Category: Part 5: Optics

Problem Statement Solve the oscillation/wave problem: Two lasers emit at $\nu_1 = 4.74\times10^{14}$ Hz and $\nu_2 = 4.74\times10^{14}+1\times10^9$ Hz. They are superimposed on a photodetector. Find the beat frequency and the period of intensity oscillation. $$\nu_{beat} = |\nu_2-\nu_1| = 1\times10^9\text{ Hz} = \boxed{1\text{ GHz}}$$ $$T_{beat} = 1/\n Given Information Mass $m$ and spring constant $k$…

Problem Statement Solve the oscillation/wave problem: Write the Jones matrix for a quarter-wave plate with fast axis horizontal. Apply it to a linearly polarized beam at $45°$ to find the output polarization state. Jones matrix (fast axis horizontal, QWP introduces $-\pi/2$ to slow $y$ component): $$J_{QWP} = e^{i\pi/4}\begin{pmatrix}1 0\\0 -i\end{pmat Given Information Mass $m$ and…

Problem Statement Solve the optics problem: In a frustrated TIR experiment, a beam undergoes TIR at a glass-air interface ($n_{glass} = 1.5$) at $\theta_i = 50°$. A second glass prism is brought to within $\delta = 200$ nm of the first. For $\lambda = 633$ nm, estimate the transmitted fraction. The evanescent wave decays as…

Problem Statement Michelson measured the angular diameter of Betelgeuse using a stellar interferometer. The fringes first disappeared when the mirror separation was $d = 3.07$ m. For $\lambda = 550$ nm, find the angular diameter $\theta$. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts…

Problem Statement Solve the optics problem: A camera lens of $f/8$ (f-number 8) is used with $\lambda = 550$ nm. Find the diffraction-limited spot size and compare with a typical CCD pixel size of $5\;\mu$m. $f\text{-number} = f/D$, so $D = f/8$. Airy disk radius $r = 1.22\lambda/D \times f = 1.22\lambda\times f\text{-number}$. $$r =…

Problem Statement Solve the oscillation/wave problem: A glass plate of thickness $t = 5.0$ mm ($n = 1.5$) produces a channelled spectrum (Fabry-Perot fringes) in transmission. Find the fringe spacing in wavenumber $\Delta\tilde{\nu}$ (in cm$^{-1}$). Channel spacing in wavenumber: $$\Delta\tilde{\nu} = \frac{1}{2nt} = \frac{1}{2\times1.5\times0.50\text{ Given Information Mass $m$ and spring constant $k$ (or equivalent), or…

Problem Statement Solve the optics problem: A solution has specific rotation $[\alpha]_D = 66°$ per dm at $\lambda = 589$ nm (sodium D line). The same solution has $[\alpha] = 170°$ per dm at $\lambda = 313$ nm (UV). Write the approximate dependence $[\alpha] \propto \lambda^{-2}$. $$\frac{[\alpha]_{313}}{[\alpha]_{589}} = \left(\frac{589}{31 Given Information Refractive index $n$ or…

Problem Statement Light is described by $E_x = A\cos\omega t$, $E_y = B\cos(\omega t + \delta)$ with $A = 2$, $B = 1$, $\delta = \pi/4$. Find the Stokes parameters. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This…

Problem Statement Solve the optics problem: In a uniaxial crystal ($n_o = 1.66$, $n_e = 1.49$), find the angle $\theta$ that the ray direction makes with the optic axis when the o-ray and e-ray have equal phase velocities. Phase velocity of e-ray: $v_e(\theta) = c/n_e(\theta)$ where $1/n_e^2(\theta) = \cos^2\theta/n_o^2 + \sin^2\theta/n_e^2$. Given Information Refractive index…

Problem Statement A microscope objective ($f = 2.0$ mm, NA $= 0.85$) images a grating with spatial frequency $f_0 = 800$ lines/mm. For $\lambda = 450$ nm, determine whether this grating is resolvable. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts…