EE535: OPTICS & PHOTONICS

LECT. 04 2025 01 26
LECTURE OUTLINE

Topics
•Ch. 1: Ray Optics
•1.4: Transfer matrices for ray optics
•Ch. 2: Wave Optics
Notes
• Some suggested problems will be suggested at the end
of each chapter.
1.4: TRANSFER MATRICES FOR RAY OPTICS
THE RAY-TRANSFER MATRIX

 So, to trace a ray as it travels around the optical axis (𝑧-axis),

we need to check how 𝑦(𝑧) and 𝜃(𝑧) change.
 In principle, we should be able to relate 𝑦 𝑧1 , 𝜃 𝑧1
to 𝑦 𝑧2 , 𝜃 𝑧2 by considering the optical medium
between 𝑧1 & 𝑧2 .
 𝑦 𝑧2 = 𝑓 𝑦 𝑧1 , 𝜃 𝑧1

 𝜃 𝑧2 = 𝑔 𝑦 𝑧1 , 𝜃 𝑧1

 If the medium is linear, the relations are reduced to:

𝑦 𝑧2 = 𝐴 𝑦 𝑧1 , +𝐵 𝜃 𝑧1 𝑦 𝐴 𝐵 𝑦
ቋ⟹ 𝑧2 = 𝑧 ⟹ 𝒘 𝑧2 = 𝐌 𝒘 𝑧1
𝜃 𝑧2 = 𝐶 𝑦 𝑧1 , +𝐷 𝜃 𝑧1 𝜃 𝐶 𝐷 𝜃 1
1.4: TRANSFER MATRICES FOR RAY OPTICS
MATRICES OF SIMPLE OPTICAL COMPONENTS

1. Propagation over a distance 𝑑:

 No change in the angle: 𝜃2 = 𝜃1
 What about 𝑦?
 Clearly, 𝑦2 = 𝑦1 + Δ𝑦 = 𝑦1 + 𝑑 tan 𝜃1 ≈ 𝑦1 + 𝑑 𝜃1
1 𝑑
 So, 𝐌 =
0 1
2. Refraction at a Planar Boundary:
 Clearly, 𝑦2 = 𝑦1 .
𝑛1
 By Snell’s law: 𝑛2 sin 𝜃2 = 𝑛1 sin 𝜃1 → 𝜃2 ≈ 𝜃
𝑛2 1

1 0
 So, 𝐌 = 0 𝑛1
𝑛2
1.4: TRANSFER MATRICES FOR RAY OPTICS
MATRICES OF SIMPLE OPTICAL COMPONENTS

3. Refraction at a Spherical Boundary:

 Let’s start with the easy part. Clearly, 𝑦2 = 𝑦1 .
 Previously, we considered spherical boundaries and found that:
𝑛1 𝑛1 𝑛1 𝑛2 − 𝑛1 𝑦
𝜃2 ≈ 𝜃1 + − 1 𝜃0 ≈ 𝜃1 −
𝑛2 𝑛2 𝑛2 𝑛2 𝑅
1 0
 So, 𝐌 = − (𝑛2−𝑛1) 𝑛1
𝑛2 𝑅 𝑛2

 What if the boundary becomes flat?

 𝑅→∞

1 0 1 0
 (𝑛2 −𝑛1 ) 𝑛1 → 0 𝑛1
− 𝑛2
𝑛2 𝑅 𝑛2
1.4: TRANSFER MATRICES FOR RAY OPTICS
MATRICES OF SIMPLE OPTICAL COMPONENTS

4. Transmission Through a Thin Lens:

 Again, the easy part; 𝑦2 = 𝑦1 .
𝑦
 As learned last Tuesday: 𝜃2 = 𝜃1 −
𝑓

1 0
 So, 𝐌 = − 1 1
𝑓

5. Reflection from a Planar Mirror:

 What is changing?
 The optical axis got reflected.
 In this case, 𝑦 and 𝜃 remained unchanged.

