## Abstract

Numerical methods for simulating etching and deposition processes were combined with electromagnetic modeling to design guided-mode resonance (GMR) filters with accurately positioned resonances and study how fabrication affects their optical behavior. GMR filters are highly sensitive to structural deformations that arise during fabrication, making accurate placement of their resonances very difficult without active tuning while in operation. Inspired by how thin film resistors are trimmed during fabrication, the numerical tools were used to design a method for adjusting position of GMR resonances at the time of fabrication.

©2007 Optical Society of America

## 1. Introduction

In 1902, R. W. Wood discovered “anomalies,” or abrupt changes, in diffraction efficiency from gratings when a parameter was varied over a narrow range [1]. A first type of anomaly was explained by Lord Rayleigh in 1907 as a sudden redistribution of energy when additional diffracted orders appear or disappear [2]. In1965, Hessel presented a new theory describing a second anomaly as a resonance type due to a guided wave phenomenon [3]. Mashev used the resonance anomaly to demonstrate a narrow-band reflection filter [4]. Since that time, guided-mode resonance (GMR) devices have found applications in astronomy, WDM filters [5], dichroic reflectors in lasers [6,7], polarizers [7], sensors [8], security marks and optical tags [9], pulse shaping and dispersion compensation [10,11], frequency selective surfaces [12], and more.

GMR filters are simple structures comprised of just a grating and a waveguide [13,14]. When precise phase-matching conditions are met, externally propagating waves can be coupled into guided modes by the grating. The guided modes are “leaky” modes due to index modulation in the waveguide. This causes the guided modes to slowly leak from the waveguide and recombine with the incident wave to establish a resonance. This highly sensitive phenomenon enables filters to be constructed with extremely narrow passbands or stopbands. In fact, linewidth can usually be made arbitrarily narrow just by reducing contrast of the grating.

GMR filters are highly compact, easy to fabricate, and typically perform better than conventional thin film stacks for sub-nanometer filters. Filter response can be symmetric with virtually no ripple outside of the passband for both transmission and reflection type filters. Efficiency can approach 100% and they may be tuned over a very large range of wavelengths.

Filters with multiple resonances have also been realized. Designs are most often obtained from direct parametric searches or by genetic optimization [15]. While rigorous methods must be used to analyze GMR filters, experimental results are typically in excellent agreement with theoretical predictions.

GMR filters tend to be very sensitive to polarization, angle of incidence, refractive index, and structural deviations due to the high sensitivity of the phase-matching condition. Crossed gratings and even some ruled grating configurations have been shown to operate independent of polarization [16–20]. Doubly periodic structures have been used to improve tolerance to oblique angles of incidence [21].

This paper outlines an approach for mitigating the high sensitivity to structural deviations and imprecise control of refractive index so that a resonance can be placed accurately at the time of fabrication. The method was inspired by how thin film resistors are trimmed to achieve an accurate value of resistance. Many GMR devices will require a stable temperature to ensure refractive index remains constant during operation.

## 2. Theory of guided-mode resonance

There are two interacting mechanisms that must be understood to fully explain the operation of a GMR. The first mechanism is diffraction of an incident wave by a grating. Inside the grating region, amplitude of the propagating field takes on the same variations as the grating. When it is periodic, the electric field can be expressed as a Fourier series where each term corresponds to a different diffracted order. The amplitude and phase of each diffracted mode must be calculated using Maxwell’s equations, but the direction can be determined by the famous grating equation [22].

Parameter *n _{g}* is the refractive index of the grating
region computed from the numerical average of the relative permittivity in the
grating region as ${n}_{g}=\sqrt{{\epsilon}_{g}}$
. Parameter

*n*

_{1}is the refractive index above the grating,

*θ*

_{inc}is the angle of incidence of the external wave,

*λ*

_{0}is the free space wavelength, Λ is the period of the grating,

*m*is an integer representing the order of the diffracted mode, and

*θ*is the angle of the

_{m}*m*

^{th}diffracted mode inside the grating region.

