Appendix B: Matrix Multiplication in Optics
“In mathematics you don’t understand things. You just get used to them.”
John von Neumann
While matrix multiplication may initially seem like abstract algebra, it becomes an indispensable tool in optics. Matrices elegantly describe how light transforms as it propagates through optical systems, making complex calculations surprisingly manageable. From polarization analysis to ray tracing, matrices reveal the underlying mathematical structure of optical phenomena.
Matrix multiplication provides one of the most powerful and systematic approaches to analyzing optical systems. Far from being merely computational shortcuts, matrices encode the fundamental transformations that light undergoes, revealing deep connections between seemingly different optical phenomena. This appendix will guide you through matrix operations with a focus on their physical meaning in optics, building from basic concepts to sophisticated applications in polarization and ray optics.
1 What Are Matrices and Why Do We Need Them in Optics? ¶ Most optical phenomena can be described as linear transformations —processes that take input light and produce output light in a predictable, systematic way. Consider some common examples:
A polarizer transforms unpolarized light into linearly polarized light
A wave plate changes the polarization state of light
A lens transforms a parallel beam into a converging beam
A prism separates white light into its component colors
Each of these transformations can be represented mathematically as a matrix operation, where the input state is multiplied by a transformation matrix to yield the output state.
1.2 Matrix Basics: Structure and Notation ¶ A matrix is a rectangular array of numbers arranged in rows and columns. For optics, we primarily work with 2×2 matrices (for polarization) and 2×2 matrices (for ray optics):
M = ( m 11 m 12 m 21 m 22 ) \mathbf{M} = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} M = ( m 11 m 21 m 12 m 22 ) 1.3 Vectors: Representing Optical States ¶ In matrix optics, we represent physical states as column vectors :
Polarization States (Jones vectors):
E ⃗ = ( E x E y ) \vec{E} = \begin{pmatrix} E_x \\ E_y \end{pmatrix} E = ( E x E y ) Ray States (ray vectors):
r ⃗ = ( y θ ) \vec{r} = \begin{pmatrix} y \\ \theta \end{pmatrix} r = ( y θ ) where y y y θ \theta θ
1.4 Why Matrix Multiplication Works for Optics ¶ The reason matrices are so powerful in optics is that optical systems are compositional : if you have two optical elements in series, the combined effect is the product of their individual effects. Mathematically:
If element A transforms state s ⃗ 1 \vec{s}_1 s 1 s ⃗ 2 \vec{s}_2 s 2 s ⃗ 2 = A s ⃗ 1 \vec{s}_2 = \mathbf{A}\vec{s}_1 s 2 = A s 1 s ⃗ 2 \vec{s}_2 s 2 s ⃗ 3 \vec{s}_3 s 3 s ⃗ 3 = B s ⃗ 2 \vec{s}_3 = \mathbf{B}\vec{s}_2 s 3 = B s 2
Then the combined transformation is: s ⃗ 3 = B ( A s ⃗ 1 ) = ( B A ) s ⃗ 1 \vec{s}_3 = \mathbf{B}(\mathbf{A}\vec{s}_1) = (\mathbf{B}\mathbf{A})\vec{s}_1 s 3 = B ( A s 1 ) = ( BA ) s 1
The combined system has matrix B A \mathbf{BA} BA
2 Matrix Multiplication: Rules and Mechanics ¶ 2.1 The Fundamental Rule ¶ Matrix multiplication follows a specific pattern. For matrices A \mathbf{A} A B \mathbf{B} B
( A B ) i j = ∑ k A i k B k j (\mathbf{AB})_{ij} = \sum_{k} A_{ik}B_{kj} ( AB ) ij = k ∑ A ik B kj In plain English: to get element ( i , j ) (i,j) ( i , j ) i i i j j j
2.2 Step-by-Step Process for 2×2 Matrices ¶ For two 2×2 matrices:
A = ( a 11 a 12 a 21 a 22 ) , B = ( b 11 b 12 b 21 b 22 ) \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} A = ( a 11 a 21 a 12 a 22 ) , B = ( b 11 b 21 b 12 b 22 ) The product C = A B \mathbf{C} = \mathbf{AB} C = AB
C = ( a 11 b 11 + a 12 b 21 a 11 b 12 + a 12 b 22 a 21 b 11 + a 22 b 21 a 21 b 12 + a 22 b 22 ) \mathbf{C} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix} C = ( a 11 b 11 + a 12 b 21 a 21 b 11 + a 22 b 21 a 11 b 12 + a 12 b 22 a 21 b 12 + a 22 b 22 ) 2.3 Matrix-Vector Multiplication ¶ When multiplying a matrix by a vector:
A v ⃗ = ( a 11 a 12 a 21 a 22 ) ( v 1 v 2 ) = ( a 11 v 1 + a 12 v 2 a 21 v 1 + a 22 v 2 ) \mathbf{A}\vec{v} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} a_{11}v_1 + a_{12}v_2 \\ a_{21}v_1 + a_{22}v_2 \end{pmatrix} A v = ( a 11 a 21 a 12 a 22 ) ( v 1 v 2 ) = ( a 11 v 1 + a 12 v 2 a 21 v 1 + a 22 v 2 ) This is how we transform optical states—the matrix represents the optical element, and the vector represents the light state.
