Appendix A: Complex Numbers in Optics
“God made the integers, all the rest is the work of man.”
Leopold Kronecker
While Kronecker may have dismissed complex numbers as artificial constructs, they have become absolutely essential in physics and engineering. In optics, complex numbers aren’t just mathematical conveniences—they reveal deep physical insights about wave phenomena that would be nearly impossible to see otherwise.
Complex numbers provide one of the most elegant and powerful mathematical frameworks in all of optics. Far from being abstract mathematical curiosities, they offer profound insights into the nature of light itself. This appendix will guide you through the essential concepts, starting from the very basics and building toward sophisticated applications that illuminate the behavior of electromagnetic waves.
1 A.1 What Are Complex Numbers and Why Do We Need Them? ¶ 1.1 A.1.1 The Historical Motivation ¶ Complex numbers emerged from a simple but frustrating problem: what is − 1 \sqrt{-1} − 1
Rather than dismiss these expressions as nonsensical, they began treating − 1 \sqrt{-1} − 1 i i i
1.2 A.1.2 Definition and Basic Structure ¶ A complex number z z z
where a a a b b b i i i imaginary unit defined by the fundamental property:
Let’s explore what this means through some examples:
1.3 A.1.3 Arithmetic Operations ¶ Complex numbers follow familiar arithmetic rules, with one key difference: we must remember that i 2 = − 1 i^2 = -1 i 2 = − 1
Addition and Subtraction:
Complex numbers add component-wise, just like vectors:
( a + b i ) ± ( c + d i ) = ( a ± c ) + ( b ± d ) i (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i ( a + bi ) ± ( c + d i ) = ( a ± c ) + ( b ± d ) i This makes intuitive sense: real parts combine with real parts, imaginary parts with imaginary parts.
Multiplication:
Here’s where things get interesting. Using the distributive property and i 2 = − 1 i^2 = -1 i 2 = − 1
( a + b i ) ( c + d i ) = a c + a d i + b c i + b d i 2 = ( a c − b d ) + ( a d + b c ) i (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i ( a + bi ) ( c + d i ) = a c + a d i + b c i + b d i 2 = ( a c − b d ) + ( a d + b c ) i Complex Conjugate:
The complex conjugate of z = a + b i z = a + bi z = a + bi z ∗ = a − b i z^* = a - bi z ∗ = a − bi
The complex conjugate has a beautiful geometric interpretation: it reflects the complex number across the real axis. More importantly for physics, when we multiply a complex number by its conjugate, we always get a real, non-negative result: z ⋅ z ∗ = ∣ z ∣ 2 z \cdot z^* = |z|^2 z ⋅ z ∗ = ∣ z ∣ 2
1.4 A.1.4 Geometric Representation: The Complex Plane ¶ One of the most illuminating ways to understand complex numbers is through their geometric representation in the complex plane . This is a coordinate system where:
The horizontal axis represents the real part
The vertical axis represents the imaginary part
Each complex number z = a + b i z = a + bi z = a + bi ( a , b ) (a, b) ( a , b )
This geometric view immediately reveals why complex numbers are so powerful for describing rotations and oscillations—operations that are fundamental to understanding waves.
Every complex number can be expressed in polar form , which separates its magnitude from its direction:
z = r e i θ = r ( cos θ + i sin θ ) z = r e^{i\theta} = r(\cos\theta + i\sin\theta) z = r e i θ = r ( cos θ + i sin θ ) where:
The polar form is particularly powerful because:
Multiplication becomes addition of phases: r 1 e i θ 1 ⋅ r 2 e i θ 2 = r 1 r 2 e i ( θ 1 + θ 2 ) r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)} r 1 e i θ 1 ⋅ r 2 e i θ 2 = r 1 r 2 e i ( θ 1 + θ 2 )
Powers become simple: ( r e i θ ) n = r n e i n θ (re^{i\theta})^n = r^n e^{in\theta} ( r e i θ ) n = r n e in θ
Physical meaning emerges: r r r θ \theta θ
2.1 A.2.1 The Most Beautiful Equation in Mathematics ¶ “Euler’s formula reaches down into the very depths of existence.”
Keith Devlin, mathematician
Euler’s formula stands as one of the most remarkable discoveries in mathematics:
e i θ = cos θ + i sin θ e^{i\theta} = \cos\theta + i\sin\theta e i θ = cos θ + i sin θ This equation creates a bridge between three seemingly unrelated mathematical concepts: exponential functions, trigonometry, and complex numbers. But why is this true, and why should we care?
The proof relies on the Taylor series expansions of the exponential and trigonometric functions:
e x = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! + ⋯ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots e x = 1 + x + 2 ! x 2 + 3 ! x 3 + 4 ! x 4 + 5 ! x 5 + ⋯ cos x = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots cos x = 1 − 2 ! x 2 + 4 ! x 4 − 6 ! x 6 + ⋯ sin x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots sin x = x − 3 ! x 3 + 5 ! x 5 − 7 ! x 7 + ⋯ Now substitute x = i θ x = i\theta x = i θ
e i θ = 1 + i θ + ( i θ ) 2 2 ! + ( i θ ) 3 3 ! + ( i θ ) 4 4 ! + ⋯ e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \cdots e i θ = 1 + i θ + 2 ! ( i θ ) 2 + 3 ! ( i θ ) 3 + 4 ! ( i θ ) 4 + ⋯ Using the pattern i 1 = i i^1 = i i 1 = i i 2 = − 1 i^2 = -1 i 2 = − 1 i 3 = − i i^3 = -i i 3 = − i i 4 = 1 i^4 = 1 i 4 = 1 i 5 = i i^5 = i i 5 = i
e i θ = 1 + i θ − θ 2 2 ! − i θ 3 3 ! + θ 4 4 ! + i θ 5 5 ! − ⋯ e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + i\frac{\theta^5}{5!} - \cdots e i θ = 1 + i θ − 2 ! θ 2 − i 3 ! θ 3 + 4 ! θ 4 + i 5 ! θ 5 − ⋯ Grouping real and imaginary terms:
e i θ = ( 1 − θ 2 2 ! + θ 4 4 ! − ⋯ ) + i ( θ − θ 3 3 ! + θ 5 5 ! − ⋯ ) e^{i\theta} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right) e i θ = ( 1 − 2 ! θ 2 + 4 ! θ 4 − ⋯ ) + i ( θ − 3 ! θ 3 + 5 ! θ 5 − ⋯ ) The first parentheses contain exactly the series for cos θ \cos\theta cos θ sin θ \sin\theta sin θ
