Expert-level optics knowledge. Use when working with geometric optics, wave optics, interference, diffraction, polarization, lasers, fiber optics, optical instruments, or photonics. Also use when the user mentions 'reflection', 'refraction', 'Snell law', 'lens', 'mirror', 'interference', 'diffraction', 'polarization', 'laser', 'fiber optic', 'holography', 'aberration', or 'optical instrument'.
You are a world-class physicist with deep expertise in optics covering geometric optics, wave optics, interference, diffraction, polarization, lasers, nonlinear optics, fiber optics, and optical instruments.
Law of Reflection:
θᵢ = θᵣ (angle of incidence = angle of reflection)
Both measured from normal to surface.
Snell's Law (Refraction):
n₁sinθ₁ = n₂sinθ₂
n = c/v = refractive index (n ≥ 1)
Common refractive indices:
Vacuum/air: n = 1.000
Water: n = 1.333
Glass: n = 1.5
Diamond: n = 2.42
Total Internal Reflection:
Occurs when light goes from dense to less dense medium.
Critical angle: sinθc = n₂/n₁ (n₁ > n₂)
θᵢ > θc → total reflection, no transmitted ray.
Dispersion:
n = n(λ) — different wavelengths refract differently.
Prism separates white light into spectrum.
Cauchy equation: n(λ) = A + B/λ²
Spherical mirror equation:
1/f = 1/do + 1/di
f = R/2 (focal length = half radius of curvature)
M = -di/do (magnification, negative = inverted)
Sign conventions:
do > 0: object in front of mirror
di > 0: real image (in front), di < 0: virtual image (behind)
f > 0: concave mirror, f < 0: convex mirror
Flat mirror: f = ∞ → di = -do (virtual, upright, same size)
Concave: converging — real images when do > f
Convex: diverging — always virtual, upright, reduced image
Thin lens equation:
1/f = 1/do + 1/di
M = -di/do = hi/ho
Lensmaker's equation:
1/f = (n-1)[1/R₁ - 1/R₂]
R > 0: center of curvature on transmission side
R < 0: center of curvature on incidence side
Lens types:
Converging (convex): f > 0
Diverging (concave): f < 0
Three principal rays for ray tracing:
1. Parallel to axis → passes through focal point F'
2. Through focal point F → emerges parallel to axis
3. Through optical center → undeviated
Power: P = 1/f (diopters, f in meters)
Combined lenses: 1/f_total = 1/f₁ + 1/f₂ - d/(f₁f₂)
Every point on a wavefront acts as a source of secondary wavelets.
New wavefront = envelope of all secondary wavelets.
Explains reflection, refraction, diffraction.
Plane wave: E = E₀cos(kx - ωt)
Spherical wave: E = (E₀/r)cos(kr - ωt)
k = 2π/λ (wave number)
Phase velocity: v = ω/k = c/n
Two-source interference (Young's double slit):
Path difference: Δ = d·sinθ ≈ d·y/L
Constructive (bright): Δ = mλ m = 0,±1,±2,...
Destructive (dark): Δ = (m+½)λ
Fringe spacing: Δy = λL/d
Intensity pattern:
I = 4I₀cos²(πdsinθ/λ)
I = 4I₀ at maxima, 0 at minima
Conditions for interference:
Coherence: stable phase relationship between sources
Coherence length: Lc = λ²/Δλ
Coherence time: τc = 1/Δf
Thin film interference:
Path difference: 2nt (for normal incidence)
Phase shift: π if reflecting from higher n medium
Constructive: 2nt = (m+½)λ (one phase shift)
Destructive: 2nt = mλ (one phase shift)
→ Anti-reflection coatings: t = λ/4n
→ Soap bubbles, oil films, optical coatings
Single slit diffraction:
Minima at: a·sinθ = mλ m = ±1,±2,...
Central maximum width: 2λ/a
Intensity: I = I₀[sin(α)/α]²
α = πa·sinθ/λ
Double slit (combined):
I = 4I₀cos²(δ/2)[sin(α)/α]²
δ = 2πd·sinθ/λ (interference term)
α = πa·sinθ/λ (diffraction envelope)
Diffraction grating:
Principal maxima: d·sinθ = mλ
d = grating spacing, m = order
Resolving power: R = λ/Δλ = mN (N = number of slits)
Rayleigh criterion (resolution limit):
θmin = 1.22λ/D (circular aperture)
Two sources just resolved when one max falls on other's min.
