## Abstract

We present analysis and design of microring resonators integrated with reflective elements to obtain custom wavelength-selective devices. We introduce a graphical method that transforms the complicated design problem of the integrated structure into a simple task of designing a reflective element possessing an appropriate reflection profile. Configurations for obtaining a comb mirror, a single peak mirror, an ultranarrow band transmission filter, and a sharp transition mirror are presented as examples.

©2010 Optical Society of America

## 1. Introduction

In recent years, microring resonators have gained much interest for their compact design and wide versatility. For the specific application of realizing wavelength-selective reflectors, there have been numerous different approaches [1,2,3,4,5,6,7,8,9] using multiple rings or other optical components in conjunction with the ring, resulting in intricate designs and an increase in the overall size of the mirror. Further, multiple ring configurations may be sensitive to fabrication issues, such as wafer scale variations that cause non-uniform inter-ring coupling coefficients, variations in microring resonances due to unequal circumferences, and variations in quality factor of adjacent rings.

In this paper, we study a mirror that consists of a single microring with an integrated reflective element and describe a graphical method to engineer its overall response without detailed numerical computations. In particular, we extend the analysis of the microring resonator with an integrated distributed Bragg reflector presented in [10, 11] to account for a more general reflective element. The method can be applied to the analysis of the structures presented in [5, 12, 13, 14]. Here, we examine configurations of a comb reflector, a single peak reflector, an ultranarrow transmission filter, and a sharp transition mirror. These devices have typical dimensions of a microring resonator with no additional external optical elements, allowing dense integration as planar photonic devices.

## 2. Analysis

Consider a microring with an integrated reflective element having the total circumference *L _{t}* =

*L*+

*L*, where

_{s}*L*and

*L*correspond to the length of the ring and the reflective element respectively, as depicted in Fig. 1. All fields in this paper are assumed to be normalized in power. For simplicity, the coupling and transmission coefficients

_{s}*κ, τ*of the bus coupler and the round trip field attenuation in the ring

*α*are assumed to be frequency independent in this paper. One can easily extend the following analysis to obtain a more accurate model which involves spectrally varying coefficients, e.g. to account for refractive index and waveguide dispersions. Here, we will only consider a passive case,

*i.e.*, 0 <

*α*≤ 1. The effect of incorporating gain in a similar structure is studied in [15]. Let the scattering matrix of the integrated reflective element be given by

If we assume that the reflective element is lossless and reciprocal, we can write its scattering matrix as [16]

which satisfies the lossless condition **S**
^{†}
**S** = **I** and the reciprocity condition *S*
_{12} = *S*
_{21}. We use *e ^{−iωT}* time convention throughout this paper. Here,

*r*and

*t*are the magnitude of the reflection and transmission coefficients of the integrated reflective element, respectively, such that

*r*

^{2}+

*t*

^{2}= 1, and

*ϕ, ψ*are phase terms.

Steady state solutions of the transmitted and reflected fields normalized to the input signal of the integrated microring are [10, 11]

where
$\mathbf{K}=\left(\begin{array}{cc}\tau & i\kappa \\ i\kappa & \tau \end{array}\right)$
is the coupling matrix [17], and *a*
^{+}
_{1} = 1, *b*
^{−}
_{1} = 0, and lossless coupling condition *κ*
^{2} + *τ*
^{2} = 1 are assumed. Note that

are the transmission phase shifts of the reflective element and the ring portion, respectively, and *β* is the modal propagation constant in the ring waveguide. We define

to be the total detuned round trip phase shift in the integrated microring, where the subscript 0 represents the quantity at the design wavelength throughout the paper. Unless otherwise specified, we assume *L* is chosen to satisfy *θ*
_{0} + *ϕ*
_{0} = 2*mπ* for some integer *m* (resonance at the design wavelength), and thus *a*
^{−}
_{1} (*θ* + *ϕ*) = *a*
^{−}
_{1} (Θ) and *b*
^{+}
_{1} (*θ* + *ϕ*) = *b*
^{+}
_{1} (Θ) hold.

