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.

©2010 Optical Society of America

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 [46]. 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 [79].

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 [1016] and is used in many applications, such as laser attenuator [10], Fabry-Perot etalon [1113], 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 [1114]. 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 [1725].

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)=E0ex2y2w2ei(kzz2πνt)
where E0 is the amplitude of the electric field, w represents the width of the wave packet in the x-y plane [26] and kz 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 θo 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:

Δx=dsin(2θ0)n12sin2θ0.

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 cos2θmax = 2n1[(n1 2-1)1/2-n1] + 1. The incidence angle θmax, 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 θmax, 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 Es,p(x,y) can be written as (the superscript s and p refer to s- and p-polarized wave packets, respectively):

Es,p(x,y)=E0r01s,pex2y2w2+E0[1(r01s,p)2]r12s,peiδn=0[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
where rij 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 Is,p of s- and p-polarized wave packets can be defined as:
Is,p=++Es,pEs,pdxdy++EEdxdy.
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).
Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e(n+1)2Δx22w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e(2q)2Δx22w2+2(R2s,p)2k=0q=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e(2q+1)2Δx22w2
where
R1s,peiϕ1s,p=r01s,p,R2s,peiϕ2s,p=[1(r01s,p)2]r12s,peiδ,R3s,peiϕ3s,p=r01s,pr12s,peiδ
Under the condition in which Δx << w or Δx≈0, the exponential terms in the expression of Is,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:
Is,p=(R1s,p)2+(R2s,p)2+2R1s,pR2s,pcos(ϕR1s,pϕR2s,p)2R1s,pR2s,pR3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
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 Is,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):

Is,p=(R1s,p)2+(R2s,p)21(R3s,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].

rs,p=r01s,p+r12s,peiδ1+r01s,pr12s,peiδ,rprs=tanΨeiΔ.

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 Ψshift and Δshift can be given as:

tanΨshift=IpIs,cosΔshift=IpsIsIp.
where Ip,s and Ips are the measured intensity of reflected light for the pure and cross p-s components, respectively, with consideration of the lateral shift effect. Ip,s can be obtained from Eq. (5), and the cross term of p-s intensity Ips is defined as:

Ips=++Re[Es(Ep)]dxdy++EEdxdy.

The standard deviation M expressed by Eq. (12) can be used to check the consistency of the calculated ellipsometric data of Ψcal and Δcal with comparison to the experimental ones of Ψexp and Δexp in data reduction procedure.

M=(ΨexpΨcal)2+(ΔexpΔcal)2.

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

Ips=R1sR1pcos(ϕR1sϕR1p)+R1sR2pn=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e(n+1)2Δx22w2+R1pR2sn=0(R3s)ncos(ϕR1pϕR2snϕR3s)e(n+1)2Δx22w2+R2sR2pk=0(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]+R2sR2pk=0q=1k(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]e(2q)2Δx22w2+R2sR2pk=0q=0k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]e(2q+1)2Δx22w2+R2sR2pk=0q=0k(R3s)kq(R3p)k+q+1cos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]e(2q+1)2Δx22w2

