ECE318 — Fundamentals of Optics

Course focuses on the physics of light, optics, and photonics. Spiritually follows ECE221 — Electric and Magnetic Fields. We cover geometric optics and wave optics.

Very cool course! But super trippy.

Concepts covered

Geometric optics

• Fundamental laws
  • Rectilinear propagation
  • Law of reflection
  • Snell’s law (law of refraction)
• Physical parameters
  • Refractive index
  • Optical path length
• Fermat’s principle
• Optical imaging system
  • Spherical interface
  • Thin lens, Lensmakers’ formula
  • Spherical mirror
  • Gaussian formula
  • Practical imaging systems
    • Telescope
    • Microscope
• System parameters
  • Magnification
  • Magnifying power
  • Angular resolution
• Cardinal points
  • Focal length
  • Principal point
  • Nodal point
• Matrix optics

Wave optics

• Wave
  • Maxwell’s equations
  • Electromagnetic wave
  • Harmonic plane wave
• Poynting’s theorem
  • Intensity (irradiance)
• Wave polarisation
  • Jones vector
  • Waveplate
  • Malus’ law
• Optical boundary conditions
  • Fresnel coefficient
  • Brewster angle
  • Total internal reflection
• Interference
  • Coherence (temporal, spatial)
  • Interferometers
    • Michelson interferometer
    • Twyman-Green interferometer
    • Fabry-Perot interferometer
  • Resolving power
  • Finesse
• Diffraction and Fourier optics
  • Background information
    • Continuous-time Fourier transform
    • Bessel function
  • Heisenberg’s uncertainty principle
  • Huygens-Fresnel principle
  • Optical image processing
  • Diffraction grating

Extra notes

Final

Basic formulas:

$$\frac{\omega }{k}=\frac{c}{n}$$ $$\omega =2\pi \nu$$ $$k=\frac{2\pi n}{\lambda }$$

Conversions for temporal coherence:

$$\lambda =\frac{c}{\nu }$$ $$\frac{\Delta \nu }{c}=\frac{\Delta \lambda }{{\lambda }^2}$$

Fringe width (multi-slit diffraction):

$$\Delta \sin {\theta }_{\text{FWHM}}=\frac{\lambda }{Na}$$

Midterm III

Polarisation:

• ${\phi }_y-{\phi }_x=m\pi$ for linear polarisation.
  • $\alpha =\arctan \frac{J_y}{J_x}$ for angle of polarisation.
• $E_{xo}=E_{yo}$ and ${\phi }_y-{\phi }_x=\frac{\pi }{2}+m\pi$ for circular polarisation.
  • Describe magnitude of polarisation.
• Elliptical polarisation otherwise.
  • Describe $x,y$ magnitude.
• $\Delta \phi ={\phi }_y-{\phi }_x$, if circular/elliptical, then:
  • Positive is left-hand polarised
  • Negative is right-hand polarised

General waveplates with slow-axis aligned along degree 0 (QWP, HWP):

$$\text{QWP}=\left[\begin{array}{cc}i& 0\\ 0& 1\end{array}\right]$$ $$\text{HWP}=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$$

Fresnel coefficients:

• External reflection: $n_i<n_t$
• Internal reflection: $n_i>n_t$, i.e., $n_i$ is in the denser medium.

For any two dielectric interfaces:

$$(r_{TE}{)}_{\text{internal}}=-(r_{TE}{)}_{\text{external}}$$ $$(r_{TM}{)}_{\text{internal}}=-(r_{TM}{)}_{\text{external}}$$

TIR conditions:

• $n_i>n_t$, i.e., the incident light wave is in the denser medium.
• ${\theta }_i>{\theta }_c$, i.e., the incident angle is greater than the critical angle.
• Brewster angle: $r_{TM}=0$

TIR:

• $R=1$
• ${\theta }_c=\arcsin \left(\frac{n_t}{n_i}\right)$
• OPL phase change: $nL=OPL$. Then ${\phi }_{OPL}=\frac{2\pi }{\lambda }OPL$

Interference conditions:

• ${\vec{E}}_1,{\vec{E}}_2$ non-orthogonally polarised
• $|{\omega }_2-{\omega }_1|=0$
• ${\phi }_2-{\phi }_1$ constant

Interference:

• Bright fringes: $2m\pi$
• Dark fringes: $2m\pi +\pi$

Conversions for temporal coherence:

$$\lambda =\frac{c}{\nu }$$ $$\Delta \nu =\frac{c}{{\lambda }^2}\Delta \lambda$$

Midterm II

Matrix optics:

$$\det M=\frac{n_{\text{in}}}{n_{\text{out}}}$$

• Computations should usually be with respect to effective distance.
• Nodal points: special points where ray passing through $N$ will exit with same angle.
• Principal point: point that effective distances are wrt to.

$$\nabla ⟷i\vec{k}$$ $$\frac{\partial }{\partial t}⟷-i\omega$$

Constants:

$$n=\sqrt{{ϵ}_r{\mu }_r}\approx \sqrt{{ϵ}_r}$$ $$\mu ={\mu }_0$$ $$c=\frac{1}{\sqrt{\mu ϵ}}$$

Units:

• $\vec{E}$ in $V\cdot m^{-1}$
• $\vec{H}$ in $A\cdot m^{-1}$
• $\vec{k}$ in $\text{rad}\cdot m^{-1}$. If $-\omega t$, propagation in positive $\hat{k}$ direction.
• $\vec{r}$ in metres.
• $\omega$ in radians per second.
• ${\phi }_0$ is in radians.

Conversions:

$$v=\frac{c}{n}=\frac{\omega }{k}$$ $$\omega =2\pi f$$ $$\hat{H}=\hat{k}\times \hat{H}$$

Poynting vector:

$$\vec{S}=\vec{E}\times \vec{H}$$

Polarisation:

• ${\phi }_y-{\phi }_x=m\pi$ for linear polarisation.
  • $\alpha =\arctan \frac{J_y}{J_x}$ for angle of polarisation.
• $E_{xo}=E_{yo}$ and ${\phi }_y-{\phi }_x=\frac{\pi }{2}+m\pi$ for circular polarisation.
  • Describe magnitude of polarisation.
• Elliptical polarisation otherwise.
  • Describe $x,y$ magnitude.
• $\Delta \phi ={\phi }_y-{\phi }_x$, if circular/elliptical, then:
  • Positive is left-hand polarised
  • Negative is right-hand polarised