Lateral shift effect on the spatial interference of light wave propagating in the single-layered dielectric film

Yuan Zhao; Ming-Yu Sheng; Yu-Xiang Zheng; Min Xu; Hai-Bin Zhao; Liang-Yao Chen

doi:10.1364/OE.18.010524

Optics Express
Vol. 18,
Issue 10,
pp. 10524-10537
(2010)
•https://doi.org/10.1364/OE.18.010524

Lateral shift effect on the spatial interference of light wave propagating in the single-layered dielectric film

Yuan Zhao, Ming-Yu Sheng, Yu-Xiang Zheng, Min Xu, Hai-Bin Zhao, and Liang-Yao Chen

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Yuan Zhao, Ming-Yu Sheng, Yu-Xiang Zheng, Min Xu, Hai-Bin Zhao, and Liang-Yao Chen, "Lateral shift effect on the spatial interference of light wave propagating in the single-layered dielectric film," Opt. Express 18, 10524-10537 (2010)

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- Physical Optics
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About this Article
History
- Original Manuscript: March 22, 2010
- Revised Manuscript: April 25, 2010
- Manuscript Accepted: April 27, 2010
- Published: May 5, 2010

Abstract

Under the oblique incidence condition, the multiple reflection of wave packets in a layered film structure will have a lateral shift increasing with the film thickness. In the analysis of the spatial interference with consideration of the lateral shift effect, a set of new analytic formulae to normalize the intensity of the s-and p-polarized wave packet was obtained to reduce the error of the ellipsometry parameters significantly as the optical path difference δ is close to mπ. The principle and method developed in this work also can be applied to other film structures in more general applications.

1. Introduction

When a monochromatic polarized plane electromagnetic wave is incident onto a layered dielectric film, interference arising from the multiple-reflected waves will occur. The reflective coefficient of the electric field resulting from the interference of multiple-reflected plane electromagnetic wave can be described by the Airy formula [1].

The polarized plane electromagnetic wave is suitable for the ideal situation when an infinite space condition is assumed. In reality, the electromagnetic wave originating from radiated light sources is finite within a localized space. Therefore, more practically, the incident waves can be represented by spatially-localized wave packets rather than by simple plane waves.

When spatially-localized wave packets are totally reflected at the interface between two different dielectric films, the reflected wave packets undergo a lateral shift on the surface from the point at which the incident wave packet is reflected. One type of lateral shift is known as the Goos–Hänchen (GH) effect resulting from a tiny penetration depth of the light propagating at the interface [2,3]. When the wave packet is incident onto a single/multiple layer film, the reflected wave packet will also experience a lateral shift [4–6]. More recently, positive and negative lateral shifts have been observed in studies where the wave packet is reflected at the interface, significantly affecting the optical properties of artificially-made metamaterials and structures [7–9].

When a monochromatic localized electromagnetic wave or wave packet is incident onto a single-layer dielectric film, this spatial interference effect will occur because the multiple-reflected wave packets undergoing a lateral shift in the film with a finite thickness are not located at the same position in space. The effect of this spatial interference has been studied for film thicknesses in the range from 140nm to 2cm [10–16] and is used in many applications, such as laser attenuator [10], Fabry-Perot etalon [11–13], quarter-wave plate [14] and Faraday isolator [15]. The spatial interference, which affects the electromagnetic wave packet propagating in film-based photonic devices such as the Fabry–Perot micro cavity and quarter-wave plate, has been studied based on the equations that may not be valid in all physical conditions, for example, by taking the limitation where the thickness of the cavity goes to zero [11–14]. The spatial interference of multiple-reflections in thin film structure was also studied by using a set of partially analytical equations [16]. The spatial interference related to the one-photon and two-photon interference effect in the space-time domain is also interesting to study for understanding the behavior of quantum entanglement [17–25].

In this work, we have carefully studied the spatial interference of multiple-reflected electromagnetic wave packet both theoretically and experimentally. We have analyzed the lateral beam shifts due to the thickness of dielectric film with its effects on the spatial interference of the wave packets, which also affects the interpretation of traditional ellipsometric parameters. We give a full consideration of lateral shifts in terms of variable refractive indices, incident angles and film thickness. Then we study the space interference affected by the lateral shift in the multiple reflected wave packet with its influence on the reflectance of s- and p-polarized electromagnetic waves, where s and p represent the perpendicular and parallel component of the electric field at the interface of two media, respectively. Finally, by taking the spatial interference into consideration, we obtain the modified ellipsometric equations to extract the data from the experiment to obtain more accurate results.

2. Theory

It is assumed the localized electromagnetic wave packet with a monochromatic frequency ν is incident onto the film surface. The incident wave packet is highly collimated and propagates along the z direction with intensity of Gaussian distribution in the x-y plane. The electric field of the incident wave packet E can be expressed as:

E (x, y, z, t) = E_{0} e^{\frac{- x^{2} - y^{2}}{w^{2}}} e^{i (k_{z} z - 2 π ν t)}

where E₀ is the amplitude of the electric field, w represents the width of the wave packet in the x-y plane [26] and k_z is the wave vector along the z direction.

As shown in Fig. 1 , the electromagnetic wave packet undergoes refraction with multiple-reflections and then comes out of the SiO₂ film deposited on the Si substrate in assumption. At zero-order, the angular distribution of incident electromagnetic wave packet is negligible, implying that the incident electromagnetic wave packets are highly directional. Thus, the intensity distribution of the electric field E is still kept with the Gaussian formation in the x-y plane, but the width of the wave packet will spread slightly [27,28].

Fig. 1 The Schematic diagram of multiple-reflected wave packets propagating in the film structure. The initial light I₀ is incident at the angle θ₀ onto the air/SiO₂/Si film structure with the refraction angle θ₁ at the film side, where d is the thickness of the film; n₀, n₁ and ñ₂ are the refractive index of the medium of air, SiO₂ film and Si substrate, respectively.

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When the electromagnetic wave packet is incident at the angle θ_o onto the interface between two homogeneous isotropic media, its refraction and reflection can be described by Snell’s law and Fresnel formulas, respectively. After the multiple-reflected wave packet comes out of the SiO₂ film deposited on the Si substrate, it propagates along the z axis. The optical path difference δ between the two successive reflective wave packets can be expressed as δ = 4πn₁d(cosθ₁)/λ, where n₁ is the refractive index of the SiO₂ film, d is the film thickness, θ₁ is the refraction angle at the SiO₂ film side of the air/SiO₂ interface, and λ is the wavelength.

When the wave packet is incident obliquely onto the film surface, there will be a lateral shift along the x direction between two successive reflected wave packets. In the situation in which the localized wave packet is totally internally-reflected at the interface of two homogeneous mediums, it will undergo the well-known lateral shift called Goos-Hänchen effect at the interface [2,3,28], which is a very small quantity and can be neglected in this study. Here, the lateral shift Δx of the wave packet due to the thickness of the film can be written as:

Δ x = \frac{d \sin (2 θ_{0})}{\sqrt{n_{1}^{2} - \sin^{2} θ_{0}}} .

