Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

In FileFig 6.5a we calculate the reflected intensity R symbolically and in FileFig 6.5b we calculate it numerically. One finds values close to 1 for N 10, 20, 40 or more. Such multiple layer mirrors, consisting of a large number of thin films of alternating high and low refractive indices, are used in laser cavities to reduce losses.

FileFig 6.5a (N5STRASYMa)

Symbolic calculation of the reflected intensity for multilayer reflection coatings.

FileFig 6.5b (N5STRANUMb)

Numerical calculation for nH 2.5, nL 1.5, and N 20.

N5STRASYMa and N5STRASYMb are only on the CD.

Application 6.7. Calculate the reflected intensity (R) and transmitted intensity (T 1 − R) for N 10, 20, 40, 100 for chosen values of nL and nH .

6.3GUIDED WAVES BY TOTAL INTERNAL REFLECTION THROUGH A PLANAR WAVEGUIDE

6.3.1 Traveling Waves

When laser light was applied to telecommunication, one looked at the possibilities of light traveling through some type of guide. Travel through the open air resulted in too many losses. Long wavelength electromagnetic waves travel through metal cables, but microwaves may travel inside a rectangular waveguide. These waveguides have parallel reflecting metal surfaces, and the wavelength of a traveling mode is characterized by the dimensions of the rectangular crosssection. As the first step for propagation of modes of laser light one considered a dielectric film of refractive index n1 with refractive indices n2 and n3 equal to 1 on the outside. Later, dielectric fibers were used for guiding the light, discussed in the next section.

In Chapter 2 we discussed the modes of a Fabry–Perot. The incident light wave was traveling perpendicular to the planes, and a relation between the wavelength and the distance between the planes characterized the modes. We now look at very similar modes, formed inside a plane parallel dielectric film but traveling parallel to the boundaries of the plate.

FIGURE 6.3 Thin film of refractive index n1 larger than the indices n2 and n3 of the outside media. Propagation is in the X direction and mode formation in the Y direction. There are exponential decreasing solutions in the outside medial of indices n2 and n3.

We consider three layers of dielectric materials extending in the X to Z direction and stacked up in the Y direction. We assume that the layer in the middle has thickness d and refractive index n1. Above and below are two other dielectric materials with refractive indices n2 and n3, both smaller than n1 (Figure 6.3). A wave in the layer with refractive index n1 is totally reflected on the two interfaces, above and below, and travels effectively in the X direction.

When treating the plane parallel plate (see Chapter 2) we summed up all the light reflected and transmitted at the different interfaces and found resonance conditions corresponding to modes. Summing up all the reflected and transmitted light is called the summation method in contrast to the boundary value method, used in Section 6.2 to describe the modes for multilayer dielectric material. To treat the problem of waves traveling with internal total reflection in a dielectric layer, we apply another method, the traveling wave method. For comparison, the boundary value method is given in the appendix.

We assume a wave is traveling in the X direction in medium (1) between the two media 2 and 3. We assume that the angle to the normal θ is larger than the critical angle, in order to have total reflection. A ray is launched from a point inside the plate and after reflection on each interface, the pattern repeats. If the component of the wave traveling in the Y direction has the same wave vector after one period of travel in the X direction, we have a traveling mode. This is only possible for discrete values of the angle θ, the angle corresponding to the direction of the k vector and the normal (Figure 6.3). We use complex notation for presenting the wave traveling between internal reflections in the X, Y plane

where λ is the wavelength in free space, ω the frequency, t the time, and n1 the refractive index in medium (1). Equation (6.36) is a function of X and Y and

6.3.3 Phase Condition for Mode Formation

We define the phase change upon reflection on medium 2, (1 → 2), as 1,2 and for reflection on medium 3, (1 → 3), as 1,3. Considering a round trip in the Y direction, we have for the phase shift: on medium 2: (2πd/λ1) 1,2, and on medium 3: 2πd/λ1 + 1,3. The sum must have values of 2πm which is written

This is the resonance condition for the mode, involving the k values in the Y direction. The phase values are calculated from Fresnel’s formulas depending on the Y components of k. The resonance conditions are different for the s- polarization (TE) modes and the p-polarization (TM) modes, and are discussed separately in Sections 6.3.4 and 6.3.5.

6.3.4 (TE) Modes or s-Polarization

The reflection coefficient on the interfaces 1,2 and 1,3 are obtained from Fresnel’s formulas (see Chapter 5),

We multiply by 2π/λ and introduce the definitions of the k vectors. Since the

or using the formula tan−1 A + tan−1 B tan−1(A + B)/(1 − AB) we may write

For the numerical calculation we prefer to write the condition of Eq. (6.49) as

In FileFig 6.6 we calculate the condition of Eq. (6.50) using

and at the crossover point of the graph the resonance condition is fulfilled. The crossover point indicates the angle θ for the mode with mode number m. The characterization of the mode depends on the refractive indices, thickness d, and wavelength λ.

Resonance condition for s-polarization (TE). The crossover point indicates the angle θ of the mode with mode number m depending on the refraction indices, thickness d, and wavelength λ. The lowest number of the mode corresponds to the lowest curve.