Fresnel vs Fraunhofer:
Fraunhofer: far field (L >> a²/λ) — parallel rays
Fresnel: near field — curved wavefronts
Linear polarization: E oscillates in single plane
Circular polarization: E rotates — equal amplitudes, 90° phase diff
Elliptical polarization: general case
Malus's Law:
I = I₀cos²θ (intensity through polarizer at angle θ)
Brewster's Angle (polarization by reflection):
tanθB = n₂/n₁
At θB: reflected light is completely s-polarized
Transmitted light: partially p-polarized
Birefringence:
Two different refractive indices: no and ne
Ordinary ray (o-ray): follows Snell's law
Extraordinary ray (e-ray): does not
Wave plates:
Quarter-wave plate (QWP): Δφ = π/2
Linear → circular polarization (and vice versa)
Half-wave plate (HWP): Δφ = π
Rotates linear polarization by 2θ
Jones vectors and matrices:
Horizontal: [1,0], Vertical: [0,1]
Right circular: [1,-i]/√2
HWP matrix: [[cos2θ, sin2θ],[sin2θ, -cos2θ]]
Laser = Light Amplification by Stimulated Emission of Radiation
Key concepts:
Spontaneous emission: random photon emission
Stimulated emission: incident photon triggers identical photon
Population inversion: more atoms in excited state than ground state
→ Required for amplification (not thermal equilibrium)
Three/four level systems:
Three-level: difficult — must depopulate ground state
Four-level: easier — lasing transition between two excited states
Ruby laser (3-level), Nd:YAG (4-level)
Laser cavity:
Two mirrors (one partially transmitting) → standing wave modes
Mode spacing: Δν = c/2L
Longitudinal modes: integer wavelengths fit in cavity
Gaussian beam:
w(z) = w₀√(1 + (z/zR)²)
zR = πw₀²/λ (Rayleigh range)
w₀ = beam waist (minimum radius)
Beam divergence: θ = λ/πw₀ (diffraction limited)
Laser properties:
Coherence: long coherence length
Monochromaticity: narrow linewidth
Directionality: small divergence
High intensity: concentrated beam
Common laser types:
HeNe: λ = 632.8 nm (red, gas laser)
CO₂: λ = 10.6 μm (infrared, cutting)
Nd:YAG: λ = 1064 nm (solid state, pulsed)
Diode: various λ (semiconductor, compact)
Ti:Sapphire: tunable (ultrafast pulses)
Total internal reflection guides light in fiber core.
Core: higher n, Cladding: lower n
Numerical aperture: NA = √(n_core² - n_clad²) = n·sinθmax
Step-index fiber:
Sharp boundary between core and cladding.
Multimode: large core, many propagating modes.
Single-mode: small core (~8μm), one mode, low dispersion.
Graded-index fiber:
n decreases gradually from center.
Reduces intermodal dispersion.
Dispersion types:
Modal: different modes travel at different speeds
Chromatic: different wavelengths travel at different speeds
Material: from dn/dλ
Waveguide: from fiber geometry
Attenuation:
Measured in dB/km
Minimum at λ = 1550 nm (~0.2 dB/km for silica)
Telecom windows: 1310 nm, 1550 nm
Applications:
Long-distance communication, endoscopy, sensors
def microscope_magnification(objective_mag, eyepiece_mag,
tube_length=160, f_obj=None):
total_mag = objective_mag * eyepiece_mag
return {
'objective': objective_mag,
'eyepiece': eyepiece_mag,
'total': total_mag,
'resolution': '0.2 μm (visible light limit)',
'note': 'Resolution limited by diffraction: d = 0.61λ/NA'
}
def telescope_magnification(f_objective, f_eyepiece):
mag = f_objective / f_eyepiece
return {
'magnification': mag,
'f_objective': f_objective,
'f_eyepiece': f_eyepiece,
'note': 'Larger aperture → better resolution and light gathering'
}
def camera_depth_of_field(f_number, focal_length, distance, coc=0.03):
"""
Depth of field calculation.
coc = circle of confusion (mm)
"""
import math
hyp = focal_length**2 / (f_number * coc)
near = (hyp * distance) / (hyp + distance - focal_length)
far = (hyp * distance) / (hyp - distance + focal_length)
dof = far - near
return {
'near_limit': round(near, 2),
'far_limit': round(far, 2),
'dof': round(dof, 2),
'hyperfocal': round(hyp, 2)
}
Monochromatic aberrations (Seidel):
Spherical aberration: marginal rays focus differently than paraxial
Coma: off-axis point sources form comet shape
Astigmatism: different focal lengths in two planes
Field curvature: flat object focuses on curved surface
Distortion: magnification varies with field height
Chromatic aberration:
Longitudinal: different colors focus at different distances
Transverse: different colors have different magnifications
Correction: achromatic doublet (crown + flint glass)
Zernike polynomials:
Mathematical basis for describing wavefront aberrations.
Used in adaptive optics and ophthalmology.
def optics_calculator():
return {
'thin_lens': '1/f = 1/do + 1/di',
'magnification': 'M = -di/do',
'snell': 'n1*sin(θ1) = n2*sin(θ2)',
'critical_angle': 'sinθc = n2/n1',
'brewster': 'tanθB = n2/n1',
'young_fringes': 'Δy = λL/d',
'rayleigh': 'θmin = 1.22λ/D',
'single_slit_min': 'a*sinθ = mλ',
'grating': 'd*sinθ = mλ',
'malus': 'I = I0*cos²θ',
'thin_film_AR': 't = λ/4n'
}
| Pitfall | Fix |
|---|---|
| Sign convention errors in lens/mirror | Define positive direction consistently |
| Forgetting phase shift on reflection | π phase shift when reflecting from denser medium |
| Rayleigh vs Abbe resolution | Rayleigh for telescopes, Abbe for microscopes |
| Ignoring coherence for interference | Interference only visible with coherent sources |
| Paraxial approximation failure | Valid only for small angles sinθ ≈ θ |
| Confusing focal length and focal point | f is distance, F is the point |