Triple-layer guided-mode resonance Brewster filter ... · A triple-layer guided-mode resonance Brewster filter consisting of a homogeneous layer and two identical gratings with their

Triple-layer guided-mode resonance Brewster

filter consisting of a homogenous layer and

coupled gratings with equal refractive index

Xin Liu,1 Shuqi Chen,

2 Weiping Zang,

1,3 and Jianguo Tian

1,2,*

1School of Physics, Nankai University, Tianjin 300071, China 2The Key Laboratory of Weak Light Nonlinear Photonics, Ministry of Education, Teda Applied Physics School, Nankai

University, Tianjin 300457, China [email protected] *[email protected]

Abstract: A triple-layer guided-mode resonance Brewster filter consisting of

a homogeneous layer and two identical gratings with their refractive indices

equal to that of the homogeneous layer is presented. The spectral properties

of this filter are analyzed based on the coupling modulation of two identical

binary gratings at Brewster angle for a TM-polarized wave. The grating layer

between substrate and homogeneous layers can significantly change the

linewidth and resonant mode position, which are due to the asymmetric field

distribution inside the grating layers. The tunability of the resonance can be

altered on different resonant channels and a practical filter can be obtained in

TM2 waveguide mode. Variation of filling factor can alter the field

localization in the grating structure and significantly adjust the linewidth of

the filter.

©2011 Optical Society of America

OCIS codes: (050.0050) Diffraction and gratings; (310.2790) Guided waves; (120.2440) Filters.

References and links

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#142436 - $15.00 USD Received 10 Feb 2011; revised 27 Mar 2011; accepted 28 Mar 2011; published 14 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8233

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1. Introduction

The narrow linewidth of guided-mode resonance (GMR) is typically observed in substrates

with geometrically tailored multilayer thin films of grating and waveguide layers [1]. This new

type of optical element [2,3] combines principles of diffraction by periodic structures with

waveguide properties and antireflection (AR) thin-film characteristics to yield filters with 100%

reflectance at a desired wavelength. By appropriate choice of multilayer waveguide-grating

parameters, such as thickness and refractive index, the high diffractive order waves yielded by

grating layer can be coupled into the guided-mode, which can propagate in the waveguide layer.

This interesting anomalous effect resulting in the energy exchange between reflection and

transmission wave has been theoretically and experimentally reported in microwave region [4],

infrared region [5–8], and visible region [9,10]. Filters based on GMR can be easily extended

into several areas, such as optical bistable devices [11], dense wavelength division multiplexing

systems in optical communication [12–14] and sensors [15]. The concept of resonant Brewster

filters of TM mode was put forward in 1998 by Magnusson et al [16]. The filters can obtain

high-efficiency reflection at Brewster angle where traditional TM reflection will vanish. The

suppression of filter sidebands with absentee layers was advanced by Shin et al [17]. Later, the

double-layer GMR filter with multiple channels at Brewster angle was presented by Wang et al

[13] and Sang et al [14]. In their studies, the filters consist of homogeneous layer with a

refractive index equal to that of grating layer. The fluctuation of the reflectance with variation

of homogeneous layer thickness can be distinctly restrained due to equality of the refractive

index of grating and homogeneous layer. The resonant center wavelength and linewidth can be

controlled without lacking low-reflection sideband features by tuning thickness of homogenous

layer, filling factor of grating, and index of substrate layer. Recently, related devices based on

tunable characteristic due to the interaction of two GMR elements and the tunability of coupled

GMR effects were studied, and wideband tuning range has been demonstrated [18–21].

However, the previous resonant filters based on the interaction of two GMR elements were

studied under normal incidence [18–20] or general oblique incidence [21], and less attention is

paid to the Brewster angel incidence on this kind of GMR filters.


Besides the simpleness of the GMR filter consisting of single grating layer, it has been

proved by Li [22] that the truncated propagation equations of the original differential theory in

the previous rigorous coupled-wave analysis (RCWA) method [23,24] no longer preserve the

continuity of the appropriate field components across the discontinuities of the permittivity

function. Using theorems of Fourier factorization, Li proposed three appropriate factorization

rules, which are concerned with any numerical work in science that requires Fourier

factorization and the rules are called the fast Fourier factorization (FFF) method. In this paper,

we proposed a triple-layer guided-mode resonant Brewster filter consisting of two binary

identical gratings and a homogeneous layer with refractive index equal to that of the gratings.

