1. Introduction

It is very important for increasing its recording density and improving its performance to develop a rigorous electromagnetic theory for a multilayered optical disc. In 1979, Hopkins created a classical scalar diffraction theory for a CD-ROM where bits were regarded as a period structure [1]. When bits become smaller and the numerical aperture of the objective becomes larger, Hopkins theory [1] will no longer give accurate results. As a result, various vector diffraction theories [1-7] were proposed. Utilizing the expansion of a vector frequency spectrum instead of a scalar frequency spectrum, Cheng, et al., [2] extended Hopkins theory to the vector theory. Brok, et al., [3] described a model of the readout of a DVD disk through decomposing the incident field into a sum of quasi-periodic incident fields and then computing the scattering of these fields by a one dimensional periodic grating. Yeh, et al., [4] calculated the effective groove depth in an optical disk with a rigorous diffraction grating program DELTA. The characteristics of these theories [1-4] are that bits are viewed as a oneor two-dimensional grating. In order to consider the nonperiodic case, a finite-difference timeor frequency-domain method was developed for calculating the electromagnetic fields in the neighborhood of bits [5,6]. In 2006, Nes, et al., [7] briefly introduced a volume integral method using dyadic Green’s functions, but the detection part was still described by a scalar treatment. Recently, Guo, et al., [8] created an electromagnetic theory for a waveguide multilayered optical memory and indicated its application in a conventional multilayered storage where the storage medium is homogenous and bits are recorded in the storage medium by layer when neglecting the fields reflected by the upper and lower surface of the storage medium.

For a multilayered optical disc, the storage medium is a planar multilayered media and the reflected fields due to the interface can not be omitted because the detected signal is equal to the coherent superposition at the detector of the reflected electric fields due to the interface and the scattering electric fields of bits. In this paper, we first derive in detail the vector diffraction models of the read-in and the readout systems, where the reflected electric fields resulting from the interface are considered, and then develop a full and rigorous vector diffraction model of a multilayered optical disc utilizing these vector diffraction models of the read-in and the readout systems together with some conclusions of Ref. [8]. As an example, the detected power is computed when the reading spot is scanned over the disc.

2. Vector diffraction model of a multilayered optical disc

As shown in Fig. 1, lenses L₁L₂ and L₂L₃ separately compose the read-in and the readout systems of a multilayered optical disc. For L₁L₂, the incident light is focused on the detected bit A in the multilayered optical disc that together with the air layers is divided into (N+1) layers by N planes perpendicular to the z axis. Dielectric interfaces are placed at z=z ₀,…, Z _N-1 and the 0th and Nth layers are semiinfinite homogenous. Bits may locate any layer of the multilayered optical disc. The laser is viewed as a point source O_o with the electric field δ(r)e _o(0)=x e _ox+y e_oy+z e _oz and the time dependence exp(jωt). Assume that the coordinate of the detected bit A is (x _o, 0, Z _A). As a stratified media results in spherical aberrations, the focus of the lens L₂ should be at the point B(x _o, 0, Z _B) where the size of the dispersed spot formed by virtually extended rays of the incident light is the smallest. At this time, only the virtual plane z=z _B and the source plane z=O _o are corrected for L₁L₂ so as to obey the sine condition, and for the virtual plane z=z _B the image space of L₁L₂ is homogenous. Therefore, one may first find the vector plane-wave spectrum (VPWS) of the incident fields at the virtual plane z=z _B that have been given by Eqs. (19) and (21) in Ref. [9], and then from which derive the VPWS of the incident electric field immediate to the Z=Z _N-1 interface at the Nth layer, namely

where d_s=Z _N-1-Z _B and the spherical polar coordinates with the origin at B have been used with 0≤θ_N ≤Φ_N and 0≤φ≤2π so s _N=(-sin θ_N cosφ, -sin θ_N sinφ, cosθ_N). Φ_N is the angular semiaperture on the image side of L₁L₂. θ_o is the emergent angle of the emergent ray s _o=x s _ox+y s _oy+z s _oz with the origin at O_o in the object space and cosθ _o=[1-(M ₁₂ n_N sinθ_N sin θ_N/n_o)²]^1/2, where M ₁₂=-(n_od_N)/(n_Nd_o) is the nominal magnification of the read-in system L₁L₂. For details, the reader is referred to Ref. [9]. Note that the positive z axis in Fig. 1 is opposite to that in Fig. 1 of Ref. [9]. In addition, the VPWS of the incident magnetic field at the plane z=z _N-1 is determined by the relation

 

Fig. 1. Schematic diagram of the read-in and the readout optical systems of a multilayered disc

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By Eq. (1) and the equation H(r)=j(ωµ ₀)^-1∇×E(r) the electromagnetic fields immediate to the z=z _N-1 interface at the Nth layer can be denoted as

where λ_N and k_N are separately the wavelength and the wave number at the Nth layer. The above fields can be viewed as the equivalent surface electric and magnetic current density [10]

For an arbitrary point P(x_p, y_p, z_p) in the mth layer of the multilayered optical disc, the electric field at the plane z=z_p due to J _s and M _s may be denoted as the following expression by the Green formalism:

where G⃡_EJ (r;r′) and G⃡_EM (r;r′) separately represent the electric dyadic Green’s function (DGF) at r(x,y,z_p) arising from unit point electric and magnetic current sources at r′(x′, y′, z _N-1) that here belong to the Nth layer.