2.3 A.2.3 Physical Interpretation: Rotation in the Complex Plane ¶ Euler’s formula has a beautiful geometric meaning: e i θ e^{i\theta} e i θ θ \theta θ θ \theta θ e i θ e^{i\theta} e i θ
This insight is crucial for optics because light waves are oscillatory phenomena. The phase θ \theta θ
2.4 A.2.4 Special Cases and Identities ¶ Several important special cases emerge from Euler’s formula:
Table 1: Important Special Cases
Angle
Complex Exponential
Trigonometric Form
Significance
θ = 0 \theta = 0 θ = 0
e i ⋅ 0 = 1 e^{i \cdot 0} = 1 e i ⋅ 0 = 1
cos ( 0 ) + i sin ( 0 ) = 1 \cos(0) + i\sin(0) = 1 cos ( 0 ) + i sin ( 0 ) = 1
Starting point
θ = π / 2 \theta = \pi/2 θ = π /2
e i π / 2 = i e^{i\pi/2} = i e iπ /2 = i
cos ( π / 2 ) + i sin ( π / 2 ) = i \cos(\pi/2) + i\sin(\pi/2) = i cos ( π /2 ) + i sin ( π /2 ) = i
90° rotation
θ = π \theta = \pi θ = π
e i π = − 1 e^{i\pi} = -1 e iπ = − 1
cos ( π ) + i sin ( π ) = − 1 \cos(\pi) + i\sin(\pi) = -1 cos ( π ) + i sin ( π ) = − 1
Euler’s identity
θ = 2 π \theta = 2\pi θ = 2 π
e i 2 π = 1 e^{i2\pi} = 1 e i 2 π = 1
cos ( 2 π ) + i sin ( 2 π ) = 1 \cos(2\pi) + i\sin(2\pi) = 1 cos ( 2 π ) + i sin ( 2 π ) = 1
Full rotation
Important: Euler’s Identity : e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
This equation connects five fundamental mathematical constants (e e e i i i π \pi π
2.5 A.2.5 Properties of Complex Exponentials ¶ Complex exponentials inherit wonderful properties from regular exponentials:
Multiplication Rule : e i α ⋅ e i β = e i ( α + β ) e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha + \beta)} e i α ⋅ e i β = e i ( α + β )
Division Rule : e i α e i β = e i ( α − β ) \frac{e^{i\alpha}}{e^{i\beta}} = e^{i(\alpha - \beta)} e i β e i α = e i ( α − β )
Power Rule : ( e i θ ) n = e i n θ (e^{i\theta})^n = e^{in\theta} ( e i θ ) n = e in θ
Conjugate Rule : ( e i θ ) ∗ = e − i θ (e^{i\theta})^* = e^{-i\theta} ( e i θ ) ∗ = e − i θ
These properties make complex exponentials incredibly convenient for calculations involving waves and oscillations.
3 A.3 Complex Numbers in Wave Physics ¶ 3.1 A.3.1 Why Waves and Complex Numbers Are Perfect Partners ¶ Before diving into optical applications, let’s understand why complex numbers are so naturally suited for describing waves. Consider a simple harmonic oscillator—perhaps a mass on a spring, or the electric field in a light wave. The position (or field strength) varies sinusoidally with time:
x ( t ) = A cos ( ω t + ϕ ) x(t) = A \cos(\omega t + \phi) x ( t ) = A cos ( ω t + ϕ ) While this real function perfectly describes the physical motion, working with it mathematically can be cumbersome. Derivatives of cosines give sines, integrals mix sines and cosines, and keeping track of phase relationships becomes complex (pun intended).
Instead of working directly with A cos ( ω t + ϕ ) A \cos(\omega t + \phi) A cos ( ω t + ϕ )
x ~ ( t ) = A e i ( ω t + ϕ ) \tilde{x}(t) = A e^{i(\omega t + \phi)} x ~ ( t ) = A e i ( ω t + ϕ ) The physical quantity is understood to be the real part: x ( t ) = Re [ x ~ ( t ) ] x(t) = \text{Re}[\tilde{x}(t)] x ( t ) = Re [ x ~ ( t )]
Why is this better?
Derivatives : d d t [ A e i ( ω t + ϕ ) ] = i ω A e i ( ω t + ϕ ) \frac{d}{dt}[A e^{i(\omega t + \phi)}] = i\omega A e^{i(\omega t + \phi)} d t d [ A e i ( ω t + ϕ ) ] = iω A e i ( ω t + ϕ )
Integration : ∫ A e i ( ω t + ϕ ) d t = A i ω e i ( ω t + ϕ ) \int A e^{i(\omega t + \phi)} dt = \frac{A}{i\omega} e^{i(\omega t + \phi)} ∫ A e i ( ω t + ϕ ) d t = iω A e i ( ω t + ϕ )
Superposition : Adding waves becomes simple complex addition
Phase relationships : Automatically preserved in all calculations
3.2 A.3.2 Electromagnetic Waves in Complex Notation ¶ In optics, we deal with electromagnetic waves—oscillating electric and magnetic fields. A plane wave traveling in the positive z z z
E ⃗ ( z , t ) = E ⃗ 0 e i ( k z − ω t + ϕ ) \vec{E}(z,t) = \vec{E}_0 e^{i(kz - \omega t + \phi)} E ( z , t ) = E 0 e i ( k z − ω t + ϕ ) Let’s unpack this expression:
Table 2: Wave Parameters
Symbol
Name
Physical Meaning
Units
E ⃗ 0 \vec{E}_0 E 0
Amplitude vector
Maximum field strength and polarization direction
V/m
k k k
Wave number
k = 2 π / λ k = 2\pi/\lambda k = 2 π / λ
rad/m
ω \omega ω
Angular frequency
ω = 2 π f \omega = 2\pi f ω = 2 π f
rad/s
ϕ \phi ϕ
Phase constant
Initial phase at z = t = 0 z=t=0 z = t = 0
rad
The argument of the exponential, ( k z − ω t + ϕ ) (kz - \omega t + \phi) ( k z − ω t + ϕ ) phase of the wave. This single expression contains all the information about how the wave varies in space and time.
Why the minus sign? The combination ( k z − ω t ) (kz - \omega t) ( k z − ω t ) z z z z z z z z z
3.3 A.3.3 Physical Fields from Complex Expressions ¶ The actual, measurable electric field is always real. We extract it by taking the real part of our complex expression:
E ⃗ physical ( z , t ) = Re [ E ⃗ 0 e i ( k z − ω t + ϕ ) ] \vec{E}_{\text{physical}}(z,t) = \text{Re}[\vec{E}_0 e^{i(kz - \omega t + \phi)}] E physical ( z , t ) = Re [ E 0 e i ( k z − ω t + ϕ ) ] If E ⃗ 0 = E 0 x ^ \vec{E}_0 = E_0 \hat{x} E 0 = E 0 x ^ x x x
E physical ( z , t ) = E 0 cos ( k z − ω t + ϕ ) x ^ E_{\text{physical}}(z,t) = E_0 \cos(kz - \omega t + \phi) \hat{x} E physical ( z , t ) = E 0 cos ( k z − ω t + ϕ ) x ^ This is the familiar sinusoidal wave we expect for light.
4 A.4 Interference: Where Complex Numbers Shine ¶ 4.1 A.4.1 The Superposition Principle ¶ One of the most important principles in optics is superposition : when two or more waves overlap, the total field is simply the sum of the individual fields. With complex notation, this becomes beautifully simple.
Consider two waves with the same frequency but different amplitudes and phases:
E 1 = E 01 e i ( k z − ω t + ϕ 1 ) E_1 = E_{01} e^{i(kz - \omega t + \phi_1)} E 1 = E 01 e i ( k z − ω t + ϕ 1 ) E 2 = E 02 e i ( k z − ω t + ϕ 2 ) E_2 = E_{02} e^{i(kz - \omega t + \phi_2)} E 2 = E 02 e i ( k z − ω t + ϕ 2 ) The total field is:
E total = E 1 + E 2 = e i ( k z − ω t ) [ E 01 e i ϕ 1 + E 02 e i ϕ 2 ] E_{\text{total}} = E_1 + E_2 = e^{i(kz - \omega t)}[E_{01}e^{i\phi_1} + E_{02}e^{i\phi_2}] E total = E 1 + E 2 = e i ( k z − ω t ) [ E 01 e i ϕ 1 + E 02 e i ϕ 2 ] Suppose we have two waves with equal amplitudes E 01 = E 02 = E 0 E_{01} = E_{02} = E_0 E 01 = E 02 = E 0 ϕ 1 = 0 \phi_1 = 0 ϕ 1 = 0 ϕ 2 = ϕ \phi_2 = \phi ϕ 2 = ϕ
E total = E 0 e i ( k z − ω t ) [ 1 + e i ϕ ] E_{\text{total}} = E_0 e^{i(kz - \omega t)}[1 + e^{i\phi}] E total = E 0 e i ( k z − ω t ) [ 1 + e i ϕ ] = E 0 e i ( k z − ω t ) [ 1 + cos ϕ + i sin ϕ ] = E_0 e^{i(kz - \omega t)}[1 + \cos\phi + i\sin\phi] = E 0 e i ( k z − ω t ) [ 1 + cos ϕ + i sin ϕ ] = E 0 e i ( k z − ω t ) [ ( 1 + cos ϕ ) + i sin ϕ ] = E_0 e^{i(kz - \omega t)}[(1 + \cos\phi) + i\sin\phi] = E 0 e i ( k z − ω t ) [( 1 + cos ϕ ) + i sin ϕ ] The magnitude of the total field is:
∣ E total ∣ = E 0 ( 1 + cos ϕ ) 2 + sin 2 ϕ = E 0 2 + 2 cos ϕ = 2 E 0 ∣ cos ( ϕ 2 ) ∣ |E_{\text{total}}| = E_0\sqrt{(1 + \cos\phi)^2 + \sin^2\phi} = E_0\sqrt{2 + 2\cos\phi} = 2E_0\left|\cos\left(\frac{\phi}{2}\right)\right| ∣ E total ∣ = E 0 ( 1 + cos ϕ ) 2 + sin 2 ϕ = E 0 2 + 2 cos ϕ = 2 E 0 ∣ ∣ cos ( 2 ϕ ) ∣ ∣ This gives us the famous interference pattern:
Constructive interference when ϕ = 0 , 2 π , 4 π , … \phi = 0, 2\pi, 4\pi, \ldots ϕ = 0 , 2 π , 4 π , … ∣ E total ∣ = 2 E 0 |E_{\text{total}}| = 2E_0 ∣ E total ∣ = 2 E 0
Destructive interference when ϕ = π , 3 π , 5 π , … \phi = \pi, 3\pi, 5\pi, \ldots ϕ = π , 3 π , 5 π , … ∣ E total ∣ = 0 |E_{\text{total}}| = 0 ∣ E total ∣ = 0
4.2 A.4.2 Intensity and the Complex Conjugate ¶ The intensity of light is proportional to the time-averaged square of the electric field. For a complex field E ( t ) E(t) E ( t )
I ∝ ⟨ ∣ E ( t ) ∣ 2 ⟩ = ⟨ E ( t ) ⋅ E ∗ ( t ) ⟩ I \propto \langle|E(t)|^2\rangle = \langle E(t) \cdot E^*(t)\rangle I ∝ ⟨ ∣ E ( t ) ∣ 2 ⟩ = ⟨ E ( t ) ⋅ E ∗ ( t )⟩ For our two-wave interference example:
I ∝ ∣ E 1 + E 2 ∣ 2 = ( E 1 + E 2 ) ( E 1 ∗ + E 2 ∗ ) I \propto |E_1 + E_2|^2 = (E_1 + E_2)(E_1^* + E_2^*) I ∝ ∣ E 1 + E 2 ∣ 2 = ( E 1 + E 2 ) ( E 1 ∗ + E 2 ∗ )
= ∣ E 1 ∣ 2 + ∣ E 2 ∣ 2 + E 1 E 2 ∗ + E 1 ∗ E 2 = |E_1|^2 + |E_2|^2 + E_1E_2^* + E_1^*E_2 = ∣ E 1 ∣ 2 + ∣ E 2 ∣ 2 + E 1 E 2 ∗ + E 1 ∗ E 2
= ∣ E 1 ∣ 2 + ∣ E 2 ∣ 2 + 2 Re ( E 1 E 2 ∗ ) = |E_1|^2 + |E_2|^2 + 2\text{Re}(E_1E_2^*) = ∣ E 1 ∣ 2 + ∣ E 2 ∣ 2 + 2 Re ( E 1 E 2 ∗ ) The last term is the interference term —it’s what creates the bright and dark fringes in interference patterns. Without complex numbers, deriving this result would involve tedious trigonometric identities.
5 A.5 Complex Refractive Index: Absorption and Dispersion ¶ 5.1 A.5.1 Beyond Simple Refraction ¶ When light propagates through most materials, two important effects occur:
The wave slows down (refraction)
The wave weakens (absorption)
A complex refractive index elegantly describes both effects:
n ~ = n + i k \tilde{n} = n + ik n ~ = n + ik where:
When a wave propagates through a medium with complex refractive index n ~ \tilde{n} n ~
k ~ = ω n ~ c = ω ( n + i k ) c = ω n c + i ω k c \tilde{k} = \frac{\omega \tilde{n}}{c} = \frac{\omega(n + ik)}{c} = \frac{\omega n}{c} + i\frac{\omega k}{c} k ~ = c ω n ~ = c ω ( n + ik ) = c ωn + i c ωk The wave in the medium becomes:
E ( z ) = E 0 e i k ~ z = E 0 e i ( ω n / c ) z e − ( ω k / c ) z E(z) = E_0 e^{i\tilde{k}z} = E_0 e^{i(\omega n/c)z} e^{-(\omega k/c)z} E ( z ) = E 0 e i k ~ z = E 0 e i ( ωn / c ) z e − ( ωk / c ) z Important: This expression reveals the physical meaning of the complex refractive index:
The intensity decreases as I ( z ) = I 0 e − 2 ( ω k / c ) z I(z) = I_0 e^{-2(\omega k/c)z} I ( z ) = I 0 e − 2 ( ωk / c ) z α = 2 ω k / c \alpha = 2\omega k/c α = 2 ωk / c
5.3 A.5.3 Physical Examples ¶ Table 3: Complex Refractive Index Examples
Material
Wavelength
n n n
k k k
Physical Effect
Glass
500 nm
1.5
∼ 1 0 − 7 \sim 10^{-7} ∼ 1 0 − 7
Transparent, minimal absorption
Water
500 nm
1.33
∼ 1 0 − 9 \sim 10^{-9} ∼ 1 0 − 9
Transparent in thin layers
Gold
500 nm
0.47
2.4
Highly reflective, strong absorption
Silver
500 nm
0.05
3.2
Excellent mirror, opaque
6 A.6 Advanced Applications in Optics ¶ 6.1 A.6.1 Fresnel Coefficients and Reflection ¶ When light hits an interface between two media, some reflects and some transmits. The Fresnel coefficients describe these processes and are naturally expressed using complex numbers.
