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Plano–Concave (Plano–Convex) Fabry–Pérot Cavity Equations

1. Geometry and Stability

Flat mirror: - $R_1 = \infty$

Curved mirror: - $R_2 = R$

Optical cavity length: - $L$

Wavelength: - $\lambda$

g-parameters:

$$ g_1 = 1 - \frac{L}{R_1} = 1 $$

$$ g_2 = 1 - \frac{L}{R} $$

Stability condition:

$$ 0 < g_1 g_2 < 1 $$

For plano–concave:

$$ 0 < 1 - \frac{L}{R} < 1 $$

$$ 0 < L < R $$

2. Waist Location

For a plano–concave cavity, the waist is located at the flat mirror.

Thus:

  • Flat-side waist: $w_{\text{flat}} = w_0$
  • Curved-side spot: $w(L)$

3. Rayleigh Range and Flat-Side Waist

Rayleigh range:

$$ z_R = \sqrt{L(R - L)} $$

Waist size:

$$ w_0 = \sqrt{\frac{\lambda}{\pi} z_R} $$

Equivalent explicit form:

$$ w_0 = \sqrt{\frac{\lambda}{\pi}} \, [L(R - L)]^{¼} $$

Or:

$$ w_0^2 = \frac{\lambda}{\pi} \sqrt{L(R - L)} $$

4. Spot Size on the Curved Mirror

Gaussian propagation:

$$ w(z) = w_0 \sqrt{1 + \left(\frac{z}{z_R}\right)^2} $$

At $z = L$:

$$ w_{\text{curved}} = w(L) = w_0 \sqrt{1 + \left(\frac{L}{z_R}\right)^2} = w_0 \sqrt{\frac{R}{R - L}} $$

Direct expression:

$$ w_{\text{curved}}^2 = \frac{\lambda}{\pi} \sqrt{L(R - L)} \cdot \frac{R}{R - L} $$

5. Wavefront Curvature

Gaussian wavefront curvature:

$$ \mathcal{R}(z) = z \left[1 + \left(\frac{z_R}{z}\right)^2\right] $$

Self-consistency at curved mirror:

$$ \mathcal{R}(L) = R $$

Using:

$$ z_R^2 = L(R - L) $$

6. Free Spectral Range

$$ \mathrm{FSR} = \frac{c}{2L} $$

7. Transverse Mode Spacing

General expression:

$$ \Delta \nu_{mn} = \frac{\mathrm{FSR}}{\pi} (m+n) \arccos!\big(\sqrt{g_1 g_2}\big) $$

For plano–concave:

$$ \Delta \nu_{mn} = \frac{\mathrm{FSR}}{\pi} (m+n) \arccos!\left(\sqrt{1 - \frac{L}{R}}\right) $$

8. Radius of Curvature from Etch Geometry

Given: - aperture radius $a$ - sagitta (center depth) $h$

Exact spherical-cap relation:

$$ R = \frac{a^2 + h^2}{2h} $$

Small-sag approximation ($h \ll a$):

$$ R \approx \frac{a^2}{2h} $$

Parabolic approximation:

If:

$$ z® \approx \frac{r^2}{2R} $$

and $z(r_m) = h$, then:

$$ R \approx \frac{r_m^2}{2h} $$

9. Optical Length vs Mechanical Gap

If mechanical gap is $L_{\text{mech}}$ and center recess is $h$:

$$ L \approx L_{\text{mech}} - h $$

Including coating phase penetration:

$$ L_{\text{eff}} = L_{\text{geom}} + \delta_1 + \delta_2 $$

Use $L_{\text{eff}}$ in all Gaussian-mode equations when precision matters.

Cavity anti-node

1. What replaces “E = 0 at the mirror”?

For a DBR, the mirror is not a hard boundary. It produces a complex reflection coefficient

$r(\omega) = |r| e^{i\phi_r(\omega)}$

At high reflectivity, $|r| \approx 1$, but the reflection phase $\phi_r$ is not necessarily $\pi$ (which would correspond to a PEC).

The standing wave condition inside a Fabry–Pérot cavity is:

$2 k L + \phi_{r1} + \phi_{r2} = 2\pi m$

where: - $k = 2\pi/\lambda$, - $L$ is the physical air gap, - $\phi_{r1}, \phi_{r2}$ are the reflection phases of the two DBRs.

So the cavity resonance depends on reflection phase, not on a node at a geometric surface.

2. Field penetration into a DBR

A key concept:

A DBR has a finite penetration depth.

The field does not abruptly vanish. Instead, it decays exponentially into the multilayer:

$E(z) \sim e^{-z/\ell_p}$

where $\ell_p$ is the mirror penetration depth.

For an ideal quarter-wave stack at its design wavelength:

$\ell_p \sim \frac{\lambda}{4\pi} \frac{n_H n_L}{n_H^2 - n_L^2}$

(approximate scaling)

Physically:

  • The reflection “happens” over several layer pairs.
  • The effective reflection plane is located inside the mirror.
  • This creates an effective cavity length larger than the physical air gap.