In terms of the formula (44) for the electric DGF in Ref. [11] and the assumption that the Nth layer is semiinfinite homogenous, for propagation in a negative z direction, after a straight forward algebra operation, another form of the electric DGF may be written as

with

with

where ϒ^< _m,N refers to the effective transmission coefficient, and the double prime (″) and the single prime (′) separately denote TE and TM mode. The presentation of the actual form of the variables in Eq. (8) is omitted for brevity. For details, the reader is referred to Ref. [11]. According to Eqs. (27a) and (27c) in Ref. [12], for propagation in a negative z direction, we find readily G̃^< _EM(s _N)=-j G̃^< _EJ(s _N)×k _N with k _N=x k_x+y k_y-z κ_N. Therefore,

Upon substituting Eqs. (2)~(5) and (7) into Eq. (6), after a straight forward algebra operation and exchanging the order of integral, the VPWS of E _r(x, y, z _p) may be easily obtained, that is

where (·) represents (s _N; z _N-1) and the spectrum functions Ẽ _rx(·), Ẽ _ry(·), H̃_rx(·) and H̃_ry(·) are the transverse Cartesian components of the VPWSs Ẽ_r (s _N; z _N-1) and H̃_r (s _N; z _N-1). In terms of the boundary condition, the identity k ₀ s _0t=k ₁ s _1t=…=k_N s _Nt holds, so substitution of Eqs. (8) and (10), Ẽ_rx(·), Ẽ_ry(·), H̃_rx(·) and H̃_ry(·) into Eq. (11) yields readily the VPWS

Then for the point P(x_p, y_p, z_p,) with cylindrical polar coordinate (ρ, β, z_p) in the mth layer of the multilayered optical disc, a well-known procedure described in many prior publications [9,13] results in the following expressions for the Cartesian coordinate components of electric fields of the point P

with C ₁₂=-πλ ^-2 _N M ₁₂ and $A_n={\int }_0^{{\mathrm{\Phi }}_N}{\cos }^{-\frac{1}{2}}{\theta }_o{\cos }^{\frac{1}{2}}{\theta }_N\sin {\theta }_N{B}_n{J}_n\left(k_N\rho \sin {\theta }_N\right)\exp \left(j{k}_N\cos {\theta }_N{d}_s\right)\mathrm{d}{\theta }_N,$ with B ₀ = cos θ ₀ ϒ^″< _m,N ϕ ^″< _m+ + cos θ_N ϒ^′< _m,N ϕ^′< _m+, B ₁ = cos^-1 θ _m cosθ _N sin θ_m ϒ^′< _m,N ϕ ^′< _m-, B ₂ = cosθ ₀ ϒ^″< _m,N ×ϕ ^″< _m+ - cosθ _N ϒ^′< _m,N ϕ ^′< _m+, and J_n(•) being the Bessel function of the first kind and order n. Formulae (13) describe the incident electric field of the read-in system L₁L₂ at an arbitrary point in the multilayered optical disc. Next we shall find the electric field at the detector.

As indicated in Ref. [8], bits may be viewed as perturbations with permittivity Δε_p (r) confined to the inside of the multilayered optical disc. Hence the electric field of an arbitrary point in the multilayered optical disc may be computed from Lippman-Schwinger equation whose three-dimensional discretization is denoted as [8]

where W_k represents the volume of the kth discretized element, N_p denotes the discretized numbers of all bits, and V(r)=-k ² _mΔε^p(r) describes the perturbations. E _r(r) is the incident electric field determined by Eq. (13). In order to calculate Eq. (14), an iterative scheme based on the parallel use of Lippman-Schwinger and Dyson’s equations has been introduced in detail in Ref. [13]. Equation (14) is implicit for all points located inside the perturbations. Once the electric field E _p(r) inside the perturbations has been computed, it can be generated explicitly for the electric field at the upper surface z=z _N-1 of the multilayered optical disc of lights propagating in positive z direction by the expression E(r)|=E _r0(r)+E _s(r) with