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. Is,p and Ips 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)=E0ex2+y2w2ei(kzz2πνt)
and the intensity of incident wave packets is defined as:
Iin=++EEdxdy=I0w2π/2
where
I0=++E0E0dxdy.
The electric field Eres,pof reflected wave packets can be expressed as.
Es,p(x,y)=E0r01s,pex2y2w2+E0n=0[1(r01s,p)2]r12s,peiδ[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Then, the normalized intensities of the pure and cross p-s polarized wave packet, Ip,s and Ips, can be defined as
Is,p=1Iin++Es,p(Es,p)dxdy,Ips=1Iin++Re[Es(Ep)]dxdy.
In order to get the value of the integration of Eq. (A5), Eq. (A4) is departed into two parts:
E1s,p=E0r01s,pex2y2w2E2s,p=E0n=0[1(r01s,p)2]r12s,peiδ[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Since
Es,p=E1s,p+E2s,p,
thus the first integrand in Eq. (A5) can be written as
Es,pEs,p*=|E1s,p|2+E1s,p(E2s,p)+(E1s,p)*E2s,p+|E2s,p|2
Since
Re[Es(Ep)]=12[Es(Ep)+(Es)Ep]
the second integrand in Eq. (A5) can be written as
Re[Es(Ep)]=12[E1s(E1p)+(E1s)E1p+E2s(E1p)+(E2s)E1p)]+12[E1s(E2p)+(E1s)E2p+E2s(E2p)+(E2s)E2p)]
Defined the coefficient
R1s,peiϕ1s,p=r01s,p,R2s,peiϕ2s,p=[1(r01s,p)2]r12s,peiδ,R3s,peiϕ3s,p=r01s,pr12s,peiδ.
Using Eq. (A6), the terms in Eq. (A9) can be written as:
|E1s,p|2=(R1s,p)2e2(x2+y2)w2
E1s,pE2s,p*+E1s,p*E2s,p=ex2+y2w2n=02R1s,pR2s,p(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e[x(n+1)Δx]2y2w2
We defined
NL=R2s,pexp(iϕR2s,p)[R3s,pexp(iϕR3s,p)]Le(x(L+1)Δx)2y2w2
then
|E2s,p|2=L=0NL×q=0Nq
Using the sum formula [13]
L=0NL×q=0Nq=k=0NkNk+k=0q=1k(NkqNk+q+Nk+qNkq)+k=0q=0k(Nk+1+qNkq+NkqNk+q+1)
we get the last term of Eq. (A9)
|E2s,p|2=k=0(R2s,p)2(R3s,p)2ke2[x(k+1)Δx]2+2y2w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2+2(R2s,p)2k=0p=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e[x(k+q+2)Δx]2+y2w2e[x(kq+1)Δx]2+y2w2]
Using Eq. (A6), the terms in Eq. (A10) can be written as:
E1sE1p+E1sE1p=2R1sR1pe2(x2+y2)w2cos(ϕR1sϕR1p)
E2sE1p+E2sE1p=2R1pR2sex2+y2w2n=0(R3s)ncos[ϕR1p+ϕR2s+nϕR3s]e[x(n+1)Δx]2+y2w2
E1sE2p+E1sE2p=2R1sR2pex2+y2w2n=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e[x(n+1)Δx]2+y2w2
We defined
ML=R2sexp(iϕR2s)[R3sexp(iϕR3s)]Le[x(L+1)Δx]2+y2w2Nq=R2pexp(iϕR2p)[R3pexp(iϕR3p)]qe[x(q+1)Δx]2+y2w2
then the last term of Eq. (A10) can be expressed as
E2sE2p+E2sE2p=L=0ML×q=0Nq*+L=0ML×q=0Nq
Using the sum formula [13]
L=0ML×q=0Nq*=k=0[MkNk*]+k=0q=1k(MkqNk+q+Mk+qNkq)+k=0q=0k(Mk+q+1Nkq+MkqNk+q+1)L=0ML×q=0Nq=k=0[NkMk*]+k=0q=0k(NkqMk+q+Nk+qMkq)+k=0q=0k(Nk+1+qMkq+NkqMk+q+1)
the last term in Eq. (A10) can be written as:
E2sE2p+E2sE2p=k=02R2sR2p(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]e2[x(k+1)Δx]2+2y2w2+2R2sR2pk=0q=1k{(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]×e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2}+2R2sR2pk=0q=1k{(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]×e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2+2R2sR2pk=0q=0k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]×e(x(kq+1)Δx)2+y2w2e(x(k+q+2)Δx)2+y2w2+2R2sR2pk=0q=0k(R3p)k+p+1(R3s)kpcos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]×e(x(kq+1)Δx)2+y2w2e(x(k+q+2)Δx)2+y2w2
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:
Is,p=1Iin[++|E1s,p|2dxdy+++(E1s,pE2s,p+E1s,pE2s,p)dxdy+++|E2s,p|2dxdy]Ireps=1Iin[++(E1sE1p+E1sE1p)dxdy+++(E2sE1p+E2sE1p)dxdy++++(E1sE2p+E1sE2p)dxdy+++(E2sE2p+E2sE2p)dxdy]
Using the following standard integrals
exp(2x2w2)dx=wπ2exp(ax2)cos(2bx)dx=πaexp(b2a)exp(ax2)sin(2bx)dx=0exp(2x2+y2w2)dxdy=w2π2
the normalized reflected intensities of the pure and cross p-s polarized wave packets can be written as:

Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e(n+1)2Δx22w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e(2q)2Δx22w2+2(R2s,p)2k=0q=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e(2q+1)2Δx22w2
Ips=R1sR1pcos(ϕR1sϕR1p)+R1sR2pn=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e(n+1)2Δx22w2+R1pR2sn=0(R3s)ncos(ϕR1pϕR2snϕR3s)e(n+1)2Δx22w2+R2sR2pk=0(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]+R2sR2pk=0q=1k(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]e(2q+1)2Δx22w2+R2sR2pk=0q=1k(R3s)kq(R3p)k+q+1cos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]e(2q+1)2Δx22w2

Appendix B

This appendix details the calculation of the normalized reflected intensity Ip,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 Is,p can be expressed as:

Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k.
Using the equation:
k=0(R3s,p)2k=11(R3s,p)2,
the normalized reflected intensity Is,p can be expressed as:
Is,p=(R1s,p)2+(R2s,p)21(R3s,p)2.
When Δx ≈0, the exponential terms in Eq. (5) becomes unity, and then the normalized reflected intensity Is,p becomes
Is,p=(R1s,p)2+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)+(R2s,p)2k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)+2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]
Because
n=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)=12n=0(R3s,p)n[exp[i(ϕR1s,pϕR2s,pnϕR3s,p)]+12n=0(R3s,p)n[exp[i(ϕR1s,pϕR2s,pnϕR3s,p)]
the first sum term in Eq. (B4) can be simplified as
n=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)]=cos(ϕR1s,pϕR2s,p)R3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
From Eq. (A22), we can get
n=0[R3s,pexp(iϕR3s,p)]n×m=0[R3s,pexp(iϕR3s,p)]m=k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)]+k=0[2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]
Then the second sum term in Eq. (B4) can be simplified as
k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)+2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]=112R3s,pcos(ϕR3s,p)+(R3s,p)2
Using Eqs. (B2), (B6) and (B8), therefore, the normalized reflected intensity Is,p becomes
Is,p=(R1s,p)2+(R2s,p)2+2R1s,pR2s,pcos(ϕR1s,pϕR2s,p)2R1s,pR2s,pR3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
which can be reduced to the Airy formula of Eq. (9) and can be written as Is,p = |rs,p|2 and:

rs,p=R1s,pexp(iϕR1s,p)R1s,pR3s,pexp[i(ϕR1s,p+ϕR3s,p)]+R2s,pexp(iϕR2s,p)1R3s,pexp(iϕR3s,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).

References and links

1. M. Born, and E. Wolf, Principles of Optics, 6th. ed., (Pergamon, UK, 1980), Chaps. 1 and 7.

2. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]  

3. K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 437(1-2), 87–102 (1948). [CrossRef]  

4. T. Tamir and H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61(10), 1397–1413 (1971). [CrossRef]  

5. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3(4), 558–565 (1986). [CrossRef]  

6. F. Falco and T. Tamir, “Improved analysis of nonspecular phenomena in beams reflected from stratified media,” J. Opt. Soc. Am. A 7(2), 185–190 (1990). [CrossRef]  

7. Z. M. Zhang and B. J. Lee, “Lateral shift in photon tunneling studied by the energy streamline method,” Opt. Express 14(21), 9963–9970 (2006). [CrossRef]   [PubMed]  

8. L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005). [CrossRef]  

9. G. Dolling, M. W. Klein, M. Wegener, A. Schädle, B. Kettner, S. Burger, and S. Linden, “Negative beam displacements from negative-index photonic metamaterials,” Opt. Express 15(21), 14219–14227 (2007). [CrossRef]   [PubMed]  

10. P. Gregorčič, A. Babnik, and J. Možina, “Interference effects at a dielectric plate applied as a high-power-laser attenuator,” Opt. Express 18(4), 3871–3882 (2010). [CrossRef]   [PubMed]  

11. E. Nichelatti and G. Salvetti, “Spatial and spectral response of a Fabry–Perot interferometer illuminated by a Gaussian beam,” Appl. Opt. 34(22), 4703–4712 (1995). [CrossRef]   [PubMed]  

