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Optics models

Derive ray, wave, diffraction, polarization, and detection models, then solve twenty optical design and measurement examples.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Refraction and paraxial imaging

Definitions & inputs. n is refractive index, θ ray angle to the surface normal, s object distance, s′ image distance, f focal length.

  1. Stationary optical path, or tangential wave-vector continuity, gives Snell’s law.

    n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1=n_2\sin\theta_2

  2. Linearize angles for paraxial rays.

    θ≃tan⁡θ≃sin⁡θ(∣θ∣≪1)\theta\simeq\tan\theta\simeq\sin\theta\quad(|\theta|\ll1)

  3. Similar triangles and the thin-lens ray rules give the imaging equation and signed magnification.

    1/f=1/s+1/s′,m=−s′/s1/f=1/s+1/s^\prime,\quad m=-s^\prime/s

Interpretation. A thin-lens approximation does not include aberrations or thickness.

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2. Interference and coherence

Definitions & inputs. E1,E2 are complex field amplitudes; φ relative phase; I1,I2 intensities.

  1. Intensity comes from the squared total field.

    I∝∣E1+E2∣2=∣E1∣2+∣E2∣2+2Re⁡(E1E2∗)I\propto|E_1+E_2|^2=|E_1|^2+|E_2|^2+2\operatorname{Re}(E_1E_2^*)

  2. Optical-path difference Δ sets phase.

    I=I1+I2+2I1I2cos⁡ϕ,ϕ=2πΔ/λI=I_1+I_2+2\sqrt{I_1I_2}\cos\phi,\quad\phi=2\pi\Delta/\lambda

  3. For two narrow slits separated by d in the far field, adjacent fringes differ by one wavelength of path.

    Δ≃dy/L⇒Δy=λL/d\Delta\simeq dy/L\Rightarrow\Delta y=\lambda L/d

Interpretation. Finite bandwidth and imperfect coherence reduce fringe visibility.

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3. Apertures and Gaussian beams

Definitions & inputs. a is slit width, D circular pupil diameter, w0 Gaussian waist radius at 1/e² intensity, and zR Rayleigh range.

  1. Integrate contributions across a uniform slit.

    E(θ)∝∫−a/2a/2eikxsin⁡θdx∝sinc⁡(πasin⁡θ/λ)E(\theta)\propto\int_{-a/2}^{a/2}e^{ikx\sin\theta}dx\propto\operatorname{sinc}(\pi a\sin\theta/\lambda)

  2. The slit zeros and circular-aperture first minimum set characteristic diffraction angles.

    asin⁡θm=mλ,θAiry≃1.22λ/Da\sin\theta_m=m\lambda,\quad\theta_{\rm Airy}\simeq1.22\lambda/D

  3. A Gaussian beam spreads with propagation; waist size and divergence are linked.

    zR=πw02/λ,w(z)=w01+(z/zR)2z_R=\pi w_0^2/\lambda,\quad w(z)=w_0\sqrt{1+(z/z_R)^2}

Interpretation. Diffraction sets a reference resolution, while sampling and aberrations can worsen it.

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4. Polarization, attenuation, and photodetection

Definitions & inputs. θ is polarizer angle, α absorption coefficient, P optical power, η quantum efficiency, q elementary charge.

  1. Project the electric field onto the analyzer axis, then square.

    I=I0cos⁡2θI=I_0\cos^2\theta

  2. Integrate attenuation through a homogeneous path.

    dI/dz=−αI⇒I=I0e−αzdI/dz=-\alpha I\Rightarrow I=I_0e^{-\alpha z}

  3. Photon energy converts power into photon rate and collected charge rate.

    N˙γ=P/(hc/λ),Iphoto=qηN˙γ\dot N_\gamma=P/(hc/\lambda),\quad I_{\rm photo}=q\eta\dot N_\gamma

Interpretation. Detector dark current, read noise, and saturation require additional models.

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Graphical worked example

Malus law for 10 W linearly polarized light and an ideal analyzer. X axis: Analyzer angle (degrees). Y axis: Transmitted power (W).
Malus law for 10 W linearly polarized light and an ideal analyzer. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Refraction into glass

Definitions & inputs. n1=1, n2=1.5, θ1=30°.

  1. Choose the governing model and isolate the requested quantity.

    θ2=arcsin⁡[(n1/n2)sin⁡θ1]\theta_2=\arcsin[(n_1/n_2)\sin\theta_1]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    θ2=arcsin⁡[(1/1.5)(0.5)]\theta_2=\arcsin[(1/1.5)(0.5)]

  3. Evaluate the expression; the result uses the units shown.

    Result=19.47122 ∘\mathrm{Result}=19.47122\ {}^\circ

Interpretation. The ray bends toward the normal.

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Example 02. Critical angle

Definitions & inputs. Light exits n1=1.5 glass into n2=1 air.

