## Peter Blood

Print publication date: 2015

Print ISBN-13: 9780199644513

Published to Oxford Scholarship Online: November 2015

DOI: 10.1093/acprof:oso/9780199644513.001.0001

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# (p.361) D Multilayer Bragg reflector

Source:
Quantum Confined Laser Devices
Publisher:
Oxford University Press

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 d1 and d2 of materials of index n1 and n2 having N complete periods and a final layer of material with index n1. 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 ith interface the Fresnel equation for the amplitude reflection going from material of index n1 to index n2 is

(D.1)
$Display mathematics$

Fig. D.1 Schematic diagram of a multilayer Bragg reflector stack made up of alternating layers of materials of index n1 and n2. The amplitude reflectivity of the incident light is r.

Fig. D.2 Illustration of the processes of reflection and transmission at the first three layers of a Bragg stack when $n0n2$ with light incident from a medium of index n0. The layer thicknesses are one-quarter of the wavelength in the layer, producing a phase change on transmission of $π/2$. The overall phase change between incident and reflected light is $(2m+1)π$, where

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 n0 that is less than the index of the first layer, n1, 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: n1n2. If the thickness of layer 1 is one-quarter of the wavelength of light in medium 1, $λ/(4n1)$, the phase change on propagation from the surface to the interface is $π/2.$1 In the stack $n1>n2$, 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 $π/2$. The total phase change between incident light and light emerging from the stack is π‎.

• Interface 2–3: n2n1. 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 $π/2+π/2=π$. Since $n2, there is a phase change of π‎ on reflection at this interface and a further phase change of $π/2+π/2=π$ on transmission to (p.362) the surface of the stack. The total phase change on emerging from the stack is $3π$.

• Interface 3–4: n1n2. 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 $π/2+π/2+π/2=3π/2$ on transmission to the interface, zero on reflection at the interface, and $π/2+π/2+π/2=3π/2$ on transmission back to the surface, making a total of $3π$.

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 2π‎ 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 $n0>n1, 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 $±π/2$ (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.

Fig. D.3 Illustration of square wave index modulation of a Bragg grating and its first cosine Fourier component as used in coupled-mode theory. The phase change for light incident at the first interface from a high-index material is zero. The phase origin of the cosine is shifted $π/2$ relative to this interface.

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 $cos(2βBy)$ 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 $π/2$ relative to the plane of the first interface.

Fig. D.4 Reflections at a Bragg mirror, following Fig. D.3. The heavy solid line is light incident on the first interface with zero amplitude at the interface. The phase shift on reflection is zero and the reflected wave is shown by the lighter line. The open circles show the incident wave with its phase referenced to the plane $y=0$ as in coupled-mode theory. The phase change on reflection is $−π/2$, so the reflected wave, also shown by open circles, is the same as that reflected at the interface. With thanks to Coldren and Corzine 1995 for inspiration.

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 $−π/2$; 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 2π‎.

(2) Set $Δβ≪κc$; then $S=κc$, so when $κcLg≫1$ to achieve high reflectivity, $tanϕg→−∞$ and $ϕg=−π/2$.