2.4 Important Properties of Matrix Multiplication ¶ Table 1: Matrix Multiplication Properties
Property
Mathematical Statement
Physical Meaning
Associative
( A B ) C = A ( B C ) (\mathbf{AB})\mathbf{C} = \mathbf{A}(\mathbf{BC}) ( AB ) C = A ( BC )
Order of grouping doesn’t matter
Not Commutative
A B ≠ B A \mathbf{AB} \neq \mathbf{BA} AB = BA
Order of optical elements matters!
Identity Element
A I = I A = A \mathbf{AI} = \mathbf{IA} = \mathbf{A} AI = IA = A
Identity represents “do nothing”
Distributive
A ( B + C ) = A B + A C \mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC} A ( B + C ) = AB + AC
Superposition principle
3 Special Matrices in Optics ¶ 3.1 The Identity Matrix ¶ The identity matrix represents “no change”:
I = ( 1 0 0 1 ) \mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} I = ( 1 0 0 1 ) For any matrix A \mathbf{A} A v ⃗ \vec{v} v
Physical meaning : An optical element that doesn’t change the light state (like a perfect window or empty space).
3.2 Rotation Matrices ¶ A rotation matrix rotates vectors by angle θ \theta θ
R ( θ ) = ( cos θ − sin θ sin θ cos θ ) \mathbf{R}(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} R ( θ ) = ( cos θ sin θ − sin θ cos θ ) For θ = 45 ° \theta = 45° θ = 45°
R ( 45 ° ) = ( 1 2 − 1 2 1 2 1 2 ) \mathbf{R}(45°) = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} R ( 45° ) = ( 2 1 2 1 − 2 1 2 1 ) Applying this to a horizontal vector ( 1 0 ) \begin{pmatrix} 1 \\ 0 \end{pmatrix} ( 1 0 )
R ( 45 ° ) ( 1 0 ) = ( 1 2 1 2 ) \mathbf{R}(45°)\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix} R ( 45° ) ( 1 0 ) = ( 2 1 2 1 ) This rotates the vector from the x-axis to 45° above the x-axis.
3.3 Inverse Matrices ¶ The inverse of matrix A \mathbf{A} A A − 1 \mathbf{A}^{-1} A − 1
A A − 1 = A − 1 A = I \mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I} A A − 1 = A − 1 A = I For a 2×2 matrix:
A − 1 = 1 det ( A ) ( a 22 − a 12 − a 21 a 11 ) \mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \begin{pmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{pmatrix} A − 1 = det ( A ) 1 ( a 22 − a 21 − a 12 a 11 ) where det ( A ) = a 11 a 22 − a 12 a 21 \det(\mathbf{A}) = a_{11}a_{22} - a_{12}a_{21} det ( A ) = a 11 a 22 − a 12 a 21 determinant .
Physical meaning : The inverse represents the “reverse” operation—if a matrix transforms state A to state B, its inverse transforms state B back to state A.