It should be noted that although Eqs. (3) and (4) are solved assuming the reflective element is positioned at the center of the ring, |*a*
^{−}
_{1} | and |*b*
^{+}
_{1} | are independent of relative position of the integrated element in the ring, provided that the element is not directly coupled to the bus waveguide. In fact, only the phase of the reflection coefficient ∠*a*
^{−}
_{1} is affected by the relative position of the reflective element. The transmission phase ∠*b*
^{+}
_{1} is not affected.

#### 2.1. Maximum Reflection Condition

In this section, we first study the reflection from the integrated microring given in Eq. (4). Let us assume that the attenuation coefficient *α* of the microring and the transmission coefficient *τ* of the bus coupler are given parameters of the microring. In Eqs. (3) and (4), the variables we can engineer are *r*(Θ) and *t*(Θ). Note that we may neglect *ψ* because it does not affect the output power. For a lossless reflective element, we have
$t=\sqrt{1-{r}^{2}}$
, and therefore we only need to examine Eqs. (3) and (4) in the Θ-*r* domain. We find the condition for obtaining maximum reflection that can be realized from the integrated microring for given *α* and *τ* by solving ∇_{Θ,r}|*a*
^{−}
_{1} (Θ,*r*)|^{2} = **0**, which yields the reflective element’s critical reflection coefficient profile

for $-\frac{\pi}{2}\le \Theta +2m\pi \le \frac{\pi}{2}$ . For other values of Θ, there is no solution. The resultant maximum reflection amplitude is evaluated by substituting Eq. (8) into Eq. (4):

Note that for constant *α* and *τ*, this maximum reflectance |*a*
^{−}
_{1c}|^{2} exists and is the same for continuous points in the Θ-*r* plane. At resonance Θ = 2*mπ*, Eq. (8) reduces to

which is the minimum field reflection amplitude necessary to produce the maximum reflectance from the microring. For *α*
^{2} ≈ 1 and *κ*
^{2} ≪ 1, Eq. (10) reduces to
${r}_{c}\left(0\right)\approx \frac{{\kappa}^{2}}{2}$
, which agrees with [12].

Note that the reflectance |*a*
^{−}
_{1} |^{2} is proportional to the field buildup intensity of the — mode in the ring. A large value of |*a*
^{−}
_{1} |^{2} is evidence that the mode is in resonance in the structure. Therefore, the branching off of the peak reflection condition *r _{c}*(Θ) in Fig. 2 for

*r*(Θ) >

_{c}*r*(0) corresponds to the resonance-splitting [18, 19, 20, 21, 22], and the separation between resonant peaks can be easily obtained by calculating the corresponding Θ from Eq. (8) provided that

_{c}*α*,

*r*, and

*τ*are known.

We define the reflectance amplification factor

which represents the maximum enhancement in the reflected power due to the field buildup in the ring. The factor *A* can be quite large for low loss devices and thus can be used for high sensitivity sensor applications [12, 23].

#### 2.2. Graphical Solutions

Figure 2 shows contour plots of |*a*
^{−}
_{1} (Θ,*r*)|^{2} for different values of *τ*
^{2} and *α*
^{2} on the Θ-*r* plane. As α or τ decreases, we observe that the full width half maximum (FWHM) of each individual peak increases.

Figure 3 depicts contour lines of the minimum critical reflection *r _{c}*(0) and the resultant reflectance from the ring |

*a*

^{−}

_{1c}|

^{2}on the

*τ*

^{2}-

*α*

^{2}plane. For example, if

*τ*

^{2}= 0.9 and

*α*

^{2}= 1, we obtain

*r*(0) = 0.053 and |

_{c}*a*

^{−}

_{1c}|

^{2}= 1 from Fig. 3, which agrees with the values from Fig. 2(a). Note that

*r*(0) increases as

_{c}*α*or

*τ*decreases, and |

*a*

^{−}

_{1c}|

^{2}decreases as

*α*decreases or

*τ*increases. Thus, increasing

*α*increases the amplification factor

*A*whereas increasing

*τ*can either increase or decrease

*A*.

