Defect Modes and Optical Cavities

Place a spacer between two Bragg mirrors and inspect a localized transmission resonance. Remove the spacer to see what the mirrors do alone—they still meet in a half-wave defect. You can also change the termination or refine the spectrum around a narrow peak. This coherent, isotropic calculation uses planar electromagnetic boundary conditions.

A resonance between two Bragg mirrors

A Bragg mirror reflects strongly across a range of wavelengths, its stop band. More pairs make a finite mirror reflect more strongly there; the band gap of an infinite periodic stack depends only on the layers. Put two of them face to face with a gap in between, and light at one wavelength inside that band can bounce between them, build up and leak out through the far side. The spectrum then shows a narrow transmission peak in a band that otherwise reflects. The default example, Symmetric Bragg cavity, has five quarter-wave pairs ( $n=2.3$ and $1.5$ , designed for 550 nm) on each side of a 275 nm air spacer, with air outside.

The round-trip condition

A wave in the spacer gains phase crossing it, and again when each mirror reflects it. It reinforces itself when a full round trip brings it back in phase:

2\,q\,\frac{2\pi}{\lambda}\,d+\varphi_1+\varphi_2=2\pi m

Here $m$ is a positive integer, $\lambda$ is the vacuum wavelength, $d$ the spacer thickness, $q$ its normal wave number ( $q=n\cos\theta$ in the spacer, 0.940 here at 20°; see the calculation section), and $\varphi_1,\varphi_2$ the reflection phases of the two mirrors seen from inside the spacer. At normal incidence both mirror phases are zero for this termination, and 275 nm is half of 550 nm, so $m=1$ lands exactly on the design wavelength. The solver agrees: transmission there is 100%.

Clearing Include cavity spacer does not quite leave plain mirrors. The two low-index layers that then meet in the middle form a 183 nm half-wave layer, which is itself a defect. It resonates at 550 nm at normal incidence and at 538.8 nm at 20°.

Linewidth, Q and stored field

On every round trip some light escapes through the mirrors. Better mirrors hold it longer and narrow the peak. The quality factor measures this:

Q=\frac{\lambda_{\text{res}}}{\Delta\lambda_{\text{FWHM}}}\approx\frac{\omega_0}{\Delta\omega}

where $\Delta\lambda_{\text{FWHM}}$ is the full width at half maximum; the two forms agree for a narrow peak. In this family, with five or more pairs, $Q$ grows roughly in proportion to $1/(1-R)$ , where $R$ is the reflectance of one mirror seen from the spacer. The Fabry–Pérot slab is the same kind of cavity with weak mirrors: a 183 nm glass half-wave layer whose surfaces reflect about 4–5%. It resonates too, but transmission never falls below about 85% (83% for TE at 20°), so there is no narrow peak and no meaningful $Q$ ; across this 50 nm scan it stays above 99%.

Light held longer also builds a stronger field. The Depth profile tab plots $|E|^2/|E_{\text{ref}}|^2$ , relative to the incident wave, at the operating wavelength. At resonance the peak is close to $(1+\sqrt R)^2/(1-R)\approx 4/(1-R)$ —83 for the default mirrors at 20°. Because $Q(1-R)$ is about 16 in this family, that is roughly $Q/4$ . At normal incidence these mirrors reflect with zero phase (0.105 rad at 20°), so the TE field has antinodes at the mirror faces and, in a half-wave spacer, a node at the centre.

Unequal mirrors

If one mirror is weaker, light leaks out faster on that side. For lossless mirrors with reflectances $R_1$ and $R_2$ seen from the spacer, the peak transmission is

T_{\max}=\frac{(1-R_1)(1-R_2)}{\left(1-\sqrt{R_1R_2}\right)^2}

It reaches 1 only when $R_1=R_2$ ; the derivation below shows where it comes from. Temporal coupled-mode theory tells the same story with lifetimes: for decay rates $1/\tau_1$ and $1/\tau_2$ through the two sides, the peak is $4\tau_1^{-1}\tau_2^{-1}/(\tau_1^{-1}+\tau_2^{-1})^2$ . Light reflected directly from the front mirror and light leaking back out of the cavity cancel exactly only when the two rates are equal.

