4. Numerical results

In this section, numerical results of electromagnetic wave transmission in a twodimensional slab waveguide based on the original and corrected FDTD methods are compared with the analytical result.

regions whose indices are nco and ncl, respectively. The core region is extended infinitely in the y- and z-directions and has width d in the x-direction. The cladding region is the rest of space. An electromagnetic wave propagates in the z-direction, and its electromagnetic field is assumed to have no y dependence. Because the system has no y dependence, it is essentially a two-dimensional system. An analyt-

> 2ux d

> > �wð Þ <sup>2</sup>jxj�<sup>d</sup>

� � sin ð Þ <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>z</sup> <sup>∣</sup>x∣ ≤ <sup>d</sup>

� � cosð Þ <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>z</sup> <sup>∣</sup>x∣ ≤ <sup>d</sup>

�w2∣x∣�<sup>d</sup>

� � sin ð Þ <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>z</sup> <sup>∣</sup>x∣ ≤ <sup>d</sup>

<sup>d</sup> sin ð Þ <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>z</sup> <sup>∣</sup>x<sup>∣</sup> <sup>&</sup>gt; <sup>d</sup>

<sup>d</sup> sin ð Þ <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>z</sup> <sup>∣</sup>x<sup>∣</sup> <sup>&</sup>gt; <sup>d</sup>

<sup>y</sup> ð Þ¼ t; x; y; z 0, (50)

<sup>d</sup> cosð Þ <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>z</sup> <sup>∣</sup>x<sup>∣</sup> <sup>&</sup>gt; <sup>d</sup>

<sup>x</sup> ð Þ¼ t; x; y; z 0, (52)

<sup>z</sup> ð Þ¼ t; x; y; z 0, (54)

2

, (49)

2

2

, (51)

2

, (53)

2

2

ical solution is known and is derived in the appendix. The solution is

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations

DOI: http://dx.doi.org/10.5772/intechopen.81338

cosð Þ u e

d

signð Þ x sin ð Þ u e

2ux d

<sup>h</sup><sup>0</sup> cosð Þ <sup>u</sup> <sup>e</sup>�wð Þ <sup>2</sup>jxj�<sup>d</sup>

Hanl

Hanl

h<sup>0</sup> cos

Eanl

coε0<sup>d</sup> sin <sup>2</sup>ux

βh<sup>0</sup> ωn<sup>2</sup> coε<sup>0</sup> cos

8 >>>><

>>>>:

2uh<sup>0</sup> ωn<sup>2</sup>

8 >>>><

>>>>:

<sup>y</sup> ð Þ¼ t; x; y; z

2wh<sup>0</sup> ωn<sup>2</sup> clε0d

> 8 >>><

> >>>:

βh<sup>0</sup> ωn<sup>2</sup> clε<sup>0</sup>

Eanl

Figure 4. Calculation domain.

Eanl

75

<sup>z</sup> ð Þ¼ t; x; y; z

Hanl

<sup>x</sup> ð Þ¼ t; x; y; z

Figure 3 shows the slab waveguide used in the computational methods, and Figure 4 shows its calculation domain. This system consists of core and cladding

Figure 3. Slab waveguide used in numerical calculation.

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations DOI: http://dx.doi.org/10.5772/intechopen.81338

Figure 4. Calculation domain.

In addition, Eq. (40) can also be rewritten as

þ ∑ m, <sup>n</sup> σ�<sup>1</sup>

> 1 2

2 

<sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup>ΔzÞ � Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

1 2 

Δx; y1; z<sup>1</sup> þ ð Þ n þ 1 ΔzÞ þ Hzðt0; x<sup>1</sup> þ m þ

<sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>zÞ � Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

Then, the algorithm of the corrected FDTD method, which is supported by the next-to-the-lowest-order approximation, can be obtained by using Eqs. (48) and

In this section, numerical results of electromagnetic wave transmission in a twodimensional slab waveguide based on the original and corrected FDTD methods are

Figure 3 shows the slab waveguide used in the computational methods, and Figure 4 shows its calculation domain. This system consists of core and cladding

