L22 · Distributed gain and feedback

DFB, DBR & VCSEL Architectures

Where should feedback sit to select a laser mode? There is no single right place; each layout picks modes in its own way. Mirrors at the chip facets, a grating spread along the gain region (DFB), separate gratings outside it (DBR) and stacked mirrors above and below a thin gain layer (VCSEL) each set different thresholds for the modes. The laser runs on whichever mode reaches threshold first under the gain curve.

Compare lasers whose feedback comes from the end facets, from a grating along the gain region, from gratings outside it, or from mirror stacks above and below. Record a passive spectrum, find the threshold modes where light only leaves the device, and see how each mode is spread along it.

Feedback structure $\text{—}$

Bragg design wavelength $\text{—}$

Poles found in search window $\text{—}$

Selected candidate rank $\text{—}$

Selected frequency relative to Bragg $\text{—}$

Selected active-region threshold $\text{—}$

Required peak gain after weighting $\text{—}$

Available gain at selected frequency $\text{—}$

Candidates reaching threshold $\text{—}$

Energy / total-loss photon lifetime $\text{—}$

Outgoing-boundary residual $\text{—}$

First-to-second peak-gain gap $\text{—}$

Lowest-candidate distinction $\text{—}$

Captured frequency spacing $\text{—}$

Captured passive-probe frequencies $\text{—}$

L22

Where should feedback sit to select a laser mode?

Active waveguide length

L_a

300\,\mathrm{\mu m}

Waveguide grating strength

\kappa L_a

2

Central one-way phase / pi

\phi/\pi

0.5

Top dielectric mirror pairs

N_t

20

Equal facet power reflectivity

R_f

0.32

Active-region intensity loss

\alpha_i

10\,\mathrm{cm^{-1}}

Imposed peak intensity gain

g_0

100\,\mathrm{cm^{-1}}

Gain full width at half maximum

\Delta\nu_g

300\,\mathrm{GHz}

Gain center relative to Bragg

\nu_g-\nu_B

0\,\mathrm{GHz}

In a uniform DFB the two lowest-threshold modes can tie. Add a quarter-wave defect at the center and compare the threshold gap before you call a design single-mode.

The Lab finds linear thresholds along the laser axis; it does not simulate a saturated laser. The facet, DFB and DBR structures use representative 1550-nanometre waveguides with a group index of 3.6. Gain is an intensity coefficient in the active region. Forward and backward envelopes, normalized to power, are carried through each uniform section by a coupled-mode matrix. The two facets reflect equally; the DFB facets and the outer DBR facets have perfect antireflection coatings. The DFB grating runs along the whole gain region. The DBR gratings are passive, and each is forty percent as long as the active section. You can add a one-way phase jump at the center; a value of half pi creates the DFB's quarter-wave defect. The VCSEL is calculated exactly at normal incidence, interface by interface and layer by layer, with indices alternating between 3.5 and 3.0 at a design wavelength of 850 nanometres. It has thirty bottom mirror pairs, an adjustable top mirror, a one-wavelength active cavity, a substrate of index 3.5 and output into air. The Fresnel factors use real indices and neglect the small interface correction caused by gain. Gain fills the whole active cavity; there is no quantum-well confinement and no solver for a transverse aperture. Layer and grating dimensions are enlarged in the drawing. A mode is a complex outgoing-wave pole: no light comes in at either end. A search from a finite set of starting points covers normalized propagation detuning up to ten for the waveguides and up to 0.22 for the VCSEL. The reported count is the number of roots found in that window, not proof that none were missed. After the poles are solved with flat gain, a Gaussian gain curve that you specify ranks them by the peak gain each one needs. A mode whose threshold is crossed is an eligible candidate in the linear model, not proof of stable single-mode lasing. The passive spectrum is a separate weak-probe calculation with the gain removed and the active-region loss kept; captured dots use 301 sample frequencies. The close probe picks a narrow window around the selected pole, using its lifetime as a guide to the resolution; it does not fit a linewidth. The mode envelope is normalized. Its lifetime is the energy stored in the active region and the mirrors divided by the escape and internal loss, so light penetrating into the mirrors is included. Optical carrier fringes, saturation, hole burning, carrier dynamics, heating, polarization and transverse competition are left out.

