Part I, Chapter 1

Geometrical Optics

In the limit where the wavelength of light is much smaller than the scale of optical elements, light propagates as rays governed by Fermat's principle and Snell's law.

1.1 Fermat's Principle

Pierre de Fermat (1662) proposed that light travels between two points along the path that takes the least time. More precisely, the optical path length is stationary (usually a minimum):

$$\delta \int_A^B n(\mathbf{r})\, ds = 0$$

where n(r) is the refractive index at position r and ds is the arc-length element. This variational principle is analogous to Hamilton's principle in mechanics, and it forms the foundation of geometrical optics. In a homogeneous medium (constant n), light travels in straight lines. In an inhomogeneous medium, rays curve toward regions of higher refractive index.

1.1.1 Derivation of Snell's Law from Fermat's Principle

Derivation: Snell's Law

Step 1. Consider a ray crossing a planar interface between media with indices n₁ and n₂. The ray travels from point A at height h₁ above the interface to point B at height h₂ below it, hitting the interface at horizontal position x.

Step 2. The total optical path length is:

$$L(x) = n_1 \sqrt{h_1^2 + x^2} + n_2 \sqrt{h_2^2 + (d-x)^2}$$

Step 3. Setting dL/dx = 0:

$$\frac{n_1 x}{\sqrt{h_1^2 + x^2}} - \frac{n_2 (d-x)}{\sqrt{h_2^2 + (d-x)^2}} = 0$$

Step 4. Recognizing sinθ₁ = x/√(h₁² + x²) and sinθ₂ = (d-x)/√(h₂² + (d-x)²):

Result: Snell's Law

$$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

1.1.2 Total Internal Reflection

When n₁ > n₂, there exists a critical angle θ_cbeyond which no refracted ray exists:

$$\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right)$$

For θ₁ > θ_c, the ray is totally internally reflected. This phenomenon is the basis of optical fiber waveguiding (Part IV) and prism-based retroreflectors.

1.2 Mirrors and Lenses

Curved mirrors and thin lenses are the workhorses of optical system design. In the paraxial approximation (small angles), the imaging properties are governed by simple algebraic relations.

1.2.1 Mirror Equation

For a spherical mirror with radius of curvature R, the focal length is f = R/2. The image and object distances satisfy:

$$\frac{1}{s'} + \frac{1}{s} = \frac{2}{R} = \frac{1}{f}$$

1.2.2 Thin Lens Equation

The lensmaker's equation relates the focal length of a thin lens to its radii of curvature and refractive index:

$$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

The thin lens imaging equation has the same form as the mirror equation: 1/s′ - 1/s = 1/f (with sign conventions). The transverse magnification is m = -s′/s.

1.2.3 Compound Systems

For two thin lenses separated by distance d, the combined focal length is:

$$\frac{1}{f_{\text{eff}}} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$$

This result can be derived systematically using the ABCD matrix formalism (next section).

1.3 ABCD Matrix Optics

In the paraxial approximation, a ray is described by its height y and angle θ relative to the optical axis. The effect of any optical element is a 2×2 matrix:

$$\begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix}$$

1.3.1 Elementary Matrices

Common ABCD Matrices

Free-space propagation (distance d):

$$M_{\text{prop}} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}$$

Thin lens (focal length f):

$$M_{\text{lens}} = \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix}$$

Curved mirror (radius R):

$$M_{\text{mirror}} = \begin{pmatrix} 1 & 0 \\ -2/R & 1 \end{pmatrix}$$

Refraction at a flat interface:

$$M_{\text{refr}} = \begin{pmatrix} 1 & 0 \\ 0 & n_1/n_2 \end{pmatrix}$$

1.3.2 System Composition

A sequence of optical elements is represented by the product of their matrices (in reverse order of traversal). For a ray passing through element 1, then 2, then 3:

$$M_{\text{sys}} = M_3 \cdot M_2 \cdot M_1$$

The imaging condition is B = 0 (all rays from a point converge regardless of angle). The stability condition for a resonator with two mirrors is:

$$0 \le g_1 g_2 \le 1, \qquad g_i = 1 - \frac{L}{R_i}$$

1.4 Aberrations

Real optical systems deviate from ideal paraxial behavior. Seidel (1856) classified the five primary monochromatic aberrations by expanding the wavefront error to fourth order in the ray coordinates.

