Because ψ is slowly varying, we neglect the second derivative of ψ with

respect to z, and this becomes

‚ψ

= ’ 2 ψ + k0 1 ’ n2 (x, y) ψ,

2

2ik0 (8.129)

x,y

‚z

which is the two-dimensional time dependent Schr¨dinger equation, but with

o

t replaced by z, the distance down the ¬bre. The wave-modes that will be

trapped and guided by the ¬bre will be those corresponding to bound states

of the axisymmetric potential

V (x, y) = k0 (1 ’ n2 (r)).

2

(8.130)

242 CHAPTER 8. SPECIAL FUNCTIONS I

If these bound states have (negative) “energy” En , then ψ ∝ e’iEn z/2k0 , and

so the actual wavenumber for frequency ω0 is

k = k0 ’ En /2k0 . (8.131)

In order to have a unique propagation velocity for signals on the ¬bre, it

is therefore necessary that the potential support one, and only one, bound

state.

If

n(r) = n1 , r < a,

= n2 , r > a, (8.132)

then the bound state solutions will be of the form

einθ eiβz Jn (κr), r < a,

ψ(r, θ) = (8.133)

Aeinθ eiβz Kn (γr), r > a,

where

κ2 = (n2 k0 ’ β 2 ),

2

(8.134)

1

γ 2 = (β 2 ’ n2 k0 ).

2

(8.135)

2

To ensure that we have a solution decaying away from the core, we need β

to be such that both κ and γ are real. We therefore require

β2

n2 > 2 > n2 . (8.136)

1 2

k0

At the interface both ψ and its radial derivative must be continuous, and so

we will have a solution only if β is such that

Jn (κa) K (γa)

=γ n

κ .

Jn (κa) Kn (γa)

This Shr¨dinger approximation to the wave equation has other applica-

o

tions. It is called the paraxial approximation.

8.3. BESSEL FUNCTIONS 243

8.3.4 Spherical Bessel Functions

Consider the wave equation

1 ‚2

2

’ 2 2 •(r, θ, φ, t) = 0 (8.137)

c ‚t

in spherical polar coordinates. To apply separation of variables, we set

• = eiωt Ym (θ, φ)χ(r),

l

(8.138)

and ¬nd that

d2 χ 2 dχ l(l + 1) ω2

’

+ χ + 2 χ = 0. (8.139)

dr 2 r2

r dr c

Substitute χ = r’1/2 R(r) and we have

ω 2 (l + 1 )2

d2 R 1 dR 2

’

+ + R = 0. (8.140)

2 2 2

dr r dr c r

1

This is Bessel™s equation with ν2 ’ (l + 2 )2 . Therefore the general solution

is

R = AJl+ 1 (kr) + BJ’l’ 1 (krr) , (8.141)

2 2

where k = |omega|/c. Now inspection of the series de¬nition of the Jν reveals

that

2

J 1 (x) = sin x, (8.142)

πx

2

2

J’ 1 (x) = cos x, (8.143)

πx

2

so these Bessel functions are actually elementary functions. This is true of

all Bessel functions of half-integer order, ν = ±1/2, ±3/2, . . .. We de¬ne the

spherical Bessel functions by2

π

jl (x) = J 1 (x), (8.144)

2x l+ 2

π

nl (x) = (’1)l+1 J 1 (x). (8.145)

2x ’(l+ 2 )

2

We are using the de¬nitions from Schi¬™s Quantum Mechanics.

244 CHAPTER 8. SPECIAL FUNCTIONS I

The ¬rst few are

1

j0 (x) = sin x,

x

1 1

sin x ’ cos x,

j1 (x) =

x2 x

3 1 3

’ sin x ’ 2 cos x,

j2 (x) =

x3 x x

1

’ cos x,

n0 (x) =

x