Finally, we note that for a given integrated reflective element, *r* and Θ are related to each other. One can construct, *r*(Θ), the reflection profile of the integrated element as a parametric function of the total detuned round trip phase from *r*(*λ*), *ϕ*(*λ*), and *θ*(*λ*). By overlaying the *r*(Θ) profile on the contour plot of |*a*
^{−}
_{1} (Θ,*r*)|^{2} on the same Θ-*r* plane, we can visualize the effect of the reflective element on the reflectance of the integrated microring resonator. In many cases, it is relatively easy to find analytical or numerical expressions for *r*(Θ) of a given element. In other words, we have a two-step process: (i) we plot |*a*
^{−}
_{1} (Θ,*r*)|^{2} on the Θ-*r* plane, and (ii) we construct *r*(Θ) and overlay it on the same plane from (i), obtaining |*a*
^{−}
_{1} (Θ)|^{2}. We may convert the result back to the *λ* domain if desired. The color code on the *r*(Θ) curve thus corresponds to the reflectance of the integrated microring reflector |*a*
^{−}
_{1}|^{2}.

## 3. Design Examples

In the following sections, we present some of the example configurations of useful wavelength-selective devices by engineering *r*(Θ) of the reflective element using the graphical solutions presented in the previous section. Throughout the analysis, we consider an integrated microring resonator with the parameters *L _{t}* =100

*λ*

_{e,0}and

*τ*

^{2}=0.9, where ${\lambda}_{e,0}=\frac{{\lambda}_{0}}{{n}_{e}}$ is the effective design wavelength and

*n*is the effective index of the waveguide.

_{e}#### 3.1. Comb of Reflection Peaks

The ideal comb reflector has high reflectance peaks spaced periodically. From the periodicity of the |*a*
^{−}
_{1}|^{2} contour plot, we can construct a comb reflector with a spectrally flat *r*(Θ). Figure 4(a) shows this simple case. The dashed line represents the desired reflection profile of the reflective element *r*(Θ) overlaid on the contour plot, and its value is constant at *r _{c}*(0) = 0.053 for

*α*

^{2}= 1. From the figure, one can easily anticipate a periodic reflection spectrum from the microring with peaks at Θ = 2

*mπ*. One of the physical candidates for the reflective element to achieve such a flat reflection profile is a low reflectivity Fabry-Pérot (FP) element. We refer to this type of integrated microring as an FP-MRR. For a low reflectivity FP element, the phase response is approximately a linear function of

*β*with the slope equal to its length

*d*; by choosing

*L*≫

*d*, we can approximate the reflection as a constant value. More detailed analysis to engineer the parameters is presented below.

Consider a simple index-contrast FP element with length ${L}_{s}=d=\frac{1}{4}{\lambda}_{e,0}$ . The reflection from the FP element is given by [24]

where *R _{int}* =

*r*

^{2}

_{int}is the reflection power from each interface. At the design wavelength, Eq. (12) can be expressed as:

For *α*
^{2} = 1, one can choose the appropriate index contrast to get *R _{int}* = 6.93×10

^{−4}such that we match the minimum critical reflection coefficient

*r*=

*r*(0).

_{c}The transmission coefficient of the FP element is given by [24]

From Eqs. (5) and (14), we observe that *ϕ* (*β*) ≈ *βd* for *R _{int}* ≪ 1, which yields
$\frac{\partial \varphi}{\partial \theta}<<1$
since

*d*≪

*L*. Therefore, we can effectively model the reflection profile of the FP element as a constant

*r*(Θ) ≈

*r*(0) near the design wavelength. Note that because the phase response in the ring is approximately linear, we choose

_{c}*L*to be an integer number of

_{t}*λ*

_{e,0}, which yields

*θ*

_{0}+

*ϕ*

_{0}≈ 200

*π*in this example.