12. F. Moreno and F. Gonzalez, “Transmission of a Gaussian beam of low divergence through a high-finesse Fabry–Perot device,” J. Opt. Soc. Am. A 9(12), 2173–2175 (1992). [CrossRef]  

13. D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry–Perot microcavity for sensing applications,” J. Opt. Soc. Am. A 22(8), 1577–1589 (2005). [CrossRef]  

14. J. Poirson, T. Lanternier, J. C. Cotteverte, A. L. Floch, and F. Bretenaker, “Jones matrices of a quarter-wave plate for Gaussian beams,” Appl. Opt. 34(30), 6806–6818 (1995). [CrossRef]   [PubMed]  

15. J. Poirson, J.-C. Cotteverte, A. Le Floch, and F. Bretenaker, “Internal reflections of the Gaussian beams in Faraday isolators,” Appl. Opt. 36(18), 4123–4130 (1997). [CrossRef]   [PubMed]  

16. M. Y. Sheng, Y. H. Wu, S. Z. Feng, Y. R. Chen, Y. X. Zheng, and L. Y. Chen, “Spatial effect on the interference of light propagated in a film structure,” Appl. Opt. 46(28), 7049–7053 (2007). [CrossRef]   [PubMed]  

17. M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999). [CrossRef]  

18. V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005). [CrossRef]  

19. M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000). [CrossRef]   [PubMed]  

20. S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004). [CrossRef]   [PubMed]  

21. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71(2), S274–S282 (1999). [CrossRef]  

22. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003). [CrossRef]   [PubMed]  

23. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000). [CrossRef]   [PubMed]  

24. Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999). [CrossRef]  

25. A. Mekis, J. C. Chen, I. Kurland I, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996). [CrossRef]   [PubMed]  

26. J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts and Company Publishs, Inc, 2005), Chap. 3.

27. J. D. Levin, An Introduction to Quantum Theory, (Cambridge University Express, 2002), Chap. 7.

28. J. D. Jackson, Classical Electrodynamics, 3rd ed., (John Wiley & Sons, Inc. 1999) p. 308.

29. L. Y. Chen, X. W. Feng, Y. Su, Y. Han, H. Z. Ma, and Y. H. Qian, “Design of a scanning ellipsometer by synchronous ro- tation of polarizer and analyzer,” Appl. Opt. 33(7), 1299–1305 (1994). [CrossRef]   [PubMed]  

30. R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1985).

31. W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Annalen der Physik und Chemie 219(6), 272–282 (1871). [CrossRef]  

