L36 · Keep the same mode

Tunable Laser & Mode-Hop-Free Scan

Why does rotating a grating alone cause mode hops? The grating peak moves away from the current cavity resonance until a neighboring mode is favored. Coordinating cavity length with the grating moves the same integer order smoothly. It must stay inside the gain and etalon passbands; a fixed selector can still support a finite scan within its bandwidth.

Tune a Littrow grating, the external cavity length, the gain center and an intracavity etalon together. Scan the frequency out and back, and find where the laser hops to another mode or stops emitting.

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 $\text{—}$

The reference wavelength is 780 nm in vacuum. The Littrow grating has 1200 grooves per millimeter, and the external cavity has an optical length of about 30 mm. External-cavity frequencies use exact integer round-trip orders and the cavity length. The grating target is set in gigahertz from the reference, and the grating angle follows the exact first-order Littrow law. Tuning to a higher frequency requires a shorter cavity. Current tracking moves the gain center by a calibrated teaching amount; the gain is Gaussian with a 12 GHz width, and temperature shifts it by minus 20 GHz per kelvin. The grating selects with a Gaussian window. The lossless Airy etalon has a 15 GHz free spectral range and adjustable reflectivity. A simplified net logarithmic merit for each mode picks one emitting mode, provided its merit is above zero. The laser stays in its previous mode as long as that mode still has positive merit and no other mode beats it by more than a set merit margin. This is an empirical memory rule, not a derived model of carrier competition or piezo hysteresis. Output is the positive merit, normalized; it is not watts or wall-plug efficiency. The recorded spectrum is the selected line broadened by a Gaussian instrument response with an 80 MHz standard deviation. The candidate merits are a separate reference, not lines measured lasing at the same time. Scan progress counts how far the actuators have moved in frequency, not physical time. Facet resonances inside the diode, device-specific current and index response, spontaneous noise, thermal lag and multimode transients are left out. The apparatus shows the recorded endpoint. Actuator stroke is magnified five hundredfold relative to the schematic cavity length; grating angles follow the physical law without angular exaggeration. Flow markers do not replay a new scan.

Start with the essentials

Focus question: Why does rotating a grating alone cause mode hops?
One-sentence intuition: A mode-hop-free scan keeps one cavity order inside the grating, gain and etalon passbands throughout the sweep.

Core mathematical model

Littrow condition for the grating

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

The grating spacing is fixed. Rotating its normal changes which wavelength returns along the incoming beam; the specular zero order goes toward a separate output collector.

Longitudinal modes of the external cavity

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

Each mode's frequency is set by its integer order and the optical length of the cavity.

Shortening the cavity in step with the scan

\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}

A fixed integer order counts the same number of half-wavelengths between the endpoints. Raising its frequency shortens those half-wavelengths, so the cavity must shorten too.

Etalon transmission (Airy function)

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

If the etalon's narrow transmission band is not moved along, it can interrupt the scan.

A simplified score for each mode

\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}

A mode can emit only when its score is positive. Saturation and absolute power are not modeled.

The laser keeps its current mode

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

Because of this empirical rule, the laser can switch modes at different points on the way out and on the way back.

Common difficulties

Tuning always means hopping

Typical misconceptionTo change the frequency, the laser has to jump to another longitudinal mode.

Better mental modelChanging the cavity length moves the same mode order smoothly to a new frequency.

A broader grating filter fixes everything

Typical misconceptionA broader grating selection guarantees a stable single-mode scan.

Better mental modelThis model always picks exactly one mode, so it cannot show whether a real laser stays single-mode. Coordinate the cavity length, gain and etalon instead.

Run the experiment

01
Rotate only the grating
The default already contains an outward and return sweep with Cavity tracking ratio at zero. Compare the steps in the frequency trajectory with the sloping target line.
What to observe: The laser frequency jumps from one fixed cavity resonance to the next.
02
Follow one mode order
Predict whether the steps will disappear, then choose Coordinated cavity. The same sweep is reacquired automatically.
What to observe: The same integer order now moves smoothly with the grating.
03
Find the next limit
Choose Fixed gain center or Stationary etalon, then choose Track every selector.
What to observe: The output can drop out even when the cavity tracking is exact.

Tunable Laser & Mode-Hop-Free Scan

Keep one cavity order inside the transmission bands