The second mechanism is wave guiding where light is made to propagate along a confined path by total internal reflection (TIR) [23]. Wave guiding can only occur when the effective index of the guided mode is greater than the surrounding media and less than the refractive index of the core itself. From this, a quantitative condition for wave guiding can be written.

*β _{m}* is the propagation constant so

*β*/

_{m}*k*

_{0}is the effective index of the

*m*

^{th}-order guided mode. Exact values are not needed for

*β*as only limits on the effective index will be considered here. From a simple ray picture of a slab waveguide [24], a guided mode can be thought of as a light ray propagating at an angle

_{m}*θ*such that

_{m}A guided-mode resonance occurs when a diffracted order exists at the same angle as a guided mode. Guided modes can be related to the incident wave by substituting Eq. (3) into Eq. (1), and the new Eq. (1) into Eq. (2). This leads to an inequality that can be used to estimate regions where guided-mode resonances may occur.

Figure 1 identifies these regions graphically as a function
of angle of incidence and normalized wavelength (normalized by the grating period,
i.e. *λ*
_{0}/Λ). It was generated using
Eq. (4) for a typical GMR filter with
*n*
_{1}=1.0, *n*
_{2}=1.5, and
*n _{g}*=2.0.

Several important aspects of GMR filters can be observed in Fig. 1. As angle of incidence moves away from the normal, resonances split. At normal incidence, symmetry dictates the two resonances occur at the same wavelength, but at oblique angles different phase matching conditions are required to couple external waves into guided modes. As wavelength is decreased, or period increased, the number of resonances increases exponentially. Most designs avoid multiple resonances, but sometimes they can be exploited. For example, Ref. [25] used a doubly resonant structure to form a device with a sharp peak in transmission instead of reflection.

## 3. Modeling fabrication

It is important to obtain an accurate description of the geometry of a GMR device due to its high sensitivity to structural deformations. For this reason, the string method [26–28] was used to model etching and deposition processes to predict geometry more accurately. The substrate surface is described by connecting a set of marker points that mark the position of the surface. As the algorithm iterates, position of each of the marker points is adjusted in a manner that best replicates the physical process being modeled.

The string method is an incredibly fast and efficient method for simulating the evolution of surfaces. It is simple to implement and can incorporate multiple physical processes. It has been used to model a variety of fabrication methods including etching [28], deposition [28], and autocloning [26]. The main drawback of the method is that it is inherently unstable [27] and therefore restricted to mostly smooth and continuous surfaces [29]. Proven techniques exist to improve stability, but they all modify the equation of motion in unobvious and unrealistic ways [30]. Due to the manner in which the method is stabilized, it can rarely be extended to model generalized three-dimensional problems.

#### 3.1 String Method Algorithm

The string method is surprisingly easy to implement and a block diagram is
provided in Fig. 3. Prior to entering the main loop, the initial
surface is described by a series of marker points placed along the surface. Each
point is described by a position vector
${\overrightarrow{x}}_{i}$
. When *N* marker
points are used, the complete set of position vectors can be written as

The first step in the main loop is to calculate how fast, and in what direction, each point on the surface must move to best model the physics of the process. This is called the rate function and is essentially a velocity associated with each marker point. This step is usually the most computationally intensive.

To update position vectors based on the rate function, a suitable time step ∆t must be chosen that is small enough to prevent particles from crossing paths. After this is calculated, position is updated according to

The string method is inherently unstable for two main reasons. First, as particles converge they may reduce accuracy of the rate function or form unstable loops in the string. To mitigate this, a “smoothing” function can be applied to the rate function, and marker points can be removed as they converge. Second, as marker points diverge they can become too sparse to adequately resolve the surface. In this case, additional marker particles must be added. Care must be taken to correctly interpolate where they are to be placed. Performing a smoothing function on the position vectors after the update can help maintain stability.