4 Jones Matrices: Polarization Optics ¶ 4.1 Representing Polarized Light ¶ In Jones calculus , we represent polarized light as a complex 2D vector:
E ⃗ = ( E x E y ) = ( E x e i ϕ x E y e i ϕ y ) \vec{E} = \begin{pmatrix} E_x \\ E_y \end{pmatrix} = \begin{pmatrix} E_x e^{i\phi_x} \\ E_y e^{i\phi_y} \end{pmatrix} E = ( E x E y ) = ( E x e i ϕ x E y e i ϕ y ) Table 2: Common Polarization States
Polarization Type
Jones Vector
Physical Description
Horizontal linear
( 1 0 ) \begin{pmatrix} 1 \\ 0 \end{pmatrix} ( 1 0 )
Electric field oscillates in x-direction
Vertical linear
( 0 1 ) \begin{pmatrix} 0 \\ 1 \end{pmatrix} ( 0 1 )
Electric field oscillates in y-direction
45° linear
1 2 ( 1 1 ) \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} 2 1 ( 1 1 )
Equal x and y components, in phase
Right circular
1 2 ( 1 − i ) \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix} 2 1 ( 1 − i )
x leads y by 90°
Left circular
1 2 ( 1 i ) \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix} 2 1 ( 1 i )
y leads x by 90°
4.2 Jones Matrices for Common Optical Elements ¶ Linear Polarizer (transmission axis at angle θ \theta θ
P ( θ ) = ( cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ ) \mathbf{P}(\theta) = \begin{pmatrix} \cos^2\theta & \cos\theta\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta \end{pmatrix} P ( θ ) = ( cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ ) Special cases:
Horizontal polarizer (θ = 0 \theta = 0 θ = 0 P H = ( 1 0 0 0 ) \mathbf{P}_H = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} P H = ( 1 0 0 0 )
Vertical polarizer (θ = 90 ° \theta = 90° θ = 90° P V = ( 0 0 0 1 ) \mathbf{P}_V = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} P V = ( 0 0 0 1 )
Quarter-Wave Plate (fast axis at angle θ \theta θ
Q ( θ ) = R ( − θ ) ( 1 0 0 i ) R ( θ ) \mathbf{Q}(\theta) = \mathbf{R}(-\theta) \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \mathbf{R}(\theta) Q ( θ ) = R ( − θ ) ( 1 0 0 i ) R ( θ ) Half-Wave Plate (fast axis at angle θ \theta θ
H ( θ ) = R ( − θ ) ( 1 0 0 − 1 ) R ( θ ) \mathbf{H}(\theta) = \mathbf{R}(-\theta) \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \mathbf{R}(\theta) H ( θ ) = R ( − θ ) ( 1 0 0 − 1 ) R ( θ ) 4.3 Worked Example: Polarization Analysis ¶ Analyze what happens when horizontally polarized light passes through:
A 45° polarizer
A vertical polarizer
Solution:
Initial state: E ⃗ 0 = ( 1 0 ) \vec{E}_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} E 0 = ( 1 0 )
Step 1: 45° polarizer
P 45 ° = ( 1 / 2 1 / 2 1 / 2 1 / 2 ) \mathbf{P}_{45°} = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix} P 45° = ( 1/2 1/2 1/2 1/2 ) E ⃗ 1 = P 45 ° E ⃗ 0 = ( 1 / 2 1 / 2 1 / 2 1 / 2 ) ( 1 0 ) = ( 1 / 2 1 / 2 ) \vec{E}_1 = \mathbf{P}_{45°}\vec{E}_0 = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1/2 \\ 1/2 \end{pmatrix} E 1 = P 45° E 0 = ( 1/2 1/2 1/2 1/2 ) ( 1 0 ) = ( 1/2 1/2 ) Step 2: Vertical polarizer
P V = ( 0 0 0 1 ) \mathbf{P}_V = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} P V = ( 0 0 0 1 ) E ⃗ 2 = P V E ⃗ 1 = ( 0 0 0 1 ) ( 1 / 2 1 / 2 ) = ( 0 1 / 2 ) \vec{E}_2 = \mathbf{P}_V\vec{E}_1 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1/2 \\ 1/2 \end{pmatrix} = \begin{pmatrix} 0 \\ 1/2 \end{pmatrix} E 2 = P V E 1 = ( 0 0 0 1 ) ( 1/2 1/2 ) = ( 0 1/2 ) Combined effect:
E ⃗ 2 = P V P 45 ° E ⃗ 0 = ( 0 0 0 1 ) ( 1 / 2 1 / 2 1 / 2 1 / 2 ) = ( 0 0 1 / 2 1 / 2 ) \vec{E}_2 = \mathbf{P}_V\mathbf{P}_{45°}\vec{E}_0 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1/2 & 1/2 \end{pmatrix} E 2 = P V P 45° E 0 = ( 0 0 0 1 ) ( 1/2 1/2 1/2 1/2 ) = ( 0 1/2 0 1/2 ) Intensity: I = ∣ E ⃗ 2 ∣ 2 = ∣ 1 / 2 ∣ 2 = 1 / 4 I = |\vec{E}_2|^2 = |1/2|^2 = 1/4 I = ∣ E 2 ∣ 2 = ∣1/2 ∣ 2 = 1/4
Without the intermediate polarizer, crossed polarizers would transmit zero intensity (Malus’s law). The 45° polarizer allows 25% transmission.