32. G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys. 69(11), 7627–7634 (1991). [CrossRef]  

References

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  • |

  1. M. Born, and E. Wolf, Principles of Optics, 6th. ed., (Pergamon, UK, 1980), Chaps. 1 and 7.
  2. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
    [Crossref]
  3. K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
    [Crossref]
  4. T. Tamir and H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61(10), 1397–1413 (1971).
    [Crossref]
  5. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3(4), 558–565 (1986).
    [Crossref]
  6. F. Falco and T. Tamir, “Improved analysis of nonspecular phenomena in beams reflected from stratified media,” J. Opt. Soc. Am. A 7(2), 185–190 (1990).
    [Crossref]
  7. Z. M. Zhang and B. J. Lee, “Lateral shift in photon tunneling studied by the energy streamline method,” Opt. Express 14(21), 9963–9970 (2006).
    [Crossref] [PubMed]
  8. L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005).
    [Crossref]
  9. G. Dolling, M. W. Klein, M. Wegener, A. Schädle, B. Kettner, S. Burger, and S. Linden, “Negative beam displacements from negative-index photonic metamaterials,” Opt. Express 15(21), 14219–14227 (2007).
    [Crossref] [PubMed]
  10. P. Gregorčič, A. Babnik, and J. Možina, “Interference effects at a dielectric plate applied as a high-power-laser attenuator,” Opt. Express 18(4), 3871–3882 (2010).
    [Crossref] [PubMed]
  11. E. Nichelatti and G. Salvetti, “Spatial and spectral response of a Fabry–Perot interferometer illuminated by a Gaussian beam,” Appl. Opt. 34(22), 4703–4712 (1995).
    [Crossref] [PubMed]
  12. F. Moreno and F. Gonzalez, “Transmission of a Gaussian beam of low divergence through a high-finesse Fabry–Perot device,” J. Opt. Soc. Am. A 9(12), 2173–2175 (1992).
    [Crossref]
  13. D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry–Perot microcavity for sensing applications,” J. Opt. Soc. Am. A 22(8), 1577–1589 (2005).
    [Crossref]
  14. J. Poirson, T. Lanternier, J. C. Cotteverte, A. L. Floch, and F. Bretenaker, “Jones matrices of a quarter-wave plate for Gaussian beams,” Appl. Opt. 34(30), 6806–6818 (1995).
    [Crossref] [PubMed]
  15. J. Poirson, J.-C. Cotteverte, A. Le Floch, and F. Bretenaker, “Internal reflections of the Gaussian beams in Faraday isolators,” Appl. Opt. 36(18), 4123–4130 (1997).
    [Crossref] [PubMed]
  16. M. Y. Sheng, Y. H. Wu, S. Z. Feng, Y. R. Chen, Y. X. Zheng, and L. Y. Chen, “Spatial effect on the interference of light propagated in a film structure,” Appl. Opt. 46(28), 7049–7053 (2007).
    [Crossref] [PubMed]
  17. M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999).
    [Crossref]
  18. V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
    [Crossref]
  19. M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
    [Crossref] [PubMed]
  20. S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
    [Crossref] [PubMed]
  21. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71(2), S274–S282 (1999).
    [Crossref]
  22. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
    [Crossref] [PubMed]
  23. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
    [Crossref] [PubMed]
  24. Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
    [Crossref]
  25. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
    [Crossref] [PubMed]
  26. J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts and Company Publishs, Inc, 2005), Chap. 3.
  27. J. D. Levin, An Introduction to Quantum Theory, (Cambridge University Express, 2002), Chap. 7.
  28. J. D. Jackson, Classical Electrodynamics, 3rd ed., (John Wiley & Sons, Inc. 1999) p. 308.
  29. L. Y. Chen, X. W. Feng, Y. Su, Y. Han, H. Z. Ma, and Y. H. Qian, “Design of a scanning ellipsometer by synchronous ro- tation of polarizer and analyzer,” Appl. Opt. 33(7), 1299–1305 (1994).
    [Crossref] [PubMed]
  30. R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1985).
  31. W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Annalen der Physik und Chemie 219(6), 272–282 (1871).
    [Crossref]
  32. G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys. 69(11), 7627–7634 (1991).
    [Crossref]

2010 (1)

2007 (2)

2006 (1)

2005 (3)

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005).
[Crossref]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
[Crossref]

D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry–Perot microcavity for sensing applications,” J. Opt. Soc. Am. A 22(8), 1577–1589 (2005).
[Crossref]

2004 (1)

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
[Crossref] [PubMed]

2003 (1)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

2000 (2)

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
[Crossref] [PubMed]

1999 (3)

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999).
[Crossref]

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71(2), S274–S282 (1999).
[Crossref]

1997 (1)

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

1995 (2)

1994 (1)

1992 (1)

1991 (1)

G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys. 69(11), 7627–7634 (1991).
[Crossref]

1990 (1)

1986 (1)

1971 (1)

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

1871 (1)

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Annalen der Physik und Chemie 219(6), 272–282 (1871).
[Crossref]

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Babnik, A.

Bertoni, H. L.

Beskrovnyy, V. N.

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
[Crossref]

Blanco, A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Bretenaker, F.

Burger, S.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Chen, L. Y.

Chen, Y. R.

Chomski, E.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Cotteverte, J. C.

Cotteverte, J.-C.

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Dolling, G.

Fabre, C.

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
[Crossref] [PubMed]

Falco, F.

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Feng, S.

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
[Crossref] [PubMed]

Feng, S. Z.

Feng, X. W.

Floch, A. L.

Foteinopolou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Gonzalez, F.

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Grabtchak, S.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Gregorcic, P.

Guo, D.

Han, Y.

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Hirlimann, C.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Hönerlage, B.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Ibisate, M.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Jellison, G. E.

G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys. 69(11), 7627–7634 (1991).
[Crossref]

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

John, S.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Kettner, B.

Klein, G.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Klein, M. W.

Kolobov, M. I.