1 0
 So , 𝐌 =
0 1
1.4: TRANSFER MATRICES FOR RAY OPTICS
MATRICES OF SIMPLE OPTICAL COMPONENTS

6. Reflection from a Spherical Mirror:

 The easy part, clearly, 𝑦2 = 𝑦1 .
 Previously, we considered spherical mirror and found that:
2𝑦
𝜃1 − 𝜃2 = 2𝜃0 =
−𝑅
1 0
 So, 𝐌 = 2
1
𝑅

 What if the boundary becomes flat?

 𝑅→∞

1 0 1 0
 2
1 →
𝑅 0 1
1.4: TRANSFER MATRICES FOR RAY OPTICS
MATRICES OF CASCADED OPTICAL COMPONENTS

 Consider a cascade on 𝑁 optical components or systems as shown in the figure:

 𝒘2 = 𝐌1 𝒘1
 𝒘3 = 𝐌2 𝒘2
 ⋮
 𝒘𝑁+1 = 𝐌𝑁 𝒘𝑁
 Can we connect 𝒘3 to 𝒘1 ?
 𝒘3 = 𝐌2 𝒘2 = 𝐌1 𝒘1 = 𝐌2 𝐌1 𝒘1
 By repeating this 𝑁 times,
𝟏

𝒘𝑁+1 = 𝐌𝑁 𝐌𝑁−1 ⋯ 𝐌2 𝐌1 𝒘1 = ෑ 𝐌𝑘 𝒘1 = 𝐌𝒘1

𝑘=𝑁
1.4: TRANSFER MATRICES FOR RAY OPTICS
MATRICES OF CASCADED OPTICAL COMPONENTS

 Exercise 1.4-2: Consider a set of 𝑁 parallel planar transparent plates of refractive indices 𝑛1 , 𝑛2 , ⋯ , 𝑛𝑁
and thicknesses 𝑑1 , 𝑑2 , ⋯ , 𝑑𝑁 , placed in air (𝑛 = 1) normal to the 𝑧-axis. Find the net transfer matrix?
 𝑁 + 1 planer interfaces:
1 0
 𝐌I,𝑘 = 0 𝑛𝑘−1
𝑛𝑘

 𝑁 Propagation over a distance:

1 𝑑𝑙
 𝐌T,𝒍 =
0 1
 The effective transfer matrix:
𝐌 = 𝐌I,𝑁+1 𝐌T,𝑁 𝐌I,𝑁 ⋯ 𝐌T,2 𝐌I,2 𝐌T,1 𝐌I,1
𝑛𝑘−1 𝑁
1 1 1 𝑑 𝑑𝑘
1 0 𝑛𝑘 𝑘
= 𝐌I,𝑁+1 ෑ 𝐌T,𝑘 𝐌I,𝑘 = ෑ 𝑛𝑘−1 = 1 ෍
𝑛𝑘
0 𝑛𝑁 𝑘=1
𝑘=𝑁 𝑘=𝑁 0
𝑛𝑘 0 1
1.4: TRANSFER MATRICES FOR RAY OPTICS

 Exercise 1.4-3: Find the effective transfer matrix for a gap followed by a thin lens.

1 0 1 𝑑 1 𝑑
 𝐌 = −1 1 0 1 = −
1
1−
𝑑
𝑓 𝑓 𝑓

 Exercise 1.4-1: Determine the needed conditions in the transfer matrix:

 Focusing system: All rays entering the system at a particular angle, whatever their position,
leave at a single position.
 𝐴=0

 Imaging system: All rays entering the system at a particular position, whatever their angle, leave
at a single position.
 𝐵=0
1.4: TRANSFER MATRICES FOR RAY OPTICS
IMAGING WITH AN ARBITRARY PARAXIAL OPTICAL SYSTEM

 Let’s consider an arbitrary paraxial system describe by some transfer matrix:

𝑦2 𝐴 𝐵 𝑦1 𝑦1 1 𝐷 −𝐵 𝑦2
𝜃2 = ⟹ =
𝐶 𝐷 𝜃1 𝜃1 det 𝐌 −𝐶 𝐴 𝜃2
 Typo in Eq. 1.4-13

 Back focal point: 𝐹

 A parallel ray comes from left crosses the optical axis at 𝐹.
𝑦1 𝑦 1
 The back focal length: 𝑓 = ≈ − 𝜃1 = − 𝐶
− tan 𝜃2 2