<sup>þ</sup> iyðt0; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup>1<sup>Þ</sup> <sup>Δ</sup>xΔ<sup>z</sup>

ð Þ x1; z1; x<sup>1</sup> þ mΔx; z<sup>1</sup> þ nΔz

ΔzÞ � Hxðt0; x1þ

mΔx; y

2 

<sup>Δ</sup>zÞ � Hxðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>m</sup> � <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup>

<sup>1</sup>; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup>

Δx; y1; z<sup>1</sup> þ nΔzÞ

ΔzÞ þ Hxðt0; x<sup>1</sup> þ ð Þ m � 1 Δx; y1; z<sup>1</sup> þ n þ

1 2 

2 

2 

ΔzÞ Δx

Δz 

2 

Δx; y1; z<sup>1</sup> þ ð Þ n � 1 Δz

Δx; y1; z<sup>1</sup> þ ð Þ n � 1 ΔzÞÞΔz

1 2 

> ΔzÞ Δx

Δz

 Δt

(48)

<sup>Δ</sup>xΔ<sup>z</sup> <sup>¼</sup> Dy <sup>t</sup><sup>0</sup> � <sup>Δ</sup>t=2; <sup>x</sup>1; <sup>y</sup>1; <sup>z</sup><sup>1</sup>

Hxðt0; x<sup>1</sup> þ mΔx; y1; z<sup>1</sup> þ n þ

Hxðt0; x<sup>1</sup> þ ð Þ m þ 1 Δx; y1; z<sup>1</sup> þ n þ

1 2 

� Hxðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>x; <sup>y</sup>1; <sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup> � <sup>1</sup>

1 2 

2 

Dy t<sup>0</sup> þ Δt=2; x1; y1; z<sup>1</sup>

Recent Advances in Integral Equations

� Hzðt0; x<sup>1</sup> þ m þ

� Hzðt0; x<sup>1</sup> þ m þ

� Hzðt0; <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> � <sup>1</sup>

<sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>3</sup> :

(47) repeatedly.

Figure 3.

74

4. Numerical results

compared with the analytical result.

Slab waveguide used in numerical calculation.

� <sup>11</sup> 12

þ 1 24

regions whose indices are nco and ncl, respectively. The core region is extended infinitely in the y- and z-directions and has width d in the x-direction. The cladding region is the rest of space. An electromagnetic wave propagates in the z-direction, and its electromagnetic field is assumed to have no y dependence. Because the system has no y dependence, it is essentially a two-dimensional system. An analytical solution is known and is derived in the appendix. The solution is

$$E\_{\mathbf{x}}^{\text{unf}}(t, \mathbf{x}, \boldsymbol{y}, \boldsymbol{z}) = \begin{cases} \frac{\beta h\_0}{\alpha m\_{co}^2 \varepsilon\_0} \cos\left(\frac{2\mu \mathbf{x}}{d}\right) \sin\left(\alpha t - \beta \mathbf{z}\right) & |\mathbf{x}| \le \frac{d}{2} \\\\ \frac{\beta h\_0}{\alpha m\_{cl}^2 \varepsilon\_0} \cos\left(u\right) e^{-\frac{\mathbf{x} \cdot (2|\mathbf{x}| - d)}{d}} \sin\left(\alpha t - \beta \mathbf{z}\right) & |\mathbf{x}| > \frac{d}{2} \end{cases},\tag{49}$$
 
$$E\_{\mathbf{y}}^{\text{unf}}(t, \mathbf{x}, \mathbf{y}, \mathbf{z}) = \mathbf{0},\tag{50}$$

