L36 · Keep the same mode

Tunable Laser & Mode-Hop-Free Scan

Coordinate a Littrow grating, external cavity, gain center and intracavity etalon. Acquire outward and return frequency scans and diagnose hops or loss of emission.

Actuator frequency travel $\text{—}$

Grating target offset $\text{—}$

Selected optical offset $\text{—}$

Selected longitudinal order $\text{—}$

Littrow angle $\text{—}$

External cavity length change $\text{—}$

Gain center offset $\text{—}$

Relative output $\text{—}$

Mode hops during emitted segments $\text{—}$

Longest continuous same-order interval $\text{—}$

Selected line minus grating target $\text{—}$

Captured scan samples $\text{—}$

Captured actuator travel $\text{—}$

Vacuum 780 nm reference, 1200 grooves per millimeter Littrow grating and an approximately 30 mm external optical cavity. External frequencies use exact integer round-trip orders and cavity length. Grating target is commanded in gigahertz relative to the reference; its angle follows the exact first-order Littrow law. Positive frequency tuning requires decreasing cavity length. Current tracking is a calibrated teaching shift of the gain center, with a 12 GHz Gaussian gain width; temperature shifts it by minus 20 GHz per kelvin. Grating selection width is Gaussian; a lossless Airy etalon has a 15 GHz free spectral range and adjustable reflectivity. A reduced net logarithmic modal merit chooses one emitting mode above zero threshold. The previous mode is retained within an explicit merit margin, a phenomenological memory rule rather than a derived carrier competition or piezo hysteresis model. Output is normalized positive merit, not watts or wall-plug efficiency. The acquired spectrum is that selected line with an 80 MHz standard-deviation Gaussian instrument response; candidate merits are a separate reference, not measured simultaneous lasing lines. Scan progress is actuator frequency travel, not physical time. Internal facet resonances, device-specific current/index response, spontaneous noise, thermal lag and multimode transients are omitted. Enlarge the apparatus without magnifying tiny actuator displacements.

Start with the essentials

Focus question: Why does rotating a grating alone cause mode hops?
One-sentence intuition: A mode-hop-free scan moves one cavity resonance together with every frequency selector.

Core mathematical model

Littrow selection

\begin{aligned}\lambda_g&=2d\sin\theta\\\nu_g&=c/\lambda_g\end{aligned}

The first diffracted order retraces the incident beam.

External longitudinal modes

\begin{aligned}\nu_q&=\frac{qc}{2L}\\\Delta\nu_{\rm FSR}&=\frac{c}{2L}\end{aligned}

Integer mode order and optical length jointly determine the frequency.

Coordinated length

\begin{aligned}L(x)&=L_0\frac{\nu_0}{\nu_0+r_Lx}\\r_L&=1\ \Rightarrow\ \nu_{q_0}=\nu_0+x\end{aligned}

Shorten the cavity when the commanded frequency rises.

Airy etalon

\begin{aligned}\mathcal T_e&=\frac{1}{1+A\sin^2[\pi(\nu-\nu_e)/D_e]}\\A&=\frac{4R_e}{(1-R_e)^2}\end{aligned}

An untracked narrow transmission band can interrupt the scan.

Reduced selection merit

\begin{aligned}M_q&=3.6-\tfrac12 u_g^2-\tfrac12 u_a^2\\&\quad+\ln\mathcal T_e\\u_g&=(\nu_q-\nu_g)/\sigma_g\\u_a&=(\nu_q-\nu_a)/\sigma_a\end{aligned}

Positive merit permits output; saturation and absolute power are not modeled.

Explicit memory rule

\begin{aligned}M_{\rm best}-M_{\rm old}&\le h\\M_{\rm old}&>0\ \Rightarrow\ q=q_{\rm old}\end{aligned}

Forward and reverse switches can differ under this phenomenological rule.

Common difficulties

Tuning means hopping

Typical misconceptionChanging optical frequency must change longitudinal order.

Better mental modelChanging the cavity length moves a fixed order continuously.

A broad filter fixes everything

Typical misconceptionBroadening the grating guarantees a stable single-mode scan.

Better mental modelThis model selects one mode by construction; it cannot certify real multimode stability. Coordinate length, gain and etalon instead.

Run the experiment

01
Rotate only the grating
Acquire an outward and return sweep with zero cavity tracking.
What to observe: Selected frequency steps between fixed external resonances.
02
Track one order
Choose coordinated cavity and repeat the same sweep.
What to observe: The same integer order moves continuously with the grating.
03
Find the next bottleneck
Hold gain or etalon fixed, then restore tracking.
What to observe: Output may disappear even when cavity tracking is exact.