Calculation of the image of an arbitrary vectorial electromagnetic field

P. R. T. Munro; P. Török

doi:10.1364/OE.15.009293

1. Introduction

In high numerical aperture (NA) imaging systems the Debye-Wolf integral [1] (see Eq. (4)) is commonly used to calculate the image of scattered light. The Debye-Wolf integral operates on the geometrical optics approximation to a field [2], which may be equated with the far-field limit, an infinite distance away from a given field. In this limit, field components become tangential to a spherical surface centered on the field source. Thus, an important step in calculating the image of a scattered field is to calculate the scattered far-field. It has previously been shown [3, 4, 5] how to calculate the image of harmonically oscillating electric and magnetic dipoles using the Debye-Wolf integral. Calculating the image of a harmonically oscillating dipole is relatively straight forward owing to the simplicity of the dipole far-field expressions. Calculating the image of an arbitrary field is however somewhat more complicated. In many cases it is possible to first perform a far-field transformation on an arbitrary field using vectorial far-field transformations [6, 7]. Note that these transformations do not reveal the far-field at infinity but may be used to calculate an approximation to this field. Then the field’s image is found by application of the Debye-Wolf integral. This solution is however not suited to all cases. For example, this solution fails to take advantage of an imaging system’s limited field of view and is also computationally demanding in the case where a field is known within a stratified medium.

Another approach to this problem is to decompose an arbitrary field into a superposition of magnetic-dipole waves, the images of which may be calculated with relative ease. In this way, existing imaging theory may be applied to calculate the image of fields which previously could not be calculated. The remainder of the paper is arranged as follows. It is first shown how to decompose an arbitrary far-field into a superposition of magnetic-dipole waves. It is then shown how to calculate the image of these magnetic-dipole waves in the absence and presence of a stratified medium. Then it is shown how to combine these techniques to calculate the image of an arbitrary field. The paper concludes with an analysis of the accuracy and performance of this method.

Note that in this paper we are concerned with the image of arbitrary electric fields. The image of arbitrary magnetic fields may however be derived in a similar way by using the e-theory of diffraction rather than the m-theory of diffraction presented below.

2. The m-theory of diffraction

Section (2) is a summary of results derived by Karczewski and Wolf [9] and in a different manner by Török [10] and Török and Sheppard [2]. Note that throughout this paper the exp(-ιωt) convention is assumed. The result of Török [10] can be stated by considering the geometry of Fig. 1. Two infinite half spaces are separated by an infinite planar surface S₀. Without loss of generality assume that all sources and sinks of radiation are situated in the half space which extends to z=-∞. The field at any point P, situated in the half space which extends to z=∞,

is given by:

E (P) = \frac{1}{2 π} \nabla_{p} \times \iint_{S_{0}} n \times E (Q) \frac{\exp (ι k_{0} r)}{r} d S

where Q=(x_Q,y_Q,z_Q) is a point on S₀, n is a surface normal directed toward the half space containing P, r is the distance from Q to P and ∇_p is used to emphasise that the curl operates on the coordinates of the observation point P. Also note that the field E must satisfy the vectorial radiation condition [11] in order for Eq. (1) to be valid. A similar result is also derived by Karczewski andWolf [9] for the case where S ₀ is not infinite and represents an aperture in a screen. This however represents a very different problem where boundary conditions of the Kirchhoff type must be assumed. The result of Török [10] however avoids making this assumption. It is from the result of Karczewski and Wolf [9] that this theory gained the name of the m-theory of diffraction.

Fig. 1. Configuration for statement of the m-theory of diffraction according to Török [10].