Using the FFF method, the main properties of reflective spectrum of the Brewster filter were

analyzed for TM polarization. The coupling characteristics between two gratings were studied

with respect to the thickness of gratings, the thickness of the homogenous layer and the lateral

alignment shift between gratings. Results show that adjusting the grating thickness and lateral

alignment can alter the linewidth of the spectral response. The dependence of coupling strength

on the lateral alignment shift can be affected by the thickness of the homogeneous layer and the

grating filling factors. The tunability of the resonance can be altered with respect to different

resonant channels and grating filling factors.

2. Structure and theory

A schematic diagram of the triple-layer waveguide grating structure under TM polarization

light at oblique incident angle is depicted in Fig. 1. The resonant part above the substrate layer

consists of two identical grating layers and a homogeneous layer between the gratings.

According to the effective media theory (EMT) [25], the second order of effective refractive

index of the grating layer under TM illumination for 0 can be written in the following

form

eff,TM eff,TM eff,TM eff,TE

222 3

(2) (0) 2 2 (0) (0)

0

1 1(1 ) ,

3 H L

F F

(1)

where and 0 are the grating period and the central resonant wavelength of the structure,

respectively. F is the grating filling factor and H and

L are high and low permittivity of the

grating materials, respectively. The zero-order permittivities under TE and TM-polarization

conditions in Eq. (1) are given by

eff,TE

eff,TM

(0)

(0)

(1 ).

/ [ (1 ) ]

H L

H L L H

F F

F F

(2)

For a high-spatial frequency waveguide grating (0/ 0 ), the second term of the

polynomial expression on right side of Eq. (1) is neglectable and the expression of the effective

index of the grating under TM-polarization can be approximately reduced as

1/2

eff,TM / [ (1 ) ] .H L L Hn F F (3)

For a fixed filling factor, the triple-layer resonant Brewster filter can be obtained through

appropriate choosing of the homogeneous layer’s refractive index, which should be equal to the

effective refractive indices of gratings calculated by Eq. (3).

In Fourier space, the basic equations of differential theory of gratings can be expressed as

follows [24]


12

12

1/

,

z

x

x

z z

Hk E

y

Ek H H

y

(4)

where 0/ ( )x xE E i (where is the circular frequency and 0 is the permeability of

vacuum). Ex and Hz are the x and z components of electric and magnetic field, respectively. k2 is

a periodic function 2 2

0( , ) ( , )k x y k x y , where k0 is the modulus of the

x

y i

1gd

ud

cn

un

s

HnLn

sn

2gdF

Incident wave Refelection

Transmission

S

Fig. 1. (Color online) Schematic diagram of the GMR Brewster filter. The high and low

refractive indices of the identical gratings are nH = 2.25 and nL = 1.8, respectively. The filling

factor of the grating is set to F = 0.5. The refractive indices of the cover, substrate and homogeneous layers are set to nc = 1.0, ns = 1.46 and nu = 1.99, respectively. The thickness of the

triple-layer structure is dg1 = dg2 = 85.6 nm, du = 30 nm. The lateral alignment shift is denoted by

S.

wave vector in vacuum. is a diagonal matrix with elements 0 2 /n n , where

0 0 sin( )ik and n . f denotes the Toeplitz matrix generated by the Fourier

coefficients of f such that its (p, q) ( , p q ) element is fp-q, and 1 denotes the matrix inverse

[22]. The numerical calculation based on differential Eq. (4) can be implemented to produce

high accurate results with rapid convergence by preserving fewer harmonic waves.