Figure 4(b) shows the reflectance spectra of the FP-MRR for *α*
^{2} = 1, 0.99, and 0.95. For each value of *α*
^{2}, the FP reflection is set to *r* = *r _{c}*(0) = 0.053, 0.058, and 0.078 respectively. Note that the peak reflection of each FP-MRR corresponds to |

*a*

^{−}

_{1c}|

^{2}=1, 0.83, and 0.45, respectively. Thus, the ring resonator amplifies the reflectance by

*A*= 360, 250, and 74, respectively. For lossy cases, a decrease in

*τ*results in increases in the critical reflection coefficient

*r*(0) and the peak reflectance |

_{c}*a*

^{−}

_{1c}|

^{2}. However, the increase in

*r*(0) requires a higher value of

_{c}*R*from Eq. (13), which may be difficult to achieve from a simple index-contrast FP element. Furthermore, a higher value of

_{int}*R*causes deviation from the linear phase approximation of the FP element.

_{int}The FWHM values of the reflection intensity of the FP-MRRs are obtained to be (2.4 × 10^{−4})λ_{0}, (2.6 × 10^{−4})*λ*
_{0}, and (3.2 × 10^{−4})*λ*
_{0} for *α*
^{2} = 1, 0.99, and 0.95, respectively at the design wavelength. In comparison, the FWHM values of buildup field intensity of the identical microrings with no internal reflection, *i.e., r*(Θ)=0, are calculated to be (1.7 × 10^{−4})*λ*
_{0}, (1.8 × 10^{−4})*λ*
_{0}, and (2.5 × 10^{−4})*λ*
_{0} for the same set of *α*
^{2} values.

The reflectance spectra of an FP-MRR resembles that of a sampled grating distributed Bragg reflector (SGDBR) [25], but the FP-MRR does not exhibit side lobes [10, 11]. Furthermore, the FWHM is essentially determined by the microring parameters *α*
^{2} and *τ*
^{2}, allowing an additional degree of freedom in engineering the peak shape. Additional advantages over the SGDBR include its more compact dimensions, simpler architecture, and reduced sensitivity to wafer scale variations due to its reuse of the same FP reflector.

#### 3.2. Single Peak Reflector

In order to design a single peak reflector at the design wavelength, one needs to suppress the reflection *r*(Θ) of the reflective element at the ring resonant wavelengths other than the design wavelength. In other words, we are looking for a reflective element whose reflection profile satisfies

One element that exhibits this response is a distributed Bragg reflector. We refer to the microring integrated with the grating as a DBR-MRR.

Consider an index-contrast *N*-period Bragg grating as a reflective element. We first note that the total length of the grating is
${L}_{s}={L}_{g}=N\Lambda \approx \frac{N{\lambda}_{e,0}}{2}$
, where Λ is the period of the grating. In the low reflectivity limit
$N\sqrt{{R}_{\mathrm{int}}}<0.2$
, we may neglect multiple reflections in the grating and obtain the reflection coefficient [24]

where Θ_{g} = (*β* – *β*
_{0})*L _{g}* is the detuned phase shift in the grating. We also note that the transmission phase shift of the low-reflectivity grating is roughly a linear function

*ϕ*(

*β*) ≈

*βL*, from which we obtain Θ ≈ (

_{g}*β*–

*β*

_{0})

*L*.

_{t}To satisfy Eq. (15), we can control the grating contrast and the fraction of the ring occupied by the grating. We define

to be the ratio between the detuned phases of the grating to the entire microring. We find

Note that *p* is fixed for a given geometry of the structure. We re-write Eq. (16) as

and imposing the first condition in Eq. (15) on Eq. (19) yields sin (*p*Θ) = 0 for all Θ = 2*mπ*, *m* ≠ 0, so 2*mpπ* should be an integer multiple of *π*. Therefore, from Eq. (18) the only possible choices for *p* are
$p=\frac{1}{2}$
or *p* = 1. The corresponding grating lengths are

Physically, the first case is where the grating occupies half of the ring (Fig. 5(a)), and the second case is where the grating occupies the entire ring (Fig. 5(b)).