4.4 Complex Polarization States ¶ Real optical systems often involve complex Jones matrices with both real and imaginary elements:
A quarter-wave plate with fast axis along x converts linear polarization to circular:
QWP matrix: Q = ( 1 0 0 i ) \mathbf{Q} = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} Q = ( 1 0 0 i )
Input: 45° linear polarization E ⃗ i n = 1 2 ( 1 1 ) \vec{E}_{in} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} E in = 2 1 ( 1 1 )
Output:
E ⃗ o u t = Q E ⃗ i n = ( 1 0 0 i ) 1 2 ( 1 1 ) = 1 2 ( 1 i ) \vec{E}_{out} = \mathbf{Q}\vec{E}_{in} = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix} E o u t = Q E in = ( 1 0 0 i ) 2 1 ( 1 1 ) = 2 1 ( 1 i ) This is left-handed circular polarization! The imaginary unit i i i
5 Ray Transfer Matrices: Geometrical Optics ¶ In geometrical optics , we can represent ray propagation using 2×2 matrices. A ray is characterized by:
r ⃗ = ( y θ ) \vec{r} = \begin{pmatrix} y \\ \theta \end{pmatrix} r = ( y θ ) An optical element transforms the ray according to:
r ⃗ o u t = ( A B C D ) r ⃗ i n = ( A B C D ) ( y i n θ i n ) \vec{r}_{out} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\vec{r}_{in} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\begin{pmatrix} y_{in} \\ \theta_{in} \end{pmatrix} r o u t = ( A C B D ) r in = ( A C B D ) ( y in θ in ) 5.2 ABCD Matrices for Common Elements ¶ Table 3: Ray Transfer Matrices
Optical Element
ABCD Matrix
Physical Effect
Free space (distance d)
( 1 d 0 1 ) \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} ( 1 0 d 1 )
Ray height changes, angle unchanged
Thin lens (focal length f)
( 1 0 − 1 / f 1 ) \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix} ( 1 − 1/ f 0 1 )
Height unchanged, angle changes
Curved mirror (radius R)
( 1 0 − 2 / R 1 ) \begin{pmatrix} 1 & 0 \\ -2/R & 1 \end{pmatrix} ( 1 − 2/ R 0 1 )
Reflection and focusing
Flat interface (n₁ to n₂)
( 1 0 0 n 1 / n 2 ) \begin{pmatrix} 1 & 0 \\ 0 & n_1/n_2 \end{pmatrix} ( 1 0 0 n 1 / n 2 )
Refraction changes angle
5.3 System Analysis Using Matrix Products ¶ For multiple elements in series, multiply their matrices in reverse order :
M t o t a l = M N M N − 1 ⋯ M 2 M 1 \mathbf{M}_{total} = \mathbf{M}_N \mathbf{M}_{N-1} \cdots \mathbf{M}_2 \mathbf{M}_1 M t o t a l = M N M N − 1 ⋯ M 2 M 1 Analyze a telescope consisting of:
Objective lens (focal length f 1 = 100 f_1 = 100 f 1 = 100
Distance d = 120 d = 120 d = 120
Eyepiece lens (focal length f 2 = 20 f_2 = 20 f 2 = 20
Solution:
Individual matrices:
Objective: L 1 = ( 1 0 − 1 / 100 1 ) \mathbf{L}_1 = \begin{pmatrix} 1 & 0 \\ -1/100 & 1 \end{pmatrix} L 1 = ( 1 − 1/100 0 1 )
Free space: D = ( 1 120 0 1 ) \mathbf{D} = \begin{pmatrix} 1 & 120 \\ 0 & 1 \end{pmatrix} D = ( 1 0 120 1 )
Eyepiece: L 2 = ( 1 0 − 1 / 20 1 ) \mathbf{L}_2 = \begin{pmatrix} 1 & 0 \\ -1/20 & 1 \end{pmatrix} L 2 = ( 1 − 1/20 0 1 )