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The result in Eq. (1) may be expanded by applying the curl operator within the integral operator. The kernel of the integral is given by:

\nabla_{p} \times [n \times E (Q) \frac{\exp (ι k_{0} r)}{r}] = \frac{\exp (ι k_{0} r)}{r} (\nabla_{p} \times (n \times E (Q))) - (n \times E (Q)) \times \nabla_{p} [\frac{\exp (ι k_{0} r)}{r}]

where r=(x_P-x_Q, y_P-y_Q, z_P-z_Q) is a vector pointing from Q to P. Since only the far-field is required for imaging, the 1/r ² term may be omitted from Eq. (2) yielding the final expression for the electric far-field as:

E (P) = \frac{1}{2 π} \iint_{S_{0}} (r \times (n \times E (Q))) \frac{\exp (ι k_{0} r)}{r^{2}} ι k_{0} d S

This reveals that each point Q can be considered the source of a spherical wave with polarisation given by r×(n×E(Q)). Note that since r·(r×(n×E(Q)))=0, these waves have only field components tangential to the spherical wavefront as expected in the far-field. Note that a result similar to Eq. (3) is derived for use in the discrete-dipole approximation [8].

The m-theory of diffraction, as expressed by the integral in Eq. (3), decomposes an arbitrary electric field into a coherent superposition of spherical waves. Interestingly, the spherical wave is equivalent to the electric field due to a harmonically oscillating magnetic dipole of moment n×E(Q). Thus, in order to find the image of an arbitrary field we must first calculate the image of an equivalent magnetic-dipole (EMD). Then the image of an arbitrary field may be found by superposition of the images of EMDs as prescribed by the m-theory of diffraction. It now remains to calculate the field at the detector due to an EMD. The coordinate systems and optical systems employed are shown in Fig. 2. Transmission and reflection cases will be considered separately.

Fig. 2. Coordinate system used to calculate image of an EMD in transmission (top) and reflection mode (bottom).

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3. The image of an equivalent magnetic dipole

It should be noted that the method used to calculate the image of an EMD is similar to the established method of calculating the image of an electric dipole by Sheppard and Wilson [3] and later by Török et al [4, 5]. For clarity, in Section (3) it is assumed that the detector and dipole spaces are homogeneous with equal refractive indices. The wavenumber in each of these spaces is k. The more complex cases of unequal refractive indices and stratified media is considered in Section (5).

3.1. Reflection

Consider initially an EMD of moment n×p positioned at the origin of coordinate system T^r ₁ as shown in Fig. 2(reflection). The Debye-Wolf integral will be employed to calculate the image of the EMD and it is presented here for reference assuming the optical system depicted in Fig. 2(reflection) is being employed:

E (r_{d}) = - \frac{ι k f_{2}}{2 π} \iint_{Ω} \frac{ε_{0} (s_{2 x}, s_{2 y})}{s_{2 z}} \exp (ι k s_{2} \cdot r_{d}) d s_{2 x} d s_{2 y}

where ε₀ is the geometrical optics approximation to the field along typical ray s₂=(s_2x, s_2y, s_2z) in the detector space of the imaging system. r _d is the position in the detector space where the field is to be evaluated, f₂ is the focal length of the detector lens, k is the wavenumber of the light in the detector space and Ω is the solid angle subtended by the convergent spherical wavefront in the detector space. Before proceeding it is necessary to note that typical rays s1 and s₂ are described using spherical coordinates as s₁=(sinθ ₁cosφ1, sinθ ₁sinϕ ₁, cosθ ₁) and s₂=(sinθ ₂cosφ ₂, sinθ ₂sinϕ ₂, cosθ ₂) respectively.

The geometrical optics field emergent from the detector lens may be found by applying the generalised Jones matrix formalism [12] as:

ε_{0} (s_{2}) = ε_{0} (s_{1}) = \sqrt{\frac{\cos θ_{2}}{\cos θ_{1}}} ℝ^{- 1} (ϕ_{1}) L (π - θ_{2}) L (π - θ_{1}) ℝ (ϕ_{1}) ε_{eq}

where ε_eq=-s₁×(n×p) and the generalised Jones matrices 𝕃 and ℝ are given by:

ℝ = [\begin{matrix} \cos ϕ & \sin ϕ & 0 \\ - \sin ϕ & \cos ϕ & 0 \\ 0 & 0 & 1 \end{matrix}]

L = [\begin{matrix} \cos θ & 0 & \sin θ \\ 0 & 1 & 0 \\ - \sin θ & 0 & \cos θ \end{matrix}]

note also that the $\sqrt{\cos θ_{2} ⁄ \cos θ_{1}}$ term arises due to the lenses satisfying the sine condition [13]. The angles θ ₁ and θ ₂ are related by sinθ ₁/sinθ ₂=f₂/f₁=β and ϕ ₁ and ϕ ₂ according to ϕ ₁=ϕ₂+π. Since it will be assumed that the plane S ₀ is defined by z=z_dp, the surface normal n will be equal to the unit vector -k̂, parallel to the negative z-axis. Expanding Eq. (5) then gives:

