must be regularized. In particular, we replace 1/a in the inversion formulas (61) and (72) by R(σ) where R is a suitable regularizer. where R is a suitable regularizer. The effect of regularization is to limit the contribution of the small singular values to the reconstruction. One way to do this is to simply cutoff all a below some cutoff σ_c. That is we set $\begin{array}{cc}R\left(\sigma \right)=\frac{1}{{\sigma }^2}\theta \left(\sigma -{\sigma }_c\right),& \left(73\right)\end{array}$

[0130] θ denoting the usual Heavyside step function. Alternatively a smooth cutoff, such as that derived by the Tikhonov method, may be employed.

[0131] To demonstrate the feasibility of the inversion using a scalar formalation, for simplicity, the reconstruction of η(r) has been obtained for a collection of spherical scatterers. This collection is representative of physical structures which may be imaged, such as a nano-structured material; the collection presents the necessary dielectric contrast to effect direct reconstruction. The forward data was calculated by considering the scattering of evanescent waves from a homogeneous sphere including multiple scattering terms by means of a partial wave expansion. Consider a sphere of radius a centered at the point r₀with refractive index n, n being related to the scattering potential or susceptibility by the expression n²=1+4πη. It may be found that $\begin{array}{cc}A\left(k_1,k_2\right)=\exp \left[i\left(k_1-k_2\right)·r_0\right]\sum _{l=0}^{\infty }\left(2l+1\right)A_l{P}_l\left({\hat{k}}_1·{\hat{k}}_2\right),& \left(74\right)\end{array}$

[0132] where A_lare the usual partial wave expansion coefficients, P_lare the Legendre polynomials, and {circumflex over (k)}=k/{square root over (k·k*)}. Since evanescent waves are considered, the argument of the Legendre polynomials in equation (74) may exceed unity. The series may nonetheless be shown to be convergent due to the rapid decay of the A_lwith increasing l.

[0133] The forward data was obtained for a collection of four spheres of radius λ/20. All scatterers are present simultaneously in the forward computation with inter-sphere scattering neglected. The incident evanescent waves were taken to have transverse wave vectors q₁on a Cartesian grid with spacing k₀/4 such that k₀≦|q|≦nk₀, n being the refractive index of the prism used to generate the incident waves. Values of q₂were also taken on the grid with spacing k₀/4 and |q₂|≦k₀, consistent with a measurement scheme in which the scattered field is measured in the far zone of the lower half-space. The spheres are arranged in two layers, one equatorial plane conincident with the z=λ/20 plane, the other with the z=λ/4 plane. In each layer, one sphere was taken to have susceptibility 4πη=0.44 (index n=1.2) and one sphere was taken to have susceptibility 4πη=0.4+0.48i (index n=1.1+0.2i). In each of the experiments, complex Gaussian noise of zero mean was added to the signal at various levels. The experiments were performed for two different prisms, namely, for FIG. 2A, the prism has an index of n=10, as might be encountered in the infrared range, and for FIG. 2B, an index n=4, as might be encountered in the visible range. The data sets are thus composed of 4980×49 points for the higher-index prism, and 752×49 data poins for the lower-index prism. The regularization parameter σ_cwas taken to be 10⁻⁶in computing the tomographic image at z λ/20 and 10⁻⁵at z=λ/4.

[0134] One can see from the reconstructions of FIGS. 2A and 2B that the real and imaginary parts of the susceptibility may be found separately and that the reconstructions are subwavelength resolved. The resolution depends both on the size of the regularization parameter, which indirectly sets the number of singular functions used in the reconstruction, and on the depth, a consequence of the fact that the probe fields decay exponentially into the sample, resulting in loss of high-frequency Fourier components of the spatial structure of the susceptibility. The tomographs at the z=λ/20 layer are more highly resolved for the higher-index prism than for the lower-index prism, but there is little difference at the z=λ/4 layer.

[0135] As depicted in high-level block diagram form in FIG. 3, system 300 is a tomography system for generating an image of an scatterer/object using measurements of scattered waves emanating from an object in response to waves illuminating the object. In particular, object 100 is shown as being under investigation. System 300 is composed of: source 320 for probing the object 100 through prism 105; data acquisition detector 330 for detecting the scattering data corresponding to the scattered waves from object 100 at one or more locations proximate to object 100; position controller 340 for controlling the locations of detectors 330 and sources 320; and computer processor 350, having associated input device 360 (e.g., a keyboard) and output device 370 (e.g., a graphical display terminal). Computer processor 350 has as its inputs positional information from controller 340 and the measured scattering data from detector 330. Even though the scatterer is shown as being present in FIG. 3, actually two sets of measurements are obtained, namely, one set with the scatterer removed, and another set with the scatterer present, to provide the necessary data for image reconstruction, as detailed above.

[0136] Computer 350 stores a computer program which implements the direct reconstruction algorithm; in particular, the stored program processes the measured scattering data to produce the image of the object or object under study using a prescribed mathematical algorithm. The algorithm is, generally, determined with reference to an integral operator relating the scattering data to the forward scattering operator as expressed by integral equation (17) or equation (38).

[0137] The methodology carried out by the present invention is set forth in high-level flow diagram 400 of FIG. 4 in terms of the illustrative system embodiment shown in FIG. 3. With reference to FIG. 4, the processing effected by block 410 enables source 320 and data acquisition detector 330 so as to measure the scattering data emanating from scatterer 100 due to illuminating waves from source 320. These measurements are passed to computer processor 350 from data acquisition detector 330 via bus 331. Next, processing block 420 is invoked to compute the kernel expressed by equation (17) or equation (38), which may for efficiency be pre-computed and stored. In turn, processing block 430 is operated to execute the reconstruction algorithm set forth in equation (17) or equation (38), thereby determining the scattering potential η(r). Finally, as depicted by processing block 440, the reconstructed tomographic image corresponding to η(r) is provided to output device 370 in a form determined by the user; device 370 may be, for example, a display monitor or a more sophisticated three-dimensional display device.

[0138] Although the present invention have been shown and described in detail herein, those skilled in the art can readily devise many other varied embodiments that still incorporate these teachings. Thus, the previous description merely illustrates the principles of the invention. It will thus be appreciated that those with ordinary skill in the art will be able to devise various arrangements which, although not explicitly described or shown herein, embody principles of the invention and are included within its spirit and scope. Furthermore, all examples and conditional language recited herein are principally intended expressly to be only for pedagogical purposes to aid the reader in understanding the principles of the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Moreover, all statements herein reciting principles, aspects, and embodiments of the invention, as well as specific examples thereof, are intended to encompass both structural and functional equivalents thereof. Additionally, it is intended that such equivalents include both currently know equivalents as well as equivalents developed in the future, that is, any elements developed that perform the function, regardless of structure.

[0139] In addition, it will be appreciated by those with ordinary skill in the art that the block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the invention.