## Abstract

We demonstrate a system consisting of a two-dimensional photonic crystal slab and two polarizers which has a tunable transmission lineshape. The lineshape can be tuned from a symmetric Lorentzian to a highly asymmetric Fano lineshape by rotating the output polarizer. We use temporal coupled mode theory to explain the measurement results. The theory also predicts tunable phase shift and group delay.

© 2013 Optical Society of America

## 1. Introduction

Controlling the lineshape of photonic resonators is of great interest for their applications in various fields [1]. For photonic crystal microcavities, lineshape tuning has been demonstrated by tuning the incident spot size [2] or polarization [3]. Similar effects have been investigated in plasmonic systems [4]. In this paper, we focus on guided resonance modes in photonic crystal slabs. For normally incident light, the transmission spectrum of a photonic crystal slab exhibits Fano lineshape features [5], which result from interference between discrete, guided resonance modes and a continuous Fabry-Pérot background [6, 7]. Past work has demonstrated that the lineshape of a guided resonance mode can be tuned by changing device dimensions [8] or incidence angle [9]. In this paper, we demonstrate that the lineshape of a guided resonance mode can be very simply tuned by rotating an external polarizer. The lineshape tuning method we present will enable better control of photonic resonance characteristics in such applications as optical filtering [10], sensing [11, 12] and signal delay.

Below, we first describe theoretically how rotating the angle of an external polarizer tunes the transmission lineshape of a photonic crystal slab from a symmetric (Lorentzian) shape to a highly asymmetric (Fano) shape. We then fabricate and characterize a Suzuki-phase lattice photonic crystal to validate the model and demonstrate lineshape tuning experimentally. We infer that the phase shift, and thus the group delay through the device, also depend on polarization. In particular, the group delay can be tuned from positive to negative by rotating the output polarizer. These effects all originate from changing the ratio between the guided-resonance transmission (indirect path) and the background, Fabry-Pérot transmission (direct path) of the photonic crystal.

## 2. Experimental system

Figure 1 shows a schematic of the system. Light propagates in the +*z*-direction. The first polarizer (P1) sets the polarization angle *θ _{i}* of light incident on the photonic crystal (PhC). The photonic crystal (PhC) is assumed to support a guided resonance mode (GRM) that is singly degenerate and couples to normally incident radiation with well-defined polarization. That is, there is a polarization direction for a linearly-polarized, normally-incident plane wave for which the overlap integral with the mode is finite, whereas the overlap integral for a plane wave polarized in the orthogonal direction is zero. We use a Suzuki-phase lattice photonic crystal for demonstration [13, 14]; other 1-D or 2-D periodic photonic crystal structures with the required mode characteristics may also be used. Due to coupling of incident light to the GRM, light transmitted through the PhC will in general not be polarized at angle

*θ*; a similar effect has previously been used to demonstrate polarization conversion in a reflective geometry [15]. We place a second polarizer (P2) after the photonic crystal with polarization angle

_{i}*θ′*=

_{p}*θ*+

_{i}*θ*, where

_{p}*θ*is the relative angle between the two polarizers. We assume that the component of transmitted light that is polarized orthogonal to

_{p}*θ′*is lost from the system (e.g. absorbed by a polarizer, or redirected elsewhere by a polarizing beam splitter).

_{p}A fully-suspended silicon PhC membrane device is fabricated in the silicon-on-insulator (SOI) platform using standard e-beam lithography and ICP-RIE dry etching. We then use wet under-etching to release the membrane and transfer it to a host carrier wafer [16]. Backside etching is used to tune the GRM to the wavelength of interest [17]. Fig. 2(a) shows the SEM picture of the fabricated device, which has a lattice constant *a* = 510 nm, hole diameter *d* = 346 nm and slab thickness *t* = 200 nm.

The simulated transmission spectrum is shown in Fig. 2(b). The spectrum is different for *x*- and *y*- polarized light; due to the rectangular unit cell of the Suzuki-phase lattice, the two directions are not equivalent. For the *x*-polarization, there is a resonance mode at 1570 nm with a quality factor of approximately 70. Simulations indicate that this mode corresponds to the fifth TE photonic band. The relatively low quality factor corresponds to a relatively wide peak, which simplifies the characterization of lineshape in experiments. For *y*-polarization, there is no resonance mode visible in the wavelength range shown.

