# Interferometers Haus Ch. 3. Matrix tools: [[optical operators]]. Fourier and correlation tools: [[fourier transform]]. ## Fabry–Perot Interferometer Two partially transmitting mirrors (S-matrices $-r_i$, $jt_i$) separated by $l$, in a medium of index $n$, at internal angle $\theta$. The one-way phase is $ \frac{\delta}{2}=\frac{\omega\,n\,l\cos\theta}{c} $ ### Sum over paths ("Feynman diagrams") Each path through the device is a product of factors: - $jt_1$ to enter - $e^{-j\delta/2}$ per one-way trip - $-r_2$, $-r_1$ per internal bounce - $jt_2$ to exit Every extra round trip multiplies by $r_1r_2e^{-j\delta}$. Summing every diagram is a geometric series: $ S_{21}=(jt_1)(jt_2)e^{-j\delta/2}\sum_{m=0}^{\infty}\left(r_1r_2e^{-j\delta}\right)^m=\frac{-t_1t_2\,e^{-j\delta/2}}{1-r_1r_2\,e^{-j\delta}} $ Power transmission: $ \mathcal T=|S_{21}|^2=\frac{t_1^2t_2^2}{(1-r_1r_2)^2+4r_1r_2\sin^2\dfrac{\delta}{2}} $ Symmetric lossless mirrors ($R=r^2$, $t^2=1-R$): $ \mathcal T=\frac{1}{1+\dfrac{4R}{(1-R)^2}\sin^2\dfrac{\delta}{2}},\qquad \mathcal R=1-\mathcal T $ ### Key numbers - **Resonance** when $\dfrac{\delta}{2}=m\pi$, i.e. $l\cos\theta$ equals an integer number of half-wavelengths in the medium: $ f_m=\frac{mc}{2nl\cos\theta} $ - **Free spectral range** (peak spacing): $ \Delta f=\frac{c}{2nl\cos\theta},\qquad \Delta\lambda\approx\frac{\lambda^2}{2nl\cos\theta} $ - **Linewidth** (FWHM, for $1-R\ll1$): $ \delta f_{1/2}\approx\frac{(1-R)\,c}{2\pi\sqrt{R}\,nl\cos\theta} $ - **Finesse**: $ \mathcal F=\frac{\Delta f}{\delta f_{1/2}}=\frac{\pi\sqrt R}{1-R} $ Unequal mirrors: $ \mathcal F=\frac{\pi(R_1R_2)^{1/4}}{1-\sqrt{R_1R_2}} $ - **Peak transmission**, unequal lossless mirrors. It equals 1 only if $R_1=R_2$ (impedance matching): $ \mathcal T_\text{max}=\frac{(1-R_1)(1-R_2)}{\left(1-\sqrt{R_1R_2}\right)^2} $ - **With a mirror loss $L$** ($t^2=1-R-L$): $ \mathcal T_\text{max}=\frac{(1-R-L)^2}{(1-R)^2} $ - **Resolving power**: two lines are resolved when separated by one FWHM. $ \frac{\lambda}{\delta\lambda}=\mathcal F\,\frac{2nl\cos\theta}{\lambda}\approx m\mathcal F $ This is typically $10^5$–$10^6$, versus $mN$ for a grating. - **Scanning FP**: moving a mirror by $\dfrac{\lambda}{2}$ shifts the whole comb of peaks by one FSR. This works as a spectrum analyzer for signals narrower than one FSR. - **Gain medium / laser**: replace $k$ by $k+j\alpha$ (amplitude gain $\alpha$). At resonance the denominator vanishes when $ r_1r_2\,e^{2\alpha l}=1 $ That is the lasing threshold: output with no input. - At resonance, the circulating internal **intensity** (each direction) is the incident intensity times $ \frac{t^2}{(1-R)^2}=\frac{1}{1-R}\quad(\text{symmetric, lossless}) $ ## Michelson Interferometer A 50/50 splitter sends light down arms $l_1$ and $l_2$; the mirrors return it and the splitter recombines it. For monochromatic light: $ I_\text{out}=\frac{I_0}{2}\Big[1\pm\cos\big(2k(l_1-l_2)\big)\Big] $ The sign depends on the splitter phases. Haus's choice gives a dark output port for equal arms, and the light then goes back out the input port, so power is conserved (Haus 3.10). It varies with mirror position much more slowly than a Fabry–Perot (cosine fringes, not sharp peaks). ## Coherence For broadband light, the output averages over frequencies. With delay $\tau=\dfrac{2(l_1-l_2)}{c}$: $ I_\text{out}(\tau)\propto\Gamma(0)\pm\mathrm{Re}\,\Gamma(\tau),\qquad \Gamma(\tau)=\langle E^*(t)\,E(t+\tau)\rangle $ - The fringe **visibility** measures the degree of coherence: $ V=\frac{I_\text{max}-I_\text{min}}{I_\text{max}+I_\text{min}}=\frac{|\Gamma(\tau)|}{\Gamma(0)} $ - Wiener–Khinchin: $\Gamma(\tau)$ and the power spectrum are a Fourier pair. So a Michelson (which measures $\Gamma$) and a scanning FP (which measures the spectrum) give the **same information** in different forms. This is Fourier-transform spectroscopy. - Coherence time $\tau_c\sim\dfrac{1}{\Delta f}$; coherence length $l_c=c\tau_c$. - Multi-mode laser (Haus 3.9): modes spaced $\dfrac{c}{2L}$ with random phases. The visibility revives whenever $2(l_1-l_2)$ is a multiple of the cavity round trip. - Twyman–Green: a Michelson with a collimated beam and a test optic in one arm. Fringes map wavefront errors, used to test flats and lenses. ## Grating Spectrometer (Haus §2.6–2.7) - Grating equation (period $\Lambda$, order $m$; the sign of $m$ depends on angle convention, so check against the figure): $ \sin\theta_m=\sin\theta_i+\frac{m\lambda}{\Lambda} $ - Angular dispersion: $ \frac{d\theta_m}{d\lambda}=\frac{m}{\Lambda\cos\theta_m} $ - Chromatic resolving power with $N$ illuminated grooves: $ \frac{\lambda}{\delta\lambda}=mN $ - Blazing (tilted facets) puts most power in one order. The higher orders are why a corrugated surface needs more than one reflected wave.