ε_{0, x} = \sqrt{\frac{\cos θ_{2}}{\cos θ_{1}}} [\frac{p_{x}}{2} [(\cos θ_{2} + \cos θ_{1}) + (\cos θ_{2} - \cos θ_{1}) \cos 2 ϕ_{1}] + \frac{p_{y}}{2} (\cos θ_{2} - \cos θ_{1}) \sin 2 ϕ_{1}]

In many cases however, the EMD will be displaced along the axial direction. This may be dealt with by including a phase term, exp(-ιkz_dp cos θ₁), in the expression for ε₀ as derived in Section (4). Then, substituting Eq. (8) into Eq. (4) gives:

E (r_{d}) = - \frac{ι f_{2}}{λ} \int_{π - α_{2}}^{π} \int_{0}^{2 π} \sqrt{\frac{\cos θ_{2}}{\cos θ_{1}}} ε_{0} (θ_{2}, ϕ_{2} + π) \exp (ι k r_{d} \sin θ_{2} \cos (ϕ_{2} - ϕ_{d})) \times

where α₂ is the semi-convergence angle of the detector lens. After expanding this and using the analytic result[1]:

\int_{0}^{2 π} \exp (ι q ϕ) \exp (ι ρ \cos (ϕ - γ)) d ϕ = 2 π ι^{q} J_{q} (ρ) \exp (ι q γ)

the following expression is obtained for the field at position r _d=(r_dt cosφ_d,r_dt sinφ,z_d) in coordinate system T^r ₂ as shown in Fig. 2(reflection):

E_{x} = p_{x} (I_{0, r}^{eq} + I_{2, r}^{eq} \cos 2 φ_{d}) + p_{y} I_{2, r}^{eq} \sin 2 φ_{d}

where a constant multiplier of ιf₂π/λ has been omitted and the I^eq functions are defined as:

I_{0, r}^{eq} = \int_{π - α_{2}}^{π} \sqrt{\frac{\cos θ_{2}}{\cos θ_{1}}} \sin θ_{2} (\cos θ_{1} + \cos θ_{2}) J_{0} (k r_{d} \sin θ_{2}) \exp (ι k (z_{d} \cos θ_{2} - z_{dp} \cos θ_{1}))) d θ_{2}

3.2. Transmission

The transmission case is similar to that of reflection. In this case, the EMD is positioned in reference frame T^t₁ of Fig. 2(transmission). Following a similar procedure to that of Section (3.1), it can be shown that the expressions for the field in transmission are the same as those for reflection except that the I^eq functions must be redefined as:

I_{0, t}^{eq} = \int_{0}^{α_{2}} \sqrt{\frac{\cos θ_{2}}{\cos θ_{1}}} \sin θ_{2} (\cos θ_{1} + \cos θ_{2}) J_{0} (k r_{d} \sin θ_{2}) \exp (ι k (z_{d} \cos θ_{2} - z_{dp} \cos θ_{1}))) d θ_{2}

It should also be noted that in the transmission case, n is given by k̂ instead of -k̂ as was the case for reflection. This leads to an additional negative sign in the field expressions which is accounted for by the order of integration in Eqs. (13). The primary difference between the reflection and transmission cases is the limit of integration over θ in the definition of the I^eq functions. This arises because the typical rays in each case oppose each other. The consequences of this are discussed further in Section (5.3).

