Moller K.D. Optics. Learning by Computing, with Examples Using Mathcad, Matlab, Mathematica, and Maple

Подождите немного. Документ загружается.

6.4. FIBER OPTICS WAVEGUIDES 269

functions

xx(k): a



· 4 · π

− k



yy(k): a



−

· 4 · π



and for the arguments

x(k):



xx(k) y(k):



yy(k).

We try the interval k : 2.65, 2.66 ...3.8 and make a graph.

Input data: radius, wavelength, and refractive indices:

a ≡ 3 λ ≡ 2.39 n1 ≡ 1.5 no ≡ 1.

From graph: ﬁrst intersection

kk : 2.66

λλ :

2 ·π

λλ  2.362.

270 6. MAXWELL II. MODES AND MODE PROPAGATION

Side calculation. If λλλ : 8,weget

kkk : 2 ·

λλλ

and kkk  0.785.

Application 6.8.

1. Change the refractive index on the outside to 1.1, 1.2, 1.3.

2. Change the radius to 2 times the wavelength and 4 times the wavelength.

APPENDIX 6.1

A6.1.1 Boundary Condition Method Applied to TE Modes of

Plane Plate Waveguide

The wave travels in the X direction, but it is sufﬁcient for mode formation to

consider only the Y direction. For the ﬁelds E and B one assumes wavelike

solutions in (1) and exponential decreasing solutions in (2) and (3). Application

of the boundary conditions matches the ﬁelds. The objective is to calculate the

characteristic equation for k

depending on

and

We have for the solutions of the wave equation in the media:

Medium (2)

 Ae

−i

(A6.1)



−i

; (A6.2)

Medium (1)

 BB cos k

Y − C sin k

Y (A6.3)

 k

BB sin k

Y − k

C cos k

Y ; (A6.4)

Medium (3)

 De

−i

(A6.5)

−

−i

. (A6.6)

Matching at the boundary at Y  0:

A  BB (A6.7)

A −k

C. (A6.8)

Matching at the boundary at Y −d:

BB cos k

d − C sin k

d  De

−i

(A6.9)

− k

B sin k

d − k

C cos k

d −

−i

. (A6.10)

6.4. FIBER OPTICS WAVEGUIDES 271

We have four linear homogeneous equations for the coefﬁcientsA, B, C, D and

therefore the determinant of the system of the four equations must be zero. The

solutions of the resulting equation determine the values of θ for which modes

are possible.

Coefﬁcients of the characteristic determinant (A.11)

ABB C D

1-1 0 0

0 k

0 cos k

d −sin k

d −e

−i

0 −k

sin k

d −k1Y cos k

−i

The exponential factors cancel out, and the determinant may be developed with

respect to the ﬁrst row into a sum of two 3 × 3 determinants



0 k

cos k

d −sin k

d −1

−k

sin k

d −k

cos k



(−1)(−1)times. (A6.12)



0 −sin k

d −1

0 −k

cos k



 0. (A6.13)

Calculation of the determinants results in

tan k

d  k

(

)/(k

−

), (A6.14)

which is the condition for the TE modes (see Eq. (6.49)). Similarly one has for

the TM modes

tan k

 n

+ n

)/(n

− n

). (A6.15)

The boundary value method gives us the same result with less effort. One only

has to assume that the k values in the media (2) and (3) are imaginary and apply

the boundary conditions to the solutions in the three sections.

FileFig A6.9 (NA1PLTE)

Calculation of the mode conditions for the TE case, which is Eq. (6.49) for the

planar waveguide.

NA1PLTE is only on the CD.

Application A6.9. Derive in a similar way the condition for the TM case, which

is the p-polarization.

272 6. MAXWELL II. MODES AND MODE PROPAGATION

See also on the CD

PN1. Rectangular Box (see p. 246).

PN2. Mirror and Fringes (see p. 250).

PN3. Plane parallel Plate (see p. 252).

PN4. Calculation of the transmitted Intensity (see p. 252).

PN5. Anti Reﬂection Coating (see p. 253).

PN6. Multi-Layer Reﬂection Coatings.

PN7. Simple Photonic Crystal.

PN8. Resonance Condition for s-Polarization (TE) of the planar Waveguide

(see p. 259).

PN9. Resonance condition for p-Polarization (TM) of the planar Waveguide

(see p. 262).

PN10. Modes in Fibers (see p. 264).