and r=(x, y, z _N-1). E _s(r) is the scattered field of bits radiating in positive z direction. Physically E _r0(r) represents the emergent electric field of lights propagating in positive z direction due to the infinite number of bounces at the multilayered optical disc. In Eqs. (13), when the point P(ρ, β, z _p) is at the upper surface z _p=z _N-1 of the multilayered optical disc, there are m=N, ϒ^< _{m, N}=1 and ϕ ^< _m±=1±R ^< _N(z _N-1) in terms of Eq. (9). Here the first term of ϕ ^< _m± denotes physically the incident electric field at the upper surface z=z _N-1 of the multilayered optical disc of read-in lights and its second term represents E _r0(r). Therefore E _r0(r) can be computed by Eq. (13) if z_p=z _N-1, m=N, ϒ^< _m,N=1, and ϕ ^< _m±=±R ^< _N(z _N-1).

Obviously, the electric field at the detector is equal to the coherent superposition of the image fields at the detector of E _r0(r) and E _s(r). In order to find it, we must first determine the VPWS of the equivalent electric field at the virtual plane z=z _B from those of E _r0(r) and E _s(r), and then derive the electric field at the detector by the vector coherent transfer function (CTF) [14] of L₂L₃. Note that this process must be strictly performed in that, for any optical system, the sine condition is only satisfied for a fixed object plane and its image plane in practical cases and the precondition of current vector image theories [9,14].

If z_p=z _N-1, m=N, ϒ^< _m,N=1, ϕ ^< _m±=±R ^< _N(z _N-1) and d_s → 2d_s, Eq. (12) will denote the VPWS of the equivalent electric field at the virtual plane z=z _B of E _r0(r). With the vector CTF, the image field at the detector of E _r0(r) may be readily obtained, which has been introduced in detail in Ref. [14]. The final results are in the entirely same form as Eq. (13) except that C ₁₂, θ_N, Φ_N, k_N and d_s are separately substituted by C ₂₃=-jπλ _i ^-2 M ₂₃ M ₁₂, θ_i, Φ_i, k_i and 2d_s where M ₂₃=-(n_Nd_i)/(n_id_N) and Φ_i are separately the nominal magnification and the angular semiaperture on the image side of the readout system L₂L₃.

As for the scattered field E _s(r), the derivations of its image field at the detector have been given in detail in Ref. [8] and the final results are Eqs. (28)~(31) in Ref. [8] except that u, θ_o and z_o are separately substituted by N-1, θ_N and z_B. For interests, the reader is referred to Ref. [8]. Note that the term cos^-3/2 θ_o should be cos^-1/2 θ_o in Eq. (29b) in Ref. [8] because of a printer’s error.

3. Detected power when the reading spot is scanned over the disc

Without loss of generality, we take an example for a monolayer optical disc shown in Fig. 2 to calculate the detected power (See Fig. 3) when the reading spot is scanned over the disc. Here, the thickness and refractive index of every layer of the monolayer optical disc are separately h ₁=h ₃=4µm, h ₂=1.5µm, n ₁=n ₃=1.406 and n ₂=1.58. A standard bit representing the binary code “1” is a cube with size of 0.5µm and the stacking factor is 0.5. Other parameters are the refractive index n=1.406 of bits, the wavelength λ=0.65µm, the numerical aperture NA=0.65 of L₂, the diameter R=4mm of L₂, M ₂₃=1/M ₁₂=40, and the output power of laser P=10 mw. The polarization of the read-in light is circularly polarized. The discretized numbers of a standard bit is2×2×2. The diameter and discretized numbers of the pinhole are separately 30µm and 20×20. The numbers of sampling are 80 along the scanning position. The relativity precision of numerical integral is 10^-6. The bits compose a binary code “0101110110”. Obviously, Fig. 3 shows that the detected power varies accurately with the sequence of the binary code “0101110110”.

 

Fig. 2. Structure of a monolayer optical disc.

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Fig. 3. The detected power when the reading spot is scanned over the disc. Here, the bits compose of a binary code “0101110110”.

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4. Conclusion

In conclusion, vector diffraction models of the read-in and the readout systems of a multilayered optical disc are introduced in detail. Utilizing these models together with some conclusions of Ref. [8], a full and rigorous vector diffraction model for a multilayered optical disc, namely the formulae (13) and the formulae (28)~(34) in Ref. [8], has been described. Compared with the previous theories [1-7], the model in this paper is fuller and more rigorous because the vector diffraction of the readout system and the reflected electric field resulting from the infinite number of bounces at the multilayered optical disc are also considered. Moreover, the nominal magnification of read-in and the readout systems are involved, which usually is not referred. The computer simulation demonstrates the validity of this model that hence can be used effectively for optimum design of the multilayered optical disc.