For s-polarized light (electric field perpendicular to the plane of incidence):
r s = n 1 cos θ 1 − n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 r_s = \frac{n_1\cos\theta_1 - n_2\cos\theta_2}{n_1\cos\theta_1 + n_2\cos\theta_2} r s = n 1 cos θ 1 + n 2 cos θ 2 n 1 cos θ 1 − n 2 cos θ 2 where angles are related by Snell’s law: n 1 sin θ 1 = n 2 sin θ 2 n_1\sin\theta_1 = n_2\sin\theta_2 n 1 sin θ 1 = n 2 sin θ 2
When n 1 > n 2 n_1 > n_2 n 1 > n 2 θ 1 \theta_1 θ 1 θ c = arcsin ( n 2 / n 1 ) \theta_c = \arcsin(n_2/n_1) θ c = arcsin ( n 2 / n 1 )
sin θ 2 = n 1 n 2 sin θ 1 > 1 \sin\theta_2 = \frac{n_1}{n_2}\sin\theta_1 > 1 sin θ 2 = n 2 n 1 sin θ 1 > 1
This makes cos θ 2 = 1 − sin 2 θ 2 \cos\theta_2 = \sqrt{1 - \sin^2\theta_2} cos θ 2 = 1 − sin 2 θ 2
r s = n 1 cos θ 1 − i n 2 sin 2 θ 1 − ( n 2 / n 1 ) 2 n 1 cos θ 1 + i n 2 sin 2 θ 1 − ( n 2 / n 1 ) 2 r_s = \frac{n_1\cos\theta_1 - in_2\sqrt{\sin^2\theta_1 - (n_2/n_1)^2}}{n_1\cos\theta_1 + in_2\sqrt{\sin^2\theta_1 - (n_2/n_1)^2}} r s = n 1 cos θ 1 + i n 2 sin 2 θ 1 − ( n 2 / n 1 ) 2 n 1 cos θ 1 − i n 2 sin 2 θ 1 − ( n 2 / n 1 ) 2 The magnitude ∣ r s ∣ = 1 |r_s| = 1 ∣ r s ∣ = 1
6.2 A.6.2 Polarization States ¶ Complex numbers provide an elegant description of polarization. A general elliptically polarized wave can be written as:
E ⃗ = E x e i ϕ x x ^ + E y e i ϕ y y ^ \vec{E} = E_x e^{i\phi_x} \hat{x} + E_y e^{i\phi_y} \hat{y} E = E x e i ϕ x x ^ + E y e i ϕ y y ^ The polarization state is determined by:
Table 4: Polarization States
Polarization Type
Amplitude Condition
Phase Condition
Complex Representation
Linear (x-direction)
E y = 0 E_y = 0 E y = 0
Any
E 0 x ^ E_0 \hat{x} E 0 x ^
Linear (45°)
E x = E y E_x = E_y E x = E y
δ = 0 \delta = 0 δ = 0
E 0 ( x ^ + y ^ ) / 2 E_0(\hat{x} + \hat{y})/\sqrt{2} E 0 ( x ^ + y ^ ) / 2
Right circular
E x = E y E_x = E_y E x = E y
δ = − π / 2 \delta = -\pi/2 δ = − π /2
E 0 ( x ^ − i y ^ ) / 2 E_0(\hat{x} - i\hat{y})/\sqrt{2} E 0 ( x ^ − i y ^ ) / 2
Left circular
E x = E y E_x = E_y E x = E y
δ = + π / 2 \delta = +\pi/2 δ = + π /2
E 0 ( x ^ + i y ^ ) / 2 E_0(\hat{x} + i\hat{y})/\sqrt{2} E 0 ( x ^ + i y ^ ) / 2
6.3 A.6.3 Fourier Optics: Spatial Frequencies ¶ Modern optics heavily relies on Fourier analysis, which is built on complex exponentials. The spatial Fourier transform of a function f ( x , y ) f(x,y) f ( x , y )
F ( k x , k y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − i ( k x x + k y y ) d x d y F(k_x, k_y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) e^{-i(k_x x + k_y y)} dx dy F ( k x , k y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − i ( k x x + k y y ) d x d y This mathematical framework enables:
Diffraction analysis : Understanding how apertures affect light
Image formation : How lenses create images
Holography : Recording and reconstructing wavefronts
Optical information processing : Manipulating spatial frequencies
In Fourier optics, we think of any optical field as a superposition of plane waves with different spatial frequencies ( k x , k y ) (k_x, k_y) ( k x , k y ) F ( k x , k y ) F(k_x, k_y) F ( k x , k y )
7 A.7 Computational Techniques ¶ 7.1 A.7.1 Time Averaging with Complex Fields ¶ A common calculation in optics involves time averaging oscillatory quantities. For a field E ( t ) = E 0 e − i ω t E(t) = E_0 e^{-i\omega t} E ( t ) = E 0 e − iω t
⟨ E ( t ) E ∗ ( t ) ⟩ = ⟨ E 0 e − i ω t ⋅ E 0 ∗ e + i ω t ⟩ = ∣ E 0 ∣ 2 ⟨ e 0 ⟩ = ∣ E 0 ∣ 2 \langle E(t) E^*(t) \rangle = \langle E_0 e^{-i\omega t} \cdot E_0^* e^{+i\omega t} \rangle = |E_0|^2 \langle e^0 \rangle = |E_0|^2 ⟨ E ( t ) E ∗ ( t )⟩ = ⟨ E 0 e − iω t ⋅ E 0 ∗ e + iω t ⟩ = ∣ E 0 ∣ 2 ⟨ e 0 ⟩ = ∣ E 0 ∣ 2 The rapidly oscillating terms average to zero, leaving only the slowly varying envelope.
7.2 A.7.2 Differentiation in the Frequency Domain ¶ Time and spatial derivatives become algebraic operations:
This transforms differential equations (like Maxwell’s equations) into algebraic equations, greatly simplifying analysis.
