# (p.361) D Multilayer Bragg reflector

# (p.361) D Multilayer Bragg reflector

This appendix provides background material on multilayer Bragg reflectors used for grating feedback devices (Chapter 14).

# D.1 Reflection by multiple layers

Figure D.1 is a schematic diagram of a Bragg stack of alternating layers of thickness *d*_{1} and *d*_{2} of materials of index *n*_{1} and *n*_{2} having *N* complete periods and a final layer of material with index *n*_{1}. The reflectivity of the whole stack is the summation of the reflection of light transmitted through each layer and interface in the stack. At a single *i*th interface the Fresnel equation for the amplitude reflection going from material of index *n*_{1} to index *n*_{2} is

Reflection and transmission at the first three interfaces are shown in greater detail in Fig. D.2. Light at normal incidence passing from material of high index to low index is reflected without phase change; for light incident on low-index material there is a phase change of *π* on reflection and *r* is negative. In the case illustrated we take the light to be incident on a medium of index *n*_{0} that is less than the index of the first layer, *n*_{1}, so reflection at the interface 0–1 produces a phase change of *π*.

The behaviour at each interface is considered in turn using the labelling on the figure:

•

**Interface 1–2**:*n*_{1}–*n*_{2}. If the thickness of layer 1 is one-quarter of the wavelength of light in medium 1, $\mathrm{\lambda}/(4{n}_{1})$, the phase change on propagation from the surface to the interface is $\mathrm{\pi}/2.$^{1}In the stack ${n}_{1}>{n}_{2}$, so there is no phase change on reflection at interface 1–2. There is a phase change on transmission back through layer 1 to the surface of $\mathrm{\pi}/2$. The total phase change between incident light and light emerging from the stack is*π*.-
•

**Interface 2–3**:*n*_{2}–*n*_{1}. If the thickness of layer 2 is also one-quarter of the wavelength of light in medium 2, the phase change on propagation from the surface to interface 2–3 is $\mathrm{\pi}/2+\mathrm{\pi}/2=\mathrm{\pi}$. Since ${n}_{2}<{n}_{1}$, there is a phase change of*π* on reflection at this interface and a further phase change of $\mathrm{\pi}/2+\mathrm{\pi}/2=\mathrm{\pi}$ on transmission to (p.362) the surface of the stack. The total phase change on emerging from the stack is $3\mathrm{\pi}$. -
•

**Interface 3–4**:*n*_{1}–*n*_{2}. Although the details are not shown on the figure, it can be seen that the phase change of light reflected from the interface between layers 3 and 4 is $\mathrm{\pi}/2+\mathrm{\pi}/2+\mathrm{\pi}/2=3\mathrm{\pi}/2$ on transmission to the interface, zero on reflection at the interface, and $\mathrm{\pi}/2+\mathrm{\pi}/2+\mathrm{\pi}/2=3\mathrm{\pi}/2$ on transmission back to the surface, making a total of $3\mathrm{\pi}$.

Therefore, if the layers all have a quarter-wavelengthoptical thickness, light emerging from the stack after reflection at each interface has phase shifts of multiples of 2*π* and is therefore in phase, and in this example the light has experienced a phase change of *π* relative to the incident light.

If the indices are such that ${n}_{0}>{n}_{1}<{n}_{2}$, the phase change on reflection by the stack is zero.

# D.2 Phase change at multilayer reflector

The multilayer Bragg stack in a VCSEL introduces a phase change of zero or *π* at the Bragg wavelength, depending on the ordering of the relative indices (Section D.1). This is derived from *transmission matrix theory*. In the account of DBR lasers using *coupled-mode theory* the phase change at the mirror at the Bragg wavelength is $\pm \mathrm{\pi}/2$ (eqn 14.12).^{2} This difference in the phase shift in the two theories arises from differences in the reference plane for zero phase, as illustrated in Fig. D.3.

In the transmission matrix approach the phase change is defined with reference to the phase of light incident at the first interface, as shown (p.363) in Fig. D.2 and the index modulation is in the form of a square wave. In coupled-mode theory it is represented by its Fourier components, of which we have considered the fundamental $cos(2{\mathrm{\beta}}_{\text{B}}y)$ in eqn 14.7. This has its origin where the cosine is 1, namely at $y=0$: in coupled-mode theory this is the plane for which the phase is zero and to which the phase shift of eqn 14.12 is referred. This origin has a phase difference of $\mathrm{\pi}/2$ relative to the plane of the first interface.

The effect is shown in Fig. D.4. An incident wave with its phase referenced to the first interface (heavy line) is reflected with zero phase change according to transmission matrix theory; this corresponds to the Bragg stack in Fig. D.2 with a hi–lo interface. The same incident wave, but with its phase referenced to the plane $y=0$ as in coupled-mode theory (see Fig. D.3), is shown by the open circles. This is reflected with a phase change of $-\mathrm{\pi}/2$; consequently the reflected wave, also shown by open circles, is the same as that reflected at the interface with no phase change. Transmission matrix theory and coupled-mode theory produce the same reflected wave.

## Notes:

(^{1})
The phase change over a distance of one wavelength is 2*π*.

(^{2})
Set $\mathrm{\Delta}\mathrm{\beta}\ll {\mathrm{\kappa}}_{\text{c}}$; then $S={\mathrm{\kappa}}_{\text{c}}$, so when ${\mathrm{\kappa}}_{\text{c}}{L}_{\text{g}}\gg 1$ to achieve high reflectivity, $tan{\mathrm{\varphi}}_{\text{g}}\to -\mathrm{\infty}$ and ${\mathrm{\varphi}}_{\text{g}}=-\mathrm{\pi}/2$.