4. Finding the image of an array of equivalent magnetic dipoles

The expressions presented in Eqs. (11)-(13) are valid for an on-axis EMD. However, finding the image of an arbitrary field requires finding the image of off-axis EMDs also. This may be done relatively simply if it is assumed that the optical system under study is shift invariant. It is possible to calculate the effect of displacing an EMD away from the focus by considering the phase of the resulting dipole wave on the Gaussian reference sphere. Consider a typical ray s1 for both in-focus on-axis and displaced EMDs as shown in Fig. 3. Note that in this diagram, y_dp and z_dp have been assumed to be zero for clarity. The ray originating at a dipole with position (x_dp, y_dp, z_dp) propagates a distance different to that of a dipole at the origin by amount Δr as shown in Fig. 3. Thus the phase term to be included in the dipole field expression must be exp(-ιkΔr). Δr may then be found as the solution to ((x_dp, y_dp, z_dp)-Δrs ₁)·_s ₁=0 which has solution: Δr=sinθ ₁ (x_dpcosϕ₁+y_dpsinϕ₁ )+z_dpcos θ ₁. Then, using the relationships sinθ ₁/sinθ ₂=f₂/f₁=β and ϕ₁=ϕ ₂+π it is possible to write the phase term of the Debye-Wolf integral (Eq. (4)) as:

\exp (- ι k Δ r) \exp (ι k s_{2} \cdot r_{d})

Thus, as long as the phase term exp(-ιkz_dpcos θ ₁) is included in Eqs. (12) and (13), the image of an EMD of moment n×p at position (x_dp,y_dp,z_dp) may be represented by:

E(r _d)=E _oa(n×p,r _d+βr_dpt)

where E _oa(n×p, r_d) is the field at detector position r _d for an on-axis EMD of moment n×p at axial position z_dp and r_dpt=(x_dp,y_dp,0).

Fig. 3. Notation for calculating the image of a laterally displaced dipole.

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It now remains to apply this result to imaging an arbitrary field E(Q) in Eq. (3). This involves finding the image of each EMD involved in the integral of Eq. (3). Assume then that the field due to an on-axis EMD of moment n×p in transmission or reflection is given by E _oa,t/r(n×p,r _d), where the subscript t/r denotes transmission (t) or reflection (r). The field at the detector due to an arbitrary field E ⁱ defined on a plane S ₀, in the vicinity of the first principal focus of the objective is given by:

E_{t ⁄ r} (r_{d}) = \frac{ι k_{0}}{2 π} {\int \int}_{S_{0}} E_{o a, t ⁄ r} (n \times E^{i} (Q), r_{d} + β q) d S

where q is a vector pointing from the intersection of S ₀ and the z-axis to point Q, the location of a particular EMD. In practice the plane S ₀ is chosen to be large enough so that “most” of the power scattered by the scattering object propagates through it unperturbed. A degree of approximation is inevitable here. A further approximation is applied is to the integral in Eq. (15) as it is evaluated as a sum over discrete EMDs. This is necessary for practical implementation.

5. The image of an equivalent magnetic dipole in a stratified medium

It is frequently necessary to calculate the image of a field distribution which is known within a stratified medium. By calculating the image of an EMD in a stratified medium, the result of Section (4) may be used to perform this task. Firstly the case of reflection is considered, then the case of transmission is inferred from the former case. This section (Section (5)) draws upon the method of Török [14] and Haeberlé et al. [15] for calculating the image of an electric dipole in a stratified medium. The image of an EMD in a stratified medium has not previously been derived.

5.1. Reflection

Consider an EMD of moment -k̂×p, where -k̂ is a unit vector directed along the negative z-axis, embedded in a stratified medium as shown in Fig. 4. The positions of the interfaces are labelled sequentially as -h₁, -h₂, …, -h_N-1 and the refractive indices of the materials are labelled sequentially as n₁, n₂, …, n_N. It is assumed that the EMD is embedded in material 1. Ray angles are labelled as shown in Fig. 4. The refractive index of the detector space is assumed to be n_d. The image of the EMD is found using a similar procedure to that of Section (4). In particular, we begin by writing the geometrical optics approximation to the field emergent from the detector lens using the generalised Jones matrix formalism as:

ε_{d} = \sqrt{\frac{\cos θ_{d}}{\cos θ_{1}}} ℝ^{- 1} (ϕ_{1}) L (π - θ_{d}) I_{E}^{- (N - 1)} L (π - θ_{N}) ℝ (ϕ_{1}) ε_{dp}

where this time, E_dp is the far-field due to an EMD given as ε_dp=-s_N×(-k̂×p), s_N is a typical ray in material N and [16, 17]

