Distributed Bragg reflector

In the GaN-based VCSEL structure, a micro cavity with a few λ in the optical

thickness and a pair of high reflectivity (above 99%) distributed Bragg reflectors (DBRs) are necessary for reducing the lasing threshold. Recently, several groups have reported optically pumped GaN-based VCSELs mainly using three different kinds of vertical resonant cavity structure forms: (1) monolithically grown vertical resonant cavity consisting of epitaxially grown III-nitride top and bottom DBRs (epitaxial DBR VCSEL), (2) vertical resonant cavity consisting of dielectric top and bottom DBR (dielectric DBR VCSEL). (3) vertical resonant cavity consists of an epitaxially grown III-nitride top DBR and a dielectric DBR (hybrid DBR VCSEL).

In order to obtain high reflectivity DBRs to reduce the threshold condition of the

VCSELs, dielectric DBRs was used in the GaN-based VCSEL in this study. Distributed Bragg reflector (DBR) consists of an alternating sequence of high and low refractive index layers with quarter-wavelength thickness, as Figure 2.2 [2.1]. Therefore, it’s necessary to know the theory of quarter-wave layer before discussing the DBRs. Now, consider the simple case of a transparent plate of dielectric material having a thickness d and refractive index n

f, as shown in Figure 2.3. Suppose that the film is nonabsorbing and that the amplitude-reflection coefficients at the interfaces are so low that only the first two reflected beams (both having undergone only one reflection) need be considered. The reflected rays are parallel on leaving the film and will interference at image plane. The optical path difference (P) for the first two reflected beam is given by

P=n_f[(AB)+(BC)]−n₁(AD) (2.1)

The corresponding phase difference (δ) associated with the optical path length difference is then just the product of the free-space propagation number and P, that is, K0P.

If the film is immersed in a single medium, the index of refraction can simply be written

as n1=n2=n. It is noted that no matter nf is greater or smaller than n, there will be a relative phase shift π radians.

Therefore,

The interference maximum of reflected light is established when δ=2mπ, in other words, an even multiple of π. In that case, eq. (2.8) can be rearranged; and the interference minimum of reflected light when δ = (2m±1) π, in other words, an odd multiple of π. In that case eq. (2.9) also can be rearranged.

⎪⎪

Therefore, for an normal incident light into thin film, the interference maximum of reflected light is established when d = λ₀/4n

f (at m=0). Based on the theory, a periodic structure of alternately high and low index quarter-wave layer is useful to be a good reflecting mirror. This periodic structure is also called Distributed Bragg Reflectors (DBRs). Therefore, the concept of DBR is that many small reflections at the interface between two layers can add up to a large net reflection. At the Bragg frequency the reflections from each discontinuity add up exactly in phase. Spectral-dependent of the reflectivity can be calculated by the transfer-matrix method [2.2]. Considering a layer if dielectric material b which is clad between two layers a and c. A transverse electromagnetic wave at normal incidence propagates throught the layer in z direction.

Taking the electric and magnetic (E and H) fields into consideration by Maxwell’s equation, a transmission matrix relating these fields can be written as

⎟⎟

In the equation, n^b is the refractive index of layer b and η^o the impedance of free space, j is the unit imaginary number, kb is the phase propagation constant in layer b,

λ π

b

b n

k = 2 ,

where λ is the wavelength in free space. Here, the absorption was not considered in this discussion. For a multilayer, a matrix Mi is formed for each layer i of thickness di in the stack. By considering the effect of all layers with summation length of each layers L, a matrix M relates to input and output fields ca be obtained,

⎪⎪

where the Y is wave admittance and o and s refer to the incident and substrate respectively. If we have a layer of index n l between layer o and s of lower under, then the reflection from interface has a phase of π radians relative to the incident wave, because of the positive index step. If the thickness of the layer is a quarter wavelength the two

second reflection. For a stack with many alternate 1/4 wave (or (n/2+1/4) wave, n integral) layers of low and high index, all interfacial reflections will add in phase.

For a Bragg reflector made from quarter wavelength layers of indices n1 and n2,as shown in Figure 2.4, the maximum reflectivity R at resonant wavelength, also denoted as Bragg wavelength (λ_B), of a stack with m non-absorbing pairs can be expressed by:

2

The no and ns in the equation are the refractive indices of incident medium and substrate, respectively and m is pair numbers of the DBR. Layer thicknesses L1,2 have to be chosen as L1,2=λ /(4n_B 1,2). The maximum reflectivity of a DBR therefore increases as the increasing difference in refractive indices and pair number of DBR. A broad spectral plateau of high reflectivity, denoted as a stop-band, appear around the Bragg wavelength, the width of which can be estimate as [2.3]

eff

. A wide stop-band provides a

larger tolerance between the designed λ and the main wavelength of the cooperated _B MQWs, this is another important reason for us to use dielectric DBRs as mirrors in our VCSEL structure. When two such high-reflectance DBRs are attached to a layer with an optical thickness integer times ofλ_c/2 (λ_c ≈λ), a cavity resonance is formed at λ_c,

If(R₁ ≈R₂ =R), ) 1 2

(1− ² ≤ ≤ R T

depending on φ. One characteristic parameter of the cavity quality is the cavity quality factor Q defines as:

2 length is λ/2, Q is the average number of round trips a photon travels inside the cavity before it escapes. Figure 2.5 shows an example of the reflection spectrum of a cavity.

The high-reflectivity or stop band of a DBR depends on the difference in refractive index of the two constituent materials, n △ (≣n1-n2). We can calculate by requiring the same optical path length normal to the layers for the DBR and the effective medium. The effective refractive index is then given by

1

For DBRs, the optical wave penetrates into the reflector by one or several quarter-wave pairs. Only a finite number out of the total number of quarter-wave pairs are effective in reflecting the optical wave. The effective number of pairs seen by the wave electric field is given by

For very thick DBRs (m→∞) the tanh function approaches unity and one obtains

2

Also, the penetration depth is given by

)

For a large number of pairs (m→∞), the penetration depth is given by