Diffraction in a stratified region of a high numerical aperture Fresnel zone plate: a simple and rigorous integral representation

Yaoju Zhang; Xiangjun Huang; Dong Zhang; Hongchang An; Yuxing Dai

doi:10.1364/OE.23.008051

Optics Express
Vol. 23,
Issue 6,
pp. 8051-8060
(2015)
•https://doi.org/10.1364/OE.23.008051

Diffraction in a stratified region of a high numerical aperture Fresnel zone plate: a simple and rigorous integral representation

Yaoju Zhang, Xiangjun Huang, Dong Zhang, Hongchang An, and Yuxing Dai

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Yaoju Zhang, Xiangjun Huang, Dong Zhang, Hongchang An, and Yuxing Dai, "Diffraction in a stratified region of a high numerical aperture Fresnel zone plate: a simple and rigorous integral representation," Opt. Express 23, 8051-8060 (2015)

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About this Article
History
- Original Manuscript: December 15, 2014
- Revised Manuscript: February 2, 2015
- Manuscript Accepted: February 2, 2015
- Published: March 20, 2015

Abstract

An algorithm for calculating the field distribution of a high numerical aperture Fresnel zone plate (FZP) in stratified media is presented, which is based on the vector angular spectrum method. The diffraction problem of FZP is solved for the case of a multilayer film with planar interfaces perpendicular to the optical axis. The solution is obtained in a rigorous mathematical manner and it satisfies the homogeneous wave equations. The electric strength vector of the transmitted and reflected field in the multilayer media is obtained for any polarized beam normally incident onto a binary phase circular FZP. For radially-, azimuthally- and linearly-polarized beam, the electric field in the focal region can be simplified as double or single integral, which can be readily used for numerical computation.

1. Introduction

Fresnel zone plate (FZP), which has lens-like properties, is an important planar diffractive element. FZPs can be used for focusing and imaging of X-rays, extreme-ultraviolet radiation, and visible light [1–6]. In theory, many papers have analyzed the focusing properties of low numerical aperture (NA) FZPs using the scalar diffraction theory [7–12]. With the development of lithography techniques, high-NA FZPs with high resolution can be readily fabricated [13,14]. In recent years, great attention has been paid toward calculation on the diffraction field of high-NA FZPs [15–20]. Richards-Wolf’s vector diffraction theory [21], Rayleigh-Sommerfeld vector diffraction theory [22], and vector angular spectrum method [23, 24] are used for analyzing the diffraction field distribution of high-NA FZPs [15–20].

The above theoretical calculations focus on the focusing properties of FZP in a single homogeneous medium [6–12,15–20]. In general, the sample is placed in a multilayered environment, such as, in biological microscopy with an oil-immersion lens and in near-field optical data storage with a solid immersion lens [25, 26], where one or more medium transitions should be taken into account. In our previous paper [27], the near-field focusing feather of a low-NA FZP through a single planar interface was analyzed using the scalar diffraction theory. In this paper, we use the vector angular spectrum method to obtain the simple formulae which describe the field distribution of a high-NA FZP in a stratified media. Our approach leads to an easy and transparent way to take into account the changes in magnitude and orientation of the electric field vector due to the transition through the FZP system and through the assembly of layers. The cylindrical coordinates system is used, which leads to obtain an appreciable simplification of the calculation on the transmitted and reflected fields in the multilayer media. For a cylindrical-vector beam of incidence we are left with double integral to be evaluated numerically and for a linearly-polarized beam of incidence we are left with single integral to calculate the diffraction field of a FZP.

2. Angular spectrum representation for diffraction field

Without loss of generality, we consider an imaging system with a binary phase circular FZP as focusing element, as depicted in Fig. 1. The FZP's zone boundaries can be expressed as

r_{j} = \sqrt{j λ f_{d} + {(j λ / 2)}^{2}}, j = 1, 2, \dots, 2 N + 1

where λ is the wavelength of incident light and f_d is a parameter which represents the designed focal length of the FZP in air. N is the number of etched zones. Under the condition of normal incidence, the FZP's transmission function at the exit plane of z = 0 is independent of the polarization, which can be written as

t (r) = {\begin{matrix} \equiv t_{rid} = \frac{t_{12} t_{23} \exp (i k n w)}{1 + r_{12} r_{23} \exp (2 i k n w)}, r_{2 m} \leq r < r_{2 m + 1} \\ \equiv t_{gro} = \frac{2}{1 + n_{1}} \exp (i k w), r_{2 m - 1} \leq r < r_{2 m} \end{matrix}

Here n is the refractive index of the FZP and n₁ = (ε₁)^1/2 is the first medium in image space. r₁₂, r₂₃, t₁₂ and t₂₃ are the Fresnel coefficients of ridge zone film, given by

r_{12} = \frac{1 - n_{1}}{1 + n_{1}}, r_{23} = \frac{n - n_{1}}{n + n_{1}}, t_{12} = \frac{2}{1 + n}, t_{23} = \frac{2 n}{n + n_{1}}