The Five Seidel Aberrations

1. Spherical aberration: Marginal rays focus closer to the lens than paraxial rays. Proportional to h&sup4; where h is the ray height.
2. Coma: Off-axis points produce comet-shaped images. Different zones of the lens produce circles of different size.
3. Astigmatism: Off-axis points have different foci for tangential and sagittal rays.
4. Field curvature (Petzval): The image of a flat object lies on a curved surface.
5. Distortion: Magnification varies with field position, producing barrel or pincushion distortion.

1.4.1 Chromatic Aberration

Because refractive index depends on wavelength (dispersion), white light produces color-dependent focal lengths. The Abbe number quantifies dispersion:

$$V_d = \frac{n_d - 1}{n_F - n_C}$$

where n_d, n_F, n_C are the refractive indices at the Fraunhofer d, F, and C spectral lines. An achromatic doublet combines a crown-glass lens with a flint-glass lens to cancel chromatic aberration at two wavelengths.

1.5 Python Simulation: ABCD Matrix Ray Tracing

This simulation traces rays through a two-lens optical system using the ABCD matrix formalism. It visualizes the ray paths, demonstrates image formation, and computes the system properties.

ABCD Matrix Ray Tracing Through a Two-Lens System

Python

script.py159 lines

import numpy as np
import matplotlib.pyplot as plt

# --- ABCD Matrices ---
def prop_matrix(d):
    return np.array([[1, d], [0, 1]])

def lens_matrix(f):
    return np.array([[1, 0], [-1/f, 1]])

# --- System parameters ---
f1 = 50.0     # focal length of lens 1 (mm)
f2 = 30.0     # focal length of lens 2 (mm)
d_sep = 100.0 # separation between lenses (mm)
s_obj = -80.0 # object distance from lens 1 (negative = left)

# System matrix: propagate to L1, refract, propagate gap, refract at L2
M_sys = lens_matrix(f2) @ prop_matrix(d_sep) @ lens_matrix(f1)
A, B, C, D = M_sys[0,0], M_sys[0,1], M_sys[1,0], M_sys[1,1]

# Effective focal length and image distance
f_eff = -1/C
# Image distance from lens 2
s_img = (A * s_obj + B) / (C * s_obj + D)
magnification = 1 / (C * s_obj + D)

# --- Ray tracing ---
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# Plot 1: Ray diagram
ax1 = axes[0, 0]
n_rays = 7
y_max = 15.0
z_positions = [s_obj, 0, d_sep, d_sep + s_img]

for i, y0 in enumerate(np.linspace(-y_max, y_max, n_rays)):
    # From object to lens 1
    theta0 = -y0 / abs(s_obj)  # aim at axis at lens 1
    ray = np.array([y0, theta0])

zs = [s_obj]
    ys = [y0]

# Propagate to lens 1
    ray = prop_matrix(abs(s_obj)) @ ray
    zs.append(0); ys.append(ray[0])

# Lens 1
    ray = lens_matrix(f1) @ ray
    zs.append(0); ys.append(ray[0])

# Propagate to lens 2
    ray = prop_matrix(d_sep) @ ray
    zs.append(d_sep); ys.append(ray[0])

# Lens 2
    ray = lens_matrix(f2) @ ray
    zs.append(d_sep); ys.append(ray[0])

# Propagate to image
    ray = prop_matrix(s_img) @ ray
    zs.append(d_sep + s_img); ys.append(ray[0])

color = plt.cm.autumn(i / n_rays)
    ax1.plot(zs, ys, '-', color=color, alpha=0.7, linewidth=1)

# Draw lenses
for z_l, f_l, label in [(0, f1, f'L1 (f={f1})'), (d_sep, f2, f'L2 (f={f2})')]:
    ax1.plot([z_l, z_l], [-y_max*1.2, y_max*1.2], 'w-', linewidth=2, alpha=0.5)
    ax1.annotate(label, (z_l, y_max*1.3), color='#f59e0b', fontsize=8, ha='center')

ax1.axhline(0, color='gray', linewidth=0.5, linestyle='--')
ax1.set_xlabel('z (mm)', color='white')
ax1.set_ylabel('y (mm)', color='white')
ax1.set_title('Ray Diagram: Two-Lens System', color='#fbbf24', fontsize=13)
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#f59e0b')