From Eq. (2), the lateral shift increases linearly with the thickness d of the film and decreases with the refractive index of the dielectric film. Considering the thickness of the film d = 400nm and the refractive index of the SiO₂ film n₁ = 1.478 on the condition of the incident wavelength λ = 640nm and the width of wave packet w = λ, the lateral shift changes with the incident angle and have a maximum value when the incident angle is equal to about 50°, as shown in the inset of Fig. 2 . For a single film with a given thickness, the maximum lateral shift will occur under the condition of cos2θ_max = 2n₁[(n₁ ²-1)^1/2-n₁] + 1. The incidence angle θ_max, at which the maximum lateral shift occurs, rapidly increases from 45° to 90° as the refractive index n₁ of the SiO₂ film decreases from 5 to 1 as shown in Fig. 2. The normalized lateral shift Δx/w changing with the incident angle is also shown in the inset of Fig. 2.

Fig. 2 The incidence angle θ_max, at which the maximum lateral shift occurs, decreases with refractive index increasing for the single-layered film with a given thickness. The inset shows the normalized lateral shift Δx/w changing with the incident angle under the condition in which the thickness of the film d = 400nm and the refractive index of the SiO₂ film n₁ = 1.478, the incident wavelength λ = 640nm with the assumption that the width of wave packet w = λ.

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The reflected electric field is the superposition of the successive reflected wave packets out of the single film structure. By taking the lateral shift effect into consideration, the complex amplitude of the reflected electric field E^s,p(x,y) can be written as (the superscript s and p refer to s- and p-polarized wave packets, respectively):

E^{s, p} (x, y) = E_{0} r_{01}^{s, p} e^{\frac{- x^{2} - y^{2}}{w^{2}}} + E_{0} [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ} \sum_{n = 0}^{\infty} {[- r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}]}^{n} e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}}

where r_ij is the reflection coefficient for the wave packets going from the medium i to the medium j. In Eq. (3), all the reflected wave packets will in approximation have the same packet size after emerging from the film structure. The normalized reflected intensity I^s,p of s- and p-polarized wave packets can be defined as:

I_{}^{s, p} = \frac{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E_{}^{s, p} E_{}^{s, p}^{*} d x d y}{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E E^{*} d x d y} .

Using Eqs. (1)-(4), the normalized reflected intensity of s- and p-polarized wave packets can be expressed as (see Eqs. (A1)-(A28) in appendix A).

\begin{array}{c} I_{}^{s, p} = (R_{1}^{s, p})^{2} + (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} + \\ 2 R_{1}^{s, p} R_{2}^{s, p} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) e^{- \frac{{(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 q ϕ_{R 3}^{s, p}) e^{- \frac{{(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 q + 1) ϕ_{R 3}^{s, p}] e^{- \frac{{(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

where

\begin{array}{c} R_{1}^{s, p} e^{i ϕ_{1}^{s, p}} = r_{01}^{s, p}, & R_{2}^{s, p} e^{i ϕ_{2}^{s, p}} = [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ}, & R_{3}^{s, p} e^{i ϕ_{3}^{s, p}} = - r_{01}^{s, p} r_{12}^{s, p} e^{- i δ} \end{array}

Under the condition in which Δx << w or Δx≈0, the exponential terms in the expression of I^s,p becomes unity, and a full interference without the lateral shift effect within the multiple reflected wave packets will occur at the film surface. The normalized intensity of reflected wave packets can be reduced to:

\begin{array}{c} I^{s, p} = (R_{1}^{s, p})^{2} \\ + \frac{{(R_{2}^{s, p})}^{2} + 2 R_{1}^{s, p} R_{2}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p}) - 2 R_{1}^{s, p} R_{2}^{s, p} R_{3}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} + ϕ_{R 3}^{s, p})}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}^{2}} \end{array}

which is equivalent to the Airy formula (see Eqs. (B1)-(B10) in appendix B). Meanwhile, under the condition of Δx >> w, the exponential terms in the expression of I^s,p will vanish and a full de-interference will happen due to the lateral shift effect with non-overlap of reflected wave packets in the space and the normalized intensity can be written as (see appendix B):

I^{s, p} = {(R_{1}^{s, p})}^{2} + \frac{{(R_{2}^{s, p})}^{2}}{1 - {(R_{3}^{s, p})}^{2}} .

However, under the condition of Δx ~w, the partial interference including the lateral shift effect will happen in the multiple reflected wave packets. The normalized intensity reaches the maximum as the optical path difference δ = πm, where m is an integer number.

When s- and p-polarized electromagnetic wave packets are incident onto the single-layered film, the normalized reflected intensity changes with optical path difference, as illustrated in Fig. 3 for wave packets with the width w = λ and w = 3λ, respectively. The normalized reflective intensity oscillates with optical path difference with period π and reaches the maximum value as the optical path difference δ = mπ with or without consideration of the lateral shift effect. It can be clearly seen in Fig. 3 that due to the lateral shift effect, the normalized reflective intensity will be rapidly reduced to be equivalent to the intensity without the interference effect as the optical path difference increases. It also can be seen that the de-interference due to the lateral shift effect with lower w/λ ratio will be more significant than that with higher w/λ ratio.

Fig. 3 Normalized reflected intensity changing with optical path difference for s- and p-polarized wave packets with and without consideration of the lateral shift effect. In calculation, assuming that the incident angle θ₀ = 70°, the refractive index of the SiO₂ film n = 1.478 at the wavelength λ = 640nm and wave packets with the width w = λ and w = 3λ, respectively.

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3. Results and discussion

Single-layered SiO₂ film samples, which are deposited onto a Si substrate by electron beam evaporation, are used to study the spatial interference. Five SiO₂ film samples with different thickness were deposited onto the crystalline Si substrate. The film thickness for the five samples (A, B, C, D and E) is 140nm, 200nm, 380nm, 400nm and 700nm, respectively. The thickness was controlled by a quartz oscillator as in situ monitor in the vacuum chamber. The films are grown in sequence using a multiple-sample holder to make sure that the set of films has identical optical constants under the same film growth conditions.

The ellipsometric parameters of Ψ and Δ were measured for the samples by using a spectroscopic ellipsometer [29] at different incidence angles and different wavelengths to extract the information of the optical refractive index n₁ and thickness d based on the Airy formula and conventional ellipsometry equation given in Eq. (9) [30].

\begin{array}{c} r^{s, p} = \frac{r_{01}^{s, p} + r_{12}^{s, p} e^{- i δ}}{1 + r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}}, & \frac{r^{p}}{r^{s}} = \tan Ψ e^{i Δ} \end{array} .

According to Sellmeier’s model [31], the optical refractive index n₁ of the transparent SiO₂ film in the visible range will slightly change with the wavelength. The physical thickness d of the film, however, should be independent of the wavelength.

The ellipsometric parameters have been measured at the fixed incident angle 70° for three thinner and thicker samples in the 300-800nm and 275.5-413.3nm wavelength ranges, respectively. Based on the conventional ellipsometry analysis, the measured thickness is 134.04nm, 188.07nm and 370.68nm, 393.67nm and 686.95nm for five (A-E) film samples to show the thickness of the film in close agreement with that measured by the in situ quartz oscillator. The refractive index of the SiO₂ thin film slightly increases with decreasing film thickness, which is in agreement with Jellison’s measurement [32].

By taking the spatial interference into consideration, the modified ellipsometric parameters Ψ_shift and Δ_shift can be given as:

\begin{array}{c} \tan Ψ_{s h i f t} = \sqrt{\frac{I^{p}}{I^{s}}}, & \cos Δ_{s h i f t} = \frac{I^{p s}}{\sqrt{I^{s} I^{p}}} . \end{array}

where I^p,s and I^ps are the measured intensity of reflected light for the pure and cross p-s components, respectively, with consideration of the lateral shift effect. I^p,s can be obtained from Eq. (5), and the cross term of p-s intensity I^ps is defined as:

I^{p s} = \frac{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} Re [E_{}^{s} {(E_{}^{p})}^{*}] d x d y}{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E E_{}^{*} d x d y} .

