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Description
Integrated photonics needs film lasers as thin as possible, yet an asymmetric planar waveguide guides light only above the cutoff thickness $h_c$ of its fundamental mode. We examine lasing in the critical regime, $h\approx h_c$, where the TE$_0$ effective index $N$ approaches the substrate index $n_s$, and show that Lummer–Gehrcke (LG) modes [1] are its sub-cutoff special case.
The samples are sol–gel mesostructured TiO$_2$/Pluronic P123 and SiO$_2$/P123 films doped with rhodamine 6G on glass ($n_s=1.515$). Dye in $\sim$12-nm micellar pores avoids concentration quenching up to 5 mmol/L, while the inorganic network sets $n_f$ from 1.45–1.52 (SiO$_2$) to 1.56–1.69 (TiO$_2$). For $n_f=1.565$, $h_c=291$ nm ($\lambda=580$ nm), so TiO$_2$ films of 200 nm ($0.69h_c$) and 340 nm ($1.17h_c$) bracket the cutoff, while SiO$_2$ films ($n_f\lesssim n_s$) have no guided mode.
Films were pumped transversely (532 nm, 15 ns, $\approx3\times0.5$ mm$^2$ stripe). Imaging the end face across the spectrograph slit gave film and substrate spectra simultaneously; far-field spots, end masking and substrate immersion completed the measurements.
The 340-nm film is single-mode: $N-n_s=1.6\times10^{-3}$, confinement factor 0.26, 73.5% of the mode power in the substrate. It lases with a sharp threshold (FWHM 60$\to$5 nm) into a beam with interference fringes, whose divergence normal to the film, $6.3^\circ$, is fixed by the evanescent tail, $\theta_\perp\approx2\sqrt{N^2-n_s^2}$. Hence $N-n_s\approx1.0\times10^{-3}$: the mode runs only $0.15^\circ$ above the critical angle, and the divergence gauges cutoff proximity.
The 200-nm film shows a sharp threshold (FWHM 39$\to$6.6 nm, twofold rebroadening above it), emitting from both film and substrate ends. Both channels narrow identically, whereas lateral emission stays ASE-like. As the pump grows 20-fold (0.007$\to$0.15 MW/mm$^2$), the film output rises superlinearly and the substrate output linearly, concentrating energy in the film; immersion, which removes the reflection at the free substrate face, suppresses the substrate peak. The SiO$_2$ film needs an order of magnitude higher pump.
We attribute the substrate channel to LG modes. Below cutoff the film, bounded by total reflection at the air side and partial near-critical reflection at the substrate side, behaves as an LG plate: along the pumped stripe it releases coherent partial beams into the substrate at the grazing angle $\theta_s$ that maximises their gain path in the film [2,3]. Transfer-matrix calculations show that, as $h\to h_c$, this gain path grows from $\approx14h$ (200 nm) to $\approx100h$ ($0.96h_c$), while the beams' exit angle from the substrate end, $\arcsin\sqrt{n_s^2-N^2}$ with $N=n_s\sin\theta_s$, falls from $12^\circ$ to $\approx1^\circ$; above $h_c$ the LG mode turns continuously into TE$_0$. Near $h_c$ the two lobes become collinear and mutually coherent, which explains the fringes and the narrow ($9$–$11^\circ$) substrate-end spot of the 340-nm film.
The critical regime combines a nanoscale gain layer, mirrorless feedback and an output direction programmable by $h$ or $n_f$; we discuss ultrathin lasers with substrate output coupling, divergence-based metrology of $N-n_s$ and index sensing at the substrate side.
[1] E.A. Tikhonov et al., Laser Phys. 28, 045801 (2018).
[2] A. Penzkofer et al., Opt. Commun. 229, 279 (2004).
[3] J. Peter et al., J. Appl. Phys. 117, 015301 (2015).
| Figure caption | Fig. 1. (a) Guided TE₀ and Lummer–Gehrcke (LG) emission channels near cutoff. (b) Calculated TE₀ divergence and LG exit angle versus h/hc (TiO₂, nf=1.565); filled dot – measured 340-nm film, open dot – predicted 200-nm film. |
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