Design of resonant mirrors with negative group delay

Antonio Gellineau; Yu-Po Wong; Olav Solgaard

doi:10.1364/OE.22.029213

1. Introduction

Controlling the spectral phase of the optical field is crucial in a large number of applications, including phase-matching in non-linear optics, dispersion control in ultrafast laser pulse technology, and determining interference conditions in resonant cavities. The spectral phase response of mirrors is therefore an important factor in design of advanced optical systems. The phase response is relatively weak in metal mirrors so their contributions to the overall phase response of the system can in most cases be ignored, but that is not the case for Bragg reflectors and photonic crystal mirrors [1,2]. These resonant structures have significant energy storage that in some cases leads to strong phase effects. For the majority of applications, ultrafast lasers, the important consideration is the variation of the group delay, which should be minimized to allow dispersion-free, broad-band reflection [3,4]. In such applications, the design goal is to synthesize the desired second derivative of the phase with respect wave number (or wave length), which gives the group delay variation. The group delay itself is given by the first derivative or the dispersion of the phase response, and it is important for phase matching and determining the resonance condition of optical cavities.

The focus of this paper is the synthesis of mirrors with the desired phase response for such applications. Our investigations show that the group delay can be given a wide range of values, including negative, over limited ranges of bandwidths. This enables several unique optical device concepts, e.g. Fabry-Perot cavities with one of more mirrors with negative group delay enables “zero and negative optical lengths.” Such mirrors allow short cavities that combine low wavelength dependence with high mirror position sensitivity. In the other extreme, long group delays can be achieved, leading to optical cavities with optical lengths that are very much longer than their physical lengths. Such cavities allow compact sensors with high quality factors. The desired phase response of a mirror will depend on the application, but in general we would like to maximize the absolute values of the reflectivity and the useful bandwidth.

2. Effective thickness

To clarify the role of the reflected phase and simplify the discussion, we introduce the effective thickness of a mirror. It is defined in terms of the derivative of the spectral phase response $ϕ_{r}$ with respect to wavenumber ( $β = 2 π / λ$ ):

t_{e f f} = \frac{1}{2} \frac{d ϕ_{r} (β)}{d β} .

where teff is the effective thickness and

ϕ_{r}

, is the reflected phase of the light incident on the mirror. At a specific wavelength, the effective thickness defined in Eq. (1) can be set to zero by choosing the correct reference plane for reflected phase, but this can only be done over a limited range of wavelengths where

d ϕ_{r} (β) / d β = 0.

In discussions of broadband phenomena, this approach is not particularly useful, so we chose the reference plane for the phase to be the front surface of the mirror as shown in Fig. 1.

Fig. 1 Mirror schematic and its effective thickness (teff).

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A general method to control spectral phase of reflections is to use a Heritage-Weiner optical spectral synthesizer [5]. In this device, the wavelengths of the incident light are separated in a dispersive element, and each wavelength component is individually modulated in amplitude and phase. This structure is ideal for controlling the reflected spectral phase response, but it is challenging to miniaturize, and is difficult to access both reflected and transmitted light, which is required in Fabry-Perot applications. We therefore focus on solutions to controlling the spectral phase in simple mirrors, including photonic crystals, gratings, and Bragg reflectors.

In order to gain insight into the limits of phase dispersion, it is instructive to describe the mirrors in terms of their scattering matrices [6]. In the scattering matrix formalism, the optical fields are represented as normalized mode amplitudes with a magnitude equal to the square root of the power flow. If the electric field is expressed

ℰ (x, y, z, t) = {\vec{E}}_{0} U (x, y) e^{j (ω t - β z)}

then we define the input mode amplitudes of the input fields as:

s_{i} = \frac{E_{0}}{\sqrt{2 η_{i}}} e^{- j β z}

where

η_{i}

is the mode impedance, i.e. ratio of the electric to the magnetic field of the mode. The mode profile U(x,y) is normalized such that

\int^{​} {| U |}^{2} d x d y = 1

, which means that

s_{i}^{+} s_{i}^{+}^{*}

equals the power flowing in the positive z direction in the mode. The output mode amplitudes,

s_{i}^{-}

, are similarly defined. With these definitions, the following scattering matrix relationship holds for our mirrors with two input modes and two output modes:

[\begin{matrix} s_{1}^{-} \\ s_{2}^{-} \end{matrix}] = S_{m i r r o r} [\begin{matrix} s_{1}^{+} \\ s_{2}^{+} \end{matrix}] = [\begin{matrix} r_{12} & t_{21} \\ t_{12} & r_{21} \end{matrix}] [\begin{matrix} s_{1}^{+} \\ s_{2}^{+} \end{matrix}]

Here r12 is the phasor that represents the total field reflection at the interface between media 1 and the mirror, and t12 represents the transmission from media 1 to 2. The quantities r21 and t21 have similar definitions. The phases of the field reflections are referred to the front and back surfaces of the mirror respectively. The phases of the transmission elements are the total phase delays experienced by the light on transmission from the front to the back surface (which equals the phase delay on transmission from the back to the front surface). In this simple model, the mirror has only two input and output ports, thus no diffraction or separate modes are considered. Media 1 and 2 extend to infinity, providing no further reflections.

First we assume that the mirrors are lossless. Together with the reciprocity requirement, this leads to the following well-known relationships between the scattering matrix elements found in [6]:

t_{12} = t_{21}

| r_{12} | = | r_{21} |

r_{12}^{*} t_{21} + t_{12}^{*} r_{21} = 0

Our focus is on the phase response, so we rewrite Eq. (7) to give an explicit condition on the reflected and transmitted phases:

\frac{r_{21}}{r_{12}^{*}} = - \frac{t_{21}}{t_{12}^{*}} \to e^{j (ϕ_{r 21} + ϕ_{r 12})} = e^{j (π + 2 ϕ_{t 12})} \to ϕ_{r 21} + ϕ_{r 12} = π + 2 ϕ_{t 12}

To place this in terms of effective thickness, we take the differential of Eq. (8):

\frac{d ϕ_{r_{21}}}{d β} + \frac{d ϕ_{r_{12}}}{d β} = 2 \frac{d ϕ_{t}}{d β}

t_{e f f 21} + t_{e f f 12} = \frac{d ϕ_{t}}{d β}

The transmitted phase can be expressed in terms of the effective index (neff) and the distance (L) between the input and output reference planes of the mirror:

ϕ_{t} = \frac{2 π n_{e f f} L}{λ} = n_{e f f} β L

t_{e f f 21} + t_{e f f 12} = (\frac{d n_{e f f}}{d β} β + n_{e f f}) \frac{L}{2}

This simple relationship, in the form of Eq. (9), (10), or (12), follows directly from the well-known Eqs. (5)-(7), and gives important insight into the design of mirrors with negative effective thickness. Materials with negative group velocity $c / (β d n_{e f f} / d β + n_{e f f})$ will certainly enable negative effective thickness mirrors, and several low loss meta-materials with negative index and index gradients have been proposed [7,8]. Such materials still represent substantial challenges, however, so in this paper, we will focus on solutions that do not require negative group velocity. Eq. (12), that is valid for lossless mirrors, shows that in the absence of negative group velocity, mirrors with negative thickness must be asymmetric (so that not both teff21 and teff12 must be negative). In earlier work we have demonstrated that negative effective thickness is possible in lossless, highly asymmetrical mirrors [9]. In the next section we demonstrate the mechanisms that lead to negative effective thickness.

3. Effective thickness in resonant mirrors

Resonant mirrors provide a means of controlling spectral phase in compact devices without significant loss. In particular, photonic crystal (PC) mirrors enable negative teff. PC mirrors support guided resonances (GR) [10] that create indirect and wavelength dependent paths through the crystal. Due to the conclusion drawn from our scattering-matrix analysis, we are particularly interested in asymmetric PCs in our search for mirrors with negative effective thickness. PCs have been modelled by Coupled Mode Theory (CMT) [11,12]. In these works, the focus was on symmetrical PCs. Here we extend the possible solutions to include asymmetric slab modes ( $r_{12} \neq r_{21}$ ), and we create a generic model for the possible reflectance and effective thickness of a PC mirror due to the guided resonances of the PC structure. Suh et al. [12] showed that we can describe the resonances by two characteristic Eqs. (13) and (14) with restrictions for a lossless system [Eq. (15)].

