MATLAB实现的光波传播数值模拟

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"Numerical Simulation of Optical Wave Propagation" 是一本基于MATLAB的光学波传播数值模拟书籍，由Jason D. Schmidt撰写。这本书提供了一系列实际应用的例子，涵盖了光学数学、波动运动理论的数学模型以及MATLAB编程技术。光学是研究光的行为和性质的科学领域，而数值计算在现代光学研究中起着至关重要的作用。通过数值模拟，科学家和工程师能够预测和解释光在不同介质中的传播行为，这对于设计光学系统如激光器、光纤通信、成像设备等至关重要。本书深入探讨了如何利用MATLAB这一强大的计算工具进行光学波传播的数值模拟。 MATLAB是一种高级编程语言和交互式环境，特别适合数值计算、数据分析和可视化。在书中，作者Jason Schmidt可能详细介绍了如何构建数学模型来描述光学现象，如衍射、干涉、散射等，并展示了如何用MATLAB代码实现这些模型。这不仅有助于读者理解基础理论，还能提高他们在实践中解决问题的能力。书中可能包含以下关键知识点： 1. 光学基本原理：包括光的直线传播、反射、折射、干涉和衍射等基本概念。 2. 数值方法：如有限差分法、傅里叶变换、格林函数等用于解决光学问题的数值技术。 3. MATLAB编程技巧：如何设置网格、选择合适的数值解法、优化算法效率等。 4. 实例分析：书中可能包含了各种实际光学问题的案例，如光束传播在不同光学组件中的行为，比如透镜、反射镜等。 5. 应用领域：可能涵盖了光纤通信、光学成像、激光物理、非线性光学等多个领域的应用实例。 6. 参考文献与索引：提供了进一步学习和研究的资源。该书对于正在学习光学、从事光学工程或需要进行光学数值模拟的科研工作者来说，是一份宝贵的参考资料。通过学习，读者可以掌握如何使用MATLAB进行复杂光学问题的数值模拟，从而增强其在光学领域的理论和实践能力。

Taking the curl of Eq. (1.27) yields

∇ × (

∇×E)=−µ

∂

∂t

(∇

× H) . (1.33)

Then, substituting in Eq. (1.28) gives

∇×(∇×E)=−µ

∂

∂t

E. (1.34)

w, applying the vector identity ∇×(∇×E)=∇(∇· E) −∇

E leads to

∇(∇· E) −∇

E = −µ

∂

∂t

E. (1.35)

Finally

, substituting in Eqs. (1.17) and (1.23), and keeping in mind that ϵ is inde-

pendent of position results in a wave differential equation:

∇

E − µ

∂

∂t

E = 0. (1.36)

Similar

manipulations beginning with the curl of Eq. (1.28) yield

∇

B − µ

∂

∂t

B = 0. (1.37)

When

the Laplacian is used on the Cartesian components of E and B, the result is

six uncoupled but identical equations of the form

∇

− µ

∂

∂t

U (x,

y, z)=0, (1.38)

where the scalar U (x, y, z) stands for any of the x-, y-orz- directed components

of the vector ﬁelds E and B.

At this point, we can deﬁne index of refraction

n =

(1.39)

and the

vacuum speed of light

c =

√

(1.40)

so that

∇

−

∂

∂t

U (x, y,z)=0. (1.41)

The electric and magnetic ﬁelds that compose light are traveling wave ﬁelds. There-

fore, ﬁelds with harmonic time dependence exp (−i2πν t) (where ν is the wave

Chapter 1

frequency) are the types of solutions sought for the purposes of this book. When

this is

substituted into Eq. (1.41), the result is

∇

2πnν

U = 0

. (1.42)

Typically, the wavelength is given by λ = c/ν, and the wavenumber is deﬁned as

k =2π/λ so that

∇

+ k

(

U =0. (1.43)

This is the Helmholtz equation, and it appears in many other branches of physics

including thermodynamics and quantum mechanics. At this point, we can dispense

with the time dependence since it is the same for all solutions of the Helmholtz

equation. From this point forward, the ﬁeld U (x, y, z) refers to the phasor por-

tion of the optical ﬁeld (i.e, no time dependence). Further, we deﬁne the units of

U (x, y, z) to be square-root watts per meter (1 W = 1 J/s = 1 N m/s) so that optical

irradiance I = |U|

is in units of watts per meter squared. The value of the electric

ﬁeld or magnetic induction can always be obtained by a simple conversion of units.

1.2.2 Simple traveling-wave ﬁelds

There are several simple traveling-wave ﬁelds that are useful in this book. These are

planar, spherical, and Gaussian-beam waves. With each of these solutions, the ﬁeld

at all points always maintains its planar, spherical, or Gaussian-beam form, and pa-

rameters like radius of curvature change in a simple manner as the wave propagates.

The next section on scalar diffraction theory handles more general cases.

A planar wave is the simplest possible traveling wave. It has uniform amplitude

and phase in any plane perpendicular to its direction of propagation. More gener-

ally, when the optical axis is not along the direction of propagation, a planar wave

ﬁeld is given by

(r)=A exp (ik · r) , (1.44)

where A is the amplitude of the wave and

k =

2π

(αˆ

x + βˆy + γˆz ) (1.45)

is the wavevector with direction cosines given by α, β,andγ. Then, making the

direction cosines more explicit,

(r)=A exp

)

2π

(α

x + βy + γz)

. (1.46)

This wave travels at an angle cos

−1

α from the x-axis and cos

−1

β from the y-axis

as shown in Fig. 1.1.

Foundations of Scalar Diffraction Theory

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