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
[Crossref]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
[Crossref] [PubMed]

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999).
[Crossref]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Lanternier, T.

Le Floch, A.

Lee, B. J.

Leonard, S. W.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Lin, R.

Linden, S.

Lopez, C.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Ma, H. Z.

Mandel, L.

L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71(2), S274–S282 (1999).
[Crossref]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Meseguer, F.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Miguez, H.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Mondia, J. P.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Moreno, F.

Možina, J.

Nichelatti, E.

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Ozin, G. A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Petit, S.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Pfister, O.

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
[Crossref] [PubMed]

Poirson, J.

Qian, Y. H.

Salvetti, G.

Schädle, A.

Sellmeier, W.

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Annalen der Physik und Chemie 219(6), 272–282 (1871).
[Crossref]

Sheng, M. Y.

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

Su, Y.

Tamir, T.

Toader, O.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

van Driel, H. M

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Vlasov, Y. A.

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Wang, L.-G.

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005).
[Crossref]

Wang, W.

Wegener, M.

Wu, Y. H.

Zhang, Z. M.

Zheng, Y. X.

Zhu, S.-Y.

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005).
[Crossref]

Ann. Phys. (2)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

Annalen der Physik und Chemie (1)

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Annalen der Physik und Chemie 219(6), 272–282 (1871).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

L.-G. Wang and S.-Y. Zhu, “Large negative lateral shifts from the Kretschmann–Raether configuration with left-handed materials,” Appl. Phys. Lett. 87(22), 221102 (2005).
[Crossref]

J. Appl. Phys. (1)

G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys. 69(11), 7627–7634 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Nature (1)

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000).
[Crossref] [PubMed]

Opt. Express (3)

Phys. Rev. A (1)

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of superresolution in reconstruction of optical objects,” Phys. Rev. A 71(4), 043802 (2005).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

Y. A. Vlasov, S. Petit, G. Klein, B. Hönerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 1030–1035 (1999).
[Crossref]

Phys. Rev. Lett. (4)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Hight transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, S. Foteinopolou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003).
[Crossref] [PubMed]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85(18), 3789–3792 (2000).
[Crossref] [PubMed]

S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92(20), 203601 (2004).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71(2), S274–S282 (1999).
[Crossref]

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999).
[Crossref]

Other (5)

M. Born, and E. Wolf, Principles of Optics, 6th. ed., (Pergamon, UK, 1980), Chaps. 1 and 7.

R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, Amsterdam, 1985).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts and Company Publishs, Inc, 2005), Chap. 3.

J. D. Levin, An Introduction to Quantum Theory, (Cambridge University Express, 2002), Chap. 7.

J. D. Jackson, Classical Electrodynamics, 3rd ed., (John Wiley & Sons, Inc. 1999) p. 308.

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Figures (4)

Fig. 1
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.
Fig. 2
Fig. 2 The incidence angle θmax, 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.
Fig. 3
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.
Fig. 4
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).

Equations (51)