\frac{d a}{d t} = (j Ω - Γ) a + D^{T} | s_{+} 〉

| s_{-} 〉 = C | s_{+} 〉 + D a

C D^{*} = - D; D^{†} D = 2 Γ

In the Eqs. above, C is the system matrix that describes the direct path for the normalized input field $| s_{+} 〉$ to enter and $| s_{-} 〉$ exit the photonic crystal. This input field can also couple to resonant modes inside the PC through the coupling matrix, D. Each mode is described by its decay (Г), its resonant frequencies (Ω) and the amplitude and phase of each resonator is represented by the complex vector a. The system matrix, C, is modelled by a stack of films that best match the absence of a resonant mode. The main difference for non-symmetrical C matrices is that the resonators do not couple in perfect phase or anti-phase through reflection and transmission ( $D_{1 k} \neq \pm D_{2 k}$ ). Instead, $D_{2 k} = A_{k} D_{1 k}$ where Ak has the relationship of Eq. (16) and $D_{m k}$ is the coupling of resonator k to medium m.

A_{k} = - \frac{(C_{11} e^{j θ_{k}} - C_{22} e^{- j θ_{k}}) + \sqrt{{(C_{11} e^{j θ_{k}} - C_{22} e^{- j θ_{k}})}^{2} + 4 C_{12} C_{21}}}{2 C_{12}} e^{j θ_{k}}

Given the decays, Гkk = $γ_{k}$ , resonant frequencies, Ωkk = $ω_{k}$ , and the resonators phase offset between output ports, θk, one can calculate the frequency dependent matrices D, $a$ and Г km, the off diagonal decay constants representing the decay of mode k to decay into mode m. We can now state the field of a single guided resonator in the PC $(a_{k})$ and the reflectivity of the field that couples to the reflected wave ( $r_{12, k}$ ) using Eqs. (13) and (14) where $ω$ is the frequency of the incident light:

j ω a_{k} = (j ω_{k} - γ_{k}) a_{k} + D^{T} | s_{+} 〉

a_{k} = \frac{D^{T} | s_{+} 〉}{j (ω - ω_{k}) + γ_{k}}

r_{12, k} = \frac{D_{1 k}^{2}}{j (ω - ω_{k}) + γ_{k}}

If we ignore the mixing of modes (Г km = 0) the total reflectivity of the PC consists of a simple sum of the field due to these resonators and the direct mode ( $C_{11}$ ):

r_{12} = C_{11} + \sum_{k} \frac{D_{1 k}^{2}}{j (ω - ω_{k}) + γ_{k}}

When resonances are close and interact strongly, we cannot use this approximation and must instead use the full matrix form:

r_{12} = C_{11} - {[D {(j (Ω - I ω) - Γ)}^{- 1} D^{T}]}_{11}

The strong phase effects of the resonant mirror comes from sharp phase transitions of the individual guided resonances $(ω = ω_{k})$ and their combinations in Eqs. (20) and (21). The Lorentzian shapes of the resonances in Eq. (20) will always lead to a positive phase slope and therefore a positive effective thickness, however the combination of a direct path and one or more Lorentzian resonances may in some cases change this [Eq. (21)]. Complicated structures can lead to strong positive or negative effective thickness due to a single resonator as the coupling $(D)$ is a function of the direct mode, but in most of the examples we investigate in this paper, negative effective thickness is the result of destructive interference between two or more resonances. This mechanism is illustrated in Fig. 2, where the part of the sum in Eq. (20) is plotted as phasors, and small changes in wavenumber are represented by rotations of the phasor. In Fig. 2, a rotation (transition from solid to dashed arrow) shows a change in phase of field—clockwise rotations represent a positive phase slope and positive effective thickness, while counterclockwise rotations represent negative effective thickness. The figure demonstrates that to achieve negative effective thickness, given that all constituent phasors have positive effective thickness, it is necessary that at least one phasor is interfering destructively with the remaining fields.

Fig. 2 Phasor diagram representing the effective thickness possible for the reflected field $r_{12}$ in Eq. (20).

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This insight has both favorable and challenging consequences for design of mirrors with negative effective thickness. On the favorable side we have shown that negative effective thickness can be created in lossless mirrors. On the down side we have the fact that at least one resonance has to operate in negative interference with the rest, which means that the reflectivity of the mirror is reduced in the wavelength range of negative thickness. This is clearly demonstrated in the following section. It should also be noted that the greater the dependence of reflected phase on wavelength, the greater we expect the dependence on incident angle. Thus resonant mirrors also display a large gradient with respect to incident angle [13].

4. Resonant mirror examples

To illustrate how the effective thicknesses of PCs [Fig. 3(a, b)] are influenced by their resonances, we simulate Si PCs using RCWA (solid lines) and find the closest match to the reflected fields obtained by a least squares curve fit using CMT [Fig. 3(a, c, d)]. For this fit, the lengths are fixed at values from the RCWA model while the indices and resonator parameters are floated. These examples demonstrate that Eq. (21) contains all the necessary physics to accurately compute the phase response of PC mirrors.