$$E\_x^{\text{anl}}(t, \mathbf{x}, \mathbf{y}, \mathbf{z}) = \begin{cases} \frac{2uh\_0}{\alpha m\_{\text{oc}}^2 \varepsilon\_0 d} \sin\left(\frac{2\mu \mathbf{x}}{d}\right) \cos\left(\alpha t - \beta \mathbf{z}\right) & |\mathbf{x}| \le \frac{d}{2} \\\\ \frac{2wh\_0}{\alpha m\_{\text{cl}}^2 \varepsilon\_0 d} \text{sign}(\mathbf{x}) \sin\left(\mathbf{z}\right) e^{-w\frac{2\alpha t - d}{d}} \cos\left(\alpha t - \beta \mathbf{z}\right) & |\mathbf{x}| > \frac{d}{2} \end{cases},\tag{51}$$

$$H\_{\mathbf{x}}^{ml}(t,\mathbf{x},\mathbf{y},\mathbf{z}) = \mathbf{0},\tag{52}$$

$$H\_{\mathcal{Y}}^{\text{anl}}(t, \mathbf{x}, \mathbf{y}, \mathbf{z}) = \begin{cases} h\_0 \cos\left(\frac{2u\mathbf{x}}{d}\right) \sin\left(\alpha t - \beta \mathbf{z}\right) & |\mathbf{x}| \le \frac{d}{2} \\\\ h\_0 \cos\left(u\right) e^{-u\frac{(2|\mathbf{x}| - d)}{d}} \sin\left(\alpha t - \beta \mathbf{z}\right) & |\mathbf{x}| > \frac{d}{2} \end{cases} \tag{53}$$

Hanl <sup>z</sup> ð Þ¼ t; x; y; z 0, (54) where "anl" indicates that this is an analytical solution. u and w satisfy

$$w = \left(\frac{n\_{cl}}{n\_{co}}\right)^2 u \tan\left(u\right),\tag{55}$$

The blue curve shows Eq. (56). At each intersection of curves Eqs. (55) and (56), there is an independent symmetric mode satisfying Eq. (58), and at each of the curves Eq. (59) and (56), there is an independent antisymmetric mode satisfying Eq. (60). The mode with the lowest u is called fundamental mode. The number of modes of the

In the computational methods, the system parameters are set with the wavelength λ as 0:30m, the core width d as 0:30m, the same with the wavelength, the core index nco as 2:0, and the cladding index ncl as 1:0. The lengths of the cell edges Δx and Δz are both λ=20, and the time step Δt is 10�<sup>12</sup> s. With these parameter values, the parameter values of the analytical solution in Eqs. (49)–(54) can be

> u ¼ 1:50, (61) w ¼ 5:23, (62) v ¼ 5:44, (63)

<sup>2</sup><sup>π</sup> <sup>¼</sup> <sup>1</sup>:94, (64)

<sup>d</sup> sin ð Þ <sup>ω</sup><sup>t</sup> <sup>∣</sup>x<sup>∣</sup> <sup>&</sup>gt; <sup>d</sup>

2

, (65)

2

system is determined by V and increases by one with respect to each π=2.

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations

DOI: http://dx.doi.org/10.5772/intechopen.81338

βλ

2ux d

<sup>h</sup>0signð Þ <sup>x</sup> cosð Þ <sup>u</sup> <sup>e</sup>�wð Þ <sup>2</sup>jxj�<sup>d</sup>

are integer multiples of 2π. In these figures, violet curves represent Hy=h0, and green curves represent core and cladding regions. The region in which the value is 0.5 is the core region with index 2.0, and the region in which the value is �0.5 is the cladding region with index 1.0. In the figures, time goes downward. The time values are 1:0, 5:0, 10:0, and 20:0 ns. The left-hand column is calculated using the original

h<sup>0</sup> cos

8 >>><

>>>:

with the parameter values in Eqs. (61) and (62).

Hy calculated using original FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

where the LHS of Eq. (64) is called the effective index, a value between ncl and nco. These parameter values show that the solution is the fundamental mode. A

Figures 6–14 are numerical and analytical results at times at which the ωt values

� � sin ð Þ <sup>ω</sup><sup>t</sup> <sup>∣</sup>x∣ ≤ <sup>d</sup>

derived as

Figure 6.