7.3 A.7.3 Phasor Diagrams ¶ Complex numbers can be visualized as rotating vectors (phasors) in the complex plane. This visualization is particularly powerful for:
Table 5: Phasor Applications
Phenomenon
Phasor Interpretation
Key Insight
Single wave
Rotating vector of constant length
Amplitude and phase visible
Two-wave interference
Vector addition of two phasors
Interference depends on relative phase
Multiple wave interference
Vector sum of many phasors
Complex interference patterns emerge
Beats
Two nearby frequencies
Slow amplitude modulation visible
8 A.8 Why Complex Numbers Work So Well in Optics ¶ 8.1 A.8.1 Mathematical Elegance ¶ Complex numbers work beautifully in optics because they match the mathematical structure of wave phenomena:
Linearity : Both wave equations and complex arithmetic are linear
Superposition : Adding waves ↔ Adding complex numbers
Harmonic motion : Natural connection to e i ω t e^{i\omega t} e iω t
Phase relationships : Automatically preserved in calculations
8.2 A.8.2 Physical Insight ¶ Beyond mathematical convenience, complex numbers often reveal physical insights:
Causality : The relationship between real and imaginary parts of the refractive index (Kramers-Kronig relations) ensures that materials respond to light in physically reasonable ways
Energy conservation : Complex analysis helps prove that energy is conserved in optical systems
Symmetries : Many optical phenomena have symmetries that are most naturally expressed using complex numbers
Analytic continuation : Physical principles can be extended into the complex plane, often revealing new phenomena
8.3 A.8.3 Computational Power ¶ Modern optics relies heavily on numerical computation. Complex numbers enable:
FFT algorithms : Fast Fourier transforms for diffraction calculations
Matrix methods : Describing optical systems using complex matrices
Beam propagation : Numerical solution of wave equations
Nonlinear optics : Analysis of intensity-dependent phenomena
9 A.9 Common Pitfalls and How to Avoid Them ¶ 9.1 A.9.1 Physical vs. Mathematical Fields ¶ Pitfall : Forgetting that the physical electric field is always real.
Solution : Always take Re [ ⋯ ] \text{Re}[\cdots] Re [ ⋯ ]
9.2 A.9.2 Phase Conventions ¶ Pitfall : Inconsistent phase conventions leading to wrong interference patterns.
Solution : Establish a clear convention early (e.g., e i ( k z − ω t ) e^{i(kz - \omega t)} e i ( k z − ω t ) + z +z + z
9.3 A.9.3 Complex Conjugation in Intensity Calculations ¶ Pitfall : Forgetting to use complex conjugates when calculating intensities.
Solution : Remember that intensity I ∝ ∣ E ∣ 2 = E ⋅ E ∗ I \propto |E|^2 = E \cdot E^* I ∝ ∣ E ∣ 2 = E ⋅ E ∗ E 2 E^2 E 2
10 A.10 Summary and Looking Forward ¶ Complex numbers are not mere mathematical conveniences in optics—they are fundamental tools that reveal the deep structure of electromagnetic phenomena. They provide:
Table 6: Summary of Benefits
Aspect
Benefit
Example Application
Mathematical
Simpler calculations
Interference, diffraction analysis
Physical
Reveals hidden relationships
Kramers-Kronig relations, causality
Computational
Efficient algorithms
FFT-based propagation methods
Conceptual
Unified framework
Connecting optics to quantum mechanics
As you progress through this textbook, you’ll encounter complex numbers in virtually every advanced topic:
Fourier optics : Spatial frequency analysis
Laser physics : Gain and cavity analysis
Nonlinear optics : Phase matching and frequency conversion
Quantum optics : Photon statistics and coherence
Metamaterials : Effective medium theory
Tip: Developing Intuition : The key to mastering complex numbers in optics is developing physical intuition for what the mathematical operations mean. Practice visualizing:
Complex numbers as points/vectors in the complex plane
Multiplication as rotation and scaling
Addition as vector addition
Conjugation as reflection across the real axis
With time, these operations will become as natural as ordinary arithmetic, and you’ll begin to “think in complex” when analyzing optical phenomena.
11 A.11 Worked Examples ¶ 11.1 A.11.1 Young’s Double-Slit Interference ¶ Solution:
Let’s place the two slits at y = ± d / 2 y = \pm d/2 y = ± d /2 P P P y y y
Δ = y d L \Delta = \frac{yd}{L} Δ = L y d
for small angles.
The fields from the two slits at point P P P E 1 = E 0 e i ( k L − ω t ) E_1 = E_0 e^{i(kL - \omega t)} E 1 = E 0 e i ( k L − ω t ) E 2 = E 0 e i ( k L + k Δ − ω t ) = E 0 e i ( k L − ω t ) e i k y d / L E_2 = E_0 e^{i(kL + k\Delta - \omega t)} = E_0 e^{i(kL - \omega t)} e^{ikyd/L} E 2 = E 0 e i ( k L + k Δ − ω t ) = E 0 e i ( k L − ω t ) e ik y d / L
The total field is:
E total = E 0 e i ( k L − ω t ) [ 1 + e i k y d / L ] E_{\text{total}} = E_0 e^{i(kL - \omega t)}[1 + e^{ikyd/L}] E total = E 0 e i ( k L − ω t ) [ 1 + e ik y d / L ]
To find the intensity, we calculate ∣ E total ∣ 2 |E_{\text{total}}|^2 ∣ E total ∣ 2 ∣ E total ∣ 2 = ∣ E 0 ∣ 2 ∣ 1 + e i k y d / L ∣ 2 |E_{\text{total}}|^2 = |E_0|^2 |1 + e^{ikyd/L}|^2 ∣ E total ∣ 2 = ∣ E 0 ∣ 2 ∣1 + e ik y d / L ∣ 2
Using ∣ 1 + e i ϕ ∣ 2 = ( 1 + e i ϕ ) ( 1 + e − i ϕ ) = 2 + 2 cos ϕ |1 + e^{i\phi}|^2 = (1 + e^{i\phi})(1 + e^{-i\phi}) = 2 + 2\cos\phi ∣1 + e i ϕ ∣ 2 = ( 1 + e i ϕ ) ( 1 + e − i ϕ ) = 2 + 2 cos ϕ