System matrix:
M = L 2 D L 1 \mathbf{M} = \mathbf{L}_2 \mathbf{D} \mathbf{L}_1 M = L 2 D L 1 Step 1: D L 1 = ( 1 120 0 1 ) ( 1 0 − 1 / 100 1 ) = ( − 0.2 120 − 0.01 1 ) \mathbf{D}\mathbf{L}_1 = \begin{pmatrix} 1 & 120 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -1/100 & 1 \end{pmatrix} = \begin{pmatrix} -0.2 & 120 \\ -0.01 & 1 \end{pmatrix} D L 1 = ( 1 0 120 1 ) ( 1 − 1/100 0 1 ) = ( − 0.2 − 0.01 120 1 )
Step 2: M = ( 1 0 − 1 / 20 1 ) ( − 0.2 120 − 0.01 1 ) = ( − 0.2 120 − 0.0105 − 5 ) \mathbf{M} = \begin{pmatrix} 1 & 0 \\ -1/20 & 1 \end{pmatrix}\begin{pmatrix} -0.2 & 120 \\ -0.01 & 1 \end{pmatrix} = \begin{pmatrix} -0.2 & 120 \\ -0.0105 & -5 \end{pmatrix} M = ( 1 − 1/20 0 1 ) ( − 0.2 − 0.01 120 1 ) = ( − 0.2 − 0.0105 120 − 5 )
Analysis:
Magnification: M = − A = 0.2 M = -A = 0.2 M = − A = 0.2
Angular magnification: M θ = − 1 / A = 5 M_θ = -1/A = 5 M θ = − 1/ A = 5
5.4 Physical Meaning of ABCD Elements ¶ Each element of the ABCD matrix has physical significance:
( y o u t θ o u t ) = ( A B C D ) ( y i n θ i n ) \begin{pmatrix} y_{out} \\ \theta_{out} \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\begin{pmatrix} y_{in} \\ \theta_{in} \end{pmatrix} ( y o u t θ o u t ) = ( A C B D ) ( y in θ in ) A : Height magnification (y o u t / y i n y_{out}/y_{in} y o u t / y in θ i n = 0 \theta_{in} = 0 θ in = 0
B : Height displacement per unit input angle (mm/mrad)
C : Angle change per unit input height (optical power, m⁻¹)
D : Angle magnification (θ o u t / θ i n \theta_{out}/\theta_{in} θ o u t / θ in y i n = 0 y_{in} = 0 y in = 0
6 Practical Applications and Examples ¶ 6.1 Polarization State Analysis ¶ Design a system to convert right-handed circular polarization to 30° linear polarization.
Solution:
Input: E ⃗ i n = 1 2 ( 1 − i ) \vec{E}_{in} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix} E in = 2 1 ( 1 − i )
Target: E ⃗ t a r g e t = ( cos 30 ° sin 30 ° ) = ( 3 / 2 1 / 2 ) \vec{E}_{target} = \begin{pmatrix} \cos 30° \\ \sin 30° \end{pmatrix} = \begin{pmatrix} \sqrt{3}/2 \\ 1/2 \end{pmatrix} E t a r g e t = ( cos 30° sin 30° ) = ( 3 /2 1/2 )
Step 1: Convert circular to linear using a quarter-wave plate
QWP with fast axis at -45°: Q − 45 ° = ( ( 1 + i ) / 2 ( 1 − i ) / 2 ( 1 − i ) / 2 ( 1 + i ) / 2 ) \mathbf{Q}_{-45°} = \begin{pmatrix} (1+i)/2 & (1-i)/2 \\ (1-i)/2 & (1+i)/2 \end{pmatrix} Q − 45° = ( ( 1 + i ) /2 ( 1 − i ) /2 ( 1 − i ) /2 ( 1 + i ) /2 )
Step 2: Rotate the linear polarization using a half-wave plate
HWP at appropriate angle to rotate from resulting linear state to 30°
This systematic approach using matrix multiplication makes complex polarization manipulations manageable.
6.2 Optical System Design ¶ Design a Galilean beam expander with 3× magnification using lenses with focal lengths 50 mm and -150 mm.