Fig. 3 a) CMT Model b) RCWA Model c) Resulting reflectance and effective thickness of both models. d) Individual Reflectance and phase of guided modes (G.R.) and the direct mode (D.M.). e) Phasors of the resonators present from CMT.

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The reflectance and phase of the photonic crystal can be described by two guided resonances (G.R. 1 $ω_{1} = 2 π c / 1356.1 n m$ , $γ_{1} = 0.27 / p s$ , $θ_{1} = {64.3}^{o}$ and G.R. 2 $ω_{2} = 2 π c / 1425.6 n m$ , $γ_{2} = 0.03 / p s$ , $θ_{2} = - {97.9}^{o}$ ) with the direct mode modelled by two films on a silicon substrate $(n_{1} = 3.8, n_{2} = 1.0, L_{1} = 323.0 n m, L_{2} = 1060.0 n m) .$ The resonator model and RCWA results agree closely, showing a minimum reflectance of 65%, a range of 5nm and a minimum teff of −7µm. This shows that the two guided resonances stated are responsible for the reflectance and effective thickness we observe in the simulation of the physical PC.

Examining the resulting structure with CMT verifies the insight, given by Eq. (20), into how the GRs enable large effective thickness with high reflectivity. Each resonator has phase that increases with wavenumber, showing that if either were dominant only positive effective thickness would be possible. Observing the phasors involved and their sum [Fig. 3(d)], shows that the two resonators coupling to the output mode cannot be in phase and enable negative effective thickness as all modes are rotating CCW with increasing wavenumber so the vector sum can only be CW when the narrower mode (G.R. 1) is 180o out of phase with the sum of the remaining modes [Fig. 3(e)]. As a result, the lowest resulting reflectivity and teff occur where G.R. 1 is out of phase and the range of useful effective thickness is closely related to G.R. 1. We note that under these circumstances the direct mode is responsible for most of the reflectance.

By including more guided resonances in the desired wavenumber range we could broaden this effect. We do this with two layers in a simple PC made of circular holes removed from Si in a rectangular pattern. The required asymmetry is generated by using different layers [Figs. 4(a) and 4(b)]. The resulting spectrum is broader (10nm) for the same reflectivity as the single layer case [Fig. 4(c)] and matches a CMT resonator model (G.R. 1 $ω_{1} = 2 π c / 1440.1 n m$ , $γ_{1} = 0.10 / p s$ , $θ_{1} = {139.4}^{o}$ , G.R. 2 $ω_{2} = 2 π c / 1450.9 n m$ , $γ_{2} = 0.30 / p s$ , $θ_{2} = {81.8}^{o}$ , G.R. 3 $ω_{3} = 2 π c / 1467.5 n m$ , $γ_{3} = 0.19 / p s$ , $θ_{2} = {101.3}^{o}$ , $n_{1} = 3.6, n_{2} = 2.1, L_{1} = 378.0 n m, L_{2} = 198.0 n m, L_{3} = 448.0 n m) .$ In this example, most of the reflectance comes from the direct mode while there are three resonators responsible for a broader negative teff range [Fig. 4(d)]. More layers allow for more overlap of resonances and greater wavelength dependent coupling. As a result, the fields of the resonators that exit the PC are not simple Lorentzian shapes and can also exhibit negative phase gradients. However, the strongest negative effective thickness similarly occurs when all individual phasors have positive slope. As a result, loss still accompanies the strongest effective thickness in the figure which again occurs when the fast varying GRs are opposed (180o difference) to the remaining fields [Fig. 4(d)].

Fig. 4 a) CMT Model b) RCWA Model c) Resulting reflectance and effective thickness of both models. d) Individual Reflectance and phase of guided resonances and the direct mode.

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It is also possible to use more layers to increase reflectance instead of broadening bandwidth. This is shown in Fig. 5 where the gap between the two layers are further apart (G.R. 1 $ω_{1} = 2 π c / 1500.0 n m$ , $γ_{1} = 0.06 / p s$ , $θ_{1} = {176.2}^{o}$ , G.R. 2 $ω_{2} = 2 π c / 1449.0 n m$ , $γ_{2} = 0.43 / p s$ , $θ_{2} = - {130.7}^{o}$ , $n_{1} = 1.2, n_{2} = 3.2,$ $L_{1} = 280.0 n m, L_{2} = 400.0 n m, L_{3} = 323.0 n m) .$ Similar to the single layer case, the width of G.R. 2 sets the range of high effective thickness and the lowest reflectivity and teff occurs when G.R. 2 is out of phase with the remaining fields. In this case there are only two resonances but GR. 2 is much smaller in magnitude which allows higher reflectivity for less effective thickness. With additional layers however, G.R. 1 shows slight negative effective thickness and an overall large negative effective thickness can be achieved.