The standard deviation M expressed by Eq. (12) can be used to check the consistency of the calculated ellipsometric data of Ψ_cal and Δ_cal with comparison to the experimental ones of Ψ_exp and Δ_exp in data reduction procedure.

M = \sqrt{{(Ψ_{\exp} - Ψ_{c a l})}^{2} + {(Δ_{\exp} - Δ_{c a l})}^{2}} .

Using Eqs. (1)-(3), the cross term of the p-s intensity I^ps can be further written as (see Eqs. (A1)- (A28) in appendix A):

\begin{array}{c} I^{p s} = R_{1}^{s} R_{1}^{p} \cos (ϕ_{R 1}^{s} - ϕ_{R 1}^{p}) + \\ R_{1}^{s} R_{2}^{p} \sum_{n = 0}^{\infty} {(R_{3}^{p})}^{n} \cos (ϕ_{R 1}^{s} - ϕ_{R 2}^{p} - n ϕ_{R 3}^{p}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{1}^{p} R_{2}^{s} \sum_{n = 0}^{\infty} {(R_{3}^{s})}^{n} \cos (ϕ_{R 1}^{p} - ϕ_{R 2}^{s} - n ϕ_{R 3}^{s}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} {(R_{3}^{s} R_{3}^{p})}^{k} c o s [ϕ_{R 2}^{s} - ϕ_{R 2}^{p} + k (ϕ_{R 3}^{s} - ϕ_{R 3}^{p})] + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) + (k - q) ϕ_{R 3}^{s} - (k + q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k + q} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) + (k - q) ϕ_{R 3}^{p} - (k + q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s})}^{k + q + 1} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) - (k + q + 1) ϕ_{R 3}^{s} + (k - q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q + 1} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) - (k + q + 1) ϕ_{R 3}^{p} + (k - q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

In order to further verify the modified equations, the proper optical path difference values of δ which is the function of the incidence angle, film thickness, refractive index and wavelength are varied in the range of 0.9-12π (rad). The ellipsometric parameters of the three thinner samples are measured with incident angles changing from 50° to 75° within the accuracy of 0.01° and at the wavelengths of 640nm (for sample A), 310nm and 440nm (for sample B) and 430nm (for sample C) corresponding to the optical path difference in the range of 0.9-4.4π (rad.). The ellipsometric parameters of two thicker samples are measured in the 275.5-413.3nm wavelength range (3.5-4eV energy range)under the condition in which the incidence angle is 50° and 65° for sample D and E, respectively, corresponding to optical path differences in the range of 4-12π (rad.). In terms of the minimum of the standard deviation M, ellipsometric parameters are calculated by using the conventional and modified film equations, respectively. The width of the wave packet is kept constant with the best fitting value of w = 5λ for all samples in the δ range of 0.9-12π (rad). The comparison between the data based on the conventional and modified film equations is shown in Fig. 4 . It can be seen that the experimental data fitting to the calculated data is much better when using the modified equations with consideration of the lateral shift effect. More importantly, when the optical path difference δ reaches to about mπ, the new modified formula can give a better correction to the error arising from the conventional film equation as shown in Eq. (9), implying that the modified ellipsometric equations can be applied to determine the film thickness with higher accuracy.

Fig. 4 Comparisons of experiment (blue square dot) and simulated data by considering the interference with (red dash line) and without (green line) lateral shift effect. The optical path difference in the 0.95-1.1π (rad.) range is corresponding to the incidence angle changing from 50° to 75° at the incident wavelength λ = 640nm for sample A, in the 1.9-2.2π (rad.) and 2.8-3.2π (rad.) ranges is corresponding to the incident angle changing from 50° to 75° at the wavelengths λ = 440nm and 310nm for sample B, in the 3.8-4.4π (rad.) range is corresponding to the incident angle changing from 50° to 75° at the wavelength λ = 430nm for sample C, and in the 4-12π (rad.) range is corresponding to the incidence angle 50° (sample D) and 65° (sample E) and in the 275.5-413.3nm wavelength range. The width of the wave packet is kept constant with the best fitting value of w = 5λ for all samples in the δ range of 0.9-12π (rad).

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4. Conclusion

In conclusion, we have analyzed the spatial interference of reflected wave packets for the film structure by considering the lateral shifts between successive reflected wave packets. We have shown that the spatial interference due to the lateral shift effect of the wave packets will affect the conventional ellipsometry analysis results. Under the oblique incidence condition, the lateral shift increases with the film thickness, resulting in the normalized reflective intensity of the s- and p-polarized wave packets oscillating and will gradually reduce in intensity without the interference effect as the optical path difference increases. In terms of the data analysis, it can be seen that without consideration of the lateral shift effect of the wave packets, the experimental data fitting to the ellipsometric parameters will be less accurate as the thickness of film increases. The data error will reach a maximum as the optical path difference δ is close to mπ for m in the range of 1-14 and for a single-layered SiO₂/Si structure. By taking the lateral shift effect into consideration, the modified ellipsometric equation can be used to fit the experiment data with higher accuracy. The principle and method given in this work can also be applied to either periodic or non-periodic film structures in general applications.

Appendix A

This appendix details the calculation of the normalized reflected intensity. I^s,p and I^ps are for the pure and cross p-s polarized wave packet in the single film structure, respectively. The electric field E of incident wave packet can be expressed as:

E (x, y, z, t) = E_{0} e^{\frac{- x^{2} + y^{2}}{w^{2}}} e^{i (k_{z} z - 2 π ν t)}

and the intensity of incident wave packets is defined as:

I_{i n} = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E E^{*} d x d y = I_{0} w^{2} π / 2

where

I_{0} = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E_{0} E_{0}^{*} d x d y

The electric field

E_{r e}^{s, p}

of reflected wave packets can be expressed as.

\begin{array}{c} E_{}^{s, p} (x, y) = E_{0} r_{01}^{s, p} e^{\frac{- x^{2} - y^{2}}{w^{2}}} \\ + E_{0} \sum_{n = 0}^{\infty} [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ} {[- r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}]}^{n} e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}} \end{array}

Then, the normalized intensities of the pure and cross p-s polarized wave packet, I^p,s and I^ps, can be defined as

\begin{array}{c} I_{}^{s, p} = \frac{1}{I_{i n}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E_{}^{s, p} {(E_{}^{s, p})}^{*} d x d y, & I_{}^{p s} = \frac{1}{I_{i n}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} Re [E^{s} {(E^{p})}^{*}] d x d y \end{array}

In order to get the value of the integration of Eq. (A5), Eq. (A4) is departed into two parts:

\begin{array}{l} E_{1}^{s, p} = E_{0} r_{01}^{s, p} e^{\frac{- x^{2} - y^{2}}{w^{2}}} \\ E_{2}^{s, p} = E_{0} \sum_{n = 0}^{\infty} [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ} {[- r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}]}^{n} e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}} \end{array}

Since

E_{}^{s, p} = E_{1}^{s, p} + E_{2}^{s, p}

thus the first integrand in Eq. (A5) can be written as

E_{}^{s, p} E_{}^{s, p *} = {| E_{1}^{s, p} |}^{2} + E_{1}^{s, p} (E_{2}^{s, p})^{*} + (E_{1}^{s, p})^{*} E_{2}^{s, p} + {| E_{2}^{s, p} |}^{2}