77

magnetic field is excited at z ¼ 0 as

Hyð Þ¼ t; x; 0; 0

$$
\omega^2 = \mu^2 + \omega^2 = \frac{(n\_{co}^2 - n\_{cl}^2)d^2\pi^2}{\lambda^2},
\tag{56}
$$

where λ is the wavelength in a vacuum, ω is the angular frequency, and β is the propagation constant. The propagation constant is the propagation directional component of wave number vector and calculated as

$$\beta = \frac{2\pi\sqrt{n\_{co}^2 w^2 + n\_{cl}^2 u^2}}{v\lambda}. \tag{57}$$

v defined by Eq. (56) is called the V-parameter, which is determined by the parameters defining the system. u and w are determined using Figure 5. In the figure, red curves represent Eq. (55) which is symmetric under the parity transformation x↦ � x as

$$H\_{\mathcal{Y}}(t, -\infty, y, z) = H\_{\mathcal{Y}}(t, \infty, y, z). \tag{58}$$

Brown curves represent

$$w = -\frac{n\_{cl}^2}{n\_{co}^2} u \cot(u),\tag{59}$$

which is antisymmetric under the parity transformations x↦ � x as

$$H\_{\mathcal{Y}}(\mathbf{t}, -\mathbf{x}, \mathbf{y}, \mathbf{z}) = -H\_{\mathcal{Y}}(\mathbf{t}, \mathbf{x}, \mathbf{y}, \mathbf{z}).\tag{60}$$

Figure 5. Graphs of Eqs. (55), (59), and (56) to determine u and w.

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations DOI: http://dx.doi.org/10.5772/intechopen.81338

The blue curve shows Eq. (56). At each intersection of curves Eqs. (55) and (56), there is an independent symmetric mode satisfying Eq. (58), and at each of the curves Eq. (59) and (56), there is an independent antisymmetric mode satisfying Eq. (60). The mode with the lowest u is called fundamental mode. The number of modes of the system is determined by V and increases by one with respect to each π=2.

In the computational methods, the system parameters are set with the wavelength λ as 0:30m, the core width d as 0:30m, the same with the wavelength, the core index nco as 2:0, and the cladding index ncl as 1:0. The lengths of the cell edges Δx and Δz are both λ=20, and the time step Δt is 10�<sup>12</sup> s. With these parameter values, the parameter values of the analytical solution in Eqs. (49)–(54) can be derived as

$$
u = \text{1.50},\tag{61}$$

$$w = 5.23,\tag{62}$$

$$v = 5.44,\tag{63}$$

$$\frac{\beta\lambda}{2\pi} = 1.94,\tag{64}$$

where the LHS of Eq. (64) is called the effective index, a value between ncl and nco. These parameter values show that the solution is the fundamental mode. A magnetic field is excited at z ¼ 0 as

$$H\_{\mathcal{T}}(t, \mathbf{x}, \mathbf{0}, \mathbf{0}) = \begin{cases} h\_0 \cos\left(\frac{2\mu\mathbf{x}}{d}\right) \sin\left(\alpha t\right) & |\mathbf{x}| \le \frac{d}{2} \\\\ h\_0 \text{sign}(\mathbf{x}) \cos\left(\mathbf{u}\right) e^{-\frac{\mathbf{u}(2\mathbf{x}) - d}{d}} \sin\left(\alpha t\right) & |\mathbf{x}| > \frac{d}{2} \end{cases},\tag{65}$$

with the parameter values in Eqs. (61) and (62).

Figures 6–14 are numerical and analytical results at times at which the ωt values are integer multiples of 2π. In these figures, violet curves represent Hy=h0, and green curves represent core and cladding regions. The region in which the value is 0.5 is the core region with index 2.0, and the region in which the value is �0.5 is the cladding region with index 1.0. In the figures, time goes downward. The time values are 1:0, 5:0, 10:0, and 20:0 ns. The left-hand column is calculated using the original

Figure 6. Hy calculated using original FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

where "anl" indicates that this is an analytical solution. u and w satisfy

co � <sup>n</sup><sup>2</sup> cl � �d<sup>2</sup>

where λ is the wavelength in a vacuum, ω is the angular frequency, and β is the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

clu<sup>2</sup>

Hyð Þ¼ t; �x; y; z Hyð Þ t; x; y; z : (58)

