ENEE 684 Chapter 5 | Geometrical Optics (Part 1)

Overview

Ergun Simsek introduces paraxial geometrical optics as a practical alternative to solving Maxwell’s equations when the wavelength is far smaller than lenses, mirrors, and other optical components. He derives and applies ABCD ray-transfer matrices to propagation, dielectric slabs, lenses, curved interfaces, and mirrors, then covers practical component selection and explains a quantum-imaging setup using 532 nm pump light and 810 nm/1550 nm photon pairs.

Key takeaways

For optical components much larger than the wavelength, paraxial ray optics replaces full electromagnetic analysis with calculations on ray height and angle.
An ABCD matrix maps [r, θ]ᵀ through an optical component, and a cascade must be multiplied in physical order with the first component on the right.
Free-space propagation over L uses [[1, L], [0, 1]], while a thin lens uses [[1, 0], [-1/f, 1]] under the lecture’s sign convention.
Beam-splitter selection must account for wavelength, polarization, split ratio, substrate material, and laser damage threshold; fused silica is often preferable to BK7 for high-power use.
Dichroic mirrors can separate an 800 nm input from its 400 nm frequency-doubled output, enabling shorter-wavelength experiments and finer fabrication.
In the quantum-imaging example, a 1550 nm photon probes the object while its correlated 810 nm partner reaches the camera, so image information is obtained without detecting the object-interacting photon.

Chapters

0:00 Why Chapter 5 Moves from Fiber Optics to Geometrical Optics

Ergun Simsek shifts from Chapter 4’s mathematics and fiber-optics focus to geometrical optics, which supports practical optical-lab design.
The planned examples include lens placement, magnification, focusing, beam expansion, and optical resonators.
The opening also previews quantum imaging as a newer application to discuss later in the lecture.

4:00 When Ray Optics Replaces Full Wave Analysis

For a 633 nm laser and centimeter-scale optics, the wavelength is much smaller than the components, so tracking rays is often sufficient.
Geometrical optics focuses mainly on ray position and propagation direction rather than polarization and field amplitude.
An optical system can be represented as a sequence of simple 2 × 2 transfer matrices.

6:00 Paraxial Rays and the Small-Angle Approximation

A ray is described by its radial displacement r from the optical axis and its angle θ relative to that axis.
Positive θ denotes an upward tilt and negative θ a downward tilt; paraxial rays stay near the optical axis.
For small angles, sin θ, tan θ, and θ in radians are approximately equal, simplifying Snell’s law to n₁θ₁ ≈ n₂θ₂.
The approximation is strongest at a few degrees, though Simsek says it can remain useful around 10° to 15° with some loss of accuracy.

10:00 Ray Vectors, ABCD Matrices, and Optical Design Software

The ray vector is written as [r, θ]ᵀ, and a component maps the incoming ray to the outgoing ray through a 2 × 2 ABCD matrix.
The output equations are r₂ = Ar₁ + Bθ₁ and θ₂ = Cr₁ + Dθ₁; the lecture uses centimeters for r and radians for θ.
Simsek identifies Zemax as widely used for detailed commercial optical design, including complex camera lens systems.
The ABCD method is presented as a convenient first design method, not a replacement for full, high-accuracy optical software.

15:00 Finding ABCD Coefficients and Checking the Determinant

A ray with θ₁ = 0 reveals A and C by showing how an initially parallel, displaced ray changes height and angle.
A ray with r₁ = 0 reveals B and D by showing how an initially on-axis, tilted ray changes height and angle.
For the free-space ray-transfer convention used in the lecture, the overall matrix determinant should be 1; a different value can signal an algebra error.
At a refractive-index boundary, the determinant follows the refractive-index ratio under the chosen ray-vector convention.

20:00 Alignment Errors, Beam Expansion, and Matrix Order

A ray can be angularly aligned with the optical axis, θ = 0, yet still be displaced from it, r ≠ 0.
A beam expander uses a diverging lens followed by a converging lens; a 1 mm beam can be designed to emerge about 10 mm wide.
The lens spacing and focal lengths must be chosen together to produce the desired output beam.
For cascaded components, multiply matrices in physical sequence with the first component on the right, applying the product from right to left.