Since

Re [E^{s} {(E^{p})}^{*}] = \frac{1}{2} [E^{s} {(E^{p})}^{*} + {(E^{s})}^{*} E^{p}]

the second integrand in Eq. (A5) can be written as

\begin{array}{c} Re [E^{s} (E^{p})^{*}] = \frac{1}{2} [E_{1}^{s} (E_{1}^{p})^{*} + (E_{1}^{s})^{*} E_{1}^{p} + E_{2}^{s} (E_{1}^{p})^{*} + (E_{2}^{s})^{*} E_{1}^{p})] + \\ \frac{1}{2} [E_{1}^{s} (E_{2}^{p})^{*} + (E_{1}^{s})^{*} E_{2}^{p} + E_{2}^{s} (E_{2}^{p})^{*} + (E_{2}^{s})^{*} E_{2}^{p})] \end{array}

Defined the coefficient

\begin{array}{c} R_{1}^{s, p} e^{i ϕ_{1}^{s, p}} = r_{01}^{s, p}, & R_{2}^{s, p} e^{i ϕ_{2}^{s, p}} = [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ}, \end{array} R_{3}^{s, p} e^{i ϕ_{3}^{s, p}} = - r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}

Using Eq. (A6), the terms in Eq. (A9) can be written as:

{| E_{1}^{s, p} |}^{2} = {(R_{1}^{s, p})}^{2} e^{\frac{2 (x^{2} + y^{2})}{w^{2}}}

E_{1}^{s, p} E_{2}^{s, p *} + E_{1}^{s, p *} E_{2}^{s, p} = e^{- \frac{x^{2} + y^{2}}{w^{2}}} \sum_{n = 0}^{\infty} 2 R_{1}^{s, p} R_{2}^{s, p} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}}

We defined

N_{L} = R_{2}^{s, p} \exp (i ϕ_{R 2}^{s, p}) {[R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})]}^{L} e^{\frac{- {(x - (L + 1) Δ x)}^{2} - y^{2}}{w^{2}}}

then

{| E_{2}^{s, p} |}^{2} = \sum_{L = 0}^{\infty} N_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*}

Using the sum formula [13]

\begin{array}{c} \sum_{L = 0}^{\infty} N_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*} = \sum_{k = 0}^{\infty} N_{k} N_{k}^{*} + \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} (N_{k - q} N_{k + q}^{*} + N_{k + q} N_{k - q}^{*}) + \\ \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (N_{k + 1 + q} N_{k - q}^{*} + N_{k - q} N_{k + q + 1}^{*}) \end{array}

we get the last term of Eq. (A9)

\begin{array}{c} {| E_{2}^{s, p} |}^{2} = \sum_{k = 0}^{\infty} {(R_{2}^{s, p})}^{2} {(R_{3}^{s, p})}^{2 k} e^{- \frac{2 {[x - (k + 1) Δ x]}^{2} + 2 y^{2}}{w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (- 2 q ϕ_{R 3}^{s, p}) e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k + q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 q + 1) ϕ_{R 3}^{s, p}] e^{- \frac{{[x - (k + q + 2) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}]} \end{array}

Using Eq. (A6), the terms in Eq. (A10) can be written as:

E_{1}^{s} E_{1}^{p *} + E_{1}^{s *} E_{1}^{p} = 2 R_{1}^{s} R_{1}^{p} e^{- \frac{2 (x^{2} + y^{2})}{w^{2}}} \cos (ϕ_{R 1}^{s} - ϕ_{R 1}^{p})

E_{2}^{s} E_{1}^{p *} + E_{2}^{s *} E_{1}^{p} = 2 R_{1}^{p} R_{2}^{s} e^{- \frac{x^{2} + y^{2}}{w^{2}}} \sum_{n = 0}^{\infty} {(R_{3}^{s})}^{n} \cos [- ϕ_{R 1}^{p} + ϕ_{R 2}^{s} + n ϕ_{R 3}^{s}] e^{- \frac{{[x - (n + 1) Δ x]}^{2} + y^{2}}{w^{2}}}

E_{1}^{s} E_{2}^{p *} + E_{1}^{s *} E_{2}^{p} = 2 R_{1}^{s} R_{2}^{p} e^{- \frac{x^{2} + y^{2}}{w^{2}}} \sum_{n = 0}^{\infty} {(R_{3}^{p})}^{n} \cos (ϕ_{R 1}^{s} - ϕ_{R 2}^{p} - n ϕ_{R 3}^{p}) e^{- \frac{{[x - (n + 1) Δ x]}^{2} + y^{2}}{w^{2}}}

We defined

\begin{array}{l} M_{L} = R_{2}^{s} \exp (i ϕ_{R 2}^{s}) [R_{3}^{s} \exp (i ϕ_{R 3}^{s})]^{L} e^{- \frac{{[x - (L + 1) Δ x]}^{2} + y^{2}}{w^{2}}} \\ N_{q} = R_{2}^{p} \exp (i ϕ_{R 2}^{p}) [R_{3}^{p} \exp (i ϕ_{R 3}^{p})]^{q} e^{- \frac{{[x - (q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} \end{array}

then the last term of Eq. (A10) can be expressed as

E_{2}^{s} E_{2}^{p *} + E_{2}^{s *} E_{2}^{p} = \sum_{L = 0}^{\infty} M_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*} + \sum_{L = 0}^{\infty} M_{L}^{*} \times \sum_{q = 0}^{\infty} N_{q}

Using the sum formula [13]

\begin{array}{c} \sum_{L = 0}^{\infty} M_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*} = \sum_{k = 0}^{\infty} [M_{k} N_{k}^{*}] + \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} (M_{k - q} N_{k + q}^{*} + M_{k + q} N_{k - q}^{*}) + \\ \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (M_{k + q + 1} N_{k - q}^{*} + M_{k - q} N_{k + q + 1}^{*}) \\ \sum_{L = 0}^{\infty} M_{L}^{*} \times \sum_{q = 0}^{\infty} N_{q} = \sum_{k = 0}^{\infty} [N_{k} M_{k}^{*}] + \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (N_{k - q} M_{k + q}^{*} + N_{k + q} M_{k - q}^{*}) + \\ \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (N_{k + 1 + q} M_{k - q}^{*} + N_{k - q} M_{k + q + 1}^{*}) \end{array}

the last term in Eq. (A10) can be written as:

\begin{array}{l} E_{2}^{s} E_{2}^{p *} + E_{2}^{s *} E_{2}^{p} \\ = \sum_{k = 0}^{\infty} 2 R_{2}^{s} R_{2}^{p} {(R_{3}^{s} R_{3}^{p})}^{k} \cos [ϕ_{R 2}^{s} - ϕ_{R 2}^{p} + k (ϕ_{R 3}^{s} - ϕ_{R 3}^{p})] e^{- \frac{2 {[x - (k + 1) Δ x]}^{2} + 2 y^{2}}{w^{2}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {{(R_{3}^{s})}_{}^{k - q} {(R_{3}^{p})}_{}^{k + q} \cos [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) + (k - q) ϕ_{R 3}^{s} - (k + q) ϕ_{R 3}^{p}] \times \\ e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k + q + 1) Δ x]}^{2} + y^{2}}{w^{2}}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {{(R_{3}^{s})}_{}^{k + q} {(R_{3}^{p})}_{}^{k - q} \cos [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) + (k - q) ϕ_{R 3}^{p} - (k + q) ϕ_{R 3}^{s}] \times \\ e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k + q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s})}_{}^{k + q + 1} {(R_{3}^{p})}_{}^{k - q} \cos [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) - (k + q + 1) ϕ_{R 3}^{s} + (k - q) ϕ_{R 3}^{p}] \times \\ e^{- \frac{{(x - (k - q + 1) Δ x)}^{2} + y^{2}}{w^{2}}} e^{- \frac{{(x - (k + q + 2) Δ x)}^{2} + y^{2}}{w^{2}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{p})}_{}^{k + p + 1} {(R_{3}^{s})}_{}^{k - p} \cos [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) - (k + q + 1) ϕ_{R 3}^{p} + (k - q) ϕ_{R 3}^{s}] \times \\ e^{- \frac{{(x - (k - q + 1) Δ x)}^{2} + y^{2}}{w^{2}}} e^{- \frac{{(x - (k + q + 2) Δ x)}^{2} + y^{2}}{w^{2}}} \end{array}

Using Eqs. (A5), (A8) and (A10), the normalized intensity of reflected electric field for pure and cross p-s polarized wave packets can be expressed as:

\begin{array}{c} I_{}^{s, p} = \frac{1}{I_{i n}} [\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {| E_{1}^{s, p} |}^{2} d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{1}^{s, p} E_{2}^{s, p}^{*} + E_{1}^{s, p}^{*} E_{2}^{s, p}) d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {| E_{2}^{s, p} |}^{2} d x d y] \\ I_{r e}^{p s} = \frac{1}{I_{i n}} [\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{1}^{s} E_{1}^{p *} + E_{1}^{s *} E_{1}^{p}) d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{2}^{s} E_{1}^{p *} + E_{2}^{s *} E_{1}^{p}) d x d y + \\ + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{1}^{s} E_{2}^{p *} + E_{1}^{s *} E_{2}^{p}) d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{2}^{s} E_{2}^{p *} + E_{2}^{s *} E_{2}^{p}) d x d y] \end{array}

Using the following standard integrals

\begin{array}{l} \int_{- \infty}^{\infty} \exp (- \frac{2 x^{2}}{w^{2}}) d x = w \sqrt{\frac{π}{2}} \\ \int_{- \infty}^{\infty} \exp (- a x^{2}) \cos (2 b x) d x = \sqrt{\frac{π}{a}} \exp (- \frac{b^{2}}{a}) \\ \int_{- \infty}^{\infty} \exp (- a x^{2}) \sin (2 b x) d x = 0 \\ \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp (- 2 \frac{x^{2} + y^{2}}{w^{2}}) d x d y = w^{2} \frac{π}{2} \end{array}

the normalized reflected intensities of the pure and cross p-s polarized wave packets can be written as:

\begin{array}{c} I_{}^{s, p} = (R_{1}^{s, p})^{2} + (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} + \\ 2 R_{1}^{s, p} R_{2}^{s, p} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) e^{- \frac{{(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 q ϕ_{R 3}^{s, p}) e^{- \frac{{(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 q + 1) ϕ_{R 3}^{s, p}] e^{- \frac{{(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

\begin{array}{c} I^{p s} = R_{1}^{s} R_{1}^{p} \cos (ϕ_{R 1}^{s} - ϕ_{R 1}^{p}) + \\ R_{1}^{s} R_{2}^{p} \sum_{n = 0}^{\infty} {(R_{3}^{p})}^{n} \cos (ϕ_{R 1}^{s} - ϕ_{R 2}^{p} - n ϕ_{R 3}^{p}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{1}^{p} R_{2}^{s} \sum_{n = 0}^{\infty} {(R_{3}^{s})}^{n} \cos (ϕ_{R 1}^{p} - ϕ_{R 2}^{s} - n ϕ_{R 3}^{s}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} {(R_{3}^{s} R_{3}^{p})}^{k} c o s [ϕ_{R 2}^{s} - ϕ_{R 2}^{p} + k (ϕ_{R 3}^{s} - ϕ_{R 3}^{p})] + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) + (k - q) ϕ_{R 3}^{s} - (k + q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k + q} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) + (k - q) ϕ_{R 3}^{p} - (k + q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k + q + 1} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) - (k + q + 1) ϕ_{R 3}^{s} + (k - q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q + 1} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) - (k + q + 1) ϕ_{R 3}^{p} + (k - q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

Appendix B

This appendix details the calculation of the normalized reflected intensity I^p,s under the condition of Δx >> w and Δx ≈ 0. As Δx >> w, the exponential terms in Eq. (5) will vanish, and the normalized reflected intensity I^s,p can be expressed as:

I^{s, p} = {(R_{1}^{s, p})}^{2} + {(R_{2}^{s, p})}^{2} \sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} .

Using the equation:

\sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} = \frac{1}{1 - {(R_{3}^{s, p})}^{2}},

the normalized reflected intensity I^s,p can be expressed as:

I^{s, p} = {(R_{1}^{s, p})}^{2} + \frac{{(R_{2}^{s, p})}^{2}}{1 - {(R_{3}^{s, p})}^{2}} .

When Δx ≈0, the exponential terms in Eq. (5) becomes unity, and then the normalized reflected intensity I^s,p becomes

\begin{array}{c} I^{s, p} = (R_{1}^{s, p})^{2} + \\ 2 R_{1}^{s, p} R_{2}^{s, p} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) + \\ (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} [{(R_{3}^{s, p})}^{2 k} + 2 \sum_{p = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 p ϕ_{R 3}^{s, p}) + 2 \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 p + 1) ϕ_{R 3}^{s, p}] \end{array}

Because

\begin{array}{c} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) = \frac{1}{2} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} [\exp [i (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p})] + \\ \frac{1}{2} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} [\exp [- i (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p})] \end{array}

the first sum term in Eq. (B4) can be simplified as

\sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p})] = \frac{\cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p}) - R_{3}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} + ϕ_{R 3}^{s, p})}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}_{}^{2}}

From Eq. (A22), we can get

\begin{array}{c} \sum_{n = 0}^{\infty} [R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})]^{n} \times \sum_{m = 0}^{\infty} {[R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})]}^{m} = \sum_{k = 0}^{\infty} [{(R_{3}^{s, p})}^{2 k} + 2 \sum_{p = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 p ϕ_{R 3}^{s, p})] + \\ \sum_{k = 0}^{\infty} [2 \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 p + 1) ϕ_{R 3}^{s, p}] \end{array}

Then the second sum term in Eq. (B4) can be simplified as

\begin{array}{l} \sum_{k = 0}^{\infty} [{(R_{3}^{s, p})}^{2 k} + 2 \sum_{p = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 p ϕ_{R 3}^{s, p}) + 2 \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 p + 1) ϕ_{R 3}^{s, p}] \\ = \frac{1}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}_{}^{2}} \end{array}

Using Eqs. (B2), (B6) and (B8), therefore, the normalized reflected intensity I^s,p becomes

\begin{array}{c} I^{s, p} = (R_{1}^{s, p})^{2} \\ + \frac{{(R_{2}^{s, p})}^{2} + 2 R_{1}^{s, p} R_{2}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p}) - 2 R_{1}^{s, p} R_{2}^{s, p} R_{3}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} + ϕ_{R 3}^{s, p})}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}^{2}} \end{array}

which can be reduced to the Airy formula of Eq. (9) and can be written as I^s,p = |r^s,p|² and:

r^{s, p} = \frac{R_{1}^{s, p} \exp (i ϕ_{R 1}^{s, p}) - R_{1}^{s, p} R_{3}^{s, p} \exp [i (ϕ_{R 1}^{s, p} + ϕ_{R 3}^{s, p})] + R_{2}^{s, p} \exp (i ϕ_{R 2}^{s, p})}{1 - R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})} .