In order to obtain the field distribution embedded in a multilayer system, we start our derivation with the field distribution in a single homogeneous medium with refractive index n₁. According to the angular spectrum representation, the time-independent electric field in z>0 can be expressed as [19,23,24]

\begin{array}{l} E_{x} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{\infty} A_{x} (k_{x}, k_{y}) \exp [i (k_{x} x + k_{y} y + k_{z} z)] d k_{x} d k_{y}, \\ E_{y} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{\infty} A_{y} (k_{x}, k_{y}) \exp [i (k_{x} x + k_{y} y + k_{z} z)] d k_{x} d k_{y}, \\ E_{z} (ρ, θ, z) = - \int_{0}^{\infty} \int_{0}^{\infty} [\frac{k_{x}}{k_{z}} A_{x} (k_{x}, k_{y}) + \frac{k_{y}}{k_{z}} A_{y} (k_{x}, k_{y})] \exp [i (k_{x} x + k_{y} y + k_{z} z)] d k_{x} d k_{y} . \end{array}

In this representation,

k_{z} = \sqrt{k^{2} - k_{x}^{2} - k_{y}^{2}}

, where k = 2πn₁/λ is the wave vector in the medium. A_x and A_y are the x and y components of the angular spectrum in the exit plane of the FZP, which is written as

\begin{array}{l} A_{x} = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{\infty} E_{x} (r, φ) t (r) \exp [- i (k_{x} x + k_{y} y)] d x d y, \\ A_{y} = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{\infty} E_{y} (r, φ) t (r) \exp [- i (k_{x} x + k_{y} y)] d x d y, \end{array}

where E_x(r, φ) and E_y(r, φ) are the components of the incident electric field in x and y directions, respectively. In above formulas, (r,φ) are the polar coordinates at a point in the exit plane of the FZP and (ρ, θ, z) are the cylindrical coordinates at a observation point in z>0 space. Equation (4) with Eq. (5) is equivalent to Rayleigh diffraction formulas of the first kind [22], which is a solution to Maxwell’s equations. The electric field [Eq. (4)] in the half-space of z>0, can be determined exactly from the knowledge of E_x(r,φ)t(r) and E_x(r,φ)t(r) in Eq. (5).

Fig. 1 Cross-section diagram of the studied focusing system with a binary phase circular FZP. The FZP's pattern is etched in a glass film and the etching depth is w. The exit pupil is located in the plane of z = 0. The image space is a multilayer structure of plane parallel film where several medium transitions can be encountered at z = d_i. The first medium at the exit pupil has electric permittivity ε₁ and the final medium has electric permittivity ε_L. All of materials are nonmagnetic. The origin of coordinates is positioned at the center of the exit pupil.

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Due to the rotational symmetry of the FZP considered and for convenience in describing the transmission effect of transverse electric (TE) and transverse magnetic (TM) waves through the interface between two nearest neighboring films, we transform electric-field components in Eq. (4) along the Cartesian-coordinate axes into their cylindrical-coordinate counterparts. After some operations, we obtain the expression for the radial, azimuthal and longitudinal components, giving

\begin{array}{l} E_{ρ} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{2 π} [A_{ξ} \cos (k_{η} - θ) - A_{η} \sin (k_{η} - θ)] \exp [i ρ k_{ξ} \cos (k_{η} - θ)] \exp (i k_{ξ} z) k_{ξ} d k_{ξ} d k_{η}, \\ E_{θ} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{2 π} [A_{ξ} \sin (k_{η} - θ) + A_{η} \cos (k_{η} - θ)] \exp [i ρ k_{ξ} \cos (k_{η} - θ)] \exp (i k_{z} z) k_{ξ} d k_{ξ} d k_{η}, \\ E_{z} (ρ, θ, z) = - \int_{0}^{\infty} \int_{0}^{2 π} \frac{k_{ξ} A_{ξ}}{k_{z}} \exp [i ρ k_{ξ} \cos (k_{η} - θ)] \exp (i k_{z} z) k_{ξ} d k_{ξ} d k_{η}, \end{array}

where

k_{z} = \sqrt{k^{2} - k_{ξ}^{2}}

k_{ξ} = \sqrt{k_{x}^{2} + k_{y}^{2}}

and

k_{η} = arc \tan (k_{y} / k_{x})

. The radial and azimuthal components (A_ξ and A_η) of the angular spectrum can be expressed as

\begin{array}{l} A_{ξ} (k_{ξ}, k_{η}) = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{2 π} [E_{r} (r, φ) \cos (φ - k_{η}) - E_{φ} (r, φ) \sin (φ - k_{η})] \\ \begin{matrix}  \end{matrix} \times t (r) \exp [- i r k_{ξ} \cos (φ - k_{η})] r d r d φ, \\ A_{η} (k_{ξ}, k_{η}) = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{2 π} [E_{r} (r, φ) \sin (φ - k_{η}) + E_{φ} (r, φ) \cos (φ - k_{η})] \\ \begin{matrix}  \end{matrix} \times t (r) \exp [- i r k_{ξ} \cos (φ - k_{η})] r d r d φ . \end{array}

We now put our attention to the derivation of electric field within the multilayer film shown in Fig. 1. Appling the angular spectrum representation again, the electric field within the jth film can be expressed as

\begin{array}{l} E_{ρ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η, j} - θ) \\ \begin{matrix}  \end{matrix} - ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η, j} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ, j} \cos (k_{η, j} - θ)] k_{ξ, j} d k_{ξ, j} d k_{η, j}, \\ E_{θ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η, j} - θ) \\ \begin{matrix}  \end{matrix} - ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η, j} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ, j} \cos (k_{η, j} - θ)] k_{ξ, j} d k_{ξ, j} d k_{η, j}, \\ E_{z, j} = - \int_{0}^{\infty} \int_{0}^{2 π} ([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times \frac{k_{ξ, j}}{k_{z, j}} \exp [i ρ k_{ξ, j} \cos (k_{η, j} - θ)] k_{ξ, j} d k_{ξ, j} d k_{η, j}, \end{array}

where

k_{z, j} = \sqrt{k_{j}^{2} - k_{ξ, j}^{2}}

and k_j = 2πn_j/λ is the wave vector in the jth film.