To meet the second condition in Eq. (15), we choose the appropriate grating contrast to set *R _{int}* and satisfy
$2N\sqrt{{R}_{\mathrm{int}}}={r}_{c}\left(0\right)$
. The reflection profile for each case is overlaid on the contour plot in Fig. 6(a). The full grating has extra nulls,

*i.e., r*= 0 at Θ =

*mπ*for all nonzero integers

*m*. Figure 6(b) depicts the resultant reflectance spectra of the two DBR-MRRs for

*α*

^{2}= 1. It should be noted, however, that for the full DBR-MRR, the grating at the coupling region may cause some reflection and scattering, which is not considered in this paper.

There is competition between the field buildup due to resonance and decaying reflection from the grating near Θ = 2*mπ*, *m* ≠ 0. However, the overall microring reflection will approach zero if the grating reflection goes to absolute zero, thus suppressing reflection at adjacent ring resonances. As a result, we observe only a single peak at the design wavelength as desired.

The FWHM of the reflection intensity is (2.4 × 10^{−4})*λ*
_{0} for both the half DBR-MRR and the full DBR-MRR at the design wavelength for the lossless case. Compared to a continuous DBR, the DBR-MRR possesses several advantages such as a more compact structure, suppression of side mode ripples, the ability to engineer the FWHM from the microring parameters, and reduced sensitivity to wafer scale variations. Although there have been several studies on a cavity consisting of a ring resonator and Bragg grating [8, 14, 26], the DBR-MRR is unique in that the grating is physically inside the ring and has carefully designed parameters to suppress the reflection peaks at adjacent ring resonances.

#### 3.3. Ultranarrow Transmission Filter

If the interface reflectivities of the FP element are increased, it is possible to construct an ultranarrow bandpass filter by employing a sharp phase change of the FP element near its resonance condition. Physically, it is difficult to form a highly reflective FP element with a single material interface; instead, one can employ DBRs to form a DBR etalon. We refer to a microring resonator integrated with a DBR etalon as a DBR-E-MRR. In this regime, the phase response of the reflective element is no longer linear; that is, the approximation *ϕ* (*β*) ≈ *βL _{s}* no longer holds.

Consider an integrated microring structure which consists of two identical *N*-period DBRs on the left and the right half of the ring and gaps of length *d* and *L*. We assume that *d* is an integer multiple of the grating period Λ. To make the analysis simpler, we re-define the DBR structure to be symmetric; that is, we exclude the last half-period portion at the bottom, as shown in Fig. 7. The length of the new DBR is then
${\overline{L}}_{g}=(N-\frac{1}{2})\Lambda $
. By doing so, we impose an additional condition *S*
_{11} = *S*
_{22} on Eq. (2) and obtain the grating’s scattering matrix

where *r _{g}* and

*t*are the magnitude of reflection and transmission coefficients of the grating, and

_{g}*ϕ*corresponds to the phase of the reflection coefficient. The + or − sign is taken when the last index of the DBR (orange) is high or low, respectively. Note that with the symmetric DBR definition, the length of the ring portion

_{g}*L̄*=

*L*+ Λ as shown in Fig. 7. We define

*θ*=

*βL̄*.

We find the transmission coefficient of the upper DBR-etalon structure of length *d* + 2*L̄ _{g}* as [24]

where
$\Gamma =\frac{1+{r}_{g}^{2}}{1-{r}_{g}^{2}}$
and *r*
^{2}
* _{g}*+

*t*

^{2}

*= 1. From Eq. (23), we obtain $\beta d+{\varphi}_{g}=\mathrm{arctan}\left[\frac{1}{\Gamma}\mathrm{tan}(\varphi -{\varphi}_{g})\right]$ where*

_{g}*ϕ*= ∠

*S*

_{12}. The reflection power from the DBR-etalon is given by [24]

where *R _{g}* =

*r*

^{2}

_{g}. Using the trigonometric identity $\mathrm{sin}\left(\mathrm{arctan}x\right)=\frac{x}{\sqrt{1+{x}^{2}}}$ , Eq. (24) simplifies to

which is the reflection profile of the integrated DBR-etalon structure.