Acknowledgements

This work was supported by the National Science Foundation (NSF) project of China with the contract numbers: #60938004 and by the STCSM project of China (Grant No. 08DJ1400302).

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Equations (51)

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E (x, y, z, t) = E_{0} e^{\frac{- x^{2} - y^{2}}{w^{2}}} e^{i (k_{z} z - 2 π ν t)}

Δ x = \frac{d \sin (2 θ_{0})}{\sqrt{n_{1}^{2} - \sin^{2} θ_{0}}} .

E^{s, p} (x, y) = E_{0} r_{01}^{s, p} e^{\frac{- x^{2} - y^{2}}{w^{2}}} + E_{0} [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ} \sum_{n = 0}^{\infty} {[- r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}]}^{n} e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}}

I_{}^{s, p} = \frac{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E_{}^{s, p} E_{}^{s, p}^{*} d x d y}{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E E^{*} d x d y} .

\begin{array}{c} I_{}^{s, p} = (R_{1}^{s, p})^{2} + (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} + \\ 2 R_{1}^{s, p} R_{2}^{s, p} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) e^{- \frac{{(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 q ϕ_{R 3}^{s, p}) e^{- \frac{{(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 q + 1) ϕ_{R 3}^{s, p}] e^{- \frac{{(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

\begin{array}{c} R_{1}^{s, p} e^{i ϕ_{1}^{s, p}} = r_{01}^{s, p}, & R_{2}^{s, p} e^{i ϕ_{2}^{s, p}} = [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ}, & R_{3}^{s, p} e^{i ϕ_{3}^{s, p}} = - r_{01}^{s, p} r_{12}^{s, p} e^{- i δ} \end{array}

\begin{array}{c} I^{s, p} = (R_{1}^{s, p})^{2} \\ + \frac{{(R_{2}^{s, p})}^{2} + 2 R_{1}^{s, p} R_{2}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p}) - 2 R_{1}^{s, p} R_{2}^{s, p} R_{3}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} + ϕ_{R 3}^{s, p})}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}^{2}} \end{array}

I^{s, p} = {(R_{1}^{s, p})}^{2} + \frac{{(R_{2}^{s, p})}^{2}}{1 - {(R_{3}^{s, p})}^{2}} .

\begin{array}{c} r^{s, p} = \frac{r_{01}^{s, p} + r_{12}^{s, p} e^{- i δ}}{1 + r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}}, & \frac{r^{p}}{r^{s}} = \tan Ψ e^{i Δ} \end{array} .

\begin{array}{c} \tan Ψ_{s h i f t} = \sqrt{\frac{I^{p}}{I^{s}}}, & \cos Δ_{s h i f t} = \frac{I^{p s}}{\sqrt{I^{s} I^{p}}} . \end{array}

I^{p s} = \frac{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} Re [E_{}^{s} {(E_{}^{p})}^{*}] d x d y}{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E E_{}^{*} d x d y} .

M = \sqrt{{(Ψ_{\exp} - Ψ_{c a l})}^{2} + {(Δ_{\exp} - Δ_{c a l})}^{2}} .

\begin{array}{c} I^{p s} = R_{1}^{s} R_{1}^{p} \cos (ϕ_{R 1}^{s} - ϕ_{R 1}^{p}) + \\ R_{1}^{s} R_{2}^{p} \sum_{n = 0}^{\infty} {(R_{3}^{p})}^{n} \cos (ϕ_{R 1}^{s} - ϕ_{R 2}^{p} - n ϕ_{R 3}^{p}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{1}^{p} R_{2}^{s} \sum_{n = 0}^{\infty} {(R_{3}^{s})}^{n} \cos (ϕ_{R 1}^{p} - ϕ_{R 2}^{s} - n ϕ_{R 3}^{s}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} {(R_{3}^{s} R_{3}^{p})}^{k} c o s [ϕ_{R 2}^{s} - ϕ_{R 2}^{p} + k (ϕ_{R 3}^{s} - ϕ_{R 3}^{p})] + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) + (k - q) ϕ_{R 3}^{s} - (k + q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k + q} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) + (k - q) ϕ_{R 3}^{p} - (k + q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s})}^{k + q + 1} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) - (k + q + 1) ϕ_{R 3}^{s} + (k - q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q + 1} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) - (k + q + 1) ϕ_{R 3}^{p} + (k - q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

E (x, y, z, t) = E_{0} e^{\frac{- x^{2} + y^{2}}{w^{2}}} e^{i (k_{z} z - 2 π ν t)}

I_{i n} = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E E^{*} d x d y = I_{0} w^{2} π / 2

I_{0} = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E_{0} E_{0}^{*} d x d y

\begin{array}{c} E_{}^{s, p} (x, y) = E_{0} r_{01}^{s, p} e^{\frac{- x^{2} - y^{2}}{w^{2}}} \\ + E_{0} \sum_{n = 0}^{\infty} [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ} {[- r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}]}^{n} e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}} \end{array}

\begin{array}{c} I_{}^{s, p} = \frac{1}{I_{i n}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} E_{}^{s, p} {(E_{}^{s, p})}^{*} d x d y, & I_{}^{p s} = \frac{1}{I_{i n}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} Re [E^{s} {(E^{p})}^{*}] d x d y \end{array}

\begin{array}{l} E_{1}^{s, p} = E_{0} r_{01}^{s, p} e^{\frac{- x^{2} - y^{2}}{w^{2}}} \\ E_{2}^{s, p} = E_{0} \sum_{n = 0}^{\infty} [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ} {[- r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}]}^{n} e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}} \end{array}

E_{}^{s, p} = E_{1}^{s, p} + E_{2}^{s, p}

E_{}^{s, p} E_{}^{s, p *} = {| E_{1}^{s, p} |}^{2} + E_{1}^{s, p} (E_{2}^{s, p})^{*} + (E_{1}^{s, p})^{*} E_{2}^{s, p} + {| E_{2}^{s, p} |}^{2}

Re [E^{s} {(E^{p})}^{*}] = \frac{1}{2} [E^{s} {(E^{p})}^{*} + {(E^{s})}^{*} E^{p}]

\begin{array}{c} Re [E^{s} (E^{p})^{*}] = \frac{1}{2} [E_{1}^{s} (E_{1}^{p})^{*} + (E_{1}^{s})^{*} E_{1}^{p} + E_{2}^{s} (E_{1}^{p})^{*} + (E_{2}^{s})^{*} E_{1}^{p})] + \\ \frac{1}{2} [E_{1}^{s} (E_{2}^{p})^{*} + (E_{1}^{s})^{*} E_{2}^{p} + E_{2}^{s} (E_{2}^{p})^{*} + (E_{2}^{s})^{*} E_{2}^{p})] \end{array}

\begin{array}{c} R_{1}^{s, p} e^{i ϕ_{1}^{s, p}} = r_{01}^{s, p}, & R_{2}^{s, p} e^{i ϕ_{2}^{s, p}} = [1 - {(r_{01}^{s, p})}^{2}] r_{12}^{s, p} e^{- i δ}, \end{array} R_{3}^{s, p} e^{i ϕ_{3}^{s, p}} = - r_{01}^{s, p} r_{12}^{s, p} e^{- i δ}