B_{j}^{s / p \pm}

represent the amplitudes of individual TE and TM plane waves in the jth film, which are derived in Appendix A. The superscript ± denotes the sign of the propagation direction and s and p denote the cases for the TE and TM waves. Note that for a single homogeneous medium, for which the total amount of layers N = 1, we find

B_{1}^{p +} = A_{ξ}

B_{1}^{s +} = A_{η}

, and

B_{1}^{p -} = B_{1}^{s -} = 0

, resulting in the same expression as in Eq. (6).

In terms of the boundary conditions that the tangential components of the electric and magnetic fields are continuous in two sides of an interface, we can deduce the tangential components of wave vector are also continuous, that is,

k_{ξ, j + 1} = k_{ξ, j} \equiv k_{ξ}, k_{η, j + 1} = k_{η, j} \equiv k_{η} .

Taking all these considerations into account, we rewrite the diffraction integrals in Eq. (8) as

\begin{array}{l} E_{ρ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η} - θ) \\ \begin{matrix}  \end{matrix} - ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ} \cos (k_{η} - θ)] k_{ξ} d k_{ξ} d k_{η}, \\ E_{θ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η} - θ) \\ \begin{matrix}  \end{matrix} + ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ} \cos (k_{η} - θ)] k_{ξ} d k_{ξ} d k_{η}, \\ E_{z, j} = - \int_{0}^{\infty} \int_{0}^{2 π} \frac{k_{ξ}}{k_{z, j}} ([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ} \cos (k_{η} - θ)] k_{ξ} d k_{ξ} d k_{η}, \end{array}

where

k_{z, j} = \sqrt{k_{j}^{2} - k_{ξ}^{2}}

. It is important to emphasize that, since the boundary conditions of electromagnetic field are used and the diffraction integral representation based on the angular spectrum method are exact solutions of the homogeneous wave equation, our formulas for the electric vector [Eq. (10)] in the multilayer medium also satisfy the homogeneous wave equation. Therefore, we have successfully obtained a simple and analytical model of vector formalism, which can be used to rigorously calculate the focusing field of a binary phase circular FZP in the multilayer media.

3. Diffraction field for several special polarized illuminations

Equation (10) is a general formula of calculating the diffraction field of light focused by a binary phase circular FZP into a multilayer film. For a given polarization of incidence it can be further simplified to a simple form.

3.1. Radially polarized illumination

For a radially-polarized beam (E_r = 1 and E_φ = 0) normally illuminating the FZP, Eq. (7) reduces to

A_{ξ} = \frac{i}{2 π} C (k_{ξ}), A_{η} = 0,

where

C (k_{ξ}) = t_{rid} \sum_{m = 0}^{N} \int_{r_{2 m}}^{r_{2 m + 1}} J_{1} (r k_{ξ}) r d r + t_{gro} \sum_{m = 1}^{N} \int_{r_{2 m - 1}}^{r_{2 m}} J_{1} (r k_{ξ}) r d r .

In the process of calculations, the integral representation of the Bessel function of the first kind [28]

\int_{0}^{2 π} \cos (n τ) \exp [i r k_{ξ} \cos τ] d τ = 2 {πi}^{n} J_{n} (r k_{ξ}),

has been used. Substituting Eqs. (11) and (A7) into Eq. (10) yields

\begin{array}{l} E_{ρ, j} (ρ, z) = \int_{0}^{\infty} C (k_{ξ}) {T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ}, \\ E_{z, j} (ρ, z) = - i \int_{0}^{\infty} \frac{k_{ξ} C (k_{ξ})}{k_{z, j}} {T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} J_{0} (ρ k_{ξ}) k_{ξ} d k_{ξ} . \end{array}

where the azimuthal component is zero anywhere in image space and both radial and axial components are nonzero. It may readily be verified that the circular FZP illuminated by a radially-polarized beam produces a rotationally symmetric intensity distribution around the optical axis in image space and the on-axis intensity is not null. When NA of the FZP is high (NA→1), the intensity of the longitudinal component is larger than that of the radial component, which is useful for high-resolution microscopy and optical trapping for micro-particles [29, 30].

3.2 Azimuthally polarized illumination

For an azimuthally-polarized beam (E_r = 0 and E_φ = 1), Eq. (7) reduces to

A_{ξ} = 0, A_{η} = \frac{i}{2 π} C (k_{ξ}) .

Substituting Eqs. (15) and (A7) into Eq. (10) yields

E_{θ, j} (ρ, z) = \int_{0}^{\infty} C (k_{ξ}) {T_{j}^{s +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{s -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ} .

where both radial and axial components are zero. As expected, the azimuthal field propagates as a purely transverse polarization through the entire diffraction region. Such a manner is consistent with Maxwell's equations. Equation (16) with Eq. (12) implies that the circular FZP illuminated by an azimuthally-polarized beam produces a rotationally symmetric hollow intensity distribution.