## 3. Analytical model for transmission lineshape

We can write an analyical expression for the transmission lineshape by adding the transmission coefficients of the direct path (Fabry-Pérot resonance) (**t*** _{d}*) and guided resonance (

**t**

*).*

_{g}The transmission coefficients of the photonic crystal slab are [6]:

*t*

_{d0}is the transmission coefficient through a homogeneous dielectric slab, and is the transmission coefficient of the guided resonance.

*ε*= (

*ω*−

*ω*

_{0})/

*γ*is the normalized detuning,

*ω*

_{0}is the resonance frequency,

*γ*is the damping coefficient of the resonance, and

*j*is the imaginary unit. The prefactor

*f*is equal to −(

*t*

_{d0}±

*r*

_{d0}) to ensure energy conservation. The plus and minus signs correspond to resonant modes that are even and odd with respect to the

*xy*-mirror plane. The mode we consider in this paper is even, so we keep only the plus sign in the following derivation. Moreover, since the bandwidth of the guided resonance is much narrower than the background Fabry-Pérot features in the spectrum, we can neglect the dispersion of

*t*

_{d0}and assume it to be a complex constant.

After the second polarizer,

**p̂**is the unit vector aligned with the second polarizer. The total transmission coefficient is thus:

The transmission spectrum *T* is described by the Fano resonance formula:

Because *r*_{d0}/*t*_{d0} is always purely imaginary [6], the first term of Eq. (8) is purely real, the second term is purely imaginary, and therefore, the asymmetry factor *q* is a complex number. This is in contrast to the system without the second polarizer, for which the asymmetry factor is real.

## 4. Comparison of theory and experiment

We measure the transmission spectrum while rotating the second polarizer at 4° intervals over 180°. We plot the measured data as a 2D function of frequency and the relative angle between the polarizers in Fig. 3(a).

We use Eq. (7) to fit the measured data. There are four unknown fitting parameters corresponding to the model: the transmission coefficient |*t*_{d0}|, the guided resonance frequency *ω*_{0}, the resonance damping coefficient *γ*, and the incident polarization *θ _{i}* (i.e. the angle between the first polarizer and the mode polarization). The reflection coefficient can be calculated by
$\left|{r}_{d0}\right|=\sqrt{1-{\left|{t}_{d0}\right|}^{2}}$. An additional fitting parameter

*F*is used to take into account measurement normalization. Figure 3(b) shows the fitted function of Fig. 3(a), with fitting parameters

*F*= 1.90, |

*t*

_{d0}| = 0.633,

*ω*

_{0}

*a*/2

*πc*= 0.319,

*γa*/2

*πc*= 0.0238 and

*θ*= 122°. Visually, the fitted spectrum reproduces the features of the experimental spectrum quite well, while smoothing out experimental noise.

_{i}To clearly illustrate lineshape tuning, we select several different angles from Fig. 3 and plot the corresponding spectra. Figure 4(a) and Fig. 4(b) show measured and fitted spectra, respectively. As the relative angle between the polarizers increases, the transmission spectrum changes from an asymmetric Fano lineshape at 10° to a symmetric Lorentzian lineshape at 90°. Further rotation of the second polarizer again yields a Fano lineshape. At *θ _{p}* = 150°, the transmission spectrum is a straight line. This occurs because the guided resonance path is perfectly cancelled by the second polarizer. The fitting result shows that

*θ*is approximately 120°, and

_{i}*θ′*=

_{p}*θ*+

_{i}*θ*≈ 270°, indicating that the second polarizer is nearly aligned with the

_{p}*y*-axis, perpendicular to the mode polarization. The non-zero transmission arises from the Fabry-Pérot direct path.

## 5. Discussion

To gain better physical insight into the tunable lineshape, we may recast the transmission in the form of a transfer function for a first order linear system [18]. In particular, the transmission coefficient of Eq. (6) can be written as

where*T*is a complex factor which represents the transmission coefficient far from resonance, and ${\epsilon}_{p}={\epsilon}_{p}^{R}+j{\epsilon}_{p}^{I}$ and ${\epsilon}_{z}={\epsilon}_{z}^{R}+j{\epsilon}_{z}^{I}$ are the complex pole and zero of the system. By comparing Eq. (6) and Eq. (9),

As define above, *ε* is real. However, we are free to consider Eq. (9) as a complex function of a complex variable *ε* = *ε ^{R}* +

*jε*. The response along the real axis then gives the physical behavior of the system.