  1. Choose the governing model and isolate the requested quantity.

    θc=arcsin⁡(n2/n1)\theta_c=\arcsin(n_2/n_1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    θc=arcsin⁡(1/1.5)\theta_c=\arcsin(1/1.5)

  3. Evaluate the expression; the result uses the units shown.

    Result=41.81031 ∘\mathrm{Result}=41.81031\ {}^\circ

Interpretation. Larger incident angles permit total internal reflection.

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Example 03. Normal-incidence Fresnel reflection

Definitions & inputs. n1=1, n2=1.5.

  1. Choose the governing model and isolate the requested quantity.

    R=[(n1−n2)/(n1+n2)]2R=[(n_1-n_2)/(n_1+n_2)]^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    R=[(1−1.5)/(1+1.5)]2R=[(1-1.5)/(1+1.5)]^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.04 \mathrm{Result}=0.04\

Interpretation. Four percent of incident intensity reflects at one interface.

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Example 04. Image distance

Definitions & inputs. f=100 mm; s=300 mm.

  1. Choose the governing model and isolate the requested quantity.

    s′=(1/f−1/s)−1s^\prime=(1/f-1/s)^{-1}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    s′=(1/100−1/300)−1s^\prime=(1/100-1/300)^{-1}

  3. Evaluate the expression; the result uses the units shown.

    Result=150 mm\mathrm{Result}=150\ {\rm mm}

Interpretation. The positive image distance is a real image.

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Example 05. Signed magnification

Definitions & inputs. s=300 mm; s′=150 mm.

  1. Choose the governing model and isolate the requested quantity.

    m=−s′/sm=-s^\prime/s

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    m=−150/300m=-150/300

  3. Evaluate the expression; the result uses the units shown.

    Result=−0.5 \mathrm{Result}=-0.5\

Interpretation. The image is inverted and half-size.

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Example 06. Lens optical power

Definitions & inputs. f=0.25 m.

  1. Choose the governing model and isolate the requested quantity.

    P=1/f\mathcal P=1/f

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    P=1/0.25\mathcal P=1/0.25

  3. Evaluate the expression; the result uses the units shown.

    Result=4 m−1\mathrm{Result}=4\ {\rm m}^{-1}

Interpretation. This is +4 dioptres.

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Example 07. Two-slit fringe spacing

Definitions & inputs. λ=500 nm, L=2 m, d=0.5 mm.

  1. Choose the governing model and isolate the requested quantity.

    Δy=λL/d\Delta y=\lambda L/d

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δy=(500×10−9)(2)/(0.5×10−3)\Delta y=(500\times10^{-9})(2)/(0.5\times10^{-3})

  3. Evaluate the expression; the result uses the units shown.

    Result=0.002 m\mathrm{Result}=0.002\ {\rm m}

Interpretation. Adjacent bright fringes are 2 mm apart.

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Example 08. Interference at phase π/2

Definitions & inputs. I1=I2=1 W/m²; φ=π/2.

  1. Choose the governing model and isolate the requested quantity.

    I=I1+I2+2I1I2cos⁡ϕI=I_1+I_2+2\sqrt{I_1I_2}\cos\phi

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    I=1+1+2cos⁡(π/2)I=1+1+2\cos(\pi/2)

  3. Evaluate the expression; the result uses the units shown.

    Result=2 W m−2\mathrm{Result}=2\ {\rm W\,m}^{-2}

Interpretation. At quadrature the interference term vanishes.

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Example 09. Single-slit first minimum

Definitions & inputs. a=100 µm, λ=500 nm.

  1. Choose the governing model and isolate the requested quantity.

    θ1=arcsin⁡(λ/a)\theta_1=\arcsin(\lambda/a)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    θ1=arcsin⁡(0.005)\theta_1=\arcsin(0.005)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.005000021 rad\mathrm{Result}=0.005000021\ {\rm rad}

Interpretation. The central lobe extends between the positive and negative first minima.

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Example 10. Circular-aperture angular scale

Definitions & inputs. D=0.10 m, λ=550 nm.

  1. Choose the governing model and isolate the requested quantity.

    θ=1.22λ/D\theta=1.22\lambda/D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    θ=1.22(550×10−9)/0.10\theta=1.22(550\times10^{-9})/0.10

  3. Evaluate the expression; the result uses the units shown.

    Result=6.71×10−6 rad\mathrm{Result}=6.71\times10^{-6}\ {\rm rad}

Interpretation. This first-zero angle is about 1.38 arcseconds.

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Example 11. Airy radius at a focal plane

Definitions & inputs. f=0.20 m, D=0.05 m, λ=500 nm.

  1. Choose the governing model and isolate the requested quantity.

    rA=1.22λf/Dr_A=1.22\lambda f/D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    rA=1.22(500×10−9)(0.20)/0.05r_A=1.22(500\times10^{-9})(0.20)/0.05

  3. Evaluate the expression; the result uses the units shown.

    Result=2.44×10−6 m\mathrm{Result}=2.44\times10^{-6}\ {\rm m}

Interpretation. The first dark ring lies 2.44 µm from the centre.