3.3 Linearly polarized illumination

For a linearly x-polarized beam of incidence (E_r = cosφ and E_φ = −sinφ), Eq. (7) can be simplified as

A_{ξ} = \frac{\cos k_{η}}{2 π} D (k_{ξ}), A_{ξ} = - \frac{\sin k_{η}}{2 π} D (k_{ξ}),

where

D (k_{ξ}) = t_{rid} \sum_{m = 0}^{N} \int_{r_{2 m}}^{r_{2 m + 1}} J_{0} (r k_{ξ}) r d r + t_{gro} \sum_{m = 1}^{N} \int_{r_{2 m - 1}}^{r_{2 m}} J_{0} (r k_{ξ}) r d r .

Equation (18) can be solved as

D (k_{ξ}) = t_{rid} \sum_{m = 0}^{N} [r_{2 m + 1}^{2} \frac{J_{1} (r_{2 m + 1} k_{ξ})}{r_{2 m + 1} k_{ξ}} - r_{2 m}^{2} \frac{J_{1} (r_{2 m} k_{ξ})}{r_{2 m} k_{ξ}}] + t_{gro} \sum_{m = 1}^{N} [r_{2 m}^{2} \frac{J_{1} (r_{2 m} k_{ξ})}{r_{2 m} k_{ξ}} - r_{2 m - 1}^{2} \frac{J_{1} (r_{2 m - 1} k_{ξ})}{r_{2 m - 1} k_{ξ}}] .

Substituting Eqs. (19) and (A7) into Eq. (10) we can obtain

\begin{array}{l} E_{ρ, j} (ρ, θ, z) = \frac{1}{2} \cos θ \int_{0}^{\infty} {(T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) - J_{2} (ρ k_{ξ})] + (T_{j}^{s +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{s -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) + J_{2} (ρ k_{ξ})]} D (k_{ξ}) k_{ξ} d k_{ξ}, \\ E_{θ, j} (ρ, θ, z) = - \frac{1}{2} \sin θ \int_{0}^{\infty} {(T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) + J_{2} (ρ k_{ξ})] + (T_{j}^{s +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{s -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) - J_{2} (ρ k_{ξ})]} D (k_{ξ}) k_{ξ} d k_{ξ}, \\ E_{z, j} (ρ, θ, z) = - i \cos θ \int_{0}^{\infty} {T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} \\ \begin{matrix}  \end{matrix} \times \frac{k_{ξ} D (k_{ξ})}{k_{z, j}} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ} . \end{array}

Transforming these electric-field components in Eq. (20) along the cylindrical-coordinate axes into their Cartesian-coordinate counterparts, we can obtain a simple expression

E_{x, j} = I_{0} + I_{2} \cos 2 θ, E_{y, j} = I_{2} \sin 2 θ, E_{z, j} = - 2 i I_{1} \cos θ

where

\begin{array}{l} I_{0} = \frac{1}{2} \int_{0}^{\infty} {[T_{j}^{s +} + T_{j}^{p +}] \exp [i k_{z, j} (z - d_{j - 1})] + [T_{j}^{s -} + T_{j}^{p -})] \exp [- i k_{z, j} (z - d_{j - 1})]} D (k_{ξ}) J_{0} (ρ k_{ξ}) k_{ξ} d k_{ξ} \\ I_{2} = \frac{1}{2} \int_{0}^{\infty} {[T_{j}^{s +} - T_{j}^{p +}] \exp [i k_{z, j} (z - d_{j - 1})] + [T_{j}^{s -} - T_{j}^{p -}] \exp [- i k_{z, j} (z - d_{j - 1})]} D (k_{ξ}) J_{2} (ρ k_{ξ}) k_{ξ} d k_{ξ} \\ I_{1} = \frac{1}{2} \int_{0}^{\infty} {T_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]} \frac{k_{ξ} D (k_{ξ})}{k_{z, j}} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ} . \end{array}

The expression of Eq. (21) with Eq. (22) is similar in form to that of the electric field in image space when light focused by a spherical refractive lens [31], but I₀, I₁, and I₂ in the present expression are based on the vector angular representation which rigorously satisfies the Maxwell's equations. For the special case in absence of stratified medium, we can obtain the point spread function on the optical axis of the binary phase FZP illuminated by the linearly-polarized beam, giving

I (0, 0, z) = {| \frac{1}{2} \int_{0}^{\infty} \exp (i k_{z} \sqrt{k^{2} - k_{ξ}^{2}} z) D (k_{ξ}) k_{ξ} d k_{ξ} |}^{2} .

This expression is in form similar to that obtained by López et al. for a binary amplitude FZP in a single medium [17].

4. Comparison with FDTD simulation results

With the purpose of validating the model developed in Sections 2 and 3, we numerically compute the field distribution in the focal region of the binary phase FZP with a plane parallel plane film and compare the numerical results with the ones obtained by the finite-difference time-domain (FDTD) method. In the following numerical calculations, we consider a binary π-phase-shifted FZP with N = 20 zones fabricated in a dielectric film with the refractive index of 1.52. An unit-amplitude linearly x-polarized beam with the wavelength of λ = 0.6328 μm is normally incident onto the FZP. The FZP's designed parameter is f_d = 10 μm and its numerical aperture is NA = 0.9 in air. Assume the thickness and refractive index of the solid dielectric film immersing the FZP are d = 10 μm and n₁ = 1.52, respectively. The ambient medium of the solid immersion FZP is air (n₂ = 1). In the FDTD simulation, mesh size is set to be 40 × 40 × 40 nm³ and perfectly matched layer is used as the boundary conditions.

