Biophotonics_Concepts_to_Applications

As noted in Fig.1.5, the frequencies of optical waves run from 1× 1012 to
3 × 1016 Hz, which corresponds to maximum and minimum wavelengths of
300 μm and 10 nm, respectively, in the optical spectrum.
The waveform also can be expressed by the following complex function

U(r;t)¼A(rÞexpfi½xtþuð ފg ¼r UðÞr expðixtÞð 2 : 2 Þ

where the time-independent factor UðÞ¼r AðÞr exp½iuðrފis the complex ampli-
tude of the wave. The waveform u(r, t) is found by taking the real part of U(r,t)

uðÞ¼r;t Re UfgðÞr;t ¼

1

2

½ŠðUðÞþr;t U*ðÞr;t 2 : 3 Þ

where the symbol * denotes the complex conjugate.
Theoptical intensityI(r, t), which is defined as the optical power per unit area
(for example, in units such as W/cm^2 or mW/mm^2 ), is equal to the average of the
squared wavefunction. That is, by letting the operation xhidenote the average of a
generic function x, then

IðÞ¼r;t U^2 ðÞr;t

¼jUðÞjr^2 hi 1 þcosð 2 ½ 2 pmtþuð ފÞr ð 2 : 4 Þ

When this average is taken over a time longer than the optical wave period T, the
average of the cosine function goes to zero so that the optical intensity of a
monochromatic wave becomes

IðÞ¼jr UðÞjr^2 ð 2 : 5 Þ

Thus the intensity of a monochromatic wave is independent of time and is the
square of the absolute value of its complex amplitude.

Example 2.1Consider a spherical wave given by

UrðÞ¼

A 0

r

expði2pr=kÞ

where A 0 = 1.5 W1/2is a constant and r is measured in cm. What is the

intensity of this wave?

Solution: From Eq. (2.5),

IðÞ¼jr UðÞjr^2 ¼ðÞA 0 =r^2 ¼ 1 :5W^1 =^2



=rincmðÞ

hi 2

¼ 2 :25 W=cm^2

28 2 Basic Principles of Light