Derivation from Maxwell's equations

Start from the macroscopic Maxwell equations for a linear, isotropic, nonmagnetic medium with no free charge. Fields vary as $\exp(-i\omega t)$ , units are SI, $\varepsilon(\omega)$ is the relative permittivity, $k_0=\omega/c$ is the vacuum wave number and $Y_0=\sqrt{\varepsilon_0/\mu_0}$ is the vacuum admittance.

\begin{aligned}\nabla\times\mathbf E&=i\omega\mu_0\mathbf H, & \nabla\cdot(\varepsilon\mathbf E)&=0,\\ \nabla\times\mathbf H&=-i\omega\varepsilon_0\varepsilon\mathbf E, & \nabla\cdot\mathbf H&=0.\end{aligned}

In a uniform region $\varepsilon$ is constant, so $\nabla\cdot\mathbf E=0$ . Taking the curl of Faraday's law, substituting the Ampère–Maxwell law and using $\nabla\times\nabla\times\mathbf E=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$ gives the Helmholtz equation, with $n^2=\varepsilon$ :

\nabla^2\mathbf E+\varepsilon k_0^2\mathbf E=0

Light at an Interface derives the normal wave number $q$ , the admittance $Y$ and the Fresnel amplitude $r$ ; the thin-film page sums the multiple reflections in one film, and the Bragg mirror page computes a mirror's reflection. Here each mirror is just a reflection and a transmission amplitude.

Field in the spacer. Put the spacer at $0<z<d$ . Its tangential electric field is a forward and a backward wave, $E_\parallel(z)=A\,e^{iqk_0z}+B\,e^{-iqk_0z}$ . Take reference planes at the two spacer faces. Seen from inside, the left mirror reflects with $r_1=\sqrt{R_1}\,e^{i\varphi_1}$ and the right one with $r_2=\sqrt{R_2}\,e^{i\varphi_2}$ ; $r_1$ turns the backward wave arriving at $z=0$ into a forward wave, $r_2$ turns the forward wave arriving at $z=d$ into a backward wave, $t_1$ carries the incident tangential amplitude $E_{\text{inc}}$ at the front of the left mirror into the spacer, and $t_2$ carries the forward wave out.

Round trip. The forward wave reaches the right face as $Ae^{i\delta}$ , with one-pass phase $\delta=qk_0d$ . The right mirror sends back $r_2Ae^{i\delta}$ , which arrives at the left face as $r_2Ae^{2i\delta}$ . The left mirror reflects that and adds what it transmits from the incident wave. The boundary conditions are already inside $r$ and $t$ , so the forward amplitude must reproduce itself:

\begin{gathered}A=t_1E_{\text{inc}}+r_1r_2e^{2i\delta}A\\ \Downarrow\\ A=\frac{t_1E_{\text{inc}}}{1-r_1r_2e^{2i\delta}}\end{gathered}

The denominator is smallest when $r_1r_2e^{2i\delta}$ is real and positive, that is when the round-trip phase $\Phi=2qk_0d+\varphi_1+\varphi_2$ equals $2\pi m$ . With $k_0=2\pi/\lambda$ that is the resonance condition above.

Transmission. The right mirror passes $t_2$ of the wave arriving at its face, so $t=t_1t_2e^{i\delta}/(1-r_1r_2e^{2i\delta})$ . With $E_{\text{inc}}=1$ , the outside medium the same as the spacer (air here, so the admittances cancel in the power ratio) and lossless mirrors, which reflect equally from either side, energy conservation gives $|t_i|^2=1-R_i$ , and

T=\frac{(1-R_1)(1-R_2)}{\left|1-\sqrt{R_1R_2}\,e^{i\Phi}\right|^2}

At resonance this reduces to the peak transmission given under Unequal mirrors above; for equal mirrors $T_{\max}=1$ . Write $\rho=\sqrt{R_1R_2}$ . Half maximum is where the denominator doubles its minimum, $(1-\rho)^2$ , so $\sin(\Delta\Phi_{\text{FWHM}}/4)=(1-\rho)/(2\sqrt\rho)$ . For good mirrors the sine is small, and the slope of $\Phi$ turns the phase width into a wavelength width:

\begin{gathered}\left|1-\rho\,e^{i\Phi}\right|^2=(1-\rho)^2+4\rho\sin^2\frac{\Phi}{2},\\ \Delta\Phi_{\text{FWHM}}\approx\frac{2(1-\rho)}{\sqrt\rho},\\ \Delta\lambda_{\text{FWHM}}\approx\frac{\Delta\Phi_{\text{FWHM}}}{|d\Phi/d\lambda|}\end{gathered}

If only the spacer contributed, $|d\Phi/d\lambda|=4\pi qd/\lambda^2$ , and the default cavity at 20° would have $Q\approx 63$ —more than five times too low. The mirror phases change with wavelength about 4.5 times faster than the spacer phase, because much of the stored field sits inside the mirror layers. With $d\Phi/d\lambda$ taken from the solver's own mirror phases, the estimate matches the measured widths within 0.5% from three to ten pairs and for the asymmetric cavity. That is why the widths below are measured on the spectrum rather than quoted from a closed formula.