Figure 2(a) shows the intensity distributions along the optical z axis. It is seen that the axial intensity distribution obtained by the analytical model is basically consistent with that obtained by the FDTD simulation. The focal length of 14.98 μm obtained by the model calculation is almost equal to the focal length of 14.85 μm obtained by the FDTD method. Figures 2(b) and 2(c) show the intensity distributions along the x and y axes, respectively, in the focal plane of f = 14.98 μm. From the figures it is found that, except the difference of the full-width at half maximum (FWHM) at the x direction, the distribution cure predicted by the proposed model is a good agreement with that simulated by the FDTD method. The difference in FWHM might result from the multiple reflection and refraction effect inside the FZP's structure body [19], which is not considered in the present model calculation.

Fig. 2 The normalized intensity distributions of light focused by a binary π-phase-shifted FZP through a solid film of d = 10 μm. (a) is the case along the optical z axis, (b) and (c) are the cases along the x and y axes, respectively, in the focal plane of f = 14.98 μm. The red and blue curves are obtained by the analytical model and by the FDTD method, respectively.

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5. Conclusion

In conclusion, we have presented a solution of electromagnetic diffraction for the problem of light focused by a high-NA binary phase circular FZP into a multilayer dielectric film. By using the angular spectrum method we have solved the above diffraction problem for the case of multiple planar interfaces in a rigorous mathematical manner, and the solution satisfies the homogeneous wave equation and is therefore valid everywhere in the multilayer film. The solution is in a simple form that can be directly used for numerical computation. The results obtained through our analytical model of vector formulism are validated by comparing them with FDTD simulation results, which allow us to demonstrate the good agreement between both methods.

Appendix A: Calculation of the transmission and reflection coefficients

In this appendix our purpose is to obtain the amplitudes $B_{j}^{s / p \pm}$ required in Eq. (10) for our focusing system. Assume a unit amplitude plane wave is incident upon the multilayer plane parallel dielectric film. For an arbitrary profile it is possible to divide this profile into homogeneous slabs of thickness b_j=d_j-d_j-1. Let $T_{j}^{s / p +} (k_{ξ})$ and $T_{j}^{s / p -} (k_{ξ})$ denote the amplitudes of forward and backward traveling waves at the interface of z=d_j-1 in the slab j, as shown in Fig. 3. Considering that the field contains no charges and currents, the tangential components of the electric and magnetic fields are continuous across the interface. Following the derivation as given in [32, 33], we obtain

(\begin{matrix} T_{j}^{s / p +} \\ T_{j}^{s / p -} \end{matrix}) = M_{j, j + 1}^{s / p} (\begin{matrix} T_{j + 1}^{s / p +} \\ T_{j + 1}^{s / p -} \end{matrix}),

where

M_{j, j + 1}^{s / p} = Q_{j}^{- 1} P_{j, j + 1}^{s / p}, P_{j, j + 1}^{s / p} = \frac{1}{t_{j, j + 1}^{s / p}} [\begin{matrix} 1 & r_{j, j + 1}^{s / p} \\ r_{j, j + 1}^{s / p} & 1 \end{matrix}], Q_{j} = [\begin{matrix} \exp (i k_{z, j} b_{j}) & 0 \\ 0 & \exp (- i k_{z, j} b_{j}) \end{matrix}] .

In Eq. (A2),

r_{j, j + 1}^{s / p}

and

t_{j, j + 1}^{s / p}

are the Fresnel reflection and transmission coefficients, respectively, at the interface j, given by

Fig. 3 Reflection and transmission of wave propagation through a slab j.

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\begin{array}{l} r_{j, j + 1}^{s} = \frac{k_{z, j} - k_{z, j + 1}}{k_{z, j} + k_{z, j + 1}}, \begin{matrix}  \end{matrix} t_{j, j + 1}^{s} = \frac{2 k_{z, j}}{k_{z, j} + k_{z, j + 1}}, \begin{matrix}  \end{matrix} for the TE polarization, \\ r_{j, j + 1}^{p} = \frac{ε_{j + 1} k_{z, j} - ε_{j} k_{z, j + 1}}{ε_{j + 1} k_{z, j} + ε_{j} k_{z, j + 1}}, t_{j, j + 1}^{p} = \frac{k_{j + 1}}{k_{j}} \frac{2 ε_{j + 1} k_{z, j}}{ε_{j + 1} k_{z, j} + ε_{j} k_{z, j + 1}}, for the TM polarization . \end{array}

The problem is to find the amplitude $T_{1}^{s / p -}$ of the wave reflected by the stack in the first layer film or the amplitude $T_{L}^{s / p +}$ of the wave transmitted in the substrate through the stack. Since there is no wave coming from the substrate, the equation giving $T_{L}^{s / p +}$ and $T_{1}^{s / p -}$ is

(\begin{matrix} 1 \\ T_{1}^{s / p -} \end{matrix}) = (\prod_{j = 1}^{L} M_{j, j + 1}^{s / p}) (\begin{matrix} T_{L}^{s / p +} \\ 0 \end{matrix}) = M_{}^{s / p} (\begin{matrix} T_{L}^{s / p +} \\ 0 \end{matrix}),

where L is the number of interfaces.

T_{L}^{s / p +}

is the amplitude of light transmitted into the substrate in the top of the substrate. Considering that

T_{L}^{s / p +}

is the amplitude of light transmitted into the substrate and in the top of the substrate, we define

M_{L - 1, L}^{s / p} \equiv P_{L - 1, L}^{s / p}

, which has been used in Eq. (A4). If one denotes

m_{i j}^{s / p}

the elements of the matrix product

M_{}^{s / p}

, then

T_{L}^{s / p +} = 1 / m_{11}^{s / p}, T_{1}^{s / p -} = m_{21}^{s / p} / m_{11}^{s / p} .

After obtaining $T_{L}^{s / p +}$ and $T_{1}^{s / p -}$ , $T_{j}^{s / p +}$ and $T_{j}^{s / p -}$ at the interface j can be calculated according to the following equation:

(\begin{matrix} T_{j}^{s / p +} \\ T_{j}^{s / p -} \end{matrix}) = \prod_{n = j}^{L} M_{n, n + 1}^{s / p} (\begin{matrix} T_{L}^{s / p +} \\ 0 \end{matrix}) .

It is noted that $T_{1}^{p +} = A_{ξ} (k_{ξ}, k_{η})$ and $T_{1}^{s +} = A_{η} (k_{ξ}, k_{η})$ for the considered image system. Finally, we obtain the amplitude of individual plane waves within each homogeneous slab j, giving

\begin{array}{l} B_{j}^{p \pm} (k_{ξ}, k_{η}) = A_{ξ} (k_{ξ}, k_{η}) T_{j}^{p \pm} (k_{r}), \\ B_{j}^{s \pm} (k_{ξ}, k_{η}) = A_{η} (k_{ξ}, k_{η}) T_{j}^{s \pm} (k_{r}) . \end{array}

Acknowledgments

This work was supported by the National Natural Science Foundation of China under contracts 61078023 and 61377021.

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Fig. 1 Cross-section diagram of the studied focusing system with a binary phase circular FZP. The FZP's pattern is etched in a glass film and the etching depth is w. The exit pupil is located in the plane of z = 0. The image space is a multilayer structure of plane parallel film where several medium transitions can be encountered at z = d_i. The first medium at the exit pupil has electric permittivity ε₁ and the final medium has electric permittivity ε_L. All of materials are nonmagnetic. The origin of coordinates is positioned at the center of the exit pupil.

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Fig. 3 Reflection and transmission of wave propagation through a slab j.

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Equations (30)

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r_{j} = \sqrt{j λ f_{d} + {(j λ / 2)}^{2}}, j = 1, 2, \dots, 2 N + 1

t (r) = {\begin{matrix} \equiv t_{rid} = \frac{t_{12} t_{23} \exp (i k n w)}{1 + r_{12} r_{23} \exp (2 i k n w)}, r_{2 m} \leq r < r_{2 m + 1} \\ \equiv t_{gro} = \frac{2}{1 + n_{1}} \exp (i k w), r_{2 m - 1} \leq r < r_{2 m} \end{matrix}

r_{12} = \frac{1 - n_{1}}{1 + n_{1}}, r_{23} = \frac{n - n_{1}}{n + n_{1}}, t_{12} = \frac{2}{1 + n}, t_{23} = \frac{2 n}{n + n_{1}}

\begin{array}{l} E_{x} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{\infty} A_{x} (k_{x}, k_{y}) \exp [i (k_{x} x + k_{y} y + k_{z} z)] d k_{x} d k_{y}, \\ E_{y} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{\infty} A_{y} (k_{x}, k_{y}) \exp [i (k_{x} x + k_{y} y + k_{z} z)] d k_{x} d k_{y}, \\ E_{z} (ρ, θ, z) = - \int_{0}^{\infty} \int_{0}^{\infty} [\frac{k_{x}}{k_{z}} A_{x} (k_{x}, k_{y}) + \frac{k_{y}}{k_{z}} A_{y} (k_{x}, k_{y})] \exp [i (k_{x} x + k_{y} y + k_{z} z)] d k_{x} d k_{y} . \end{array}

\begin{array}{l} A_{x} = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{\infty} E_{x} (r, φ) t (r) \exp [- i (k_{x} x + k_{y} y)] d x d y, \\ A_{y} = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{\infty} E_{y} (r, φ) t (r) \exp [- i (k_{x} x + k_{y} y)] d x d y, \end{array}

\begin{array}{l} E_{ρ} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{2 π} [A_{ξ} \cos (k_{η} - θ) - A_{η} \sin (k_{η} - θ)] \exp [i ρ k_{ξ} \cos (k_{η} - θ)] \exp (i k_{ξ} z) k_{ξ} d k_{ξ} d k_{η}, \\ E_{θ} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{2 π} [A_{ξ} \sin (k_{η} - θ) + A_{η} \cos (k_{η} - θ)] \exp [i ρ k_{ξ} \cos (k_{η} - θ)] \exp (i k_{z} z) k_{ξ} d k_{ξ} d k_{η}, \\ E_{z} (ρ, θ, z) = - \int_{0}^{\infty} \int_{0}^{2 π} \frac{k_{ξ} A_{ξ}}{k_{z}} \exp [i ρ k_{ξ} \cos (k_{η} - θ)] \exp (i k_{z} z) k_{ξ} d k_{ξ} d k_{η}, \end{array}

\begin{array}{l} A_{ξ} (k_{ξ}, k_{η}) = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{2 π} [E_{r} (r, φ) \cos (φ - k_{η}) - E_{φ} (r, φ) \sin (φ - k_{η})] \\ \begin{matrix}  \end{matrix} \times t (r) \exp [- i r k_{ξ} \cos (φ - k_{η})] r d r d φ, \\ A_{η} (k_{ξ}, k_{η}) = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{2 π} [E_{r} (r, φ) \sin (φ - k_{η}) + E_{φ} (r, φ) \cos (φ - k_{η})] \\ \begin{matrix}  \end{matrix} \times t (r) \exp [- i r k_{ξ} \cos (φ - k_{η})] r d r d φ . \end{array}