I_{E}^{(N - 1)} = [\begin{matrix} T_{p}^{(N - 1)} & 0 & 0 \\ 0 & T_{s}^{(N - 1)} & 0 \\ 0 & 0 & T_{p}^{(N - 1)} \end{matrix}]

where T^(N-1)_s,p are the transmission coefficients of the stratified medium defined as:

T_{m}^{(N - 1)} = \frac{t_{m}^{(N - 1)} \prod_{j = 1}^{N - 2} t_{m}^{(j)} \exp (ι β_{j + 1})}{D_{m}^{(N - 1)}}

where β_l=k_l(h_l-1-h_l)cosθ_l, m is either p or s and denotes polarisation. D^(N-1)_m is defined as:

D_{m}^{(2)} = 1 + r_{m}^{(1)} r_{m}^{(2)} \exp (2 ι β_{2})

and t⁽¹⁾_s, t⁽¹⁾_p, r⁽¹⁾_s, r⁽¹⁾_p are the Fresnel transmission and reflection coefficients respectively and are defined as [19]:

\begin{matrix} t_{s}^{(l)} = \frac{2 n_{l + 1} \cos θ_{l + 1}}{n_{l + 1} \cos θ_{l + 1} + n_{l} \cos θ_{l}} & t_{p}^{(l)} = \frac{2 n_{l + 1} \cos θ_{l + 1}}{n_{l} \cos θ_{l + 1} + n_{l + 1} \cos θ_{l}} \\ r_{s}^{(l)} = \frac{n_{l + 1} \cos θ_{l + 1} - n_{l} \cos θ_{l}}{n_{l + 1} \cos θ_{l + 1} + n_{l} \cos θ_{l}} & r_{p}^{(l)} = \frac{n_{l} \cos θ_{l + 1} - n_{l + 1} \cos θ_{l}}{n_{l} \cos θ_{l + 1} + n_{l + 1} \cos θ_{l}} \end{matrix}

The following quantities must be defined before proceeding:

s_d=(cosϕ_dsinθ_d, sinϕ_dsinθ_d, cos θ_d) r_d=(x_d,y_d,z_d)

s_N=(cosϕ_Nsinθ_N,sinϕ_Nsinθ_N, cosθ_N) r_dt=(x_d, y_d)=r_dt(cosφ_d,sinφ_d)

s_dt=(cosϕ_dsinθ_d,sinϕ_d sinθ_d) r_dp=(x_dp,y_dp,z_dp)

s_Nt=(cosϕ_Nsinθ_N,sinϕ_N sinθ_N)

where s_d is a typical ray in the detector space, the dipole is positioned at r _dp and the field is calculated at r d=(r_dt cosφd, r_dt sinφd,z_d). Then substituting Eq. (16) in the Debye-Wolf equation, modified for the case of a dipole embedded in a stratified medium, given as [14, 15]:

E_{d} (r_{d}) = - \frac{ι k_{d}}{2 π} {\int \int}_{Ω_{d}} \frac{ε_{d} (s_{d})}{s_{dz}} \exp (ι k_{d} r_{d} \cdot s_{d}) \exp (- ι k_{N} r_{dp} \cdot s_{N}) \exp (ι k_{0} Ψ_{i}) d s_{dx} d s_{dy}

where Ψ_i=h_N-1n_Ns_Nz-n₁h₁s_1z. The electric field, for an on-axis dipole, is then given by Eqs. (11) with the I^eq functions defined as:

I_{0, r}^{eq, s} = \int_{π - α_{d}}^{π} \sqrt{\frac{\cos θ_{d}}{\cos θ_{1}}} \sin θ_{d} (T_{s} \cos θ_{N} + T_{p} \cos θ_{d}) J_{0} (k_{d} r_{dt} \sin θ_{d}) \times

The results of Section (4) may then be employed to find the field of an off-axis EMD as well as an array of EMDs.

Fig. 4. Labelling scheme for calculating the image of an EMD embedded in a stratified medium in reflection (left) and transmission (right).

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5.2. Transmission

The field for the case of transmission follows from that of reflection. The stratified medium and nomenclature are presented in Fig. 4.

