Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo
.pdfPARAXIAL WAVE EQUATION FOR INHOMOGENEOUS MEDIA |
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12.2HELMHOLTZ EQUATION FOR INHOMOGENEOUS MEDIA
The Helmholtz equation for homogeneous media in which the index of refraction is constant was derived in Section 4.2. In this section, this is extended to inhomogeneous media. Let uðx; y; z; tÞ represents the wave field at position ðx; y; zÞ and at time t in a medium with the refractive index nðx; y; zÞ with the wavelength l and angular frequency o. According to Maxwell’s equations, it must satisfy the scalar wave equation
r2uðx; y; z; tÞ þ n2ðx; y; zÞk2uðx; y; z; tÞ ¼ 0 |
ð12:2-1Þ |
where k ¼ o=c ¼ 2p=l.
A general solution to the scalar wave equation can be written in the form of
uðx; y; z; tÞ ¼ Uðx; y; zÞ cosðot þ ðx; y; zÞÞ |
ð12:2-2Þ |
where Uðx; y; zÞ is the amplitude and ðx; y; zÞ is the phase at position ðx; y; zÞ. In complex notation, Eq. (12.2-2) is written as
uðx; y; z; tÞ ¼ Re½Uðx; y; zÞ expð jotÞ& |
ð12:2-3Þ |
where Uðx; y; zÞ is the complex amplitude equal to jUðx; y; zÞj expð j ðx; y; zÞÞ. Substituting uðx; y; z; tÞ from Eq. (12.2-3) into the wave equation (12.2-1) yields
the Helmholtz equation in inhomogeneous media:
ðr2 þ k2ðx; y; zÞÞUðx; y; zÞ ¼ 0 |
ð12:2-4Þ |
where the position-dependent wave number kðx; y; zÞ is given by
kðx; y; zÞ ¼ nðx; y; zÞk0 |
ð12:2-5Þ |
12.3 PARAXIAL WAVE EQUATION FOR INHOMOGENEOUS MEDIA
The paraxial wave equation for homogenous media was discussed in Section 5.4. In this section, it is extended to inhomogeneous media. Suppose that the variation of index of refraction is given by
nðx; y; zÞ ¼ n þ nðx; y; zÞ |
ð12:3-1Þ |
where n is the average index of refraction. The Helmholtz equation (12.2-4) becomes
½r2 þ n2k02 þ 2n nk02&U ¼ 0 |
ð12:3-2Þ |
where ð nÞ2k2 term has been neglected.
190 WAVE PROPAGATION IN INHOMOGENEOUS MEDIA
If the field is assumed to be propagating mainly along the z-direction, it can be expressed as
Uðx; y; zÞ ¼ U0 |
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ð12:3-3Þ |
ðx; y; zÞe jkz |
0ð Þ where U x; y; z is assumed to be a slowly varying function of z, and k equals nk0.
Substituting Eq. (12.3-3) in the Helmholtz equation yields
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12:3-4 |
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As U0ðx; y; zÞ is a slowly varying function of z, Eq. (12.3-3) can be approximated as
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ð12:3-5Þ |
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Equation (12.3-5) is the paraxial wave equation for inhomogeneous media. Equation (12.3-3) allows rapid phase variation with respect to the z-variable to be
factored out. Then, the slowly varying field in the z-direction can be numerically represented along z that can be much coarser than the wavelength for many problems. The elimination of the second derivative term in z in Eq. (12.3-5) also reduces the problem from a second order boundary value problem requiring iteration or eigenvalue analysis, to a first order initial value problem that can be solved by simple ‘‘integration’’ along the z-direction. This also means a time reduction of computation by a factor of at least the number of longitudinal grid points.
The disadvantages of this approach are limiting consideration to fields that propagate primarily along the z-axis (i.e., paraxiality) and also placing restrictions on the index variation, especially the rate of change of index with z. The elimination of the second derivative with respect to z also eliminates the possibility for backward traveling wave solutions. This means devices for which reflection is significant cannot be sufficiently modeled.
12.4BEAM PROPAGATION METHOD
The beam propagation method is a numerical method for approximately simulating optical wave propagation in inhomogeneous media. There are a number of different versions of the BPM method. The one discussed in this section has the constraints that reflected waves can be neglected and all refractive index differences are small [Marcuse].
The BPM can be derived by starting with the angular spectrum method (ASM) discussed in Section 4.3, which is for constant index of refraction, say, n. The BPM
BEAM PROPAGATION METHOD |
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Figure 12.1. Modeling of the inhomogeneous medium for implementing BPM.
can be conceived as a model consisting of homogeneous sections terminated by virtual lenses as shown in Figure 12.1.
The BPM is derived by assuming that the wave is governed in small sections of width z by diffraction as in a homogeneous medium of index n, but with different amounts of phase shift between the sections generated by virtual lenses, modeling the refractive index inhomogeneity. The additional phase shift can be considered to be due to ½nðx; y; zÞ n&. By allowing z to be small, two steps are generated: (1) Wave propagation in a homogeneous medium of index n, which is the average index in the current section, computed with the ASM and (2) the virtual lens effect, which is discussed below.
The effectiveness of this method depends on the choice of z at each step to be small enough to achieve the desired degree of accuracy within a reasonable amount of time.
12.4.1Wave Propagation in Homogeneous Medium with Index n
In each section, the medium is initially assumed to be homogeneous with index of
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refraction n.Then, k also equals k. The wave propagation in such a section, say, with |
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ðfx þ fy Þ so that the evanescent waves are excluded, Uðx; y; zÞ is |
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k2 4p2ð fx2 þ fy2ÞzÞ |
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expð j2pð fxx þ fyyÞÞdfxdfy |
ð12:4-1Þ |
where Að fx; fy; z zÞ is the initial plane wave spectrum given by Eq. (4.3-2). The whole process can be written as
Uðx; y; zÞ ¼ F 1fFfUðx; y; 0Þg expðjmzÞg |
ð12:4-2Þ |
WAVE PROPAGATION IN A DIRECTIONAL COUPLER |
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Figure 12.6. BPM simulation of power exchange between nonsynchronous wave guides.
12.5.2.2 Case 2: Nonsynchronous Wave guides. In the nonsynchronous case, the mismatch n1 ¼6 n2 makes b ¼ b1 b2 ¼6 0. The power of the propagating wave in wave guide 2 is obtained as
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2sin2ðgzÞ# |
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ja2ðzÞj2 ¼ jA1j2 |
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ð12:5-8Þ |
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The term g ¼ ð b=2Þ2 þ C2 does not allow the quantity in the bracket to be equal to 1. Hence, complete power exchange between the two guides cannot be accomplished at any z as shown in Figure 12.5. The corresponding BPM simulation gives the same results as shown in Figure 12.6.
These results show that the BPM gives highly accurate results as compared with the coupled mode theory in these applications. In more complicated designs such as the nonperiodic grating-assisted directional coupler, the coupled mode theory cannot be used, and the BPM is the method of choice for reliable analysis and subsequent design [Pojanasomboon, Ersoy, 2001].