{| E_{1}^{s, p} |}^{2} = {(R_{1}^{s, p})}^{2} e^{\frac{2 (x^{2} + y^{2})}{w^{2}}}

E_{1}^{s, p} E_{2}^{s, p *} + E_{1}^{s, p *} E_{2}^{s, p} = e^{- \frac{x^{2} + y^{2}}{w^{2}}} \sum_{n = 0}^{\infty} 2 R_{1}^{s, p} R_{2}^{s, p} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) e^{\frac{- {[x - (n + 1) Δ x]}^{2} - y^{2}}{w^{2}}}

N_{L} = R_{2}^{s, p} \exp (i ϕ_{R 2}^{s, p}) {[R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})]}^{L} e^{\frac{- {(x - (L + 1) Δ x)}^{2} - y^{2}}{w^{2}}}

{| E_{2}^{s, p} |}^{2} = \sum_{L = 0}^{\infty} N_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*}

\begin{array}{c} \sum_{L = 0}^{\infty} N_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*} = \sum_{k = 0}^{\infty} N_{k} N_{k}^{*} + \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} (N_{k - q} N_{k + q}^{*} + N_{k + q} N_{k - q}^{*}) + \\ \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (N_{k + 1 + q} N_{k - q}^{*} + N_{k - q} N_{k + q + 1}^{*}) \end{array}

\begin{array}{c} {| E_{2}^{s, p} |}^{2} = \sum_{k = 0}^{\infty} {(R_{2}^{s, p})}^{2} {(R_{3}^{s, p})}^{2 k} e^{- \frac{2 {[x - (k + 1) Δ x]}^{2} + 2 y^{2}}{w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (- 2 q ϕ_{R 3}^{s, p}) e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k + q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 q + 1) ϕ_{R 3}^{s, p}] e^{- \frac{{[x - (k + q + 2) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}]} \end{array}

E_{1}^{s} E_{1}^{p *} + E_{1}^{s *} E_{1}^{p} = 2 R_{1}^{s} R_{1}^{p} e^{- \frac{2 (x^{2} + y^{2})}{w^{2}}} \cos (ϕ_{R 1}^{s} - ϕ_{R 1}^{p})

E_{2}^{s} E_{1}^{p *} + E_{2}^{s *} E_{1}^{p} = 2 R_{1}^{p} R_{2}^{s} e^{- \frac{x^{2} + y^{2}}{w^{2}}} \sum_{n = 0}^{\infty} {(R_{3}^{s})}^{n} \cos [- ϕ_{R 1}^{p} + ϕ_{R 2}^{s} + n ϕ_{R 3}^{s}] e^{- \frac{{[x - (n + 1) Δ x]}^{2} + y^{2}}{w^{2}}}

E_{1}^{s} E_{2}^{p *} + E_{1}^{s *} E_{2}^{p} = 2 R_{1}^{s} R_{2}^{p} e^{- \frac{x^{2} + y^{2}}{w^{2}}} \sum_{n = 0}^{\infty} {(R_{3}^{p})}^{n} \cos (ϕ_{R 1}^{s} - ϕ_{R 2}^{p} - n ϕ_{R 3}^{p}) e^{- \frac{{[x - (n + 1) Δ x]}^{2} + y^{2}}{w^{2}}}

\begin{array}{l} M_{L} = R_{2}^{s} \exp (i ϕ_{R 2}^{s}) [R_{3}^{s} \exp (i ϕ_{R 3}^{s})]^{L} e^{- \frac{{[x - (L + 1) Δ x]}^{2} + y^{2}}{w^{2}}} \\ N_{q} = R_{2}^{p} \exp (i ϕ_{R 2}^{p}) [R_{3}^{p} \exp (i ϕ_{R 3}^{p})]^{q} e^{- \frac{{[x - (q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} \end{array}

E_{2}^{s} E_{2}^{p *} + E_{2}^{s *} E_{2}^{p} = \sum_{L = 0}^{\infty} M_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*} + \sum_{L = 0}^{\infty} M_{L}^{*} \times \sum_{q = 0}^{\infty} N_{q}

\begin{array}{c} \sum_{L = 0}^{\infty} M_{L} \times \sum_{q = 0}^{\infty} N_{q}^{*} = \sum_{k = 0}^{\infty} [M_{k} N_{k}^{*}] + \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} (M_{k - q} N_{k + q}^{*} + M_{k + q} N_{k - q}^{*}) + \\ \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (M_{k + q + 1} N_{k - q}^{*} + M_{k - q} N_{k + q + 1}^{*}) \\ \sum_{L = 0}^{\infty} M_{L}^{*} \times \sum_{q = 0}^{\infty} N_{q} = \sum_{k = 0}^{\infty} [N_{k} M_{k}^{*}] + \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (N_{k - q} M_{k + q}^{*} + N_{k + q} M_{k - q}^{*}) + \\ \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} (N_{k + 1 + q} M_{k - q}^{*} + N_{k - q} M_{k + q + 1}^{*}) \end{array}

\begin{array}{l} E_{2}^{s} E_{2}^{p *} + E_{2}^{s *} E_{2}^{p} \\ = \sum_{k = 0}^{\infty} 2 R_{2}^{s} R_{2}^{p} {(R_{3}^{s} R_{3}^{p})}^{k} \cos [ϕ_{R 2}^{s} - ϕ_{R 2}^{p} + k (ϕ_{R 3}^{s} - ϕ_{R 3}^{p})] e^{- \frac{2 {[x - (k + 1) Δ x]}^{2} + 2 y^{2}}{w^{2}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {{(R_{3}^{s})}_{}^{k - q} {(R_{3}^{p})}_{}^{k + q} \cos [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) + (k - q) ϕ_{R 3}^{s} - (k + q) ϕ_{R 3}^{p}] \times \\ e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k + q + 1) Δ x]}^{2} + y^{2}}{w^{2}}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {{(R_{3}^{s})}_{}^{k + q} {(R_{3}^{p})}_{}^{k - q} \cos [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) + (k - q) ϕ_{R 3}^{p} - (k + q) ϕ_{R 3}^{s}] \times \\ e^{- \frac{{[x - (k - q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} e^{- \frac{{[x - (k + q + 1) Δ x]}^{2} + y^{2}}{w^{2}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s})}_{}^{k + q + 1} {(R_{3}^{p})}_{}^{k - q} \cos [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) - (k + q + 1) ϕ_{R 3}^{s} + (k - q) ϕ_{R 3}^{p}] \times \\ e^{- \frac{{(x - (k - q + 1) Δ x)}^{2} + y^{2}}{w^{2}}} e^{- \frac{{(x - (k + q + 2) Δ x)}^{2} + y^{2}}{w^{2}}} + \\ 2 R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{p})}_{}^{k + p + 1} {(R_{3}^{s})}_{}^{k - p} \cos [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) - (k + q + 1) ϕ_{R 3}^{p} + (k - q) ϕ_{R 3}^{s}] \times \\ e^{- \frac{{(x - (k - q + 1) Δ x)}^{2} + y^{2}}{w^{2}}} e^{- \frac{{(x - (k + q + 2) Δ x)}^{2} + y^{2}}{w^{2}}} \end{array}