The field is given by Eq. (11) by redefining the I^eq,s functions as:

I_{0, t}^{eq, s} = \int_{0}^{α_{d}} \sqrt{\frac{\cos θ_{d}}{\cos θ_{1}}} \sin θ_{d} (T_{s} \cos θ_{N} + T_{p} \cos θ_{d}) J_{0} (k_{d} r_{dt} \sin θ_{d}) \times

5.3. Differences between reflection and transmission cases

It is possible to derive an association between Eqs. (22) and Eqs. (23) by making the substitutions $\bar{θ_{N}} = π - θ_{N}$ , $\bar{θ_{1}} = π - θ_{1}$ and $\bar{θ_{d}} = π - θ_{d}$ into Eqs. (22). Then, by noting that cos(π−θ)=−cos θ and sin(π-θ)=sinθ, it is possible to show that:

I_{0, r}^{eq, s} = I_{0, t}^{eq, s^{*}} I_{1, r}^{eq, s} = - I_{1, t}^{eq, s^{*}} I_{2, r}^{eq, s} = I_{2, t}^{eq, s^{*}}

Where * denotes complex conjugation. Substitution of these expressions into Eqs. (11) shows that when both the EMD and observation plane are positioned in-focus, the image of an EMD in reflection is simply that of the same EMD in transmission rotated about the optical axis by π. It may be shown that this asymmetry holds also for the image of an electric dipole.

6. Implementation

As Section (7) will show, it is necessary to consider a large number of EMDs in order to accurately calculate the image of an arbitrary field. This has the potential to be a very time consuming computational task especially when one considers that numerical evaluation of the I^eq,s integrals becomes more difficult as aberrations are introduced. The I^eq,s functions were evaluated using Gaussian quadrature integration. When an interface is present, a very high order of integration may be required to accurately evaluate the I^eq,s functions. This process may be sped up by creating lookup tables for I^s,eq₀, I^s,eq₁ and I^s,eq₂ as a function of r_dt. The maximum required value of r_dt may be calculated with knowledge of how large the detector is and how wide the EMD plane is. Linear interpolation is used to interpolate the values of the I^eq,s functions for all required values of r_dt. This results in a significant speed up as interpolation requires very few operations compared to performing numerical integration over several hundred sample points as is often required when stratified media are employed [18].

We conclude this section with a comment on the efficiency of the method. The alternative to this method is to perform a far-field transformation of the scattered field followed by an application of the Debye-Wolf integral. The far-field transformation requires a surface integral to be evaluated for each sample point within the pupil of the lens. The kernel of this surface integral, which includes a Green’s function, must be calculated for each pupil-observation and surface point pair. In the proposed method we need only integrate over a finite plane and we avoid calculating a Green’s function by shifting the image field of off-axis EMDs as shown in Section (4).

7. Evaluation and analysis

The method of imaging an arbitrary electric field by EMD decomposition is tested using the field of a harmonically oscillating electric dipole. The image of such a dipole may be calculated directly using theory derived by Török et al [4, 5]. The field of the electric dipole may also be used to calculate the moments of a collection of EMDs, situated on a plane in the vicinity of the focus of the objective lens, which may be used to calculate the image of the electric dipole using EMD decomposition. The accuracy of the field calculated using EMD decomposition may thus be assessed by comparison with the field calculated directly.

The configuration used is shown in Fig. 5. A dipole of moment p is positioned at the origin of coordinate system T₁. The image is to be calculated at the plane z=0 in reference frame T₂. Two factors most significantly affect the accuracy of the EMD decomposition method. One is the width of the plane of EMDs and the other is the density of EMDs on the plane. Tests were performed whereby each of these was varied.

The Rayleigh resolution criteria was used as the basis for the spacing of EMDs. It is reasonable to assume that the EMD separation must be at most less than the coherent two-point resolution of the imaging system.

