L22 · Distributed gain and feedback

DFB, DBR & VCSEL Architectures

Compare facet, distributed, separated-grating and vertical feedback. Acquire a passive spectrum, find outgoing-wave threshold modes, and inspect their spatial envelopes.

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}

A uniform DFB can have a tied lowest pair. Add a central quarter-wave defect and compare the threshold gap before calling a design single-mode.

This is a linear axial threshold workbench, not a saturated laser simulation. Facet, DFB and DBR structures use representative 1550-nanometre waveguides with a group index of 3.6. The gain is an intensity coefficient in the active region. Uniform coupled-mode segment matrices propagate power-normalized forward and backward envelopes. Facet reflectivities are equal; DFB facets and DBR end facets are perfectly antireflection coated. The DFB grating overlaps the whole gain region; the DBR gratings are passive and each is forty percent of the active length. A central one-way phase discontinuity is adjustable; half-pi makes the DFB quarter-wave defect. VCSEL uses exact normal-incidence interfaces and propagation through alternating indices 3.5 and 3.0 at a design wavelength of 850 nanometres. It has thirty bottom mirror pairs, a variable top mirror, a one-wavelength active cavity, index-3.5 substrate and air output. Real-index Fresnel factors neglect the small interface correction from gain. Gain fills the whole active cavity; there is no quantum-well confinement or transverse aperture solver. Layer and grating dimensions are enlarged in the drawing. Modes solve a complex outgoing-wave pole with zero incoming field at both ends. A finite multistart search covers normalized propagation detuning within ten for waveguides and 0.22 for VCSEL; reported counts are roots found in that window, not a completeness proof. A specified Gaussian gain envelope ranks the required peak gain after the flat-gain pole solve. Crossings are eligible linear candidates, not proof of stable single-mode lasing. The passive spectrum is a separate weak-probe calculation with gain removed and active-region loss retained; captured dots use 301 sample frequencies. The close probe chooses a narrow window around the selected pole using its lifetime as a resolution guide; it does not fit linewidth. The mode envelope is normalized, and its lifetime integrates energy in the active region and mirrors before dividing by escape and internal loss. Thus mirror penetration is included. Optical carrier fringes, saturation, hole burning, carrier dynamics, thermal effects, polarization and transverse competition are omitted.

Start with the essentials

Focus question: Where should feedback sit to select a laser mode?
One-sentence intuition: Gain and feedback must close the same physical boundary problem.

Core mathematical model

Couple the two wave envelopes

\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 generator contains active intensity gain, loss, detuning and reciprocal grating coupling.

Compose feedback regions

\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}

DFB mixes gain and feedback along the same region. DBR separates passive gratings from active propagation.

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 real and imaginary parts must vanish. DFB, DBR and VCSEL use zero facet reflections at the exterior reference planes.

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}

The power-normalized interface conserves net flux. Exact layer propagation replaces the waveguide coupled-mode approximation for VCSEL.

Rank a pole under the gain envelope

\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 threshold gap is linear selection evidence. A tie is not resolved by this solver, and a crossing does not prove stable single-mode operation.

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}

Mirror penetration contributes stored energy even though gain is restricted to the active region. This is the selected threshold-mode lifetime.

Common difficulties

One low threshold guarantees single-mode operation

Typical misconceptionThe lowest candidate is the only mode that can ever lase.

Better mental modelSaturation, spatial hole burning and transverse competition must be modeled before asserting stable single-mode output.

Every dark spectral dip is a laser mode

Typical misconceptionA passive reflection minimum directly gives lasing threshold.

Better mental modelThreshold requires a complex outgoing-wave pole and a gain balance; passive weak-probe data answers a different question.

Run the experiment

01
Find the tied edge pair
Choose uniform DFB and inspect the next candidate.
What to observe: The lowest thresholds are symmetric around the grating frequency.
02
Introduce a defect
Switch to quarter-wave DFB and compare the threshold gap and envelope.
What to observe: A lower-threshold central mode concentrates near the discontinuity.
03
Move feedback outside gain
Compare separated DBR with VCSEL; acquire each passive spectrum.
What to observe: Passive mirror regions still store mode energy.
04
Remove mirror pairs
Choose few top-mirror pairs and compare its threshold with the ordinary VCSEL.
What to observe: Escape increases; imposed gain may no longer reach threshold.