# Plot 2: Stability diagram for resonators
ax2 = axes[0, 1]
g1_vals = np.linspace(-2, 2, 400)
g2_vals = np.linspace(-2, 2, 400)
G1, G2 = np.meshgrid(g1_vals, g2_vals)
stability = (G1 * G2 >= 0) & (G1 * G2 <= 1)
ax2.contourf(G1, G2, stability.astype(float), levels=[0.5, 1.5], colors=['#f59e0b'], alpha=0.3)
ax2.contour(G1, G2, G1*G2, levels=[0, 1], colors=['#fbbf24'], linewidths=1.5)
# Mark special resonators
configs = {'Planar': (1, 1), 'Confocal': (0, 0), 'Concentric': (-1, -1), 'Hemispherical': (1, 0)}
for name, (g1, g2) in configs.items():
    ax2.plot(g1, g2, 'o', color='#fb923c', markersize=8)
    ax2.annotate(name, (g1+0.08, g2+0.08), color='white', fontsize=8)
ax2.set_xlabel('g1', color='white')
ax2.set_ylabel('g2', color='white')
ax2.set_title('Resonator Stability Diagram', color='#fbbf24', fontsize=13)
ax2.tick_params(colors='white')
ax2.set_xlim(-2, 2); ax2.set_ylim(-2, 2)
for spine in ax2.spines.values():
    spine.set_color('#f59e0b')

# Plot 3: Spherical aberration
ax3 = axes[1, 0]
h_vals = np.linspace(0, 20, 200)
f_nominal = 50.0
R = 2 * f_nominal  # for a plano-convex lens, roughly
n_glass = 1.5
# Exact refraction at a spherical surface
theta_i = np.arcsin(h_vals / R)
theta_r = np.arcsin(np.sin(theta_i) / n_glass)
# Paraxial focus
f_paraxial = f_nominal
# Marginal focus (approximate third-order)
f_marginal = f_nominal * (1 - 0.001 * h_vals**2)
aberration = f_paraxial - f_marginal

ax3.plot(h_vals, f_marginal, color='#f59e0b', linewidth=2, label='Marginal focal length')
ax3.axhline(f_paraxial, color='#fbbf24', linestyle='--', alpha=0.5, label='Paraxial focus')
ax3.fill_between(h_vals, f_marginal, f_paraxial, alpha=0.15, color='#f59e0b')
ax3.set_xlabel('Ray height h (mm)', color='white')
ax3.set_ylabel('Focal length (mm)', color='white')
ax3.set_title('Spherical Aberration', color='#fbbf24', fontsize=13)
ax3.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#f59e0b')

# Plot 4: Chromatic aberration - dispersion curve
ax4 = axes[1, 1]
wavelengths = np.linspace(400, 700, 300)  # nm
# Cauchy equation for BK7-like glass
B1, C1 = 1.5168, 4300  # Cauchy coefficients
n_lambda = B1 + C1 / wavelengths**2
f_lambda = 1 / ((n_lambda - 1) * (1/50 - 1/(-100)))  # plano-convex R1=50, R2=-100

colors_wave = plt.cm.rainbow(np.linspace(0, 1, len(wavelengths)))
for i in range(len(wavelengths)-1):
    ax4.plot(wavelengths[i:i+2], f_lambda[i:i+2], color=colors_wave[i], linewidth=2)

ax4.set_xlabel('Wavelength (nm)', color='white')
ax4.set_ylabel('Focal length (mm)', color='white')
ax4.set_title('Chromatic Aberration (Dispersion)', color='#fbbf24', fontsize=13)
ax4.tick_params(colors='white')
for spine in ax4.spines.values():
    spine.set_color('#f59e0b')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.show()

print(f"System ABCD matrix:")
print(f"  A = {A:.4f}, B = {B:.4f}")
print(f"  C = {C:.6f}, D = {D:.4f}")
print(f"Effective focal length: {f_eff:.2f} mm")
print(f"Image distance from L2: {s_img:.2f} mm")
print(f"Magnification: {magnification:.3f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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