Stored field. Take equal lossless mirrors, TE, and the same medium outside and in the spacer. (In TM the plotted $|E|^2$ also contains $E_z$ , so the peak differs away from normal incidence: 53.9 at 20° against 61.2 from the formula below.) At resonance $T=1$ , so the net flux in the spacer equals the incident flux: $|A|^2-|B|^2=|E_{\text{inc}}|^2$ . The backward wave is the reflected forward wave, $|B|=\sqrt R\,|A|$ , so $|A|^2=|E_{\text{inc}}|^2/(1-R)$ . Where the two waves add in phase,

\frac{|E|^2_{\max}}{|E_{\text{inc}}|^2}=\frac{\left(1+\sqrt R\right)^2}{1-R}\approx\frac{4}{1-R}

With zero mirror phase the waves add in phase exactly at the mirror faces, and a half-wave spacer has a node in the middle. The solver gives 71.84 at normal incidence, as the formula does. At 20° each mirror reflects with phase $\varphi=0.105$ rad, so the waves would add in phase slightly beyond the spacer faces. Inside the spacer the maximum is at the faces:

\begin{gathered}|A|^2\left|1+\sqrt R\,e^{i\varphi}\right|^2\\ =\frac{1+R+2\sqrt R\cos\varphi}{1-R}\end{gathered}

That is 83.2, which the solver reproduces, against 83.5 for zero phase.

Three things to try

The editor opens at 20° with TE polarization and a wavelength of 550 nm. The spectrum is in the Parameter dependence result tab; the Refine button is in the Resonance tab. Widths below were read at half maximum from a refined scan.

Find the peak, then refine it. At 20° the resonance has moved from 550 nm to 534.6 nm, so 550 nm sits in the stop band and transmits 0.3%. Scaling by $\cos 20^\circ$ alone would predict 516.8 nm. The mirrors' reflection phase is not fixed: part of the stored field sits inside the mirror layers, where refraction keeps light closer to the normal (8.6° in $n=2.3$ , 13.2° in $n=1.5$ ), so that part of the round trip barely changes with angle. At 534.6 nm and 20° each mirror reflects with a phase of 0.105 rad, which closes the gap in the round-trip condition. The peak is 1.55 nm wide, so $Q\approx 346$ (295 at normal incidence; about 250 for TM at 20°). Pressing Refine ±2 nm around operating wavelength as the page opens scans 548–552 nm and finds nothing above 0.4%. Open Light and type 534.6 into Wavelength (nm), then press Refine and return to Parameter dependence. The depth profile then peaks at about 83 at the spacer surfaces, with a node at its centre.

Add mirror pairs. Left mirror pairs and Right mirror pairs (0–24) set the two mirrors. The resonance barely moves: 534.4 nm with three pairs per side, 534.7 nm with ten. The width does. Six pairs give 0.63 nm and $Q\approx 840$ ; seven give 0.26 nm and $Q\approx 2050$ , with a field peak of about 490 (with the wavelength on the peak, 534.67 nm). Ten pairs give 0.018 nm, $Q\approx 29\,000$ , narrower than the 0.1 nm step of the default sweep. Its highest point is $T=0.14$ , and the depth profile at 534.7 nm shows the same 14%. With the wavelength at 534.7 nm, Refine samples every 0.004 nm and reaches 0.98. The peak was there all along; the grid stepped over it.

Unequal mirrors. The Asymmetric Bragg cavity keeps five pairs on the left and three on the right. At 20° its peak is at 534.5 nm with $T=0.498$ , 5.5 nm wide, and the field in the spacer reaches only 7. Seen from the spacer, the mirrors reflect 0.953 and 0.755. For lossless mirrors the peak transmission is exactly $(1-R_1)(1-R_2)/(1-\sqrt{R_1R_2})^2=0.498$ , which the solver reproduces. The coupled-mode formula above says the same thing qualitatively: the more unequal the leakage, the lower the peak.