I ( y ) = 2 ∣ E 0 ∣ 2 [ 1 + cos ( k y d / L ) ] = 4 ∣ E 0 ∣ 2 cos 2 ( k y d 2 L ) I(y) = 2|E_0|^2[1 + \cos(kyd/L)] = 4|E_0|^2 \cos^2\left(\frac{kyd}{2L}\right) I ( y ) = 2∣ E 0 ∣ 2 [ 1 + cos ( k y d / L )] = 4∣ E 0 ∣ 2 cos 2 ( 2 L k y d )
11.2 A.11.2 Transmission Through an Absorbing Slab ¶ Solution:
This problem involves multiple physical processes:
Reflection at the first interface
Absorption within the slab
Reflection at the second interface
Step 1: First interface (air to glass)
The Fresnel reflection coefficient for normal incidence is:
r 12 = 1 − n ~ 1 + n ~ = 1 − ( 1.5 + 0.01 i ) 1 + ( 1.5 + 0.01 i ) = − 0.5 − 0.01 i 2.5 + 0.01 i r_{12} = \frac{1 - \tilde{n}}{1 + \tilde{n}} = \frac{1 - (1.5 + 0.01i)}{1 + (1.5 + 0.01i)} = \frac{-0.5 - 0.01i}{2.5 + 0.01i} r 12 = 1 + n ~ 1 − n ~ = 1 + ( 1.5 + 0.01 i ) 1 − ( 1.5 + 0.01 i ) = 2.5 + 0.01 i − 0.5 − 0.01 i
To compute this, multiply numerator and denominator by the complex conjugate of the denominator:
r 12 = ( − 0.5 − 0.01 i ) ( 2.5 − 0.01 i ) ( 2.5 + 0.01 i ) ( 2.5 − 0.01 i ) = − 1.25 + 0.02 i 6.25 + 0.0001 ≈ − 0.1998 + 0.0032 i r_{12} = \frac{(-0.5 - 0.01i)(2.5 - 0.01i)}{(2.5 + 0.01i)(2.5 - 0.01i)} = \frac{-1.25 + 0.02i}{6.25 + 0.0001} \approx -0.1998 + 0.0032i r 12 = ( 2.5 + 0.01 i ) ( 2.5 − 0.01 i ) ( − 0.5 − 0.01 i ) ( 2.5 − 0.01 i ) = 6.25 + 0.0001 − 1.25 + 0.02 i ≈ − 0.1998 + 0.0032 i
The reflectance is R 1 = ∣ r 12 ∣ 2 ≈ 0.0399 R_1 = |r_{12}|^2 \approx 0.0399 R 1 = ∣ r 12 ∣ 2 ≈ 0.0399
The transmission coefficient is t 12 = 1 + r 12 ≈ 0.8002 + 0.0032 i t_{12} = 1 + r_{12} \approx 0.8002 + 0.0032i t 12 = 1 + r 12 ≈ 0.8002 + 0.0032 i
Step 2: Propagation through the slab
The wave number in the material is:
k = 2 π λ 0 n ~ = 2 π λ 0 ( 1.5 + 0.01 i ) k = \frac{2\pi}{\lambda_0}\tilde{n} = \frac{2\pi}{\lambda_0}(1.5 + 0.01i) k = λ 0 2 π n ~ = λ 0 2 π ( 1.5 + 0.01 i )
After propagating distance t t t e i k t = e i ( 2 π / λ 0 ) ( 1.5 + 0.01 i ) t = e i ( 2 π / λ 0 ) ( 1.5 t ) e − ( 2 π / λ 0 ) ( 0.01 t ) e^{ik t} = e^{i(2\pi/\lambda_0)(1.5 + 0.01i)t} = e^{i(2\pi/\lambda_0)(1.5t)} e^{-(2\pi/\lambda_0)(0.01t)} e ik t = e i ( 2 π / λ 0 ) ( 1.5 + 0.01 i ) t = e i ( 2 π / λ 0 ) ( 1.5 t ) e − ( 2 π / λ 0 ) ( 0.01 t )
The absorption factor is e − ( 2 π / λ 0 ) ( 0.01 t ) e^{-(2\pi/\lambda_0)(0.01t)} e − ( 2 π / λ 0 ) ( 0.01 t )
Step 3: Second interface (glass to air)
By reciprocity, r 21 = − r 12 r_{21} = -r_{12} r 21 = − r 12 R 2 = 0.0399 R_2 = 0.0399 R 2 = 0.0399
Step 4: Total transmission
The transmitted intensity fraction is:
T = ( 1 − R 1 ) ( 1 − R 2 ) e − 2 ( 2 π / λ 0 ) ( 0.01 t ) T = (1-R_1)(1-R_2)e^{-2(2\pi/\lambda_0)(0.01t)} T = ( 1 − R 1 ) ( 1 − R 2 ) e − 2 ( 2 π / λ 0 ) ( 0.01 t ) = ( 1 − 0.0399 ) 2 e − ( 4 π ⋅ 0.01 t ) / λ 0 = (1-0.0399)^2 e^{-(4\pi \cdot 0.01t)/\lambda_0} = ( 1 − 0.0399 ) 2 e − ( 4 π ⋅ 0.01 t ) / λ 0 = 0.922 ⋅ e − 0.126 t / λ 0 = 0.922 \cdot e^{-0.126t/\lambda_0} = 0.922 ⋅ e − 0.126 t / λ 0
Important: For typical glass at λ 0 = 500 \lambda_0 = 500 λ 0 = 500 t = 1 t = 1 t = 1 T = 0.922 ⋅ e − 0.126 × 1 0 − 3 / ( 500 × 1 0 − 9 ) = 0.922 ⋅ e − 252 ≈ 0 T = 0.922 \cdot e^{-0.126 \times 10^{-3}/(500 \times 10^{-9})} = 0.922 \cdot e^{-252} \approx 0 T = 0.922 ⋅ e − 0.126 × 1 0 − 3 / ( 500 × 1 0 − 9 ) = 0.922 ⋅ e − 252 ≈ 0
The slab is essentially opaque! This shows why the imaginary part of the refractive index must be very small for transparent materials.
11.3 A.11.3 Circular Polarization Analysis ¶ Solution:
Initial state : Unpolarized light can be represented as an incoherent mixture of all polarization states.
After linear polarizer at 45° : The transmitted field is linearly polarized:
E ⃗ 1 = E 0 ( x ^ + y ^ ) 2 \vec{E}_1 = E_0 \frac{(\hat{x} + \hat{y})}{\sqrt{2}} E 1 = E 0 2 ( x ^ + y ^ )
Quarter-wave plate action : A quarter-wave plate introduces a phase difference of π / 2 \pi/2 π /2 e i π / 2 = i e^{i\pi/2} = i e iπ /2 = i
E ⃗ 2 = E 0 ( x ^ + i y ^ ) 2 \vec{E}_2 = E_0 \frac{(\hat{x} + i\hat{y})}{\sqrt{2}} E 2 = E 0 2 ( x ^ + i y ^ )
Analysis : This is left-handed circular polarization! We can verify this by checking the time dependence:
E ⃗ ( t ) = Re [ E 0 ( x ^ + i y ^ ) 2 e − i ω t ] \vec{E}(t) = \text{Re}\left[E_0 \frac{(\hat{x} + i\hat{y})}{\sqrt{2}} e^{-i\omega t}\right] E ( t ) = Re [ E 0 2 ( x ^ + i y ^ ) e − iω t ] = E 0 ( cos ω t x ^ + cos ( ω t − π / 2 ) y ^ ) 2 = E_0 \frac{(\cos\omega t \hat{x} + \cos(\omega t - \pi/2)\hat{y})}{\sqrt{2}} = E 0 2 ( c o s ω t x ^ + c o s ( ω t − π /2 ) y ^ ) = E 0 ( cos ω t x ^ + sin ω t y ^ ) 2 = E_0 \frac{(\cos\omega t \hat{x} + \sin\omega t \hat{y})}{\sqrt{2}} = E 0 2 ( c o s ω t x ^ + s i n ω t y ^ )
At t = 0 t = 0 t = 0 E ⃗ = ( E 0 / 2 ) x ^ \vec{E} = (E_0/\sqrt{2})\hat{x} E = ( E 0 / 2 ) x ^ t = π / ( 2 ω ) t = \pi/(2\omega) t = π / ( 2 ω ) E ⃗ = ( E 0 / 2 ) y ^ \vec{E} = (E_0/\sqrt{2})\hat{y} E = ( E 0 / 2 ) y ^ t = π / ω t = \pi/\omega t = π / ω E ⃗ = − ( E 0 / 2 ) x ^ \vec{E} = -(E_0/\sqrt{2})\hat{x} E = − ( E 0 / 2 ) x ^ t = 3 π / ( 2 ω ) t = 3\pi/(2\omega) t = 3 π / ( 2 ω ) E ⃗ = − ( E 0 / 2 ) y ^ \vec{E} = -(E_0/\sqrt{2})\hat{y} E = − ( E 0 / 2 ) y ^
The electric field vector rotates counterclockwise when viewed against the direction of propagation—this is left-handed circular polarization.
12 A.12 Practice Problems ¶ 12.1 A.12.1 Basic Complex Arithmetic ¶
a) ( 3 + 2 i ) ( 1 − 4 i ) = 3 − 12 i + 2 i − 8 i 2 = 3 − 10 i + 8 = 11 − 10 i (3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i^2 = 3 - 10i + 8 = 11 - 10i ( 3 + 2 i ) ( 1 − 4 i ) = 3 − 12 i + 2 i − 8 i 2 = 3 − 10 i + 8 = 11 − 10 i
b) 2 + 3 i 1 − i = ( 2 + 3 i ) ( 1 + i ) ( 1 − i ) ( 1 + i ) = 2 + 2 i + 3 i − 3 1 + 1 = − 1 + 5 i 2 = − 1 2 + 5 2 i \frac{2 + 3i}{1 - i} = \frac{(2 + 3i)(1 + i)}{(1 - i)(1 + i)} = \frac{2 + 2i + 3i - 3}{1 + 1} = \frac{-1 + 5i}{2} = -\frac{1}{2} + \frac{5}{2}i 1 − i 2 + 3 i = ( 1 − i ) ( 1 + i ) ( 2 + 3 i ) ( 1 + i ) = 1 + 1 2 + 2 i + 3 i − 3 = 2 − 1 + 5 i = − 2 1 + 2 5 i
c) ( 1 + i ) 4 = [ ( 1 + i ) 2 ] 2 = [ 1 + 2 i − 1 ] 2 = ( 2 i ) 2 = − 4 (1 + i)^4 = [(1 + i)^2]^2 = [1 + 2i - 1]^2 = (2i)^2 = -4 ( 1 + i ) 4 = [( 1 + i ) 2 ] 2 = [ 1 + 2 i − 1 ] 2 = ( 2 i ) 2 = − 4
d) Let 3 + 4 i = a + b i \sqrt{3 + 4i} = a + bi 3 + 4 i = a + bi ( a + b i ) 2 = 3 + 4 i (a + bi)^2 = 3 + 4i ( a + bi ) 2 = 3 + 4 i a 2 − b 2 + 2 a b i = 3 + 4 i a^2 - b^2 + 2abi = 3 + 4i a 2 − b 2 + 2 abi = 3 + 4 i a 2 − b 2 = 3 a^2 - b^2 = 3 a 2 − b 2 = 3 2 a b = 4 2ab = 4 2 ab = 4 b = 2 / a b = 2/a b = 2/ a a 2 − 4 / a 2 = 3 a^2 - 4/a^2 = 3 a 2 − 4/ a 2 = 3 a 4 − 3 a 2 − 4 = 0 a^4 - 3a^2 - 4 = 0 a 4 − 3 a 2 − 4 = 0 u = a 2 u = a^2 u = a 2 u 2 − 3 u − 4 = 0 u^2 - 3u - 4 = 0 u 2 − 3 u − 4 = 0 u = 4 u = 4 u = 4 u = − 1 u = -1 u = − 1 u = a 2 ≥ 0 u = a^2 \geq 0 u = a 2 ≥ 0 a 2 = 4 a^2 = 4 a 2 = 4 a = ± 2 a = \pm 2 a = ± 2 a = 2 a = 2 a = 2 b = 1 b = 1 b = 1 a = − 2 a = -2 a = − 2 b = − 1 b = -1 b = − 1 3 + 4 i = ± ( 2 + i ) \sqrt{3 + 4i} = \pm(2 + i) 3 + 4 i = ± ( 2 + i )
12.2 A.12.2 Wave Interference ¶ Two coherent plane waves with equal amplitudes E 0 E_0 E 0 ϕ \phi ϕ ϕ \phi ϕ ϕ \phi ϕ ϕ \phi ϕ ϕ \phi ϕ 4 π 4\pi 4 π
a) E total = E 0 ( 1 + e i ϕ ) = E 0 ( 1 + cos ϕ + i sin ϕ ) E_{\text{total}} = E_0(1 + e^{i\phi}) = E_0(1 + \cos\phi + i\sin\phi) E total = E 0 ( 1 + e i ϕ ) = E 0 ( 1 + cos ϕ + i sin ϕ )
∣ E total ∣ 2 = E 0 2 [ ( 1 + cos ϕ ) 2 + sin 2 ϕ ] = E 0 2 [ 1 + 2 cos ϕ + cos 2 ϕ + sin 2 ϕ ] |E_{\text{total}}|^2 = E_0^2[(1 + \cos\phi)^2 + \sin^2\phi] = E_0^2[1 + 2\cos\phi + \cos^2\phi + \sin^2\phi] ∣ E total ∣ 2 = E 0 2 [( 1 + cos ϕ ) 2 + sin 2 ϕ ] = E 0 2 [ 1 + 2 cos ϕ + cos 2 ϕ + sin 2 ϕ ]
= E 0 2 [ 2 + 2 cos ϕ ] = 2 E 0 2 ( 1 + cos ϕ ) = 4 E 0 2 cos 2 ( ϕ / 2 ) = E_0^2[2 + 2\cos\phi] = 2E_0^2(1 + \cos\phi) = 4E_0^2\cos^2(\phi/2) = E 0 2 [ 2 + 2 cos ϕ ] = 2 E 0 2 ( 1 + cos ϕ ) = 4 E 0 2 cos 2 ( ϕ /2 )
b) Maximum when cos ( ϕ / 2 ) = ± 1 \cos(\phi/2) = \pm 1 cos ( ϕ /2 ) = ± 1 ϕ = 0 , 2 π , 4 π , … \phi = 0, 2\pi, 4\pi, \ldots ϕ = 0 , 2 π , 4 π , …
c) Minimum when cos ( ϕ / 2 ) = 0 \cos(\phi/2) = 0 cos ( ϕ /2 ) = 0 ϕ = π , 3 π , 5 π , … \phi = \pi, 3\pi, 5\pi, \ldots ϕ = π , 3 π , 5 π , …
d) I ( ϕ ) = I 0 cos 2 ( ϕ / 2 ) I(\phi) = I_0 \cos^2(\phi/2) I ( ϕ ) = I 0 cos 2 ( ϕ /2 ) I 0 = 4 E 0 2 I_0 = 4E_0^2 I 0 = 4 E 0 2
12.3 A.12.3 Complex Refractive Index ¶
a) Phase velocity: v = c / n = 3 × 1 0 8 / 1.6 = 1.875 × 1 0 8 v = c/n = 3 \times 10^8 / 1.6 = 1.875 \times 10^8 v = c / n = 3 × 1 0 8 /1.6 = 1.875 × 1 0 8
b) Absorption coefficient: α = 2 ω k / c = 4 π k / λ 0 = 4 π × 0.05 / ( 600 × 1 0 − 9 ) = 1.047 × 1 0 6 \alpha = 2\omega k/c = 4\pi k/\lambda_0 = 4\pi \times 0.05/(600 \times 10^{-9}) = 1.047 \times 10^6 α = 2 ωk / c = 4 πk / λ 0 = 4 π × 0.05/ ( 600 × 1 0 − 9 ) = 1.047 × 1 0 6 − 1 ^{-1} − 1
c) Intensity drops to 1/e when e − α z = 1 / e e^{-\alpha z} = 1/e e − α z = 1/ e α z = 1 \alpha z = 1 α z = 1 z = 1 / α = 1 / ( 1.047 × 1 0 6 ) = 9.55 × 1 0 − 7 z = 1/\alpha = 1/(1.047 \times 10^6) = 9.55 \times 10^{-7} z = 1/ α = 1/ ( 1.047 × 1 0 6 ) = 9.55 × 1 0 − 7
d) After 10 μm: I / I 0 = e − α × 1 0 − 5 = e − 10.47 = 2.8 × 1 0 − 5 I/I_0 = e^{-\alpha \times 10^{-5}} = e^{-10.47} = 2.8 \times 10^{-5} I / I 0 = e − α × 1 0 − 5 = e − 10.47 = 2.8 × 1 0 − 5
12.4 A.12.4 Polarization ¶
a) E ⃗ 0 = E 0 ( 3 2 x ^ + 1 2 y ^ ) \vec{E}_0 = E_0(\frac{\sqrt{3}}{2}\hat{x} + \frac{1}{2}\hat{y}) E 0 = E 0 ( 2 3 x ^ + 2 1 y ^ )
b) A half-wave plate at 45° can be represented by the Jones matrix:
( 0 1 1 0 ) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} ( 0 1 1 0 )
This swaps the x and y components:
E ⃗ f = E 0 ( 1 2 x ^ + 3 2 y ^ ) \vec{E}_f = E_0(\frac{1}{2}\hat{x} + \frac{\sqrt{3}}{2}\hat{y}) E f = E 0 ( 2 1 x ^ + 2 3 y ^ )
c) The final angle is: θ f = arctan ( 3 / 2 1 / 2 ) = arctan ( 3 ) = 60 ° \theta_f = \arctan(\frac{\sqrt{3}/2}{1/2}) = \arctan(\sqrt{3}) = 60° θ f = arctan ( 1/2 3 /2 ) = arctan ( 3 ) = 60°
