EE 485, Winter 2004, Lih Y. Lin
Chapter 2 Wave Optics
- Improves on Ray Optics by including phenomena such as interference and
diffraction.
- Limitations: (1) Cannot provide a complete picture of reflection and refraction
at the boundaries between dielectric materials. (2) Cannot explain optical
phenomena that require vector formalism, such as polarization.
2.1 Postulates of Wave Optics
The wave equation
c
Light speed in a medium:
c= 0
n
*
*
Wave function u (r , t ) [position r = ( x, y, z ) , time t] satisfies
(2.1-1)
1 ∂ 2u
Wave equation
∇ u− 2 2 =0
(2.1-2)
c ∂t
Wave
equation
is
linear.
Principle
of
superposition
applies:
*
*
*
u (r , t ) = u1 (r , t ) + u 2 (r , t )
Wave equation approximately applicable to media with position-dependent
refractive indices, provided that the variation is slow within distances of a
*
*
wavelength → Locally homogeneous, n = n(r ), c = c(r ) .
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Intensity, power, and energy
Optical intensity: optical power per unit area (watts/cm2)
*
*
I (r , t ) = 2 u 2 (r , t )
(2.1-3)
: average over a period >> 1/frequency.
Optical power (watts) flowing into an area A normal to the propagation direction:
*
P(t ) = ∫ I (r , t )dA
(2.1-4)
A
Optical energy (joules) collected in a given time interval T is ∫ P (t )dt .
T
2.2 Monochromatic Waves
*
*
*
u (r , t ) = a (r ) cos[2πft + ϕ (r )]
(2.2-1)
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EE 485, Winter 2004, Lih Y. Lin
A. Complex Representation and the Helmholtz Equation
Complex wavefunction
*
*
*
*
U (r , t ) = a (r ) exp[ jϕ (r )] exp( j 2πft ) = U (r )e jωt
*
U (r ) : complex amplitude
∇ 2U −
1 ∂ 2U
=0
c 2 ∂t 2
(2.2-2,5)
(2.2-4)
The Helmholtz equation and wavenumber
Substituting (2.2-5) into (2.2-4) obtains:
*
(∇ 2 + k 2 )U (r ) = 0
k =ω
: wavenumber
c
(2.2-7)
Optical intensity
*
* 2
I (r ) = U (r )
not a function of time.
(2.2-10)
(2.2-8)
Wavefronts
*
The wavefronts are the surfaces of equal phase, ϕ (r ) = constant.
B. Elementary Waves
The plane wave
Complex amplitude:
* *
*
U (r ) = A exp(− jk ⋅ r ) = A exp[− j (k x x + k y y + k z z )
*
k : wavevector → direction of propagation
*
k = k = wavenumber
(2.2-11)
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EE 485, Winter 2004, Lih Y. Lin
2π c
=
k
f
c is called the phase velocity of the wave.
Wavelength
λ=
In a medium of refractive index n, f is the same,
λ
c
c = 0 , λ = 0 , k = nk 0
n
n
The spherical wave
A
*
U (r ) = e − jkr
r
2
A
*
I (r ) = 2
r
(2.2-12)
(2.2-14)
(2.2-15)
The paraboloidal wave
Fresnel approximation of the spherical wave.
*
*
Examine a spherical wave originating at r = 0 at points r = ( x, y, z ) sufficiently
close to the z axis but far from the origin, so that ( x 2 + y 2 ) << z . Use paraxial
approximation, Taylor series expansion, and Fresnel approximation:

A
x2 + y 2 
*

(2.2-16)
U (r ) = exp(− jkz ) exp − jk
z
2 z 

First phase term: planar wave. Second phase term: paraboloidal wave.
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EE 485, Winter 2004, Lih Y. Lin
• Validity of Fresnel approximation
Fresnel approximation valid for points (x, y) lying within a circle of radius a
centered about the z axis at position z, if a satisfies a 4 << 4z 3λ , or
N Fθ m2
<< 1
(2.2-17)
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where θ m = a / z is the maximum angle, and
a2
NF =
λz
Fresnel number
(2.2-18)
2.4 Simple Optical Components
A. Reflection and Refraction
Laws of reflection and refraction can be verified by wave optics. Please read the
textbook.
B. Transmission through Optical Components
(Ignore reflection and absorption. Main emphasis on phase shift and associated
wavefront bending.)
Transmission through a transparent plate
• Normal incidence
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EE 485, Winter 2004, Lih Y. Lin
Complex amplitude transmittance
t ( x, y ) = exp(− jnk 0 d )
→ The plate introduces a phase shift nk 0 d = 2π
(2.4-3)
d
λ
• Oblique incidence
t ( x, y ) = exp[− jnk 0 (d cos θ1 + x sin θ1 )]
Thin transparent plate of varying thickness
Transmittance = (Transmittance in air) × (Transmittance in plate)
t ( x, y ) = exp[− jk 0 (d 0 − d ( x, y ) )]exp[− jnk0 d ( x, y )]
(2.4-4)
= exp(− jk 0 d 0 ) exp[− j (n − 1)k0 d ( x, y )]
Thin lens
Utilizing Eq. (2.4-4) results in
ª
x2 + y2 º
t ( x, y ) = h0 exp « jk 0
2 f »¼
¬
where
h0 = exp(− jnk0 d 0 )
: constant phase factor
R
: focus length of the lens
f =
n −1
(2.4-6)
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EE 485, Winter 2004, Lih Y. Lin
Diffraction grating
An optical component periodically modulates the phase or the amplitude of the
incident wave. It can be made of a transparent plate with periodically varying
thickness or periodically graded refractive index.
The diffraction grating shown above converts an incident plane wave of
wavelength λ << Λ , traveling at a small angle θ i with respect to the z axis, into
several plane waves at small angles
λ
θ q = θi + q
(2.4-9)
Λ
with the z axis.
q = 0, ±1, ±2, … : diffraction order
In general, without paraxial approximation
λ
sin θ q = sin θ i + q
(2.4-10)
Λ
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EE 485, Winter 2004, Lih Y. Lin
C. Graded-index Optical Components
Instead of varying thickness, varying refractive index.
t ( x, y ) = exp[− jn0 k 0 d ( x, y )]
Varying thickness:
t ( x, y ) = exp[− jn( x, y )k0 d 0 ]
Varying refractive index:
(2.4-11)
2.5 Interference
Linearity of the wave equation → Superposition of the wavefunctions
But not superposition of the optical intensity because of interference.
(Consider waves of the same frequency in this section.)
A. Interference of Two Waves
*
*
*
U (r ) = U1 (r ) + U 2 (r )
U 1 ≡ I1 e jϕ , U 2 ≡ I 2 e jϕ
Intensity interference equation:
2
I = U = I1 + I 2 + 2 I1 I 2 cos ϕ
1
2
ϕ ≡ ϕ 2 − ϕ1
(2.5-4,5)
Spatial redistribution of optical intensity. Power conservation still holds.
Interferometer
Waves superimpose with delay d → ϕ = 2π (d λ )
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EE 485, Winter 2004, Lih Y. Lin