UNIFORM LAYERS

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Stable scattering solution · revision 0

Calculation and evidence

Time dependence is $\exp(-i\omega t)$ . Amplitudes are tangential $E_y$ (TE) or $E_x$ (TM), with

\begin{aligned}q&=\sqrt{\varepsilon-n_0^2\sin^2\theta}\\ Y&=\begin{cases}q & \text{TE}\\ \varepsilon/q & \text{TM}\end{cases}\end{aligned}

Interfaces use Fresnel amplitudes; propagation uses $\exp(iq\,2\pi d/\lambda)$ . Reflection recursion composes scattering responses using only decaying exponentials in passive finite layers. Backward fields are anchored at each layer’s exit.

The exterior media must be transparent and positive-index. Internal layers can be passive and absorbing. All layers are coherent, uniform, isotropic and nonmagnetic. An exact zero longitudinal wave number is reported as a singular amplitude basis; nearby angles remain calculable. No roughness, incoherent thick-layer approximation or beam envelope is included.

Independent fixtures use Steven Byrnes’s tmm 0.2.0 for oblique TE/TM, absorbing films, evanescent layers, multilayers and Bragg cavities (symmetric, asymmetric and lossy). Powers agree within $10^{-8}$ in the fixtures. Checks also cover boundary fields, antireflection and zero-thickness limits. The Python fixture generator is retained with the source.

History of the method

In a 1946 abstract for a meeting of the American Physical Society, E. M. Purcell pointed out that a resonator changes spontaneous emission. For nuclear magnetic transitions at radio frequencies the free-space emission rate is negligible. Coupled to a resonant circuit, a transition instead sees one oscillator within a frequency range $\nu/Q$ , and its emission probability increases by a factor $3Q\lambda^3/4\pi^2V$ , where $V$ is the resonator volume. The same idea—a cavity reshapes the modes available to light—underlies the field enhancement shown in the depth profile here.

In the further reading of their chapter 6 (printed p. 121), Joannopoulos and co-authors identify a 1987 letter by Eli Yablonovitch in Physical Review Letters as the first suggestion that a material with a complete three-dimensional photonic band gap might be fabricated, with a related suggestion shortly afterwards by Sajeev John. They add that Yablonovitch suggested such a gap would totally inhibit spontaneous emission of atoms inside the crystal within the gap. John’s title names his route: strong localization of photons in certain disordered dielectric superlattices. The letters themselves could not be read during verification, so this account relies on the book.

The one-dimensional structure on this page needs no complete gap. Joannopoulos and co-authors treat a defect layer in a multilayer as two facing mirrors that trap light at discrete frequencies inside the gap (printed pp. 58–59). The defect adds a single peak to an otherwise empty density of states, a property they note is exploited in the dielectric Fabry–Perot bandpass filter (p. 60). Their example—a doubled low-index layer—is the structure left when the spacer is removed here. The Fabry–Pérot history is on the thin-film page.

References

John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn and Robert D. Meade, Photonic Crystals: Molding the Flow of Light, second edition, Princeton University Press (2008), chapter 4, printed pages 44–60: multilayer gaps, quarter-wave design, evanescent modes and defect modes (pp. 58–60). Open PDF on the authors’ book page, posted with Princeton University Press’s permission.

Steven J. Byrnes, Multilayer optical calculations, 2016, revision 5 (2020): field conventions, transfer matrices, flux and absorption. The browser implementation uses scattering recursion for numerical stability; the Python implementation supplies independent fixtures.

The same book by Joannopoulos et al., chapter 10, printed pages 198–203: temporal coupled-mode theory of a resonant filter, the definition $Q=\omega_0\tau/2$ , the Lorentzian transmission peak with fractional half-maximum width $1/Q$ , and 100% peak transmission only for equal decay rates. The coupled-mode peak formula above follows from its equation (10); the page uses it only qualitatively, and the Airy formula for the planar cavity is derived in this page’s disclosure. Chapter 6, Further reading, printed page 121: early suggestions of complete three-dimensional band gaps by Yablonovitch and John.

E. M. Purcell, Spontaneous emission probabilities at radio frequencies, abstract B10 in Proceedings of the American Physical Society, Physical Review 69(11–12), 681 (1946). The DOI is Crossref’s record for the Proceedings section. The full abstract was read in a one-page scan hosted by chemphys.ca.

Eli Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Physical Review Letters 58(20), 2059–2062 (1987). The publisher’s page was not accessible during verification; the letter is described through Joannopoulos et al., p. 121.

Sajeev John, Strong localization of photons in certain disordered dielectric superlattices, Physical Review Letters 58(23), 2486–2489 (1987). The publisher’s page was not accessible during verification; the letter is described through Joannopoulos et al., p. 121.

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Calculation source code · Implementation and validation notes