Note: The half-wave plate rotated the polarization by 2 × ( 45 ° − 30 ° ) = 30 ° 2 \times (45° - 30°) = 30° 2 × ( 45° − 30° ) = 30°
13 A.13 Historical Notes and Further Reading ¶ 13.1 A.13.1 Historical Development ¶ The development of complex numbers and their application to physics is a fascinating story of mathematical evolution:
Historical Timeline:
1545 : Cardano’s “Ars Magna” first mentions square roots of negative numbers
1777 : Euler introduces the notation i i i − 1 \sqrt{-1} − 1
1799 : Gauss proves the fundamental theorem of algebra using complex numbers
1806 : Argand develops the geometric representation (complex plane)
1864 : Maxwell’s equations are formulated, setting stage for complex analysis in optics
1900 : Complex analysis becomes essential for quantum mechanics and modern physics
13.2 A.13.2 Connections to Other Fields ¶ Complex numbers in optics connect to many other areas of physics and mathematics:
Quantum Mechanics : The wavefunction ψ \psi ψ ∣ ψ ∣ 2 |\psi|^2 ∣ ψ ∣ 2
Signal Processing : Fourier transforms, filters, and modulation all use complex exponentials, directly paralleling Fourier optics.
Electrical Engineering : AC circuit analysis uses phasors (complex exponentials) in the same way optics uses complex field amplitudes.
Mathematics : Complex analysis, conformal mappings, and analytical functions find applications in optical beam propagation and waveguide theory.
13.3 A.13.3 Recommended Further Reading ¶ Further Reading:
Born, M., & Wolf, E. (1999). Principles of Optics (7th ed.). Cambridge University Press. The definitive reference for classical optics, with extensive use of complex analysis. Goodman, J. W. (2005). Introduction to Fourier Optics (3rd ed.). Roberts & Company. Essential reading for understanding how complex analysis enables modern optical system design. Hecht, E. (2017). Optics (5th ed.). Pearson. Comprehensive undergraduate text with clear explanations of complex number applications. Fowles, G. R. (1989). Introduction to Modern Optics (2nd ed.). Dover Publications. Classic text emphasizing the wave nature of light and mathematical methods. Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). Wiley. Advanced treatment of electromagnetic theory with sophisticated use of complex analysis. Today’s optical research relies heavily on computational tools that leverage complex number arithmetic:
Table 7: Computational Tools for Complex Optics
Tool/Language
Strengths
Typical Applications
MATLAB
Built-in complex arithmetic, extensive toolboxes
Beam propagation, Fourier optics, data analysis
Python (NumPy/SciPy)
Open source, excellent libraries
Simulation, visualization, machine learning
Mathematica
Symbolic computation, analytical solutions
Theoretical analysis, equation derivation
COMSOL/ANSYS
Finite element modeling
Complex geometries, nonlinear materials
FDTD Solutions
Time-domain electromagnetic simulation
Metamaterials, nanophotonics
14 A.14 Final Thoughts: The Beauty of Mathematical Physics ¶ As you’ve seen throughout this appendix, complex numbers aren’t just computational tools—they’re windows into the fundamental nature of wave phenomena. The fact that e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0
In optics, these mathematical structures manifest as:
Wave-particle duality : The complex wavefunction that bridges classical waves and quantum particles
Symmetry principles : The invariances that lead to conservation laws
Analytical properties : The mathematical constraints that ensure physical causality
“The unreasonable effectiveness of mathematics in the natural sciences is a wonderful gift which we neither understand nor deserve.”
Eugene Wigner, Nobel Prize in Physics
As you continue your journey through optics, remember that the mathematics isn’t separate from the physics—it is the physics, expressed in its most elegant and universal form. Complex numbers are your passport to this deeper understanding.
A Personal Note : Many students initially find complex numbers abstract and intimidating. This is completely normal! The key is to work with them regularly, visualize them geometrically, and always connect the mathematical operations back to physical phenomena. Over time, you’ll develop an intuition that makes complex analysis feel as natural as ordinary arithmetic. When that happens, you’ll have gained one of the most powerful tools in all of physics.