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Example 12. Gaussian Rayleigh range

Definitions & inputs. w0=0.5 mm, λ=1 µm.

  1. Choose the governing model and isolate the requested quantity.

    zR=πw02/λz_R=\pi w_0^2/\lambda

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    zR=π(0.5×10−3)2/10−6z_R=\pi(0.5\times10^{-3})^2/10^{-6}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.7853982 m\mathrm{Result}=0.7853982\ {\rm m}

Interpretation. The beam area doubles at one Rayleigh range.

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Example 13. Gaussian beam radius

Definitions & inputs. w0=0.5 mm, z=2zR.

  1. Choose the governing model and isolate the requested quantity.

    w=w01+(z/zR)2w=w_0\sqrt{1+(z/z_R)^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    w=0.55w=0.5\sqrt5

  3. Evaluate the expression; the result uses the units shown.

    Result=1.118034 mm\mathrm{Result}=1.118034\ {\rm mm}

Interpretation. The reported radius uses 1/e² intensity.

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Example 14. Polarizer transmission

Definitions & inputs. I0=10 W/m², analyzer angle 60°.

  1. Choose the governing model and isolate the requested quantity.

    I=I0cos⁡2θI=I_0\cos^2\theta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    I=10cos⁡2(60∘)I=10\cos^2(60^\circ)

  3. Evaluate the expression; the result uses the units shown.

    Result=2.5 W m−2\mathrm{Result}=2.5\ {\rm W\,m}^{-2}

Interpretation. The incident light is already linearly polarized.

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Example 15. Quarter-wave optical path

Definitions & inputs. λ=600 nm.

  1. Choose the governing model and isolate the requested quantity.

    Δ=λ/4\Delta=\lambda/4

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δ=600/4\Delta=600/4

  3. Evaluate the expression; the result uses the units shown.

    Result=150 nm\mathrm{Result}=150\ {\rm nm}

Interpretation. This optical-path difference produces π/2 phase delay.

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Example 16. Beer-law transmission

Definitions & inputs. α=2 m⁻¹, path l=0.3 m.

  1. Choose the governing model and isolate the requested quantity.

    T=e−αlT=e^{-\alpha l}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    T=e−2(0.3)T=e^{-2(0.3)}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5488116 \mathrm{Result}=0.5488116\

Interpretation. About 54.9% of intensity remains.

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Example 17. Photon energy at 500 nm

Definitions & inputs. hc=1239.841984 eV nm.

  1. Choose the governing model and isolate the requested quantity.

    Eγ=hc/λE_\gamma=hc/\lambda

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Eγ=1239.841984/500E_\gamma=1239.841984/500

  3. Evaluate the expression; the result uses the units shown.

    Result=2.479684 eV\mathrm{Result}=2.479684\ {\rm eV}

Interpretation. Optical power consists of many such photons per second.

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Example 18. Photon rate

Definitions & inputs. P=1 mW, λ=500 nm, h=6.62607015×10⁻³⁴ J s.

  1. Choose the governing model and isolate the requested quantity.

    N˙=Pλ/(hc)\dot N=P\lambda/(hc)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    N˙=10−3(500×10−9)/(6.62607015×10−34⋅299792458)\dot N=10^{-3}(500\times10^{-9})/(6.62607015\times10^{-34}\cdot299792458)

  3. Evaluate the expression; the result uses the units shown.

    Result=2.517058×1015 s−1\mathrm{Result}=2.517058\times10^{15}\ {\rm s}^{-1}

Interpretation. This is incident photon rate, not detected electron rate.

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Example 19. Photodiode current

Definitions & inputs. Incident photon rate 10¹² s⁻¹, η=0.8, q=1.602176634×10⁻¹⁹ C.

  1. Choose the governing model and isolate the requested quantity.

    I=qηN˙I=q\eta\dot N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    I=(1.602176634×10−19)(0.8)(1012)I=(1.602176634\times10^{-19})(0.8)(10^{12})

  3. Evaluate the expression; the result uses the units shown.

    Result=1.281741×10−7 A\mathrm{Result}=1.281741\times10^{-7}\ {\rm A}

Interpretation. Quantum efficiency accounts for uncollected photons.

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Example 20. Shot-noise signal-to-noise

Definitions & inputs. N=10000 detected photoelectrons, negligible background.

  1. Choose the governing model and isolate the requested quantity.

    SNR=N/N=N\mathrm{SNR}=N/\sqrt N=\sqrt N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10000\sqrt{10000}

  3. Evaluate the expression; the result uses the units shown.

    Result=100 \mathrm{Result}=100\

Interpretation. Read noise and background would reduce SNR.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.