Physics tutorial

A pole is stronger evidence than a bright picture

BackgroundA semiconductor laser can get its feedback from its end facets, from a grating along the gain region (DFB), from gratings outside it (DBR), or from mirror stacks above and below the active layer (VCSEL). For each one, record a passive spectrum, find the threshold modes, where light only leaves the device, and look at how each mode is spread in space.

Why it mattersA conclusion about mode selection should rest on the modes the model actually computes, not on how the structure looks.

Start with the essentials

Focus question: Where should feedback sit to select a laser mode?
One-sentence intuition: A laser mode needs gain and feedback that together let light leave both ends with nothing coming in.

Core mathematical model

Coupled forward and backward waves

\frac{d}{dz}\begin{pmatrix}F\\B\end{pmatrix}=\begin{pmatrix}q&i\kappa\\-i\kappa&-q\end{pmatrix}\begin{pmatrix}F\\B\end{pmatrix}

The propagation matrix contains the intensity gain and loss of the active region, the detuning and the reciprocal grating coupling.

Building a laser from sections

\begin{aligned}q&=(g-\alpha_i)/2+i\delta\\ T(z)&=\exp(Az)\\ P_\phi&=\operatorname{diag}(e^{i\phi},e^{-i\phi})\end{aligned}

In a DFB laser the grating and the gain share one region. In a DBR laser the passive gratings are separate from the section where light is amplified.

A mode has output with no input

\begin{aligned}F(0)&=r_1B(0)\\ B(L)&=r_2F(L)\\ \mathcal D&=T_{21}r_1+T_{22}\\ &\quad-r_2(T_{11}r_1+T_{12})=0\end{aligned}

Both the real and the imaginary part must be zero. For DFB, DBR and VCSEL, the facet reflection at the outer reference planes is set to zero.

Quarter-wave dielectric mirrors

\begin{aligned}n_Hd_H&=n_Ld_L=\lambda_B/4\\ I_{12}&=\frac12\begin{pmatrix}s+s^{-1}&s-s^{-1}\\s-s^{-1}&s+s^{-1}\end{pmatrix}\\ s&=\sqrt{n_2/n_1}\end{aligned}

With power-normalized amplitudes, each interface conserves the net power flow. For the VCSEL, the model propagates through each layer exactly instead of using the waveguide coupled-mode approximation.

Ranking modes under the gain curve

\begin{aligned}g(\nu)&=g_0e^{-4\ln2[(\nu-\nu_g)/\Delta\nu_g]^2}\\ g_{0,\rm req}&=g_{\rm th}/w(\nu)\end{aligned}

A gap between thresholds is evidence of mode selection in the linear model. This solver cannot break a tie, and a mode whose gain crosses threshold is not proven to run stably in a single mode.

Include energy stored in the mirrors

\tau_p=\frac{\displaystyle\int\frac{n_g}{c}(|F|^2+|B|^2)\,dz}{\displaystyle P_{\rm esc}+\int_{\rm active}\alpha_i(|F|^2+|B|^2)\,dz}

Light penetrates into the mirrors and stores energy there, even though only the active region has gain. The result is the photon lifetime of the selected threshold mode.

Common difficulties

The lowest threshold guarantees a single mode

Typical misconceptionThe mode with the lowest threshold is the only one that can ever lase.

Better mental modelBefore claiming stable single-mode output, you have to model saturation, spatial hole burning and transverse competition.

Every dip in the spectrum is a laser mode

Typical misconceptionA minimum in the passive reflection spectrum directly gives the lasing threshold.

Better mental modelThreshold needs a complex outgoing-wave pole and a gain balance. A passive weak-probe spectrum answers a different question.

Run the experiment

01
Find the tied pair at the band edges
Choose Uniform DFB and press Inspect next candidate.
What to observe: The lowest thresholds sit symmetrically on either side of the grating's Bragg frequency.
02
Add a phase defect
Switch to Quarter-wave DFB and compare the threshold gap and the mode envelope.
What to observe: A central mode with a lower threshold appears, and its energy gathers around the phase jump.
03
Move the feedback out of the gain region
Compare Separated DBR with VCSEL stack, and press Acquire passive spectrum for each.
What to observe: The passive mirror regions still store part of the mode's energy.
04
Remove mirror pairs
Choose Few top-mirror pairs and compare its threshold with the ordinary VCSEL stack.
What to observe: More light escapes, and the gain you set may no longer reach threshold.