Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 3

Use mode locking to phase-lock many longitudinal cavity modes; their repeated constructive interference produces short optical pulses. A saturable absorber can select this pulsed operation. A $2\mathrm{MHz}$ fundamental repetition rate corresponds to a $500\mathrm{ns}$ round trip, or an optical round-trip length $c/R\simeq 150\mathrm{m}$; a linear air-filled cavity would therefore be about $75\mathrm{m}$ long. A compact alternative is a faster mode-locked oscillator followed by pulse selection and, if required, amplification. Ordinary Q-switching alone generally does not provide $350\mathrm{fs}$ pulses.^[ref]

The incident intensity radius is $w_L=2.5\mathrm{mm}$, half the quoted diameter. For an ideal collimated Gaussian beam focused by a thin lens, the diffraction-limited waist isThe incident Rayleigh range $\pi w_L^2/{\lambda }_0\simeq 24.5\mathrm{m}$ is much larger than $f$, justifying this approximation. The pulse energy is $U=P_{\mathrm{av}}/R=0.25\mu \mathrm{J}$. Taking its duration as an effective energy-to-peak-power width gives $P_{\mathrm{pk}}\simeq U/\tau =7.14\times 10^5\mathrm{W}$. Since $I(r)=I_0{e}^{-2r^2/w_0^2}$ has integrated power $I_0\pi w_0^2/2$,If $350\mathrm{fs}$ denotes the temporal intensity full width at half maximum of a Gaussian pulse, the time integral instead gives $P_{\mathrm{pk}}=2\sqrt{\ln 2}U/(\sqrt{\pi }\tau )$, and $I_{\mathrm{pk}}=1.03\times 10^{13}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$. The pulse shape is unspecified, so the estimate's order-one convention is explicit. Aberration, clipping and propagation through glass were excluded from this calculation as stipulated.

The contrast is weak: $(n_1-n_2)/n_2=0.01$. Use a scalar weak-guidance Helmholtz equation for a transverse field component, with longitudinal wave number $b$ and vacuum wave number $k_0=2\pi /{\lambda }_0$. Separating variables as $\psi =F_l(r)e^{il\varphi }{e}^{ibz}$ givesA bound mode requires $n_2{k}_0<b<n_1{k}_0$. DefineThe step-index fiber normalized frequency is $V$; $u$ is a dimensionless core transverse wave number. Regularity at $r=0$ selects the Bessel function of the first kind, while decay at infinity selects the Modified Bessel function of the second kind:In this scalar approximation, continuity of the field and its radial derivative givesAt cutoff the evanescent wave extends arbitrarily far into the cladding, so $b\downarrow n_2{k}_0$, $w\downarrow 0$ and $u\to V$. For the first azimuthally varying family $l=1$, the recurrence relations reduce the last equality toAs $w\to 0$, $K_1(w)\sim 1/w$ and $K_0(w)\sim -\ln w$, making the right side tend to zero. Thus the onset is at the first positive zero of $J_0$:The scalar $l=0$ fundamental family has no positive cutoff. The first additional spatial family is the $l=1$ family just found; the next radial $l=0$ and first $l=2$ families require the larger zero $j_{1,1}\simeq 3.832$. This establishes the first multimode boundary rather than merely finding one possible cutoff. The two polarizations of the fundamental family are not counted as two spatial modes. Exact vector interface conditions split some higher-mode degeneracies at finite contrast; the displayed condition is the weak-guidance result intended by the supplied scalar equations.

The raised-index region has $n_1=1.01(1.50)=1.515$, whereas $n_2=1.50$. Therefore its numerical aperture factor is $\sqrt{n_1^2-n_2^2}=0.212662$. With the printed rounded cutoff,ThusSlower scanning increases the written radius and makes the guide multimode; faster scanning makes it single-spatial-mode in this idealization, provided a physically meaningful positive core radius remains. Using $j_{0,1}=2.404825\dots$ gives $s_c\simeq 586.2\mathrm{mm}{\mathrm{s}}^{-1}$. This numerical threshold extrapolates the supplied empirical radius law; its experimental range is not given.