Solution:
System layout: Positive lens → Distance d → Negative lens
Matrices:
L 1 = ( 1 0 − 1 / 50 1 ) \mathbf{L}_1 = \begin{pmatrix} 1 & 0 \\ -1/50 & 1 \end{pmatrix} L 1 = ( 1 − 1/50 0 1 )
D = ( 1 d 0 1 ) \mathbf{D} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} D = ( 1 0 d 1 )
L 2 = ( 1 0 1 / 150 1 ) \mathbf{L}_2 = \begin{pmatrix} 1 & 0 \\ 1/150 & 1 \end{pmatrix} L 2 = ( 1 1/150 0 1 )
For 3× expansion: We need the A element = -3
System matrix: M = L 2 D L 1 \mathbf{M} = \mathbf{L}_2\mathbf{D}\mathbf{L}_1 M = L 2 D L 1
Working through the multiplication with the constraint A = -3 gives us the required spacing d.
7 Common Pitfalls and How to Avoid Them ¶ 7.1 Order of Operations ¶ Pitfall : Multiplying matrices in the wrong order.
Solution : Remember that for light traveling left to right through elements 1, 2, 3, the system matrix is M 3 M 2 M 1 \mathbf{M}_3\mathbf{M}_2\mathbf{M}_1 M 3 M 2 M 1
7.2 Sign Conventions ¶ 7.3 Physical Interpretation ¶ Pitfall : Treating matrices as pure mathematics without physical meaning.
Solution : Always ask “What does this matrix element represent physically?” Each number should have units and physical significance.
7.4 Complex vs. Real Matrices ¶ Pitfall : Mixing Jones calculus (complex) with ray matrices (real).
Solution : Jones matrices describe polarization (wave optics), ABCD matrices describe ray propagation (geometrical optics). Don’t mix the formalisms unless you’re doing sophisticated coherent ray analysis.
8 Matrix Multiplication Techniques and Shortcuts ¶ 8.1 Special Matrix Products ¶ Rotation matrices:
R ( α ) R ( β ) = R ( α + β ) \mathbf{R}(\alpha)\mathbf{R}(\beta) = \mathbf{R}(\alpha + \beta) R ( α ) R ( β ) = R ( α + β ) Translation matrices:
( 1 d 1 0 1 ) ( 1 d 2 0 1 ) = ( 1 d 1 + d 2 0 1 ) \begin{pmatrix} 1 & d_1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & d_2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & d_1 + d_2 \\ 0 & 1 \end{pmatrix} ( 1 0 d 1 1 ) ( 1 0 d 2 1 ) = ( 1 0 d 1 + d 2 1 ) Thin lens combinations:
Two thin lenses in contact: L t o t a l = L 2 L 1 \mathbf{L}_{total} = \mathbf{L}_2\mathbf{L}_1 L t o t a l = L 2 L 1
( 1 0 − 1 / f 2 1 ) ( 1 0 − 1 / f 1 1 ) = ( 1 0 − 1 / f e f f 1 ) \begin{pmatrix} 1 & 0 \\ -1/f_2 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -1/f_1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -1/f_{eff} & 1 \end{pmatrix} ( 1 − 1/ f 2 0 1 ) ( 1 − 1/ f 1 0 1 ) = ( 1 − 1/ f e ff 0 1 ) where 1 f e f f = 1 f 1 + 1 f 2 \frac{1}{f_{eff}} = \frac{1}{f_1} + \frac{1}{f_2} f e ff 1 = f 1 1 + f 2 1
8.2 Computational Tips ¶ For hand calculations:
Look for patterns (identity elements, zeros)
Factor common terms
Use special properties (orthogonal, symmetric matrices)
Check dimensions and units at each step
For computer calculations:
import numpy as np
# Define matrices
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
# Matrix multiplication
C = A @ B # or np.dot(A, B)9 Summary and Physical Insights ¶ 9.1 Why Matrix Methods Work So Well ¶ Matrix multiplication succeeds in optics because:
Table 4: Fundamental Reasons for Matrix Success
Mathematical Property
Physical Principle
Optical Example
Linearity
Superposition of waves
Interference, diffraction
Composition
Sequential operations
Multi-element systems
Group structure
Reversibility
Time-reversed paths
Representation theory
Symmetries
Crystal optics, polarization
9.2 Connections to Other Physics ¶ Quantum Mechanics : Matrix methods in optics directly parallel quantum mechanical operators acting on state vectors.
Classical Mechanics : ABCD matrices are analogous to transfer matrices in mechanical vibrations and electrical circuits.
Signal Processing : Jones matrices operate on complex signals just like digital filters operate on electronic signals.
9.3 Practical Benefits ¶ Design : Systematic approach to complex optical systems
Analysis : Reveal system properties (stability, bandwidth, etc.)
