Category: Part 5: Optics

Problem Statement Solve the optics problem: A material has complex refractive index $\tilde{n} = 1.5 + 0.3i$ at $\lambda = 500$ nm. Find (a) the absorption coefficient $\alpha$, (b) the depth at which intensity drops to $1/e$. (a) Absorption coefficient: $\alpha = 4\pi k/\lambda$ where $k = 0.3$ is the extinction coefficient. $$\alpha = \frac…

Problem Statement Solve the optics problem: An LiNbO$_3$ Mach-Zehnder electro-optic modulator has $V_{\pi} = 5$ V at $\lambda = 1550$ nm. Find the phase shift for $V = 2.5$ V and the transmission through the modulator. Phase shift: $\phi = \pi V/V_{\pi} = \pi\times2.5/5 = \pi/2$. Transmission of a Mach-Zehnder: $T = \cos^2(\phi/2) = \cos^2(\p…

Problem Statement Solve the oscillation/wave problem: Gold ($\varepsilon_{Au} = -12.4+1.2i$) supports surface plasmons at $\lambda = 633$ nm. Find the propagation constant $k_{sp}$ of the surface plasmon at a gold-air interface. $$k_{sp} = \frac{\omega}{c}\sqrt{\frac{\varepsilon_{Au}\varepsilon_{air}}{\varepsilon_{Au}+\varepsilon_{air}}} = k_0\sqrt{\fr Given Information Mass $m$ and spring constant $k$ (or equivalent), or wave parameters Initial conditions (amplitude $A$, phase $\phi$)…

Problem Statement Solve the optics problem: A step-index fiber has material dispersion $D = -100$ ps/(nm·km) at $\lambda = 1300$ nm. For a pulse with spectral width $\delta\lambda = 1$ nm, find the pulse broadening over $L = 10$ km. $$\sigma_t = |D|\cdot L \cdot \delta\lambda = 100\text{ ps/(nm·km)}\times10\text{ km}\times1\text{ nm} = \boxed Given Information…

Problem Statement A He-Ne laser has gain profile width $\Delta\nu_g = 1.5$ GHz and cavity length $L = 15$ cm. Find the number of cavity modes within the gain profile and state the condition for single-mode operation. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see…

Problem Statement A substance is illuminated with laser light at $\nu_0 = 6.0\times10^{14}$ Hz. A Raman line appears at $\nu_R = 5.9\times10^{14}$ Hz (Stokes line). Find the vibrational frequency of the molecule and the wavelength of the anti-Stokes line. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as…

Problem Statement Solve the optics problem: At total internal reflection in glass ($n = 1.5$) at angle $\theta_i = 50°$, find the 1/e penetration depth of the evanescent wave into air for $\lambda = 633$ nm. $$\kappa = \frac{2\pi}{\lambda}\sqrt{n^2\sin^2\theta_i – 1} = \frac{2\pi}{633\times10^{-9}}\sqrt{(1.5)^2\sin^2 50° – 1}$$ $$= \frac{2\pi Given Information Refractive index $n$ or focal…

Problem Statement Solve the oscillation/wave problem: Newton’s rings are formed with light at normal incidence and the 10th bright ring has diameter $D_{10} = 5.0$ mm ($R = 1.0$ m, $\lambda = 589$ nm). If the illumination angle is tilted to $\theta = 30°$ in the glass, how does the ring pattern change? At oblique…

Problem Statement A monochromator with grating (1200 lines/mm, $f = 250$ mm, exit slit $s = 0.05$ mm) is used in 1st order. Find the spectral bandwidth passed by the exit slit near $\lambda = 500$ nm. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see…

Problem Statement Solve the oscillation/wave problem: A deep-UV lithography system uses $\lambda = 193$ nm and NA $= 1.35$ (immersion). Find the minimum feature size (half-pitch). $$\text{Feature size} = k_1\frac{\lambda}{NA}$$ With $k_1 = 0.25$ (aggressive resolution enhancement): $$= 0.25\times\frac{193}{1.35} = 0.25\times143 = \boxed{35.7\text{ nm} Given Information Mass $m$ and spring constant $k$ (or equivalent), or…