CHAPTER 10



CHAPTER 10. LIGHT WAVES

What is the actual nature of light? Is light a wave phenomenon or a particle phenomenon? The essential feature of a particle is its localization and the essential feature of a wave is its non-localization. Light is a stream of particles (known as phonons). It is also an electromagnetic wave propagating in a medium.

10.1. One-Dimensional Waves

The most familiar waves and the easiest to visualize are the mechanical waves (Fig10.1).

[pic]

[pic]

Wave profile is a function of both position and time,

[pic] (10.1)

Here we limit ourselves to a wave that does not change its shape as it progresses through space. Suppose the wave pulse moves along the x-axis at a constant speed v. After a time t the pulse has moved along the x-axis a distance vt, but its shape remains unchanged. In the coordinate system S( (Fig.1) that travels along with the wave pulse at a speed v, the pulse can be thought as stationary, so that

[pic]

Since x( =x-vt, it follows that,

[pic] (10.2)

Eq.(10.2) represents the most general form of the one-dimensional wavefunction. It describes a wave having the desired profile, moving in the positive x-direction with a speed v. Similarly, if the wave travels in the negative x-direction, the wavefunction becomes,

[pic], with v>0 (10.3)

10.2. Harmonic Waves (Sinusoidal Waves)

Harmonic wave is the simplest wave form with a profile of a sine or cosine curve. Any wave shape can be synthesized by a superposition of harmonic waves.

[pic] (10.4)

where k is a positive constant known as the propagation number, A is the amplitude of the wave. The wave is periodic in both space and time. The spatial period is known as the wavelength (.

[pic] (10.5)

For harmonic wave, [pic].

[pic]

If the wave is viewed from a fixed position, it is periodic in time with a repetitive unit temporal called temporal period (.

[pic]

The inverse of the temporal period is the frequency f.

[pic]

Therefore, [pic]. The other two quantities are angular frequency [pic] and wave number or spatial frequency [pic] which is important in spectroscopy. The wave number is the number of whole wavelengths in one meter.

[pic]

The harmonic wave functions can be rewritten as

[pic] (10.6)

[pic] (10.7)

[pic] (10.8)

[pic] (10.9)

All these waves extend from [pic] to [pic]and each wave is monochromatic.

Example: Given the traveling wave function [pic], find the frequency, the wavelength, the temporal period, the amplitude, and the direction of motion.

(Ans: 6Hz; 10m; 1/6s; 2m; negative x)

Example: The speed of electromagnetic waves in vacuum is 3(108 m/s. Find the frequency of yellow-green light of wavelength 555 nm. Overhead power lines radiate electromagnetic waves at a frequency of 50 Hz. Compare the wavelength with yellow-green light. (Ans: 5.41(1014 Hz; 6(106 m)

10.3. Phase and Phase Velocity

More generally, instead of Eq.(10.4), the harmonic wave function can be written as,

[pic] (10.10)

The argument of the sine function is called phase: [pic]. (0 is called the initial phase which is the phase at t=0 and x=0. The rate-of-change of phase at any fixed location is the angular frequency of the wave,

[pic]

Similarly, the rate-of-change of phase with distance, holding t constant is,

[pic]

The speed at which the profile moves under the condition of constant phase is called phase velocity of the wave,

[pic] (10.11)

The phase velocity is the wave velocity at which the wave profile propagates. The phase velocity is accompanied by a positive sign when the wave moves in the direction of increasing x and a negative one in the direction of decreasing x.

10.4. Properties of Waves

When a wave propagates in space, the line that links all the points of the same phase is called wavefront. The propagation of a wave can be regarded as the advancing of a wavefront at the wave (or phase) velocity. The straight line in Fig.10.5 is perpendicular to all wavefronts in the direction of propagation and is called a ray.

[pic]

Light in nature is an electromagnetic wave. It is a transverse wave in which the electric and magnetic fields oscillate in a direction perpendicular to the direction of wave propagation.

At a given time, when all the surfaces on which a disturbance has constant phase form a set of planes, each generally perpendicular to the propagation direction, the waves are called planes waves. If the wavefronts form concentric spheres that increase in diameter as they expand out into the surrounding space, the wave is a spherical wave.

[pic]

[pic] [pic]

10.5. Amplitude and Intensity

The wave carries an energy that is proportional to the square of its amplitude. The amount of energy passing through unit area per second is defined as the intensity of the wave. Therefore the intensity is also proportional to the speed of light in the medium.

10.6. Wavelength and Frequency

Suppose c and ( are the speed and wavelength of light in vacuum, respectively. In a medium of refractive index n, the speed and wavelength are reduced, respectively,

[pic] and [pic] (10.12)

The frequency of light in the medium is unchanged.

[pic]

Radio Waves: (frequency) a few Hz up to about 109 Hz.

Microwaves: (frequency) 109 Hz to 3(1011 Hz.

Infrared: (wavelength) 1 mm to about 760 nm.

Visible light: (wavelength) 760 nm to about 390 nm.

Ultraviolet: (wavelength) 390 nm to 1 nm.

X-Rays: (wavelength) 1 nm to 6.0(10-12 m.

10.7. Wave Packets or Groups

A sinusoidal wave implies that the source is vibrating infinitely. In reality, the source of light can only produce interrupted waves with finite length. Such an interrupted wave is known as a wave packet, wave group, or wave train. Mathematically (from Fourier analysis), a wave packet with a finite length consists of a group of frequencies. The smaller the number of frequencies in a wave packet then the greater will be the length of the wave packet. Different frequencies in a group mean different wavelengths. It can be shown that the spread of wavelength (( about the mean wavelength ( is,

[pic] (10.13)

where N is the number of waves in the wave packet.

[pic]

10.8. Reflection and Transmission

Fresnel’s equations (10.14-17) are derived to calculate the amount of light reflected and transmitted at an interface between two optical media.

[pic] (10.14)

[pic] (10.15)

[pic] (10.16)

[pic] (10.17)

where r( is the amplitude coefficient of reflection perpendicular to the plane of incidence, whilst t( is the amplitude coefficient of transmission perpendicular to the plane of incidence. Similarly, r// is the amplitude coefficient of reflection parallel to the plane of incidence, whilst t// is the amplitude coefficient of transmission parallel to the plane of incidence.

[pic]

Using the Snell’s law, the above equations can be simplified to,

[pic] (10.18)

[pic] (10.19)

[pic] (10.20)

[pic] (10.21)

When particular values are obtained from the above Eqs.(10.14-21), a minus sign in the result for r(, t(, r// and t// simply means the electric vector direction is opposite to that indicated in Fig.10.10.

For nearly normal incidence, i and i( are very small so that the sine is approximately equal to the tangent and the cosine is equal to unity. From the Eqs.(10.14-17), we have,

[pic] and [pic] (10.22)

The reflectance is defined as the fraction of the incident intensity reflected. Recall that the intensity is proportional to the square of the amplitude and also proportional to the speed of light in the medium, we have,

[pic]

The reflectance for near normal incidence is then,

[pic] (10.23)

The transmittance is defined as the fraction of the incident intensity transmitted.

[pic]

The transmittance for near normal incidence is then,

[pic] (10.24)

From Eqs.(10.23) and (10.24) we can get,

[pic] (10.25)

This is an expected result which means that if there is no absorption, the sum of the reflected and transmitted intensities should be equal to the incident intensity.

[pic]

For n(>n, according to Snell’s law, we have i(n is commonly called external reflection. When n( ................
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