It is difficult to find the reflection profile as a function of the total detuned phase shift in the integrated ring Θ = (*θ* – *θ*
_{0})+(*ϕ* – *ϕ*
_{0}) because the grating’s phase response *ϕ _{g}* is not linear. One may employ a numerical simulation such as the transfer matrix method (TMM) to obtain the reflection profile

*r*(Θ) of the DBR-etalon as a function of the total detuned phase. In the appendix, we derive a closed form approximation of

*r*(Θ) for small Θ using the effective mirror model, which is insightful for designing the performance parameters, such as the FWHM.

Figure 8 shows the response of a DBR-E-MRR under lossless conditions with *r*
^{2}
_{g} = 0.99, *d* = *L* = 0, and *N* = 100 using the linear approximation method presented in the appendix. Figure 9 shows the transmission response of the device using TMM. The FWHM of the isolated DBR etalon *δ*Θ_{DBR-etalon} ≈ 0.34*π* is reduced to *δ*Θ_{DBR-E-MRR} ≈ 0.016*π* by integrating the DBR etalon into a microring. To find the microring response as a function of *λ*, we substitute Θ(*β*) = [(*d* + 2*L _{e}*)Γ + 2

*L*+

_{e}*L*+ Λ)](

*β*–

*β*

_{0}) from the appendix, and we note $\frac{\beta}{{\beta}_{0}}=\frac{{\lambda}_{0}}{\lambda}$ assuming dispersion is negligible for simplicity. The slope $\frac{\partial \Theta}{\partial \beta}=\frac{{4L}_{e}}{1-{R}_{g}}+\Lambda $ can be large when the effective grating length

*L*is long or the reflection

_{e}*R*is large, and this yields a sharp response as a function of

_{g}*β*or

*λ*. For example,

*δ*Θ = 0.016

*π*corresponds to $\delta \lambda =\mid \frac{\partial \lambda}{\partial \beta}\frac{\partial \beta}{\partial \Theta}\delta \Theta \mid \approx \left(\frac{1-{R}_{g}}{4{\beta}_{0}{L}_{e}}\delta \Theta \right){\lambda}_{0}\approx \left(2.4\times {10}^{-6}\right){\lambda}_{0}$ in this particular case. The FWHM in wavelength decreases rapidly as

*R*→ 1. The value of

_{g}*R*= 0.99 was chosen considering typical maximum reflectance values of planar DBRs.

_{g}Note that when *d* = 0, we get *r* = 0 from Eq. (24) if *ϕ _{g}* =

*mπ*for any integer

*m*. This corresponds to Θ = (

*β*–

*β*

_{0})Λ + 2

*mπ*because

*ϕ*(

_{g}*β*

_{0}) = 0. That is, for |(

*β*–

*β*

_{0})Λ| ≪

*π, i.e.*, $\mid \frac{\beta}{{\beta}_{0}}-1\mid <<1$ , the first term in Θ becomes negligible, and therefore the DBR-etalon has a null at the adjacent ring resonances in addition to the null at

*λ*

_{0}.