Fig. 5 a) Resulting reflectance and effective thickness of both models. b) Individual Reflectance and phase of guided modes (G.R.) and the direct mode (D.M.).

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This CMT analysis is easily extended to other PC structures. High contrast gratings (HCG) have been shown to enable high reflectivity for polarizations along the direction of periodicity [14]. As an example, we consider a double layer HCG [Figs. 6(a) and 6(b)]. Though the physical structure is very different, its behavior is well described by CMT [Eq. (19)] and the same tradeoffs for reflectivity, effective thickness and range are observed [Figs. 6(c) and 6(d)] (G.R. 1 $ω_{1} = 2 π c / 1300.0 n m$ , $γ_{1} = 0.02 / p s$ , $θ_{1} = {118.9}^{o}$ , G.R. 2 $ω_{2} = 2 π c / 1428.3 n m$ , $γ_{2} = 0.07 / p s$ , $θ_{2} = {153.0}^{o}$ , $n_{1} = 2.3, n_{2} = 3.8,$ $L_{1} = 100.0 n m, L_{2} = 533.0 n m, L_{3} = 469.0 n m) .$

Fig. 6 a) CMT Model b) RCWA Model c) Resulting reflectance and effective thickness of both models. d) Individual Reflectance and phase of guided resonances and the direct mode.

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5. Effective thickness in lossy materials

In the preceding sections we have focused on lossless mirrors that fulfill the standard Eqs. (10)-(12). For comparison we now calculate the phase response of simple metal (lossy) mirrors. To obtain high phase gradients we can use natural plasmonic or engineered Mie resonances [8,15] with or without loss. We consider the effective thickness of a bulk material in order to separate the effects of resonant structure from the material’s resonances. The effective thickness at normal incidence using Fresnel Eqs. for the boundary between air and a material with refractive index n and extinction coefficient k is described by Eq. (22).

t_{e f f} = \frac{(n^{2} - k^{2} - 1) \frac{d k}{d β} - 2 n k \frac{d n}{d β}}{{(k^{2} + n^{2} - 1)}^{2} + 4 k^{2}}

Near a material resonance, n and k vary strongly with wavenumber. To illustrate the possibilities of negative effective thickness with a plasmonic resonance we have used published data for the optical constants of Aluminum [16] in Fig. 7(a). The region of negative effective thickness is broad (100 nm bandwidth) and the reflectivity is high (90%) [Fig. 7(b)], but teff is only around −30nm, which is too small to be useful in most applications. This illustrates that material constants alone can enable negative effective thickness in simple mirrors, though the magnitude of the phase effects are relatively insignificant compared to those of resonant mirrors.

Fig. 7 a) Refractive index, n and absorption coefficient, k for Aluminum. b) Reflectance and effective thickness of the bulk material.

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6. Conclusion

Our theoretical treatment based on well-known formulas for the reflectivity and transmission of lossless two-ports shows that asymmetry is required for mirrors to feature negative effective thickness in the absence of negative group velocity materials. Given this background, we used coupled mode theory (CMT) to show that negative phase slope and negative thickness can in fact be realized in asymmetric resonant mirrors. Negative thickness results from interacting resonances, and we show examples of how to combine resonances to achieve a combination of broadband high reflectivity and strong effective thickness effects. Our CMT provides intuition into design tradeoff and specifically shows that more complicated mirrors with several resonances leads to higher reflectivity and stronger effective thickness effects. These results were verified by RCWA simulations. The theory outlined in this paper combined with simulation tools like RCWA provide the framework to understand and design resonant mirrors with applications in sensors, lasers, and resonators.

Acknowledgments

The authors gratefully acknowledge funding from the NSF Center of Probing the Nanoscale under award 282 PHY-0830228.

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Design of resonant mirrors with negative group delay

Abstract

1. Introduction

2. Effective thickness

3. Effective thickness in resonant mirrors

4. Resonant mirror examples

5. Effective thickness in lossy materials

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (22)

Optics Express