#### 3.2 Calculation of the Rate Function

In the most general setting, deposition, etching, and redeposition of etched material can all be occurring at the same time during fabrication. Calculation of the rate function is described well in Refs. [26,28], so they will only be summarized here. These mechanisms are illustrated in Fig. 4 along with “snapshots” from the string method. Arrows indicate velocity of the particular mechanism being illustrated. The overall velocity is the sum of all rate mechanisms.

During sputter deposition, neutral particles are deposited from a target to the
substrate with an angular dependence of cos^{n}
*ϕ*. The parameter *ϕ*
denotes angle from the vertical axis, and *n* characterizes the
diffusion profile. Isotropic deposition corresponds to *n* = 0 ,
while strictly vertical deposition corresponds to *n* =
∞ . While most material diffuses vertically, diffusion at oblique
angles of incidence leads to deposition on the sidewalls. The deposition rate at
any point ${\overrightarrow{x}}_{i}$
along the surface can be
computed as

The parameter *d* describes the magnitude of flux of incident
particles, θ is the angle of the surface normal from the vertical
axis, and
*V _{T}*(${\overrightarrow{x}}_{i}$
,ϕ)
is the visibility of the target at angle ϕ from point
${\overrightarrow{x}}_{i}$
. Visibility equals 1 if the
target is visible and 0 if it is not. The angular dependence on visibility leads
to horizontal surfaces evolving faster.

The rate function for an etching mechanism can be computed as

Parameter *b* denotes the etch rate of a perfectly horizontal
surface. Parameter *c* characterizes angular selectivity. The
function
sin^{2}
*θ*cos*θ* has
a maximum value near *θ* = 55°. A peak etch
rate at this angle is emphasized with increasing
∣*c*∣. This phenomenon is known to produce
faceting, or flat sloped surfaces, at the top corners of square shaped
structures like binary gratings.

Materials removed during an etching process can be redeposited back onto the surface. The path particles take between being etched and redeposited is quite complicated and difficult to model, particularly for deep trenches. This phenomena produces a net transfer of material from upper parts of trenches to lower parts. The rate function for redeposition can be expressed as an integration over the full range of angles from which particles can diffuse.

Parameter *e* characterizes the rate of redeposition. The
visibility function *V _{s}*
(${\overrightarrow{x}}_{i}$
,ϕ) indicates what
portion of the surface is visible at angle

*ϕ*for point ${\overrightarrow{x}}_{i}$ . Points at deeper parts of a groove will receive greater redeposition because a larger span of surface is visible from which to receive particles.

The overall rate function is the sum of all rate mechanisms and is in the direction of the surface normal.

In this work, etching and deposition were performed on circular holes formed in a
dielectric. Circular symmetry was used to reduce the problem to two dimensions.
This approximation is valid as long as adjacent holes are physically separate.
Parameters for modeling deposition were set to *d*=1,
*n*=5, and
*b*=*c*=*e*=0. Parameters for
modeling the etching process were set to *d*=-10,
*n*=20, and
*b*=*c*=*e*=0. These were
empirically chosen to best represent typical deposition and etching processes
observed in the lab.

## 4. Electromagnetic simulation

To simulate the optical behavior of GMR devices, frequency-domain methods are usually preferred to more efficiently resolve sharp features in the spectral response. Of these, finite-difference frequency-domain (FDFD) [31] and rigorous coupled-wave analysis (RCWA) [31–33] seem to be most popular. RCWA was used in this work due to its greater computational efficiency for three-dimensional structures.

RCWA is a semi-analytical method where Maxwell’s equations are left
analytical in the *z*-direction and Fourier transformed in the
*xy*-plane. For layered periodic structures, the field is
represented by an infinite set of spatial harmonics in each layer. To be solved
numerically, these must be truncated to a finite set of spatial harmonics. The
concept and geometry of RCWA is illustrated in Fig. 5 for rectangular symmetry.