In order to give the size of the EMD plane a physical meaning, it is useful to consider the total power radiated by a harmonically oscillating dipole of moment p and in particular, how much of it propagates through the EMD plane. The total amount of power radiated by a dipole is given by [19, 20]:

P = \frac{c^{2} Z k^{4}}{12 π} {∣ p ∣}^{2}

where c is the speed of light in the medium containing the dipole, $Z = \sqrt{μ ⁄ ϵ}$ is the material impedance and k is the wavenumber of the radiation in the medium containing the dipole. By using a moment p with components in phase and in the xy-plane, it may be assumed that power radiates equally into two half spaces separated by the plane z=0 in coordinate system T₁. It is then possible to calculate the power flowing through the EMD plane as a proportion of the total power radiating into the half space containing the plane. The power flowing through the EMD plane may be calculated according to:

P_{plane} = \iint_{S_{0}} \frac{1}{2} ℜ {E \times H^{*}} \cdot (- \hat{k}) d S

where ℜ is the real part, E and H are the electric and magnetic fields respectively due to the electric dipole, -k̂ is the unit vector directed along the negative z-axis and therefore the normal to the EMD plane situated on S ₀ and * denotes complex conjugation. This was evaluated numerically for each EMD plane. The electromagnetic field due to a harmonically oscillating dipole is given by [19, 20]

H = \frac{c k^{2}}{4 π} (\hat{n} \times p) \frac{e^{ι k r}}{r} (1 - \frac{1}{ι k r})

where n̂ is a unit vector directed from the dipole to the point of observation, and r is the distance from the dipole to the point of observation.

Fig. 5. Configuration for calculating the image of an harmonically oscillating electric dipole directly (a) and using EMD decomposition (b).

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Each test was performed using a differently configured stratified medium. In each case the dipole was positioned at the origin with moment p=(0,1,0). The stratified medium had a single interface with h ₁=2µm (see Fig. 4). Each of the three tests used n₂=1.6 and tests one and two used n ₁=1 whilst the third used n ₁=1.6. The second test employed aberration compensation.

A free space wavelength of λ₀=500nm, a dry .95 NA objective and a transverse magnification of β=100 was employed in each test. The EMD separation was varied from the Rayleigh limit to 10% of this. The EMD plane was square, situated at z=-λ₀/2 and its width was varied from 1µm to 50µm. EMDs were arranged uniformly along lines parallel to the x-and y-axes. The spacing was measured along each of these axes. Fig. 6 shows a plot of the proportion of power flowing through the EMD plane as a function of its width. Note that since the power flowing through the EMD plane is a monotonically increasing function of plane width, in what follows radiated power and plane width will be used interchangeably.

Fig. 6. Plot showing how the proportion of power flowing through the EMD plane varies with the width of the plane.

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As a starting point it is useful to show what the electric field looks like at the detector for each of the three test cases. Fig. 7 shows the electric field components for each test. As is expected, the field in test one is significantly aberrated due to the dipole being embedded in 2µm of plastic with refractive index 1.6. The differences between test two and test three are due primarily to Fresnel reflection at the interface. In order to assess the accuracy of the EMD approach to imaging an arbitrary field, an aggregate error metric defined as:

ϵ_{E} = \frac{\sum_{i = 1}^{N} {∣ E^{eq} (r_{i}) - E^{dir} (r_{i}) ∣}^{2}}{\sum_{i = 1}^{N} {∣ E^{dir} (r_{i}) ∣}^{2}}

where N is the number of points where the field is sampled, E^eq is the field calculated using EMD decomposition, E ^dir is the field calculated directly and ri is the position of the ith field value. An additional, component specific, aggregate error measure is also defined here as:

ϵ_{E}^{s} = \frac{\sum_{i = 1}^{N} {∣ (E^{eq} (r_{i}) - E^{dir} (r_{i})) \cdot s ∣}^{2}}{\sum_{i = 1}^{N} {∣ E^{dir} (r_{i}) \cdot s ∣}^{2}}

where s is a unit vector parallel to the component for which the aggregate error is to be calculated. All total and component aggregate errors were calculated over the part of the detector plane depicted in Fig. 7.

