Tài liệu Nguyên tắc cơ bản của lượng tử ánh sáng P10 ppt

CHAPTER

10

STATISTICAL

OPTICS

10.1 STATISTICAL PROPERTIES OF RANDOM LIGHT

A. Optical Intensity

B. Temporal Coherence and Spectrum

C. Spatial Coherence

D. Longitudinal Coherence

10.2 INTERFERENCE OF PARTIALLY COHERENT LIGHT

A. Interference of Two Partially Coherent Waves

B. Interference and Temporal Coherence

C. Interference and Spatial Coherence

*lo.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT

THROUGH OPTICAL SYSTEMS

A. Propagation of Partially Coherent Light

B. Image Formation with Incoherent Light

C. Gain of Spatial Coherence by Propagation

10.4 PARTIAL POLARIZATION

Principles of Optics, first published in 1959 by Max Born and Emil Wolf, brought attention to the

importance of coherence in optics. Emil Wolf is responsible for many advances in the theory of optical

coherence.

342

Fundamentals of Photonics

Bahaa E. A. Saleh, Malvin Carl Teich

Copyright © 1991 John Wiley & Sons, Inc.

ISBNs: 0-471-83965-5 (Hardback); 0-471-2-1374-8 (Electronic)

Statistical optics is the study of the properties of random light. Randomness in light

arises because of unpredictable fluctuations of the light source or of the medium

through which light propagates. Natural light, e.g., light radiated by a hot object, is

random because it is a superposition of emissions from a very large number of atoms

radiating independently and at different frequencies and phases. Randomness in light

may also be a result of scattering from rough surfaces, diffused glass, or turbulent

fluids, which impart random variations to the optical wavefront. The study of the

random fluctuations of light is also known as the theory of optical coherence.

In the preceding chapters it was assumed that light is deterministic or “co￾herent.” An example of coherent light is the monochromatic wave u(r, t) =

Re{U(r) exp(j2rrvt)}, for which the complex amplitude U(r) is a deterministic complex

function, e.g., U(r) = A exp( -jkr)/r in the case of a spherical wave [Fig. 10.0-l(a)].

The dependence of the wavefunction on time and position is perfectly periodic and

predictable. On the other hand, for random light, the dependence of the wavefunction

on time and position [Fig. 10.0-l(b)] is not totally predictable and cannot generally be

described without resorting to statistical methods.

How can we extract from the fluctuations of a random optical wave some meaning￾ful measures that characterize it and distinguish it from other random waves? Examine,

for instance, the three random optical waves whose wavefunctions at some position

vary with time as in Fig. 10.0-2. It is apparent that wave (b) is more “intense” than

wave (a) and that the envelope of wave (c) fluctuates “faster” than the envelopes of

the other two waves. To translate these casual qualitative observations into quantitative

measures, we use the concept of statistical averaging to define a number of nonrandom

measures. Because the random function u(r, t) satisfies certain laws (the wave equation

and boundary conditions) its statistical averages must also satisfy certain laws. The

theory of optical coherence deals with the definitions of these statistical averages, with

Time

dependence

X

t

Wavefronts

0 ;-

0

0

0 z

(a) lb)

Figure 10.0-l Time dependence and wavefronts of (a) a monochromatic spherical wave, which

is an example of coherent light; (b) random light.

343

344 STATISTICAL OPTICS

(a) (6) (cl

Figure 10.0-2 Time dependence of the wavefunctions of three random waves.

the laws that govern them, and with measures by which light is classified as coherent,

incoherent, or, in general, partially coherent.

Familiarity with the theory of random fields (random functions of many

variables-space and time) is necessary for a full understanding of the theory of optical

coherence. However, the ideas presented in this chapter are limited in scope, so that

knowledge of the concept of statistical averaging is sufficient.

In Sec. 10.1 we define two statistical averages used to describe random light: the

optical intensity and the mutual coherence function. Temporal and spatial coherence

are delineated, and the connection between temporal coherence and monochromaticity

is established. The examples of partially coherent light provided in Sec. 10.1 demon￾strate that spatially coherent light need not be temporally coherent, and that

monochromatic light need not be spatially coherent. One of the basic manifestations of

the coherence of light is its ability to produce visible interference fringes. Section 10.2

is devoted to the laws of interference of random light. The transmission of partially

coherent light in free space and through different optical systems, including image￾formation systems, is the subject of Sec. 10.3. A brief introduction to the theory of

polarization of random light (partial polarization) is provided in Sec. 10.4.

10.1 STATISTICAL PROPERTIES OF RANDOM LIGHT

An arbitrary optical wave is described by a wavefunction u(r, t) = Re{U(r, t)}, where

U(r, t) is the complex wavefunction. For example, U(r, t) may take the form

U(r) exp( j2rvt) for monochromatic light, or it may be a sum of many similar functions

of different u for polychromatic light (see Sec. 2.6A for a discussion of the complex

wavefunction). For random light, both functions, u(r, t) and U(r, t), are random and

are characterized by a number of statistical averages introduced in this section.

A. Optical Intensity

The intensity I(r, t) of coherent (deterministic) light is the absolute square of the

complex wavefunction U(r, t),

I(r, t) = lU(r, t)12. (10.1-l)

(see Sec. 2.2A, and 2.6A). For monochromatic deterministic light the intensity is

independent of time, but for pulsed light it is time varying.

STATISTICAL PROPERTIES OF RANDOM LIGHT 345

For random light, U(r, t) is a random function of time and position. The intensity

IU(r, t)12 is therefore also random. The average intensity is then defined as

where the symbol ( . ) now denotes an ensemble average over many realizations of the

random function. This means that the wave is produced repeatedly under the same

conditions, with each trial yielding a different wavefunction, and the average intensity

at each time and position is determined. When there is no ambiguity we shall simply

call I(r, t) the intensity of light (with the word “average” implied). The quantity

IU(r, t>12 is called the random or instantaneous intensity. For deterministic light, the

averaging operation is unnecessary since all trials produce the same wavefunction, so

that (10.1-2) is equivalent to (10.1-l).

The average intensity may be time independent or may be a function of time, as

illustrated in Figs. 10.1-l(a) and (b), respectively. The former case applies when the

optical wave is statistically stationary; that is, its statistical averages are invariant to

time. The instantaneous intensity JU(r, t)12 fluctuates randomly with time, but its

average is constant. We will denote it, in this case, by I(r). Stationarity does not

I(r)

t

t

Figure 10.1-l (a) A statistically stationary wave has an average intensity that does not vary with

time. (6) A statistically nonstationary wave has a time-varying intensity. These plots represent,

e.g., the intensity of light from an incandescent lamp driven by a constant electric current in (a>

and a pulse of electric current in (b).