In this work, the conventional algorithm for RCWA was modified to more efficiently
handle nonrectangular grating symmetries. A detailed description of the formulation
can be found in Ref. [31]. The field and dielectric function in the
*i*
^{th} grating layer were expanded into a sum of
spatial harmonics expressed in terms of the reciprocal lattice vectors of the
grating unit cell,
${\overrightarrow{K}}_{1}$
and
${\overrightarrow{K}}_{2}$
. The remainder of the formulation was
standard for RCWA. The generalized expansions were

## 5. Device design

To adjust position of a resonance at the time of fabrication, an additive or subtractive process performed on the surface of the device seems easiest to implement. Refractive index and structural dimensions are much less feasible to adjust after a device has been built. For this reason, it becomes necessary to devise a resonant structure that is sensitive to this approach. A most sensitive device should have the core exposed since position of the resonance is strongly affected by core thickness. In this manner, GMR filters can be adjusted over the greatest possible tuning range at the time of fabrication.

A GMR filter that is easy to build and satisfies requirements discussed above is
depicted in Fig. 6. A surface relief grating is formed directly onto the
surface of the guiding layer. By etching or depositing small amounts of material,
position of the resonance can be tuned by adjusting thickness of the core region
through parameter *T*. It will be shown that grating geometry changes
during the tuning process. For this reason, the initial grating depth and period
must be chosen so the final grating depth and period are near optimal.

#### 5.1 Ideal device

To design a device that can operate independent of polarization, a crossed
grating GMR consistent with that in Fig. 3 can be used. This device, along with its
reflection spectrum, is depicted in Fig. 7. The device has hexagonal symmetry so grating
features can be largest. It is an array of circular holes formed on the surface
of a high index guiding layer. The device has parameters
*n*
_{1}=1.0, *n*
_{2}=2.31,
*n _{g}*=3.91,

*a*=1.15 μm,

*f*=0.7,

*d*=230 nm, and

*T*=345 nm. Two resonances can be observed that are spaced around 100 nm apart. This is one factor that limits the overall tuning range to a 100 nm operational window highlighted in the figure.

Hexagonal symmetry was chosen for its narrower resonance and larger grating
dimensions. Tuning of this device is accomplished by adjusting thickness of core
region by deposition or etching processes. The effect on reflection spectrum
within the operational window as parameter *T* is adjusted is
depicted in Fig. 8. At the left side of this figure, position and
width of the resonance shows a nonlinear response to variations in film
thickness. From this observation, designs intended only to adjust position of
resonance should use a thicker core layer. If the core layer is too thick, the
guide can become multimode and introduce additional resonance peaks into the
reflection spectrum. It can also be observed that as resonance is adjusted,
suppression at the edges of the operational window is reduced. Nominal thickness
of the GMR filter was chosen to minimize this effect over the greatest possible
tuning range. The duty cycle and relief depth were chosen to simultaneously
optimize suppression outside of the passband and peak reflection on resonance.
Duty cycle was defined as the fractional area occupied by the air holes.

#### 5.2 Tuning by deposition

Material may be deposited onto an initial grating formed onto the surface of a substrate. By measuring resonance position at various stages during deposition, GMR resonances can be placed accurately in frequency. This process is summarized in Fig. 9.

Due to physics inherent in the deposition process, the rectangular profile of the initial grating is not maintained. Visibility from the target to the substrate prevents deposition from being uniform causing trenches to form at the base of vertical sidewalls [26]. One artifact of deposition is decreased duty cycle. Off-axis diffusion leads to deposition on vertical sidewalls. This mechanism works to reduce hole radius as material is deposited. The consequence is much poorer suppression outside of the passband and a shift in position of resonances. To mitigate this effect, duty cycle of the initial grating should be exaggerated such that after deposition, duty cycle is correct. This mechanism places an upper limit on what duty cycle may be realized based on directionality of deposition and how thin of a grating can be made.