Define:

$L_{\text{eff}} = L_{\text{air}} + \ell_{p1} + \ell_{p2}$

3. Where is the antinode?

Inside a high-reflectivity DBR cavity:

  • The field does not go to zero at the mirror surface.
  • Instead, the field maximum (antinode) is located slightly inside the cavity, not exactly at the dielectric interface.

For a symmetric cavity at the Bragg wavelength:

  • The reflection phase is approximately $\phi_r \approx \pi$.
  • The standing wave resembles that of a metal-mirror cavity.
  • But the node is shifted inside the mirror by roughly the penetration depth.

So:

The effective boundary condition is that the field node is located roughly one penetration depth inside the mirror.

Therefore, in a plano-convex DBR cavity:

  • The longitudinal antinode spacing is determined by $L_{\text{eff}}$.
  • The longitudinal mode shape is centered within the effective cavity length.
  • The first node is not at the coating surface but inside the mirror stack.

4. Practical consequences

(1) Mode volume includes penetration into the mirror:

$V_{\text{mode}} \propto L_{\text{eff}}$

(2) Radius of curvature

The cavity stability parameter uses

$L_{\text{eff}}$

not just the air gap. The $g$ parameter becomes:

$g = 1 - \frac{L_{\text{eff}}}{R}$

(3) Antinode positioning for emitters

If placing atoms, color centers, or molecules at an antinode, position them relative to the standing wave in the air region, determined by the phase condition above—not by assuming a node at the surface.

5. Exact computation procedure

  1. Use the transfer matrix method (TMM) for your DBR.
  2. Extract the reflection phase $\phi_r(\lambda)$.
  3. Compute effective cavity length from:

$L_{\text{eff}} = \frac{1}{2k} \left( 2\pi m - \phi_{r1} - \phi_{r2} \right)$

  1. Solve the full field distribution using TMM including air gap + mirrors.

This yields the true spatial location of nodes and antinodes.

6. Rule of thumb

At the Bragg wavelength:

  • The first antinode in the air gap is approximately $\lambda/4$ from the effective mirror plane.
  • The effective mirror plane is about one penetration depth inside the stack.
  • For high index contrast DBRs, penetration depth is on the order of $\lambda/(2\pi)$.

So the shift is typically tens to a few hundred nanometers at optical wavelengths.

Bottom line

In a quarter-wave stack cavity:

  • There is no hard node at the physical surface.
  • The mirror acts as a distributed reflector.
  • The cavity length must include penetration depth.
  • The antinode structure is determined by reflection phase, not by assuming $E=0$ at the surface.

Purcell factor

I assume:

  • Ideal two-mirror Fabry–Pérot cavity in air ($n = 1$)
  • Emitter on resonance, at an antinode
  • Perfect dipole alignment ($\xi = 1$)
  • Wavelength $\lambda = 780 \text{nm}$

Purcell factor formula

$F_P = \frac{3}{4\pi2}\left(\frac{\lambda}{n}\right)3 \frac{Q}{V}\,\xi$

For a Fabry–Pérot cavity:

$Q = \frac{2L\mathcal{F}}{\lambda}$

$V \approx \frac{\pi w_0^2 L}{4}$

After cancellation of $L$:

$F_P \approx \frac{6}{\pi^3}\,\mathcal{F}\left(\frac{\lambda}{n w_0}\right)^2 \xi$

With $n = 1$, $\xi = 1$, and $\lambda = 780 \text{nm}$:

$F_P \approx 0.1935\, \mathcal{F} \left(\frac{0.78}{w_0[\mu m]}\right)^2$

Numerical values

$\mathcal{F} = 10^4$

  • $w_0 = 1 \mu m$: $F_P = 1177$
  • $w_0 = 2 \mu m$: $F_P = 294$
  • $w_0 = 3 \mu m$: $F_P = 131$
  • $w_0 = 4 \mu m$: $F_P = 73.6$
  • $w_0 = 5 \mu m$: $F_P = 47.1$

$\mathcal{F} = 10^5$

  • $w_0 = 1 \mu m$: $F_P = 11{,}773$
  • $w_0 = 2 \mu m$: $F_P = 2{,}943$
  • $w_0 = 3 \mu m$: $F_P = 1{,}308$
  • $w_0 = 4 \mu m$: $F_P = 736$
  • $w_0 = 5 \mu m$: $F_P = 471$

$\mathcal{F} = 10^6$

  • $w_0 = 1 \mu m$: $F_P = 117{,}731$
  • $w_0 = 2 \mu m$: $F_P = 29{,}433$
  • $w_0 = 3 \mu m$: $F_P = 13{,}081$
  • $w_0 = 4 \mu m$: $F_P = 7{,}358$
  • $w_0 = 5 \mu m$: $F_P = 4{,}709$

Scaling rule

$F_P \propto \left(\frac{\lambda}{n}\right)^2$

To adjust for a different wavelength or refractive index:

$F_P \rightarrow F_P \times \left(\frac{\lambda / n}{780 \text{nm}}\right)^2$