N6PLSPS

Wave Traveling with Total Internal Reflection Through a Planar Waveguide

Resonance condition of s-polarization. Global definition of n1, n2, n3, d, and λ above the graph.

λ360

ys is for m 1, yys for m 2, and yyys for m 3. For these parameters the angle θ of the first three possible modes is determined:

ys(θ) : −ys1(θ) − ys3(θ) + π yys(θ) : −ys1(θ) − ys3(θ) + π · 2

yyys(θ) : −ys1(θ) − ys3(θ) + π · 3.

θ

θ

θ

θ

θ

At the crossover point of y with ys, yys, or yyys, respectively, the resonance condition is fulfilled. The functions ys, yys, and yyys are complex in the region from horizontal appearance to zero. This is shown in the next graph where the argument is plotted. The complex region has to be disregarded for the determination of the crossover point.

q

q

q

q

Application 6.6.

1.Change the refractive index and choose d and λ such that all three modes are possible.

2.Change the thickness d and choose n1 and λ such that all three modes are possible.

3.Change the wavelength λ and choose the refractive index n1 and the thickness d such that all three modes are possible.

6.3.5 (TM) Modes or p-Polarization

The coefficients of reflection on media 1 and 3 are obtained from Fresnel’s formulas. First multiply by 2π/λ and then introduce the definition of k. One can observe that the k value in the medium above and below is imaginary. One obtains complex numbers for the two reflection coefficients similarly as for the s-case

and at the crossover point of the graph the resonance condition is fulfilled. The crossover point indicates the angle θ for the mode with mode number m, depending on the refraction indices, thickness d, and wavelength λ. Given m, d, and λ, the k vector for the traveling mode is given and if real, the mode is possible.

Resonance condition for p-polarization (TM). The crossover point indicates the angle θ of the mode with mode number m, depending on the refraction indices, the thickness d, and wavelength λ. The lowest number of the mode corresponds to the lowest curve.

N7PLPPS is only on the CD.

Application 6.7.

1.Change the refractive index and choose d and λ such that all three modes are possible.

2.Change the thickness d and choose n1 and λ such that all three modes are possible.

3.Change the wavelength λ and choose the refractive index n1 and the thickness d such that all three modes are possible

4.Give an example for λ 0.00025 mm, d 0.0005 mm and show that we have for m 1 a cosine mode, and for m 2 a sine mode.

6.4FIBER OPTICS WAVEGUIDES

6.4.1 Modes in a Dielectric Waveguide

In Section 6.3 we discussed mode propagation in a dielectric film of thickness of several wavelengths and refractive index n2. The modes were guided by two outer dielectric media of refractive indices smaller than n1. The mode propagation was in the X direction. The media were stacked in the Y direction and extended in the Z direction without limits. A constant refractive index was assumed.

We now consider a dielectric fiber of radius a and homogeneous refractive index n1 in a surrounding medium of refractive index no (Figure 6.4). Following in general second Jackson (1975, p.364), we choose the direction of propagation in the z direction, but use refractive indices instead of dielectric constants. We start with the general wave equations

and call the differentiation with respect to the variables perpendicular to the direction of propagation the transverse Laplacian t − ∂2/∂z2.

Assuming periodic exponential solutions for the time dependence and the field in the z direction, one has the two constants (n1ω/c)2 and k2 after application of the second derivative. The ratio ω/c is equal to k1 in the dielectric and k is the wave vector for the wave traveling in the Z direction. For “inside” and “outside”

FIGURE 6.4 Coordinates for mode propagation in a fiber of radius a. The tranverse coordinates are ρ and φ and the modes propagate in the z-direction.

with β real. We use cylindrical coordinates ρ and φ, and assume that we have no azimuthal variation, which means there is no dependence on φ. We have two differential equations for the two transversal components Et and Bt , which have Bessel functions as solutions

We have Jo(γρ) as solutions for Et and Bt “inside,” and Ko(βρ) for Et and Bt “outside.” From Maxwell’s equations we obtain relations between the field components depending on the “transverse” coordinates ρ and φ and the components depending on z. The relations are divided into two groups, βρ and Eφ depending on Bz, and Bφ and Eρ depending on Ez.

These relations are for “inside”:

For “outside” one has a similar set of equations.

Here we treat only the TE modes, that is for a nonvanishing Bz component. One has explicitly

Eφ −(iωA/cβ)K1(βρ)

where A is a constant. Only the first two equations of Eqs. (6.70) and (6.71) are independent, and application of the boundary conditions at ρ a yields the equations:

Elimination of A results in the characteristic equation for the determination of k2, written in γ and β with both depending on k,

Since γ and β are both functions of k, we plot the right and the left sides of Eq. (6.74) on the same graph, with both depending on k. At the crossing of the curves we get the resulting value of k. This is shown in the second graph of FileFig 6.8. The “cutoff” frequency is obtained for Jo(γ a) 0, which is γ a 2.405

and the corresponding wavelength is λc {[ (n21 − n2o)a2π]/2.405}. At that wavelength β2 is 0 and k is equal to the free space value.

FileFig 6.8 (N8CWGK)

Determination of k for dielectric circular waveguide.

N8CWGK