A second artifact of deposition is increased relief depth. Portions of the grating at the bottom of the grooves are less visible to the target and receive less deposition. Portions at the top of the grooves are fully visible and receive maximum deposition. This leads to a higher growth rate at the top of the grooves than the bottom, which increases relief depth during deposition. This effect leads to poorer suppression off resonance and a shift in position of the resonance peaks. To mitigate this effect, relief depth of the initial grating should be reduced such that the final relief depth is correct.

A third artifact of deposition is a GMR that is somewhat doubly-periodic. Portions of the surface at the bottom of the grating grooves nearest to the sidewalls receive less deposition due to a shadowing effect. This causes a “hump” to appear in the middle of the grooves resembling the hump at the top of the grooves. If conditions are right, multiple resonances may appear in the reflection spectrum. It is entirely possible this mechanism could be exploited in some manner, but it was not considered here.

Figure 10 is a movie showing how the reflection spectrum of a device evolves during deposition. In this movie, it is the second resonance that is of interest when it reaches 1.54 μm. In practice, the deposition process would be stopped at this point. The initial grating was designed to minimize background suppression and optimize peak reflection at this point of operation. For clarity, only a cross section of the fully three-dimensional device is shown.

If lesser turning range is tolerable, device performance can be enhanced using larger lattice constant and adjusting film thickness. The main drawback of this additive approach is that it is difficult to realize a large grating duty cycle which is advantageous here. For this reason, it is best to use a highly directional deposition or use the subtractive process described next.

#### 5.3. Tuning by etching

As an alternative to deposition, devices may be tuned using an etching process. This offers several advantages. First, it is an easier process and less expensive to implement. Second, it does not form a structure that is doubly-periodic. Third, it produces a grating profile with some anti-reflection properties due to smoother transition from air into the dielectric. As a result, the device offers improved suppression away from resonance.

Figure 11 summarizes a tune-by-etching fabrication process. A higher refractive index layer is deposited onto a bare substrate to form the GMR core. A mask is formed onto this layer and the pattern is transfer etched into the core layer. The initial grating becomes fully exposed when the mask is removed. Subsequent etching reduces thickness of the GMR core causing resonance to move toward shorter wavelengths. The final etching process is stopped when the GMR resonance is positioned correctly.

In contrast to the tune-by-deposition process, tune-by-etching leads to an increased duty cycle and decreased relief depth. This can be compensated by starting with smaller holes and greater relief depth and allowing the tuning process to make the corrections. This mechanism places a lower limit on what duty cycle can be achieved. In addition, overall relief depth is diminished due to redeposition and shadowing effects. This too can be compensated by exaggerating the initial dimensions and allowing the etching process to reduce relief depth.

Figure 12 is a movie showing how the reflection spectrum of a device evolves during deposition. Here it is the first resonance that is of interest when it reaches 1.54 μm. In practice, the etching process would be stopped at this point. The initial grating was designed to minimize background suppression and optimize peak reflection at this point of operation. For clarity, only a cross section of the fully three-dimensional device is shown.

## 6. Conclusion

Guided-mode resonance filters are highly compact devices than can easily realize narrow pass bands of less than one nanometer. Their strong resonance makes their behavior highly sensitive to geometry and structural deformations. For this reason, it is difficult to fabricate these devices when position of the resonances must be placed very accurately in frequency. Inspired by how thin film resistors are trimmed during fabrication, a method was developed for tuning the position of GMR resonances during fabrication.

To more accurately predict their optical behavior, fabrication was simulated using the string method. The resulting geometry was imported into RCWA to account for structural deformations and determine optical behavior. Using these tools, additive and subtractive tuning processes were studied. Both were shown to be feasible, but the etching process was preferred because it led to better device performance and is often easier to implement in the lab.

Future work to improve device performance would likely involve GMR layers buried between multilayer substrates. This will require methods for tuning intermediate layers to intermediate wavelengths.

## Acknowledgments

This work was funded in part by the National Science Foundation CAREER Grant ECS 0348280.

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