\begin{array}{l} E_{ρ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η, j} - θ) \\ \begin{matrix}  \end{matrix} - ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η, j} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ, j} \cos (k_{η, j} - θ)] k_{ξ, j} d k_{ξ, j} d k_{η, j}, \\ E_{θ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η, j} - θ) \\ \begin{matrix}  \end{matrix} - ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η, j} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ, j} \cos (k_{η, j} - θ)] k_{ξ, j} d k_{ξ, j} d k_{η, j}, \\ E_{z, j} = - \int_{0}^{\infty} \int_{0}^{2 π} ([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times \frac{k_{ξ, j}}{k_{z, j}} \exp [i ρ k_{ξ, j} \cos (k_{η, j} - θ)] k_{ξ, j} d k_{ξ, j} d k_{η, j}, \end{array}

k_{ξ, j + 1} = k_{ξ, j} \equiv k_{ξ}, k_{η, j + 1} = k_{η, j} \equiv k_{η} .

\begin{array}{l} E_{ρ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η} - θ) \\ \begin{matrix}  \end{matrix} - ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ} \cos (k_{η} - θ)] k_{ξ} d k_{ξ} d k_{η}, \\ E_{θ, j} = \int_{0}^{\infty} \int_{0}^{2 π} {([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \sin (k_{η} - θ) \\ \begin{matrix}  \end{matrix} + ([B_{j}^{s +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{s -} \exp [- i k_{z, j} (z - d_{j - 1})]) \cos (k_{η} - θ)} \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ} \cos (k_{η} - θ)] k_{ξ} d k_{ξ} d k_{η}, \\ E_{z, j} = - \int_{0}^{\infty} \int_{0}^{2 π} \frac{k_{ξ}}{k_{z, j}} ([B_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + B_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times \exp [i ρ k_{ξ} \cos (k_{η} - θ)] k_{ξ} d k_{ξ} d k_{η}, \end{array}

A_{ξ} = \frac{i}{2 π} C (k_{ξ}), A_{η} = 0,

C (k_{ξ}) = t_{rid} \sum_{m = 0}^{N} \int_{r_{2 m}}^{r_{2 m + 1}} J_{1} (r k_{ξ}) r d r + t_{gro} \sum_{m = 1}^{N} \int_{r_{2 m - 1}}^{r_{2 m}} J_{1} (r k_{ξ}) r d r .

\int_{0}^{2 π} \cos (n τ) \exp [i r k_{ξ} \cos τ] d τ = 2 {πi}^{n} J_{n} (r k_{ξ}),

\begin{array}{l} E_{ρ, j} (ρ, z) = \int_{0}^{\infty} C (k_{ξ}) {T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ}, \\ E_{z, j} (ρ, z) = - i \int_{0}^{\infty} \frac{k_{ξ} C (k_{ξ})}{k_{z, j}} {T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} J_{0} (ρ k_{ξ}) k_{ξ} d k_{ξ} . \end{array}

A_{ξ} = 0, A_{η} = \frac{i}{2 π} C (k_{ξ}) .

E_{θ, j} (ρ, z) = \int_{0}^{\infty} C (k_{ξ}) {T_{j}^{s +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{s -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ} .

A_{ξ} = \frac{\cos k_{η}}{2 π} D (k_{ξ}), A_{ξ} = - \frac{\sin k_{η}}{2 π} D (k_{ξ}),

D (k_{ξ}) = t_{rid} \sum_{m = 0}^{N} \int_{r_{2 m}}^{r_{2 m + 1}} J_{0} (r k_{ξ}) r d r + t_{gro} \sum_{m = 1}^{N} \int_{r_{2 m - 1}}^{r_{2 m}} J_{0} (r k_{ξ}) r d r .

D (k_{ξ}) = t_{rid} \sum_{m = 0}^{N} [r_{2 m + 1}^{2} \frac{J_{1} (r_{2 m + 1} k_{ξ})}{r_{2 m + 1} k_{ξ}} - r_{2 m}^{2} \frac{J_{1} (r_{2 m} k_{ξ})}{r_{2 m} k_{ξ}}] + t_{gro} \sum_{m = 1}^{N} [r_{2 m}^{2} \frac{J_{1} (r_{2 m} k_{ξ})}{r_{2 m} k_{ξ}} - r_{2 m - 1}^{2} \frac{J_{1} (r_{2 m - 1} k_{ξ})}{r_{2 m - 1} k_{ξ}}] .

\begin{array}{l} E_{ρ, j} (ρ, θ, z) = \frac{1}{2} \cos θ \int_{0}^{\infty} {(T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) - J_{2} (ρ k_{ξ})] + (T_{j}^{s +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{s -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) + J_{2} (ρ k_{ξ})]} D (k_{ξ}) k_{ξ} d k_{ξ}, \\ E_{θ, j} (ρ, θ, z) = - \frac{1}{2} \sin θ \int_{0}^{\infty} {(T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) + J_{2} (ρ k_{ξ})] + (T_{j}^{s +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{s -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]) \\ \begin{matrix}  \end{matrix} \times [J_{0} (ρ k_{ξ}) - J_{2} (ρ k_{ξ})]} D (k_{ξ}) k_{ξ} d k_{ξ}, \\ E_{z, j} (ρ, θ, z) = - i \cos θ \int_{0}^{\infty} {T_{j}^{p +} (k_{ξ}) \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} (k_{ξ}) \exp [- i k_{z, j} (z - d_{j - 1})]} \\ \begin{matrix}  \end{matrix} \times \frac{k_{ξ} D (k_{ξ})}{k_{z, j}} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ} . \end{array}