^{I}The *θ _{p}* dependence of the zero point (

*ε*) of the system shown in Eq. (11) plays an important role in tuning the line shape. For simplicity, we choose

_{z}*θ*to be 45° and $\left|{t}_{d0}\right|=\sqrt{0.5}$. Figure 5 shows the pole and zeros of the transmission coefficient for different

_{i}*θ*. For all cases of

_{p}*θ*, the pole point is fixed at (0, 1) in the complex plane while the zero point varies. In Fig. 6, we choose three special cases when

_{p}*θ*is equal to 0°, 60° and 90°, respectively, to illustrate how zero position affects the system response. Different

_{p}*θ*angles are shown in different rows in Fig. 6. The first and second columns show the system amplitude and phase response in the complex plane.

_{p}Figure 6(a) shows the amplitude response for *θ _{p}* = 0°, when the polarizers are aligned. The zero is at (−0.5, 0.5) in the complex plane and is visible as an amplitude minimum (black region). The pole is at (0, 1) and is visible as an amplitude maximum (white region). The amplitude response along the real axis is shown as a superimposed red line (see red scale bar on right). The transmission first decreases and then increases as a function of frequency. The phase response is shown in Fig. 6(b). The zero and pole are connected by a line of phase discontinuity. The phase response on the real axis is again shown as a superimposed red line. Both the amplitude and phase response change with polarizer angle.

Figure 6(d) and Fig. 6(e) show the case of *θ _{p}* = 60°. The pole remains fixed at (0,1), while the zero has moved to (0.36,1.36). The real transmission and phase response reflect this change, showing inverted behavior relative to Fig. 6(a) and Fig. 6(b).

Figure 6(g) and Fig. 6(h) show the case when *θ _{p}* is equal to 90°, and the polarizers are perpendicular to one another. In this case, the zero is not visible on the plot. The amplitude lineshape is a symmetric Lorentzian, and the phase response decreases by

*π*over the resonance.

We thus see that by tuning the angle of the output polarizer, we not only can change the lineshape of the transmission spectrum, but also the phase change through the structure. The tunable phase change can be used to obtain a tunable group delay, calculated as Δ*τ* = −*dϕ/dω*. Figure 6(c), Fig. 6(f) and Fig. 6(i) show the dimensionless group delay, (Δ*τ*)*c/a*. For a fixed frequency (fixed *ε _{R}* ≈ −0.5), the group delay can be tuned from negative to positive by changing

*θ*.

_{p}The physical value of the delay can be obtained from the graph by multiplying by *a/c* = 1.7 *fs*. For the relatively low quality factor mode studied here, the delays are in the 1 to 2 *fs* range. The delay increases linearly with quality factor, *Q*. Recently, a photonic crystal lattice has been designed that supports a coupled, linearly-polarized guided resonance mode with theoretical quality factor as high as 10^{5}[19]. Such a *Q* value would increase the delay to the picosecond range.

In this paper, we have demonstrated lineshape and time delay tuning by fixing the incident polarization and tuning the collection polarization. We could also have fixed the collection polarization and tuned the incident polarization. This can be inferred from Eq. (8). Substituting *θ _{i}* with

*θ′*−

_{p}*θ*, fixing

_{p}*θ′*to be 45°, and tuning

_{p}*θ*from 0° to 180°, the zero point of the system has the same trace as shown in Fig. 5, yielding a similar tuning mechanism for the lineshape.

_{p}## 6. Conclusion

In summary, we have proposed a method for tuning the transmission lineshape through a photonic crystal slab membrane using polarization control. The principle is to tune the transmission ratio between the direct, Fabry-Pérot path and the indirect, guided resonance path. We first presented an analytical model for the transmission spectrum. We then fabricated and characterized a Suzuki-phase lattice photonic crystal slab membrane to validate the model. There is excellent agreement between theoretical predictions and experimental results. We show that the lineshape can be tuned from a highly asymmetric shape to a highly symmetric one, while the corresponding group delay is tuned from negative to positive. The tuning is achieved simply by rotating an external output polarizer. More generally, the polarization angle dependence measurement and surface fitting technique shown in this work provide an accurate method to characterize guided resonance modes. We expect our results to be useful in a range of applications of photonic crystal guided-resonance modes, including filtering and sensing.

## Acknowledgments

The authors thank Chenxi Lin and Jing Ma for their help with the fabrication process and fruitful conversation with Yunchu Li. Ningfeng Huang was supported by the Center for Energy Nanoscience, an Energy Frontiers Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0001013. Luis Javier Martínez was supported by the Army Research Office under award no. 56801-PCS. Computing resources were provided by the University of Southern California Center for High Performance Computing and Communication (http://www.usc.edu/hpcc).

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