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E(x,y,z,t)=E0ex2y2w2ei(kzz2πνt)
Δx=dsin(2θ0)n12sin2θ0.
Es,p(x,y)=E0r01s,pex2y2w2+E0[1(r01s,p)2]r12s,peiδn=0[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Is,p=++Es,pEs,pdxdy++EEdxdy.
Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e(n+1)2Δx22w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e(2q)2Δx22w2+2(R2s,p)2k=0q=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e(2q+1)2Δx22w2
R1s,peiϕ1s,p=r01s,p,R2s,peiϕ2s,p=[1(r01s,p)2]r12s,peiδ,R3s,peiϕ3s,p=r01s,pr12s,peiδ
Is,p=(R1s,p)2+(R2s,p)2+2R1s,pR2s,pcos(ϕR1s,pϕR2s,p)2R1s,pR2s,pR3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
Is,p=(R1s,p)2+(R2s,p)21(R3s,p)2.
rs,p=r01s,p+r12s,peiδ1+r01s,pr12s,peiδ,rprs=tanΨeiΔ.
tanΨshift=IpIs,cosΔshift=IpsIsIp.
Ips=++Re[Es(Ep)]dxdy++EEdxdy.
M=(ΨexpΨcal)2+(ΔexpΔcal)2.
Ips=R1sR1pcos(ϕR1sϕR1p)+R1sR2pn=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e(n+1)2Δx22w2+R1pR2sn=0(R3s)ncos(ϕR1pϕR2snϕR3s)e(n+1)2Δx22w2+R2sR2pk=0(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]+R2sR2pk=0q=1k(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]e(2q)2Δx22w2+R2sR2pk=0q=0k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]e(2q+1)2Δx22w2+R2sR2pk=0q=0k(R3s)kq(R3p)k+q+1cos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]e(2q+1)2Δx22w2
E(x,y,z,t)=E0ex2+y2w2ei(kzz2πνt)
Iin=++EEdxdy=I0w2π/2
I0=++E0E0dxdy
Es,p(x,y)=E0r01s,pex2y2w2+E0n=0[1(r01s,p)2]r12s,peiδ[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Is,p=1Iin++Es,p(Es,p)dxdy,Ips=1Iin++Re[Es(Ep)]dxdy
E1s,p=E0r01s,pex2y2w2E2s,p=E0n=0[1(r01s,p)2]r12s,peiδ[r01s,pr12s,peiδ]ne[x(n+1)Δx]2y2w2
Es,p=E1s,p+E2s,p
Es,pEs,p*=|E1s,p|2+E1s,p(E2s,p)+(E1s,p)*E2s,p+|E2s,p|2
Re[Es(Ep)]=12[Es(Ep)+(Es)Ep]
Re[Es(Ep)]=12[E1s(E1p)+(E1s)E1p+E2s(E1p)+(E2s)E1p)]+12[E1s(E2p)+(E1s)E2p+E2s(E2p)+(E2s)E2p)]
R1s,peiϕ1s,p=r01s,p,R2s,peiϕ2s,p=[1(r01s,p)2]r12s,peiδ,R3s,peiϕ3s,p=r01s,pr12s,peiδ
|E1s,p|2=(R1s,p)2e2(x2+y2)w2
E1s,pE2s,p*+E1s,p*E2s,p=ex2+y2w2n=02R1s,pR2s,p(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e[x(n+1)Δx]2y2w2
NL=R2s,pexp(iϕR2s,p)[R3s,pexp(iϕR3s,p)]Le(x(L+1)Δx)2y2w2
|E2s,p|2=L=0NL×q=0Nq
L=0NL×q=0Nq=k=0NkNk+k=0q=1k(NkqNk+q+Nk+qNkq)+k=0q=0k(Nk+1+qNkq+NkqNk+q+1)
|E2s,p|2=k=0(R2s,p)2(R3s,p)2ke2[x(k+1)Δx]2+2y2w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2+2(R2s,p)2k=0p=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e[x(k+q+2)Δx]2+y2w2e[x(kq+1)Δx]2+y2w2]
E1sE1p+E1sE1p=2R1sR1pe2(x2+y2)w2cos(ϕR1sϕR1p)
E2sE1p+E2sE1p=2R1pR2sex2+y2w2n=0(R3s)ncos[ϕR1p+ϕR2s+nϕR3s]e[x(n+1)Δx]2+y2w2
E1sE2p+E1sE2p=2R1sR2pex2+y2w2n=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e[x(n+1)Δx]2+y2w2
ML=R2sexp(iϕR2s)[R3sexp(iϕR3s)]Le[x(L+1)Δx]2+y2w2Nq=R2pexp(iϕR2p)[R3pexp(iϕR3p)]qe[x(q+1)Δx]2+y2w2
E2sE2p+E2sE2p=L=0ML×q=0Nq*+L=0ML×q=0Nq