Hyð Þ¼� t; �x; y; z Hyð Þ t; x; y; z : (60)

cow<sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup>

propagation constant. The propagation constant is the propagation directional

q

<sup>w</sup> ¼ � <sup>n</sup><sup>2</sup> cl n2 co

which is antisymmetric under the parity transformations x↦ � x as

n2

v defined by Eq. (56) is called the V-parameter, which is determined by the parameters defining the system. u and w are determined using Figure 5. In the figure, red curves represent Eq. (55) which is symmetric under the parity transfor-

π2

u tan ð Þ u , (55)

<sup>v</sup><sup>λ</sup> : (57)

u cotð Þ u , (59)

<sup>λ</sup><sup>2</sup> , (56)

<sup>w</sup> <sup>¼</sup> ncl nco � �<sup>2</sup>

<sup>v</sup><sup>2</sup> <sup>¼</sup> <sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>w</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>

β ¼ 2π

component of wave number vector and calculated as

mation x↦ � x as

Figure 5.

76

Graphs of Eqs. (55), (59), and (56) to determine u and w.

Brown curves represent

Recent Advances in Integral Equations

Figure 7. Hy calculated using corrected FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Figure 11.

Figure 10.

Figure 12.

79

Hy analytically calculated at <sup>t</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Hy calculated using corrected FDTD method at <sup>t</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations

DOI: http://dx.doi.org/10.5772/intechopen.81338

Hy calculated using original FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup> second.

Figure 8. Hy analytically calculated at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Figure 9. Hy calculated using original FDTD method at <sup>t</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations DOI: http://dx.doi.org/10.5772/intechopen.81338

Figure 10. Hy calculated using corrected FDTD method at <sup>t</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Figure 11. Hy analytically calculated at <sup>t</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Figure 12. Hy calculated using original FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup> second.

Figure 8.

Figure 7.

Figure 9.

78

Hy analytically calculated at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Hy calculated using corrected FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Recent Advances in Integral Equations

Hy calculated using original FDTD method at <sup>t</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>9</sup> second.

Figure 13.

Hy calculated using corrected FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup> second.

Figure 14. Hy analytically calculated at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup> second.

FDTD method, the middle column is calculated using the corrected FDTD method, and the right-hand column is the analytical solution wherein the region

$$
\alpha \text{ot} < \beta \text{z},
\tag{66}
$$

which shows the error between the FDTD and the analytical calculations at each

Figure 15 shows the err functions of the original and corrected FDTD methods defined by Eq. (67). As shown in the figure, almost every time except for less than 0.14 ns, the err function of the corrected FDTD method is less than that of the original. This means that the corrected method is more accurate than that of the original. In addition, when the time is greater than 6 ns, both curves begin to oscillate. The amplitude of the oscillation of the corrected FDTD method is clearly less than that of the original. This indicates that the corrected method is

time. In Eq. (67), the denominator of the right-hand side is proportional to the power of the propagating electromagnetic wave passing through the x � y plane per

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations

In this chapter, a higher-order correction to the original FDTD method supported by the next-to-the-lowest-order approximation of the integral form of Maxwell's equation was shown. The essence of this method is the approximation of integrals over a cell surface and edge using discretized electric and magnetic fields. The results of numerical calculations of an electromagnetic wave propagating in a two-dimensional slab waveguide using the corrected and original FDTD methods and analysis were also shown. The differences between the corrected and original FDTD methods were compared using the err function, and the corrected method

The author would like to thank Mr. A. Okabe, who is the collaborator of reference [2], which is based on this chapter. The author would also like to thank Enago

was found to be more accurate and reliable than the original.

(www.enago.com) for the English language review.

unit length of the y-direction.

err tð Þ of the original and corrected FDTD methods.

DOI: http://dx.doi.org/10.5772/intechopen.81338

more reliable.

Figure 15.