\begin{array}{c} I_{}^{s, p} = \frac{1}{I_{i n}} [\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {| E_{1}^{s, p} |}^{2} d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{1}^{s, p} E_{2}^{s, p}^{*} + E_{1}^{s, p}^{*} E_{2}^{s, p}) d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {| E_{2}^{s, p} |}^{2} d x d y] \\ I_{r e}^{p s} = \frac{1}{I_{i n}} [\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{1}^{s} E_{1}^{p *} + E_{1}^{s *} E_{1}^{p}) d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{2}^{s} E_{1}^{p *} + E_{2}^{s *} E_{1}^{p}) d x d y + \\ + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{1}^{s} E_{2}^{p *} + E_{1}^{s *} E_{2}^{p}) d x d y + \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (E_{2}^{s} E_{2}^{p *} + E_{2}^{s *} E_{2}^{p}) d x d y] \end{array}

\begin{array}{l} \int_{- \infty}^{\infty} \exp (- \frac{2 x^{2}}{w^{2}}) d x = w \sqrt{\frac{π}{2}} \\ \int_{- \infty}^{\infty} \exp (- a x^{2}) \cos (2 b x) d x = \sqrt{\frac{π}{a}} \exp (- \frac{b^{2}}{a}) \\ \int_{- \infty}^{\infty} \exp (- a x^{2}) \sin (2 b x) d x = 0 \\ \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp (- 2 \frac{x^{2} + y^{2}}{w^{2}}) d x d y = w^{2} \frac{π}{2} \end{array}

\begin{array}{c} I_{}^{s, p} = (R_{1}^{s, p})^{2} + (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} + \\ 2 R_{1}^{s, p} R_{2}^{s, p} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) e^{- \frac{{(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 q ϕ_{R 3}^{s, p}) e^{- \frac{{(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ 2 (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} \sum_{q = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 q + 1) ϕ_{R 3}^{s, p}] e^{- \frac{{(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

\begin{array}{c} I^{p s} = R_{1}^{s} R_{1}^{p} \cos (ϕ_{R 1}^{s} - ϕ_{R 1}^{p}) + \\ R_{1}^{s} R_{2}^{p} \sum_{n = 0}^{\infty} {(R_{3}^{p})}^{n} \cos (ϕ_{R 1}^{s} - ϕ_{R 2}^{p} - n ϕ_{R 3}^{p}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{1}^{p} R_{2}^{s} \sum_{n = 0}^{\infty} {(R_{3}^{s})}^{n} \cos (ϕ_{R 1}^{p} - ϕ_{R 2}^{s} - n ϕ_{R 3}^{s}) e^{\frac{- {(n + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} {(R_{3}^{s} R_{3}^{p})}^{k} c o s [ϕ_{R 2}^{s} - ϕ_{R 2}^{p} + k (ϕ_{R 3}^{s} - ϕ_{R 3}^{p})] + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) + (k - q) ϕ_{R 3}^{s} - (k + q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k + q} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) + (k - q) ϕ_{R 3}^{p} - (k + q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k + q + 1} {(R_{3}^{p})}^{k - q} c o s [(ϕ_{R 2}^{p} - ϕ_{R 2}^{s}) - (k + q + 1) ϕ_{R 3}^{s} + (k - q) ϕ_{R 3}^{p}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} + \\ R_{2}^{s} R_{2}^{p} \sum_{k = 0}^{\infty} \sum_{q = 1}^{k} {(R_{3}^{s})}^{k - q} {(R_{3}^{p})}^{k + q + 1} c o s [(ϕ_{R 2}^{s} - ϕ_{R 2}^{p}) - (k + q + 1) ϕ_{R 3}^{p} + (k - q) ϕ_{R 3}^{s}] e^{\frac{- {(2 q + 1)}^{2} Δ x^{2}}{2 w^{2}}} \end{array}

I^{s, p} = {(R_{1}^{s, p})}^{2} + {(R_{2}^{s, p})}^{2} \sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} .

\sum_{k = 0}^{\infty} {(R_{3}^{s, p})}^{2 k} = \frac{1}{1 - {(R_{3}^{s, p})}^{2}},

I^{s, p} = {(R_{1}^{s, p})}^{2} + \frac{{(R_{2}^{s, p})}^{2}}{1 - {(R_{3}^{s, p})}^{2}} .

\begin{array}{c} I^{s, p} = (R_{1}^{s, p})^{2} + \\ 2 R_{1}^{s, p} R_{2}^{s, p} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) + \\ (R_{2}^{s, p})^{2} \sum_{k = 0}^{\infty} [{(R_{3}^{s, p})}^{2 k} + 2 \sum_{p = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 p ϕ_{R 3}^{s, p}) + 2 \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 p + 1) ϕ_{R 3}^{s, p}] \end{array}

\begin{array}{c} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p}) = \frac{1}{2} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} [\exp [i (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p})] + \\ \frac{1}{2} \sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} [\exp [- i (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p})] \end{array}

\sum_{n = 0}^{\infty} {(R_{3}^{s, p})}^{n} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} - n ϕ_{R 3}^{s, p})] = \frac{\cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p}) - R_{3}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} + ϕ_{R 3}^{s, p})}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}_{}^{2}}

\begin{array}{c} \sum_{n = 0}^{\infty} [R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})]^{n} \times \sum_{m = 0}^{\infty} {[R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})]}^{m} = \sum_{k = 0}^{\infty} [{(R_{3}^{s, p})}^{2 k} + 2 \sum_{p = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 p ϕ_{R 3}^{s, p})] + \\ \sum_{k = 0}^{\infty} [2 \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 p + 1) ϕ_{R 3}^{s, p}] \end{array}

\begin{array}{l} \sum_{k = 0}^{\infty} [{(R_{3}^{s, p})}^{2 k} + 2 \sum_{p = 1}^{k} {(R_{3}^{s, p})}^{2 k} \cos (2 p ϕ_{R 3}^{s, p}) + 2 \sum_{p = 0}^{k} {(R_{3}^{s, p})}^{2 k + 1} \cos [(2 p + 1) ϕ_{R 3}^{s, p}] \\ = \frac{1}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}_{}^{2}} \end{array}

\begin{array}{c} I^{s, p} = (R_{1}^{s, p})^{2} \\ + \frac{{(R_{2}^{s, p})}^{2} + 2 R_{1}^{s, p} R_{2}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p}) - 2 R_{1}^{s, p} R_{2}^{s, p} R_{3}^{s, p} \cos (ϕ_{R 1}^{s, p} - ϕ_{R 2}^{s, p} + ϕ_{R 3}^{s, p})}{1 - 2 R_{3}^{s, p} \cos (ϕ_{R 3}^{s, p}) + {(R_{3}^{s, p})}^{2}} \end{array}

r^{s, p} = \frac{R_{1}^{s, p} \exp (i ϕ_{R 1}^{s, p}) - R_{1}^{s, p} R_{3}^{s, p} \exp [i (ϕ_{R 1}^{s, p} + ϕ_{R 3}^{s, p})] + R_{2}^{s, p} \exp (i ϕ_{R 2}^{s, p})}{1 - R_{3}^{s, p} \exp (i ϕ_{R 3}^{s, p})} .

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusion

Appendix A

Appendix B

Acknowledgements

References and links

Cited By

Figures (4)

Equations (51)

Optics Express