 But, 𝑓 is larger than the distance between the right edge 𝑉 and the back focal point 𝐹:
𝑦2 𝐴𝑦1 𝐴
 𝐹−𝑉 ≈ = =−
−𝜃2 −𝐶𝑦1 𝐶

1 𝐴 1
 ℎ =𝑓− 𝐹−𝑉 =− + =− 1−𝐴 = 1−𝐴 𝑓
𝐶 𝐶 𝐶
1.4: TRANSFER MATRICES FOR RAY OPTICS
IMAGING WITH AN ARBITRARY PARAXIAL OPTICAL SYSTEM

 Let’s consider an arbitrary paraxial system describe by some transfer matrix:

𝑦2 𝐴 𝐵 𝑦1 𝑦1 1 𝐷 −𝐵 𝑦2
𝜃2 = ⟹ =
𝐶 𝐷 𝜃1 𝜃1 det 𝐌 −𝐶 𝐴 𝜃2
 Type in Eq. 1.4-13

 Front focal point: 𝐹′

 A parallel ray comes from right crosses the optical axis at 𝐹 ′ .
𝑦2 𝑦 𝑦2 1
 The front focal length: 𝑓 ′ = ≈ 𝜃2 = det 𝐌 = det 𝐌 = det 𝐌 𝑓
tan 𝜃1 1 −𝐶𝑦2 −𝐶

 Again, 𝑓 ′ is larger than the distance between the left edge 𝑉 ′ and the front focal point 𝐹 ′ :
𝑦1 𝐷𝑦2 𝐷
 𝑉 ′ − 𝐹′ ≈ = =
𝜃1 −𝐶𝑦2 −𝐶

𝐷
 −ℎ′ = 𝑓 ′ − 𝑉 ′ − 𝐹 ′ ⟹ ℎ′ = −𝑓 ′ + 𝑉 ′ − 𝐹 ′ = −𝑓 ′ − = −𝑓 ′ + 𝐷𝑓
𝐶
1.4: TRANSFER MATRICES FOR RAY OPTICS
PERIODIC OPTICAL SYSTEMS

 In many cases, one ends up with repeated identical unit systems, as shown in the figure.

𝑦𝑛 𝑛 𝑦0
𝐴 𝐵
 So, clearly, 𝜃 = 𝜃0
𝑛 𝐶 𝐷
 The function of any matrix can be calculated
using its eigenvalue decomposition.
𝐴 𝐵 𝜂1 0 −1
 = 𝒗1 𝒗2 𝒗1 𝒗2
𝐶 𝐷 0 𝜂2
𝐴 𝐵 𝜂1 0 −1
 𝑓 = 𝒗1 𝒗2 𝑓 𝒗1 𝒗2
𝐶 𝐷 0 𝜂2
𝑛 𝑛
𝐴 𝐵 𝜂1 0 −1 𝜂1𝑛 0 −1
 = 𝒗1 𝒗2 𝒗1 𝒗2 = 𝒗1 𝒗2 𝑛
𝒗1 𝒗2
𝐶 𝐷 0 𝜂2 0 𝜂2
1.4: TRANSFER MATRICES FOR RAY OPTICS
PERIODIC OPTICAL SYSTEMS

𝐴 𝐵
 For, 𝐌 = ,
𝐶 𝐷
𝐴+𝐷 𝐴+𝐷 2
 𝜂1,2 = ± − det 𝐌 = 𝑏 ± 𝑏2 − 𝐹 2 = 𝑏 ± 𝑗 𝐹 2 − 𝑏 2
2 2
𝑏
 Let cos 𝜑 = .
𝐹

 So, 𝜂1,2 = 𝑏 ± 𝑗 𝐹 2 − 𝑏 2 = 𝐹 cos 𝜑 ± 𝑗𝐹 1 − cos 2 𝜑 = 𝐹 cos 𝜑 ± 𝑗 sin 𝜑 = 𝐹𝑒 ±𝑗𝜑

𝑛
 Then, 𝜂1,2 = 𝐹 𝑛 𝑒 ±𝑗𝑛𝜑
 𝐹 depends on the refractive indexes of the first and last media.
𝑛𝑚+1
 𝐹 2 = det 𝐌 =
𝑛0

 If the first and last media are the same, 𝐹 = 1.