L=0ML×q=0Nq*=k=0[MkNk*]+k=0q=1k(MkqNk+q+Mk+qNkq)+k=0q=0k(Mk+q+1Nkq+MkqNk+q+1)L=0ML×q=0Nq=k=0[NkMk*]+k=0q=0k(NkqMk+q+Nk+qMkq)+k=0q=0k(Nk+1+qMkq+NkqMk+q+1)
E2sE2p+E2sE2p=k=02R2sR2p(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]e2[x(k+1)Δx]2+2y2w2+2R2sR2pk=0q=1k{(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]×e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2}+2R2sR2pk=0q=1k{(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]×e[x(kq+1)Δx]2+y2w2e[x(k+q+1)Δx]2+y2w2+2R2sR2pk=0q=0k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]×e(x(kq+1)Δx)2+y2w2e(x(k+q+2)Δx)2+y2w2+2R2sR2pk=0q=0k(R3p)k+p+1(R3s)kpcos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]×e(x(kq+1)Δx)2+y2w2e(x(k+q+2)Δx)2+y2w2
Is,p=1Iin[++|E1s,p|2dxdy+++(E1s,pE2s,p+E1s,pE2s,p)dxdy+++|E2s,p|2dxdy]Ireps=1Iin[++(E1sE1p+E1sE1p)dxdy+++(E2sE1p+E2sE1p)dxdy++++(E1sE2p+E1sE2p)dxdy+++(E2sE2p+E2sE2p)dxdy]
exp(2x2w2)dx=wπ2exp(ax2)cos(2bx)dx=πaexp(b2a)exp(ax2)sin(2bx)dx=0exp(2x2+y2w2)dxdy=w2π2
Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)e(n+1)2Δx22w2+2(R2s,p)2k=0q=1k(R3s,p)2kcos(2qϕR3s,p)e(2q)2Δx22w2+2(R2s,p)2k=0q=0k(R3s,p)2k+1cos[(2q+1)ϕR3s,p]e(2q+1)2Δx22w2
Ips=R1sR1pcos(ϕR1sϕR1p)+R1sR2pn=0(R3p)ncos(ϕR1sϕR2pnϕR3p)e(n+1)2Δx22w2+R1pR2sn=0(R3s)ncos(ϕR1pϕR2snϕR3s)e(n+1)2Δx22w2+R2sR2pk=0(R3sR3p)kcos[ϕR2sϕR2p+k(ϕR3sϕR3p)]+R2sR2pk=0q=1k(R3s)kq(R3p)k+qcos[(ϕR2sϕR2p)+(kq)ϕR3s(k+q)ϕR3p]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q(R3p)kqcos[(ϕR2pϕR2s)+(kq)ϕR3p(k+q)ϕR3s]e(2q)2Δx22w2+R2sR2pk=0q=1k(R3s)k+q+1(R3p)kqcos[(ϕR2pϕR2s)(k+q+1)ϕR3s+(kq)ϕR3p]e(2q+1)2Δx22w2+R2sR2pk=0q=1k(R3s)kq(R3p)k+q+1cos[(ϕR2sϕR2p)(k+q+1)ϕR3p+(kq)ϕR3s]e(2q+1)2Δx22w2
Is,p=(R1s,p)2+(R2s,p)2k=0(R3s,p)2k.
k=0(R3s,p)2k=11(R3s,p)2,
Is,p=(R1s,p)2+(R2s,p)21(R3s,p)2.
Is,p=(R1s,p)2+2R1s,pR2s,pn=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)+(R2s,p)2k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)+2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]
n=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)=12n=0(R3s,p)n[exp[i(ϕR1s,pϕR2s,pnϕR3s,p)]+12n=0(R3s,p)n[exp[i(ϕR1s,pϕR2s,pnϕR3s,p)]
n=0(R3s,p)ncos(ϕR1s,pϕR2s,pnϕR3s,p)]=cos(ϕR1s,pϕR2s,p)R3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
n=0[R3s,pexp(iϕR3s,p)]n×m=0[R3s,pexp(iϕR3s,p)]m=k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)]+k=0[2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]
k=0[(R3s,p)2k+2p=1k(R3s,p)2kcos(2pϕR3s,p)+2p=0k(R3s,p)2k+1cos[(2p+1)ϕR3s,p]=112R3s,pcos(ϕR3s,p)+(R3s,p)2
Is,p=(R1s,p)2+(R2s,p)2+2R1s,pR2s,pcos(ϕR1s,pϕR2s,p)2R1s,pR2s,pR3s,pcos(ϕR1s,pϕR2s,p+ϕR3s,p)12R3s,pcos(ϕR3s,p)+(R3s,p)2
rs,p=R1s,pexp(iϕR1s,p)R1s,pR3s,pexp[i(ϕR1s,p+ϕR3s,p)]+R2s,pexp(iϕR2s,p)1R3s,pexp(iϕR3s,p).

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