CHAPTER

Blackbody Radiation,

Atomic Emission,

and Lasers

7.1 INTRODUCTION

At the end of the nineteenth century electromagnetic theory and thermodynamics

were well developed and there was the question of reuniﬁcation of these theories.

Blackbody radiation is the emission of electromagnetic radiation from a closed

cavity at temperature T through a small hole (Figure 7.1). It was found that the

frequency distribution of the emitted radiation was dependent upon the temper-

ature T . M. Planck formulated the famous blackbody radiation law in 1900. He

used electromagnetic theory and thermodynamics, but needed an assumption that

marked the beginning of quantum theory. Planck’s law involved a fundamental

physical constant, now called Planck’s constant. The emission of light from the

blackbody was interpreted as quantum emission of light. In the ﬁrst half of the

last century, the quantum emission of atoms and molecules was studiedand in

FIGURE 7.1 Schematic of a blackbody radiator. The body is surrounded by the heat bath of

temperature T . Electromagnetic radiation is emitted and the frequency distribution depends on

temperature T .

273

274

7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

the second half of the century the laser was developed. (“Laser" stands for light

ampliﬁcation by stimulated emission of radiation).

7.2 BLACKBODY RADIATON

7.2.1 The Rayleigh–Jeans Law

An example of blackbody radiation is a wire heated by electricity. First it gets

red and when the temperature increases it becomes white. Increasing the temper-

ature, one ﬁnds that the maximum of the frequency distribution of the emitted

electromagnetic radiation shifts to a shorter wavelength. In Figure 7.1 we show

a schematic of a blackbody. Radiation is emitted at thermal equilibrium and the

energy is drawn from the heat bath at temperature T . The ﬁrst, but insufﬁcient,

analysis of blackbody radiation was done by Rayleigh and Jeans, analyzing the

modes of a rectangular box in a heat bath (Figure 7.2). Oscillators were assumed

to be at the walls of the box, emitting and absorbing light. The modes of the

box are standing waves and in Figure 7.3 we show one standing wave in one

direction as we have in a Fabry–Perot. The number of standing waves in three

dimensions was analyzed and the number of modes determined with respect to

the same energy which is the same frequency or wavelength. It was assumed

that in thermal equilibrium, each mode carried the energy kT , where k is Boltz-

mann’s constant. The energy density per frequency interval du/dν was equal to

the energy dE per volume V and frequency interval dν. The energy density per

frequency interval du/dν was then calculated to be

du/dν  (1/V)dE/dν  8πkTν

. (7.1)

FIGURE 7.2 A box as cavity in the heat bath of temperature T ; see Figure 7.1.

FIGURE 7.3 Standing wave pattern in one dimension. The length is l

, the magnitude of the wave

is A, and the wavelength λ.

7.2. BLACKBODY RADIATON

275

This was the Rayleigh–Jeans law and turned out to be valid for the long wave-

length region only. In the short wavelength region, the energy density increased

by ν

and led to very high and unrealistic large energy densities, the so-called

“violet catastrophe."

The graph in FileFig 7.1 shows the Rayleigh–Jeans radiation law in the visible

spectral region, where one can observe the increase of the radiation density to

shorter wavelengths.

FileFig 7.1 (L1RAJEANS)

Graph of the Raleigh–Jean law using units of energy density per frequency

interval.

L1RAJEANS is only on the CD.

7.2.2 Planck’s Law

The radiation law of blackbody radiation was discovered by Max Planck and is

valid in both the short and the long wavelength regions. Planck’s radiation law

agreed with the Rayleigh-Jeans law for the long wavelength region. Many years

later Einstein derived Planck’s radiation law using the concept of transition prob-

abilities. One assumes ﬁctitious oscillators to be on the walls of the blackbody

cavity. These oscillators are in contact with the heat bath at temperature T . When

radiating, they get the energy from the heat bath and transform it into radiation

energy. Einstein deﬁned the processes of induced absorption, induced emission,

and spontaneous emission for the emission and absorption of radiation by the

oscillators (Figure 7.4). The probability of a transition of induced absorption

between the levels numbered 1 and 2 is called W

and is proportional to the

FIGURE 7.4 Schematic of induced absorption and induced and spontaneous emission.