As explained previously, a number of simulations were performed for each test to determine the effect of EMD density and size of the EMD plane. Fig. 8(a) shows how the aggregate error in test one varies with the density of EMDs. Each line represents a different sized EMD plane. In the region of the plot where the curves are reasonably flat, the individual curves are all ordered such that the aggregate error is inversely proportional to EMD plane size. This is indicated by a down arrow on the plot. This plot shows that an EMD density of six times that of the Rayleigh two point resolution is sufficient to achieve the optimal accuracy for a given dipole plane size. Thus there is little reason to increase dipole density beyond this. This is fortunate since computation time is proportional to the total number of EMDs. This result was consistent across tests one, two and three and so plots for tests two and three have not been included.

Fig. 7. Electric field components at the detector for tests one (top), two (middle) and three (bottom). Components were normalised by the peak magnitude of the y-component.

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Fig. 8(b) shows a plot of the aggregate errors for each of the Cartesian field components for varying sizes of the EMD plane for test one. An EMD density of six times that of the Rayleigh two point resolution was used. The first point to note from this plot is that the aggregate error doesn’t decrease appreciably until the time averaged power flowing through the EMD plane reaches 95% of that available. After this the aggregate errors drop off rapidly. It is also noticeable that the x-component has a greater aggregate error than the other components across all EMD plane dimensions. This can be understood by looking at a plot of the I^eq,s functions actually used to compute the fields. These are plotted in Fig. 9 as a function of r_dt. Fig. 9 shows that the I^s,eq₂ function has the greatest oscillations relative to its peak magnitude and the envelope of these oscillations stays larger, as a proportion of the peak magnitude. The x-field component is derived entirely from the I^s,eq₂ function and so EMDs positioned further from the axis have a more significant impact upon the x-field component than in the other field components. It thus takes a wider EMD plane for the x-field component to converge. The I^s,eq₂ has only a small effect on the accuracy of the y-field component as it is dominated by the I^s,eq₀ function which settles much more quickly. The same phenomenon is present in tests two and three.

The next point to consider is the relationship between the aggregate error of the three tests as a function of the EMD plane width. This is plotted in Fig. 10 and perhaps surprisingly, the aggregate error does not vary significantly between the three tests. This may be expected because the field in test one is significantly aberrated. The effect of the aberration is to give the I^eq,s functions a longer tail. Fig. 11 shows plots of the magnitude of the I^eq,s functions for tests one and three. As can be seen, the I^eq,s functions from test one take longer to decay to 0 and do so in a smoother, more gradual fashion. This is why the aggregate error for test one doesn’t decrease as rapidly in the region of 95 % time averaged power as it does for the other tests.

Fig. 8. Plots showing (a) a comparison of aggregate error as a function of the EMD density for test one. Each curve represents a different EMD plane width and (b) component wise aggregate errors for test one as a function of the total power propagating through the EMD plane.

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Fig. 9. Plot of I^eq,s functions for test one, normalised collectively (left) and normalised individually (right).

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8. Conclusions

This paper presented the new result of how the image of an arbitrary field may be calculated using the Debye-Wolf integral. The method was derived by employing the m-theory of diffraction. The method was extended to include the case where the field to be imaged is known within a stratified medium.

Finally, as a guide, in the case of a dipole field, it was shown that an EMD spacing of six times the Rayleigh two-point resolution provides sufficient accuracy in the calculated field. Also, a wide EMD plane is very important. In particular, the EMD plane should be wide enough such that at least 95% of the radiated power available propagates through it.

Fig. 10. Comparison of total aggregate error for tests one, two and three as a function of the total power propagating through the EMD plane.

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Fig. 11. Comparison of magnitude of I^s,eq₀, I^s,eq₁ andI^s,eq₂ for test one and test three.

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Acknowledgements

Part of this work was supported by the Engineering and Physical Sciences Research Council of the United Kingdom grant EP/C528905/1.

References and links

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Calculation of the image of an arbitrary vectorial electromagnetic field

Abstract

1. Introduction

2. The m-theory of diffraction

3. The image of an equivalent magnetic dipole

3.1. Reflection

3.2. Transmission

4. Finding the image of an array of equivalent magnetic dipoles

5. The image of an equivalent magnetic dipole in a stratified medium

5.1. Reflection

5.2. Transmission

5.3. Differences between reflection and transmission cases

6. Implementation

7. Evaluation and analysis

8. Conclusions

Acknowledgements

References and links

Cited By

Figures (11)

Equations (57)

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