5. Conclusion

Acknowledgements

81

Hz is zero. Moreover, the differences in the results between the FDTD calculations and those of the analytical ones make it clear that ∣Hy=h0∣ at some points exceeds one in the FDTD calculations, even though the values at any point are equal to or less than 1 in the analytical results. However, the differences in the calculation results between the original and corrected FDTD methods are unclear. This indicates that it is impossible to conclude whether the corrected FDTD method is better than the original one using these figures.

To compare the accuracy and reliability of the original and corrected FDTD methods, we use a function err tð Þ defined as

$$err(t) = \frac{\sum\_{p\_\succ q} \frac{H\_p^{mm}(t, p\Delta x, 0, q\Delta x) - H\_p^{ml}(t, p\Delta x, 0, q\Delta x)}{n(p\Delta x, 0, q\Delta x)^2}}{\sum\_{p\_\succ q} \frac{H\_p^{ml}(t, p\Delta x, 0, q\Delta x)^2}{n(p\Delta x, 0, q\Delta x)^2}},\tag{67}$$

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations DOI: http://dx.doi.org/10.5772/intechopen.81338

Figure 15. err tð Þ of the original and corrected FDTD methods.

which shows the error between the FDTD and the analytical calculations at each time. In Eq. (67), the denominator of the right-hand side is proportional to the power of the propagating electromagnetic wave passing through the x � y plane per unit length of the y-direction.

Figure 15 shows the err functions of the original and corrected FDTD methods defined by Eq. (67). As shown in the figure, almost every time except for less than 0.14 ns, the err function of the corrected FDTD method is less than that of the original. This means that the corrected method is more accurate than that of the original. In addition, when the time is greater than 6 ns, both curves begin to oscillate. The amplitude of the oscillation of the corrected FDTD method is clearly less than that of the original. This indicates that the corrected method is more reliable.

## 5. Conclusion

FDTD method, the middle column is calculated using the corrected FDTD method,

Hz is zero. Moreover, the differences in the results between the FDTD calculations and those of the analytical ones make it clear that ∣Hy=h0∣ at some points exceeds one in the FDTD calculations, even though the values at any point are equal to or less than 1 in the analytical results. However, the differences in the calculation results between the original and corrected FDTD methods are unclear. This indicates that it is impossible to conclude whether the corrected FDTD method is better

To compare the accuracy and reliability of the original and corrected FDTD

<sup>y</sup> <sup>ð</sup>t;pΔx;0;qΔzÞ�Hanl

Hanl

n pð Þ <sup>Δ</sup>x;0;qΔ<sup>z</sup> <sup>2</sup>

<sup>y</sup> ð Þ <sup>t</sup>;pΔx;0;qΔ<sup>z</sup> <sup>2</sup> n pð Þ <sup>Δ</sup>x;0;qΔ<sup>z</sup> <sup>2</sup>

ωt < βz, (66)

<sup>y</sup> ð Þ t;pΔx;0;qΔz

, (67)

and the right-hand column is the analytical solution wherein the region

than the original one using these figures.

Hy analytically calculated at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup> second.

Hy calculated using corrected FDTD method at <sup>t</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup> second.

Recent Advances in Integral Equations

Figure 14.

80

Figure 13.

methods, we use a function err tð Þ defined as

err tðÞ¼

∑p, <sup>q</sup>

Hnum

∑p, <sup>q</sup>

In this chapter, a higher-order correction to the original FDTD method supported by the next-to-the-lowest-order approximation of the integral form of Maxwell's equation was shown. The essence of this method is the approximation of integrals over a cell surface and edge using discretized electric and magnetic fields.

The results of numerical calculations of an electromagnetic wave propagating in a two-dimensional slab waveguide using the corrected and original FDTD methods and analysis were also shown. The differences between the corrected and original FDTD methods were compared using the err function, and the corrected method was found to be more accurate and reliable than the original.

## Acknowledgements

The author would like to thank Mr. A. Okabe, who is the collaborator of reference [2], which is based on this chapter. The author would also like to thank Enago (www.enago.com) for the English language review.