#### 3.4. Sharp Transition Mirror

If we detune *L* to *L*′ = *L* + Δ*L*, we obtain *θ*′ = *θ* +(*β* – *β*
_{0})Δ*L*, which in turn shifts Θ′ = Θ + (*β* – *β*
_{0})Δ*L*; that is, increasing *L* shifts *r*(Θ) horizontally to the right and increases the slope
$\frac{\partial \Theta \prime}{\partial \beta}=\frac{\partial \Theta}{\partial \beta}+\Delta L$
by a negligible amount. Figure 10 depicts the spectral response of the microring device under the lossless condition, *r*
^{2}
_{g} = 0.5, and Δ*L* = 0.011*λ*
_{e,0}. These values were chosen so that the etalon’s reflection coefficient curve (dashed) is approximately tangent to the critical reflection curve rc. This produces a spectrally wider high reflectance band. Adjustments to the parameters can be made to reduce the ripple at the expense of a narrower high reflectance band. Note that the null of the etalon’s reflection profile corresponds to the new “design” wavelength, which is at a different location than Θ = 0. By shifting the DBR-etalon reflection profile, we have engineered a sharp transition mirror. In the lossless case, the reflectance goes from 100% to 0% in *δλ* =(7.4 × 10 ^{−5})*λ*
_{0}. A sharper transition can be made by increasing |*r _{g}*|

^{2}and decreasing Δ

*L*at the expense of a narrower high reflectance bandwidth, more ripple, and more difficult fabrication tolerances.

## 4. Conclusion

We presented a two-step graphical method to easily engineer the response of a microring resonator integrated with a reflective element and derived the critical reflection coefficient of the element to maximize the overall reflectance. The main findings of this investigation are that:

- for fixed values of
*α*and*τ*, the maximum achievable reflectance is the same along a continuous curve in the Θ-*r*plane - in the lossless case, 100% reflectance can be obtained with weak reflective elements
- the FP-MRR can generate a periodic reflectance spectrum with peaks at the resonance wavelengths of the ring
- the DBR-MRR with the grating occupying either half or all of the ring suppresses reflection at adjacent resonance wavelengths of the ring and thereby produces a single peak profile
- the DBR-E-MRR reduces the FWHM of the DBR etalon and can be designed to function as either an ultranarrow filter or a sharp transition mirror.

Integrating reflective elements into a ring resonator leads to structures that are generally more compact than their isolated equivalents, have virtually no side mode ripple, and offer control of performance parameters such as FWHM or maximum reflectance by changing *τ*. These integrated microring devices are potential candidates for a diverse assortment of compact optical components in planar photonic integrated circuits.

## Appendix

The reflection profile *r*(Θ) for the DBR etalon can be calculated numerically using TMM, but here we derive a closed form approximation so that we can quickly vary the design and understand the effect of each design parameter. In the analysis, we will restrict ourselves to the sharp phase transition region |Θ| ≪ 1. To estimate
$\frac{\partial \varphi}{\partial \beta}$
near the design wavelength, we replace the grating mirror with the effective mirror model.

Each grating’s reflection magnitude *r _{g}* and phase

*ϕ*near the design wavelength can be effectively modeled as [24]

_{g}(28)

where

is the effective length of the mirror, and

is the effective number of periods. The magnitude of the DBR reflection *r _{g}* is assumed to be constant near its design wavelength.

To find Eq. (25) as a function of Θ, we evaluate the rate of the phase changes at the design wavelength
${\left[\frac{\partial \varphi}{\partial \beta}\right]}_{{\beta}_{0}}=(d+{2L}_{e})\Gamma +{2L}_{e}$
. We find the first order approximation *ϕ* (*β*) ≈ [(*d* + 2*L _{e}*)Γ + 2

*L*](

_{e}*β*–

*β*

_{0})+

*ϕ*

_{0}. We can re-write

*θ*=(

*β*–

*β*

_{0})

*L̄*+

*β*

_{0}

*L̄*=(

*β*–

*β*

_{0})(

*L*+Λ)+

*θ*

_{0}. From these results and Eq. (27), we relate

*ϕ*–

*ϕ*to Θ as

_{g}Plugging Eq. (31) into Eq. (25) gives the effective reflection profile *r*(Θ). Note that for *R _{g}* → 1, the fractional coefficient in Eq. (31) approaches unity.

## Acknowledgments

The authors are grateful for the startup fund support from the University of Illinois.

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