E_{x, j} = I_{0} + I_{2} \cos 2 θ, E_{y, j} = I_{2} \sin 2 θ, E_{z, j} = - 2 i I_{1} \cos θ

\begin{array}{l} I_{0} = \frac{1}{2} \int_{0}^{\infty} {[T_{j}^{s +} + T_{j}^{p +}] \exp [i k_{z, j} (z - d_{j - 1})] + [T_{j}^{s -} + T_{j}^{p -})] \exp [- i k_{z, j} (z - d_{j - 1})]} D (k_{ξ}) J_{0} (ρ k_{ξ}) k_{ξ} d k_{ξ} \\ I_{2} = \frac{1}{2} \int_{0}^{\infty} {[T_{j}^{s +} - T_{j}^{p +}] \exp [i k_{z, j} (z - d_{j - 1})] + [T_{j}^{s -} - T_{j}^{p -}] \exp [- i k_{z, j} (z - d_{j - 1})]} D (k_{ξ}) J_{2} (ρ k_{ξ}) k_{ξ} d k_{ξ} \\ I_{1} = \frac{1}{2} \int_{0}^{\infty} {T_{j}^{p +} \exp [i k_{z, j} (z - d_{j - 1})] + T_{j}^{p -} \exp [- i k_{z, j} (z - d_{j - 1})]} \frac{k_{ξ} D (k_{ξ})}{k_{z, j}} J_{1} (ρ k_{ξ}) k_{ξ} d k_{ξ} . \end{array}

I (0, 0, z) = {| \frac{1}{2} \int_{0}^{\infty} \exp (i k_{z} \sqrt{k^{2} - k_{ξ}^{2}} z) D (k_{ξ}) k_{ξ} d k_{ξ} |}^{2} .

(\begin{matrix} T_{j}^{s / p +} \\ T_{j}^{s / p -} \end{matrix}) = M_{j, j + 1}^{s / p} (\begin{matrix} T_{j + 1}^{s / p +} \\ T_{j + 1}^{s / p -} \end{matrix}),

M_{j, j + 1}^{s / p} = Q_{j}^{- 1} P_{j, j + 1}^{s / p}, P_{j, j + 1}^{s / p} = \frac{1}{t_{j, j + 1}^{s / p}} [\begin{matrix} 1 & r_{j, j + 1}^{s / p} \\ r_{j, j + 1}^{s / p} & 1 \end{matrix}], Q_{j} = [\begin{matrix} \exp (i k_{z, j} b_{j}) & 0 \\ 0 & \exp (- i k_{z, j} b_{j}) \end{matrix}] .

\begin{array}{l} r_{j, j + 1}^{s} = \frac{k_{z, j} - k_{z, j + 1}}{k_{z, j} + k_{z, j + 1}}, \begin{matrix}  \end{matrix} t_{j, j + 1}^{s} = \frac{2 k_{z, j}}{k_{z, j} + k_{z, j + 1}}, \begin{matrix}  \end{matrix} for the TE polarization, \\ r_{j, j + 1}^{p} = \frac{ε_{j + 1} k_{z, j} - ε_{j} k_{z, j + 1}}{ε_{j + 1} k_{z, j} + ε_{j} k_{z, j + 1}}, t_{j, j + 1}^{p} = \frac{k_{j + 1}}{k_{j}} \frac{2 ε_{j + 1} k_{z, j}}{ε_{j + 1} k_{z, j} + ε_{j} k_{z, j + 1}}, for the TM polarization . \end{array}

(\begin{matrix} 1 \\ T_{1}^{s / p -} \end{matrix}) = (\prod_{j = 1}^{L} M_{j, j + 1}^{s / p}) (\begin{matrix} T_{L}^{s / p +} \\ 0 \end{matrix}) = M_{}^{s / p} (\begin{matrix} T_{L}^{s / p +} \\ 0 \end{matrix}),

T_{L}^{s / p +} = 1 / m_{11}^{s / p}, T_{1}^{s / p -} = m_{21}^{s / p} / m_{11}^{s / p} .

(\begin{matrix} T_{j}^{s / p +} \\ T_{j}^{s / p -} \end{matrix}) = \prod_{n = j}^{L} M_{n, n + 1}^{s / p} (\begin{matrix} T_{L}^{s / p +} \\ 0 \end{matrix}) .

\begin{array}{l} B_{j}^{p \pm} (k_{ξ}, k_{η}) = A_{ξ} (k_{ξ}, k_{η}) T_{j}^{p \pm} (k_{r}), \\ B_{j}^{s \pm} (k_{ξ}, k_{η}) = A_{η} (k_{ξ}, k_{η}) T_{j}^{s \pm} (k_{r}) . \end{array}

Abstract

1. Introduction

2. Angular spectrum representation for diffraction field

3. Diffraction field for several special polarized illuminations

3.1. Radially polarized illumination

3.2 Azimuthally polarized illumination

3.3 Linearly polarized illumination

4. Comparison with FDTD simulation results

5. Conclusion

Appendix A: Calculation of the transmission and reflection coefficients

Acknowledgments

References and links

Cited By

Figures (3)

Equations (30)

Optics Express