276 7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

energy density du/dν,

 B

du/dν. (7.2a)

Similarly the probability of a transition of induced emission from level 2 to level

1 is called W

 B

du/dν. (7.2b)

The probability of spontaneous emission is not proportional to du/dν and is

called

 A

. (7.3)

, B

, and A

are called the Einstein probability coefﬁcients. In thermal

equilibrium, there are as many transitions up as down, and for the “down" we

have in addition spontaneous emission. One has

du/dν)  N

du/dν + A

). (7.4)

The numbers N

and N

are the occupation numbers of the two states of energyE

and E

. In thermal equilibrium N

and N

follow from the Boltzmann distribution

 N

exp −(E

/kT ) and N

 N

exp −(E

/kT ), (7.5)

where N

is a constant. In thermal equilibrium one has N

,which means

the occupation number for the lower energy states is always higher.

From Eqs. (7.4) and (7.5) one gets for the energy density per frequency interval

du/dν  A

/{B

exp[(E

− E

)/kT ] − B

}. (7.6)

The constants A

, B

, and B

are determined by considering two limiting

cases. When T →∞, one has du/dν →∞and it follows that B

 B

.We

may write for Eq. (7.6),

du/dν  A

/{B

(exp[(hν)/kT] − 1)}, (7.7)

where E

−E

 hν and h is Planck’s constant. To consider the long wavelength

region, we develop the exponential. In, the limit of long wavelengths, where

ν → 0, the energy density per frequency interval (right-hand side of Eq. (7.7))

du/dν  A

/{B

(hν/kT )}. (7.8)

This must be equal to the Rayleigh–Jeans Law, du/dν  8πkTν

(see

Eq. (7.1)) and one obtains

)  8πhν

. (7.9)

Introduction of Eq. (7.9) into Eq. (7.7) gives us Planck’s formula for the energy

density per frequency interval

du/dν  (8πhν

){1/[exp(hν/kT ) −1]}. (7.10)

7.2. BLACKBODY RADIATON

277

Graphs of Planck’s formula, depending on wavelength and frequency, are given

in FileFigs.2 and 3, respectively.

7.2.3 Stefan–Boltzmann Law

The Stefan–Boltzmann law gives the integrated energy over all wavelengths or

frequencies depending on the temperature T . Integration of Eq. (7.10) over the

frequency results in the energy density u:

u  (8/15)π

(kT )

/(hc)

. (7.11)

This is the energy density equal to the energy per unit volume in the cavity of

the blackbody. To calculate the energy emitted from the hole of the blackbody,

we introduce the radiance L

(or brightness) of the blackbody. The power dW

leaving the blackbody is calculated by the product

dW  L

da cos θd, (7.12)

where da is the area from which the power is emitted, d the solid angle into

which it travels, and  the angle between the normal of the area and the center

line of the solid angle (Figure 7.5). The radiance L

is measured by placing a

power meter before the opening of the blackbody, taking into account the area

FIGURE 7.5 (a) Power emitted from the surface element da traverses the volume element dV  ql

in time l/c; (b) the solid angle seen from dV.

278 7. BLACKBODY RADIATION, ATOMIC EMISSION, AND LASERS

and solid angle. This can be written

dW/(da cos θd)  L

. (7.13)

The radiation is emitted into a hemisphere and when integrating over the solid

angle of the hemisphere we have

W  L

daπ. (7.14)

The radiance L

is related to the energy density u of the blackbody and we want

to ﬁnd the relation between these quantities. We consider a small volume dV  l

times q (Figure 7.5a). The power dW, transmitted in the time interval dt from

the area da of the blackbody into the solid angle d, is then

dWdt  L

da cos ddt. (7.15)

Using from Figure 7.5 d  q/r

and dt  l/c and observing that dWdt 

dudV one obtains

dudV  (L

da cos θdV)/(r

c). (7.16)

Integration over the area times the solid angle, seen from dV (Figure 7.5b),

results in

u  (L

/c)[

∫

da(cos θ/r

)]  (L

/c)

∫

d

. (7.17)

The integral is over the surrounding sphere and is 4π. Using Eqs. (7.11) and

(7.14) one has

· π 

(πkT)

(hc)

. (7.18)

This is the Stefan–Boltzmann law telling us that the “total emission" is

proportional to T

. The constant σ  (2c/15)(πk)

/(hc)

has the value

5.670310

−8

W/(m

In FileFig 7.4 graphs shown of the Stefan–Boltzmann law in units of

power/area.

7.2.4 Wien’s Law

The wavelength at maximum emission of the blackbody may be calculated by

rewriting Eq. (7.10) as a function of the wavelength and setting it to zero. One

gets

T  2.8910

−3

mK. (7.19)

This is called Wien’s displacement law.

Graphs of Wien’s displacement law are shown in FileFig 7.5.