1.4: TRANSFER MATRICES FOR RAY OPTICS
PERIODIC OPTICAL SYSTEMS

𝑛
𝜂 0
 Finally, by utilizing the facts that 𝜂1 and 𝜂2 are complex conjugates, 1 is diagonal
0 𝜂2
matrix, and the elements of are real values 𝐌, it can be shown that:
𝑛 𝐴 sin(𝑛𝜑) − sin 𝑛 − 1 𝜑 𝐵 sin(𝑛𝜑)
𝐴 𝐵 1
 =
𝐶 𝐷 sin(𝜑) 𝐶 cos(𝑛𝜑) 𝐷 sin(𝑛𝜑) − sin 𝑛 − 1 𝜑
 Please see Tovar, Anthony A., and Lee W. Casperson. "Generalized Sylvester theorems for
periodic applications in matrix optics." JOSA A 12.3 (1995): 578-590..
 In terms of ray parameters:
1 𝐵 sin 𝑛𝜑
 𝑦𝑛 = 𝐴 sin 𝑛𝜑 − sin 𝑛 − 1 𝜑 𝑦0 + 𝜃0
sin(𝜑) sin(𝜑)

cos 𝑛𝜑 1
 𝜃𝑛 = 𝐶 𝑦0 + 𝐷 sin(𝑛𝜑) − sin 𝑛 − 1 𝜑 𝜃0
sin(𝜑) sin(𝜑)
1.4: TRANSFER MATRICES FOR RAY OPTICS
PERIODIC OPTICAL SYSTEMS

 Furthermore, 𝐴 sin 𝑛𝜑 − sin 𝑛 − 1 𝜑 is clearly harmonic:

 Then, 𝑦𝑛 has some upper limit depends on 𝑦0 .
 𝑦𝑛 = 𝑦max 𝐹 𝑛 sin 𝑛𝜑 + 𝜙0

 If the first and last media are

the same, 𝐹 = 1.
𝑏
 But cos 𝜑 = .
𝐹

𝑏
 To have a table system, ≤ 1.
𝐹
1.4: TRANSFER MATRICES FOR RAY OPTICS

 Example 1.4-1: A Sequence of Equally Spaced Identical Lenses.

1 0 1 𝑑 1 𝑑
 𝐌 = −1 1 0 1 = −
1
1−
𝑑
𝑓 𝑓 𝑓

 Since the first and last media are air, 𝐹 = 1

𝐴+𝐷 𝑑
 𝑏= =1−
2 2𝑓

𝑏 𝑑
 To have a stable system = 1− ≤1
𝐹 2𝑓
𝑑 𝑑
 −1 ≤ 1 − ≤1 ⟹0≤ ≤2
2𝑓 2𝑓

 0 ≤ 𝑑 ≤ 4𝑓
CHAPTER 2: WAVE OPTICS

 Light propagates in the form of waves.

 Three main domains:
 Visible: 390 nm – 760 nm
 Infrared: 760 nm – 300 𝜇m
 Ultraviolet: 10 nm – 390 nm
2.1: POSTULATES OF WAVE OPTICS
THE WAVE EQUATION

 As electromagnetic radiation, light propagates in the form of two mutually coupled vector
waves, an electric-field wave, and a magnetic-field wave. (electromagnetic optics)
 In some cases, we can ignore the coupling and treat the light as a scalar wave. (wave optics)
 What are the main quantities needed to describe waves mathematically?
 Wave parameters: frequency (𝜈), angular frequency (𝜔), phase shift (𝜑), speed (𝑐), wavelength (𝜆)
 Amplitude

 A wave of scalar 𝑢 𝒓, 𝑡 is function changing oscillatory in time and space. This is described
by the “standard” PDE of scalar waves:
2 𝑢 𝒓, 𝑡
1 𝜕
∇2 𝑢 𝒓, 𝑡 − 2 =0.
𝑐 𝜕𝑡 2