 d 
I = 2 I 0 1 + cos 2π 
 λ 

(2.5-6)
Three important examples of intereferometers: Mach-Zehnder interferometer,
Michelson interferometer, Sagnac interferometer.
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EE 485, Winter 2004, Lih Y. Lin
Intensity I is a very sensitive function of ϕ = 2π
d
nd
nfd
= 2π
= 2π
λ
λ0
c0
→ Can measure small variation of d, n, λ0, or f
Interference of two oblique plane waves
U 1 = I 0 exp(− jkz )
U 2 = I 0 exp[− jk ( z cos θ + x sin θ )]
At z = 0, ϕ = kx sin θ ,
I = 2 I 0 [1 + cos(kx sin θ )]
→
Period of interference pattern Λ = λ
sin θ
B. Multiple-wave Interference
M waves of equal amplitude and equal phase difference
U m = I 0 exp[ j (m − 1)ϕ ], m = 1, 2, ..., M
(2.5-7)
(2.5-9)
M
U = ∑U m
m =1
sin 2 (Mϕ 2 )
(2.5-10)
sin 2 (ϕ 2 )
In the graph of I as a function of ϕ , the number of minor peaks between the main
peaks = (M - 1).
2
I = U = I0
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EE 485, Winter 2004, Lih Y. Lin
Infinite number of waves of progressively smaller amplitudes and equal phase
difference
U m = I 0 r ( m−1) e j ( m−1)ϕ , m = 1, 2, 3, ...
r <1
∞
I0
U = ∑U m =
(2.5-13)
1 − re jϕ
m =1
I max
2
(2.5-15)
I=U =
2
1 + (2F π ) sin 2 (ϕ / 2 )
I0
I max ≡
where
(2.5-16)
(1 − r ) 2
π r
F ≡
: Finesse
(2.5-17)
1− r
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EE 485, Winter 2004, Lih Y. Lin
When r approaches 1, I max can be very large! → Principle of optical resonators,
lasers.
• Physical meaning of F
Consider values of ϕ near the ϕ = 0 resonance peak
I max
(2.5-18)
I≈
2
1 + (F π ) ϕ 2
Full width at half maximum (FWHM)
2π
∆ϕ =
(2.5-19)
F
→ Finesse is a measure of the sharpness of the interference function.
2.6 Polychromatic Light
Can be expressed as the sum of monochromatic waves over frequency.
The pulsed plane wave
1 ∞
*
U (r , t ) =
Aω e − jkz e jωt dω
∫
2π 0
z
= a (t − )
c
∞
1
a (t ) =
Aω e jωt dω
∫
2π 0
is a function with arbitrary shape
(2.6-8,9)
(2.6-10)
If a (t ) is of finite duration σ t in time →
Pulse width in space = cσ t
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EE 485, Winter 2004, Lih Y. Lin
Pulse width in frequency domain (spectral bandwidth) depends on pulse shape, see
Appendix A.
E.g., for Gaussian function in space, its Fourier transform is still Gaussian in
1
frequency domain. σ t = 1 ps, cσ t = 0.3 mm, σ f =
= 80 GHz.
4πσ t
Interference (beating) between two monochromatic waves
U (t ) = I1 e j 2πf t + I 2 e j 2πf t
1
2
(2.6-11)
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EE 485, Winter 2004, Lih Y. Lin
Utilizing the interference equation (2.5-4),
2
I = U = I1 + I 2 + 2 I1 I 2 cos[2π ( f1 − f 2 )t ]
f1 − f 2 ≡ Beat frequency
(2.6-12)
Interference of M monochromatic waves
Consider an odd number M = 2L + 1 waves, each with intensity I0 and frequencies
f q = f 0 + qf F , q = -L, …, 0, … L
centered about f 0 and spaced by f F << f 0 .
U (t ) = I 0
L
∑ exp[ j 2π ( f 0 + qf F )t ]
(2.6-13)
sin 2 (Mπt TF )
sin 2 (πt TF )
(2.6-14)
q =− L
Utilizing Eq. (2.5-10),
2
I = U = I0
→ A periodic sequence of pulses with period TF ≡ 1 / f F , peak intensity M 2 I 0 .
Application: Mode-locked laser for generating short laser pulses.
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