Stop band

The stop band of a periodic optical structure is the range of wavelengths over which light cannot propagate through it and is reflected instead. Every structure whose refractive index repeats with a period close to half a wavelength, or a multiple of it, has one, centered on the wavelength that satisfies the Bragg condition. Its width scales with the index contrast per period: a thin-film mirror of TiO₂ and SiO₂ designed for 550 nm reflects from 480 to 644 nm, an AlGaAs mirror for 850 nm over about 83 nm, and a fiber Bragg grating with an index modulation of $10^{-4}$ over about 0.1 nm. In one dimension the stop band is the same thing as a photonic band gap; inside it the field decays evanescently into the structure, so a long enough stack reflects nearly all the light.

Quarter-wave stacks

For a distributed Bragg reflector made of quarter-wave layers of index $n_H$ and $n_L$ at normal incidence, the relative width in frequency is

$$\frac{\Delta f}{f_0} = \frac{4}{\pi}\arcsin\!\left(\frac{n_H - n_L}{n_H + n_L}\right).$$

The band edges lie at $\lambda_0/(1 \pm x)$ with $x$ half of that ratio, so the band is symmetric in frequency and slightly lopsided toward long wavelengths. Worked values:

The GaAs/AlAs indices are representative values near 1 µm. For a small index step the formula reduces to $\Delta f/f_0 \approx 2\Delta n/(\pi \bar n)$, so the width is proportional to the contrast. Adding layer pairs raises the reflectance inside the band and steepens its edges but leaves the width unchanged.

Gratings and the coupling coefficient

In a waveguide or fiber grating the index step per period is small, and the stop band is described through the coupling coefficient $\kappa$ of coupled-mode theory. An infinitely long uniform grating reflects completely for detunings $|\delta| < \kappa$ from the Bragg condition, which in wavelength is a band of width

$$\Delta\lambda_\text{stop} \approx \frac{\lambda_B^2\,\kappa}{\pi n_g},$$

with $n_g$ the group index. A sinusoidal modulation of amplitude $\Delta n$ filling the mode has $\kappa = \pi\Delta n/\lambda$, which makes the relative width $\Delta n/n_g$. For a fiber Bragg grating at 1550 nm with $\Delta n = 10^{-4}$, $\kappa$ = 2.03 cm⁻¹ and the band is 0.105 nm wide for $n_g$ = 1.468. A grating of finite length $L$ has a reflection band that is wider than this, measured between the first zeros as

$$\Delta\lambda_0 = \frac{\lambda_B^2}{\pi n_g L}\sqrt{(\kappa L)^2 + \pi^2}.$$

For the same fiber grating made 14.8 mm long ($\kappa L$ = 3, 99% peak reflectance) this gives 0.15 nm.

Stop band of a DFB laser

A DFB laser with a uniform grating and antireflection-coated facets has no mode at the Bragg wavelength itself. Its two lowest-threshold modes sit just outside the stop band, one at each edge, with equal threshold gain, which is why many devices add a quarter-wave phase shift that places a single mode in the band center. For the grating in that entry ($\lambda_B$ = 1550 nm, $L$ = 300 µm, $\kappa L$ = 2, so $\kappa$ = 66.7 cm⁻¹, with the group index taken equal to $n_\text{eff}$ = 3.2 as in that entry's figure), the infinite-grating width is 1.6 nm and the first-zero width about 3.0 nm. With the group index of 3.6 used elsewhere on this site for InP waveguides they become 1.4 and 2.6 nm. Below threshold the stop band shows up in the amplified spontaneous emission spectrum as a dip between the two edge modes, and its measured width is a standard way to extract $\kappa$.

Measurement

A stop band is measured as a dip in transmission or a peak in reflection, with a spectrophotometer for thin-film mirrors and with a broadband source or swept laser and an optical spectrum analyzer for gratings, using a circulator to collect the reflected light. Quoted widths differ by definition: the band-edge width, the width between first reflection zeros and the full width at half maximum are all in use, and for the DFB example above the first two differ by almost a factor of two.

Angle, polarization and higher dimensions

At oblique incidence the stop band of a thin-film stack moves to shorter wavelengths and splits between s and p polarization, the behavior that dichroic mirrors and edge filters are designed around. In two- and three-dimensional photonic crystals, each direction has its own stop band, and a complete band gap exists only where those for all directions and polarizations overlap. A stop band is also the rejection band of grating-based notch filters.

Common questions

Is a stop band the same as a photonic band gap?

In one dimension, yes: both name the frequency range in which the periodic structure supports no propagating wave along the stacking axis. In two and three dimensions "stop band" usually refers to one direction, and "complete band gap" to a range that is forbidden in every direction.

Does adding layers widen the stop band of a Bragg mirror?

No. The width is fixed by the index contrast. More layers raise the peak reflectance and sharpen the edges; a wider band needs a higher contrast, a chirped stack, or two stacks designed for neighboring wavelengths.

Why is a fiber Bragg grating so much narrower than a dielectric mirror?

Its index modulation is $10^{-5}$ to $10^{-3}$, against 0.85 between TiO₂ and SiO₂ layers, and the width scales with that step.

References: H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010); A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984); R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Academic Press, 2010); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012); J. D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).