Optimization : Mathematical framework for improvement
Simulation : Direct computer implementation
Insight : Physical understanding through mathematical structure
10 Practice Problems ¶ 10.1 Basic Matrix Operations ¶ Calculate the following matrix products:
a) ( 2 1 0 3 ) ( 1 4 2 1 ) \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix} ( 2 0 1 3 ) ( 1 2 4 1 )
b) ( 1 0 − 1 / 50 1 ) ( 1 100 0 1 ) \begin{pmatrix} 1 & 0 \\ -1/50 & 1 \end{pmatrix}\begin{pmatrix} 1 & 100 \\ 0 & 1 \end{pmatrix} ( 1 − 1/50 0 1 ) ( 1 0 100 1 )
c) ( cos θ − sin θ sin θ cos θ ) 2 \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}^2 ( cos θ sin θ − sin θ cos θ ) 2
10.2 Polarization Analysis ¶ 10.3 Ray Optics System ¶ 10.4 System Design ¶ Design a 4f optical processor (two lenses separated by sum of focal lengths) using:
Input lens: f₁ = 200 mm
Output lens: f₂ = 100 mm
a) Find the distances for proper 4f configuration
b) Calculate the system ABCD matrix
c) What is the lateral magnification?
d) What happens to the angular magnification?
11 Solutions to Practice Problems ¶
a) ( 2 1 0 3 ) ( 1 4 2 1 ) = ( 4 9 6 3 ) \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 4 & 9 \\ 6 & 3 \end{pmatrix} ( 2 0 1 3 ) ( 1 2 4 1 ) = ( 4 6 9 3 )
b) ( 1 0 − 1 / 50 1 ) ( 1 100 0 1 ) = ( 1 100 − 1 / 50 1 ) \begin{pmatrix} 1 & 0 \\ -1/50 & 1 \end{pmatrix}\begin{pmatrix} 1 & 100 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 100 \\ -1/50 & 1 \end{pmatrix} ( 1 − 1/50 0 1 ) ( 1 0 100 1 ) = ( 1 − 1/50 100 1 )
c) ( cos 2 θ − sin 2 θ sin 2 θ cos 2 θ ) \begin{pmatrix} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{pmatrix} ( cos 2 θ sin 2 θ − sin 2 θ cos 2 θ ) 2 θ 2\theta 2 θ
This requires step-by-step matrix multiplication through all three elements. The QWP converts the 45° linear to circular, the HWP rotates the polarization, and the final polarizer extracts the component along 60°. The calculation involves complex arithmetic due to the QWP matrix.
The thick lens matrix is the product of three matrices:
M = M b a c k ⋅ M t h i c k n e s s ⋅ M f r o n t \mathbf{M} = \mathbf{M}_{back} \cdot \mathbf{M}_{thickness} \cdot \mathbf{M}_{front} M = M ba c k ⋅ M t hi c kn ess ⋅ M f ro n t where each matrix represents refraction at interfaces and propagation through the material.
a) Distances: 200 mm + 100 mm = 300 mm separation
b) The 4f system has the special property that A = -1, B = 0, giving perfect imaging
c) Lateral magnification = -f₂/f₁ = -0.5
d) Angular magnification = -f₁/f₂ = -2
12 Final Thoughts: The Elegance of Linear Algebra in Physics ¶ Matrix multiplication reveals deep structural relationships in optics that would be nearly impossible to see otherwise. The mathematical framework doesn’t just make calculations easier—it reveals the underlying symmetries and conservation laws that govern how light behaves.
“Mathematics is the language with which God has written the universe.”
Galileo Galilei
As you continue studying optics, you’ll discover that matrix methods appear everywhere: from the simplest polarizer to the most sophisticated laser system. The investment in understanding these mathematical tools pays dividends throughout your career in optics and photonics.
Tip: Developing Matrix Intuition : The key to mastering matrix methods is developing physical intuition for what each mathematical operation means:
Matrix multiplication = sequential operations
Matrix inverse = reverse operation
Determinant = scaling factor
Trace = system characteristic
With practice, you’ll begin to “think in matrices” when analyzing optical systems, making complex designs seem surprisingly manageable.
Matrix multiplication in optics is more than a computational tool—it’s a window into the mathematical structure of the electromagnetic world. Master these methods, and you’ll have one of the most powerful techniques in all of physics at your command.