24:00 Free-Space Propagation and Dielectric Slabs

Over a free-space distance L, the ray height changes by approximately Lθ while its angle stays constant, giving the matrix [[1, L], [0, 1]].
A plane-parallel dielectric slab preserves the external ray direction but changes the lateral displacement inside the material.
In the lecture’s approximation, the slab’s propagation term uses an effective distance L/n rather than L.
The slab model is useful when estimating beam shifts through components such as beam splitters.

30:00 Thin-Lens Refraction and the Lens Center Ray

Under the thin-lens approximation, a lens preserves ray height at the lens plane and changes the ray angle according to focal length.
The lens matrix is [[1, 0], [-1/f, 1]], with the sign of f distinguishing converging and diverging lenses.
A ray aimed through the lens center continues approximately undeviated because it meets the local surfaces near normal incidence.
The lecture cautions that real lenses can be thick and affected by aberrations, limiting the thin-lens model.

35:00 Curved Interfaces and Refraction Through Spherical Surfaces

Simsek applies the small-angle form of Snell’s law at a spherical dielectric boundary to relate the incident and transmitted ray angles.
For a single refracting surface, ray height is unchanged at the interface while curvature and refractive indices determine the angular change.
The sign convention assigns positive radius to a convex surface whose center of curvature lies to the right, and negative radius to a concave surface.
The resulting matrix offers a practical first-order model for lens surfaces and other curved optical structures.

40:00 Spherical Mirrors, Focal Length, and the Mirror Equation

For a spherical mirror, the paraxial focal length is half the radius of curvature: f = R/2.
The lecture uses positive focal length for concave mirrors and negative focal length for convex mirrors.
Two principal rays, one parallel to the optical axis and one directed toward the mirror center, locate the reflected image.
Object distance u and image distance v satisfy 1/u + 1/v = 1/f.

45:00 Beam Splitters for Polarization and Reference Paths

Beam splitters divide an optical signal into paths, such as a measurement arm and a reference arm in an interferometric system.
Polarizing beam splitters transmit and reflect different polarization states, useful when a downstream optical modulator requires vertically polarized light.
Beam-splitter designs vary by wavelength range, power handling, polarization behavior, and the number of output beams.
Simsek notes that diamond photonics can route light among multiple low-power devices with low loss.

50:00 Choosing Beam Splitters by Wavelength, Glass, and Laser Power

Check that a beam splitter supports the laser wavelength, such as 800 nm, and the required bandwidth.
BK7 is presented as a common choice for visible broadband light, while fused silica is better suited to many high-power applications.
Continuous-wave lasers are assessed using power density, while pulsed lasers require attention to pulse energy and damage thresholds.
Light’s frequency and perceived color do not change when it enters glass, although its wavelength shortens inside the medium because the refractive index is higher.

55:00 Split Ratios, Pellicle Optics, and Preventing Laser Damage

A 99:1 polarization specification describes leakage into the undesired polarization; extinction ratios similarly quantify unwanted power.
Non-polarizing splitters are appropriate when polarization is unimportant, while common power ratios include 50:50 and 90:10.
A 90:10 splitter can reserve 10% of the input for a reference or protect a sensitive detector by sending it only a fraction of the power.
High-power or ultrafast lasers can damage thin pellicle splitters; beam dumps absorb stray reflections, and minimizing thick glass reduces loss, delay, and dispersion.

1:06:00 Dichroic Mirrors Separate 400 nm and 800 nm Light

Dichroic mirrors use wavelength-dependent transmission and reflection, typically at a specified incidence angle such as 45°.
A frequency-doubling crystal can convert 800 nm laser light into 400 nm light; a dichroic mirror can then separate the two wavelengths.
Shorter wavelengths improve the diffraction-limited feature size; Simsek describes using successive conversion stages from 800 nm to 400 nm and then 200 nm.
The example applies this approach to producing roughly 100 nm-wide channels in graphene.

1:10:00 Quantum Imaging with 810 nm and 1550 nm Correlated Photons

A 532 nm pump drives a periodically poled KTP crystal, which generates correlated photons near 810 nm and 1550 nm.
The 1550 nm photon interacts with the object, while the 810 nm photon travels to a camera through wavelength-selective dichroic mirrors.
The camera can form an image from photons correlated with the object-interacting photons, even though the detected 810 nm photons did not themselves strike the object.
The setup suggests ways to transfer information from wavelengths that ordinary visible cameras cannot detect into a measurable imaging path.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Ergun Simsek.