3. Results and discussion

By using the transfer matrix method of thin film, the calculated Brewster angle of the

triple-layer filter is 57.13°. The angular response of the triple-layer waveguide grating structure

is shown in Fig. 2(a) with the operating wavelength of 800 nm. For locating the resonant peak at

Brewster angle, the periods of the identical gratings are set to 333.19 nm . The resonance

is induced because the coupling of the first evanescent diffraction order to a leaky waveguide

mode replaces the classical Brewster angle zero reflection effect, thus the zeroth reflected order

is reradiated. Here, the period of the gratings satisfies the high-spatial frequency condition

(0/ 0.42 ). The deviation of the zero-order permittivity of the grating from the second

order one is 0.02, which can be ignored if the homogeneous material is chosen. The spectral

response in Fig. 2(b) with the


0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

700 750 800 850 9000.0

0.2

0.4

0.6

0.8

1.0

Refl

ecta

nce

Incident angle (degree)

(a) (b)

Refl

ecta

nce

Wavelength (nm)

799 800 8010.0

0.5

1.0

Refl

ecta

nce

Wavelength (nm)

Fig. 2. (Color online) (a) Angular response of a triple-layer GMR filter for a TM-polarized incident wave with the operating wavelength of 800 nm. (b) Spectral response of the triple-layer

filter at the Brewster angle ( 57.13B ) indicated in (a). All the geometric parameters are

given in Fig. 1.

incident wave at Brewster angle exhibits a resonant peak at 800 nm with a linewidth of

~0.05 nm, which is much narrower than the linewidth in the previous work [14]. The linewidth

of the low sideband reflection is over ~200 nm. The resonant response with a reflectance peak

over 99.9% is zoomed in Fig. 2(b). Result shows that the line shape of the spectral response is

symmetrical as the total thickness of the triple-layer structure satisfies the half-wavelength

condition.

For multilayer resonant Brewster filters consisting of grating layers and homogeneous layer

with same effective refractive index, the line shape and linewidth of the reflection response can

be controlled by altering the thickness of the grating layer while fixing the total thickness (dg1 +

du + dg2). In the triple-layer presented in this paper, the thickness of the homogeneous layer is

thinner than that of the grating layer, and the field inside the two identical gratings is strongly

coupled. Consequently, the reflection response will be greatly affected by varying the grating

thickness. Besides, the distribution of electromagnetic field in the grating layers is different as

constructive and destructive interference of the internal diffracted fields, which are reflected

from the top and bottom surfaces of the gratings due to the different adjacent mediums of the

two identical gratings. Therefore, the dependence of the reflection response on the thickness of

upper and lower gratings should be analyzed while fixing the thickness of (dg1 + du) or (du +

dg2), which are shown in Fig. 3. As can be seen, altering thickness of the grating layers will

result in the change of the linewidth as well as the shifts of resonant peak. The resonant peak has

a red shift as increasing the thickness of the grating layers for two cases, but the linewidth

changes distinctly as varying the lower grating thickness (see in Fig. 3(a)). The key parameters

of the spectral response in Fig. 3(a) are listed in Table 1. The data in Table 1 indicates that the

effect of the lower grating layer on the spectral characteristic is larger than that of the upper one

because boundary conditions of phase matching for the two gratings are different [26], which

results in the different distribution of electromagnetic field inside the grating layers [21]. An

additional interesting point is that the central wavelength can be adjusted to desired wavelength

by slightly altering the incident angle while keeping the linewidth, line shape and sideband

features almost unchanged for same grating thickness. Figures 3(b) and (c) demonstrate that the

resonant peak can be tuned to the same resonant wavelength by slightly varying the incident

angle. The resonant incident angle can be changed by altering the thicknesses of upper and

lower grating layers while keeping the spectral characteristics unchanged. In contrast to the

previous work [14], the thickness of the lower grating can be adjusted to obtain different

linewidths


0.0

0.5

1.0

10-5

10-4

10-3

10-2

10-1

100

798 799 800 801 802

10-4

10-3

10-2

10-1

100

Designed value

dg1

=65.6 nm

dg2

=65.6 nm

dg1

=105.6 nm

dg2

=105.6 nm

Ref

lect

an

ce

(a)

(b)

(c)

Wavelength (nm)

Fig. 3. (Color online) (a) Spectral response (solid curves) of the filter in Fig. 1 for a TM-polarized

incidence at the Brewster angle (57.13°). The sum of (dg1 + du) is kept constant for (dashed

curves) dg1 = 65.6 nm and (dash-dotted curves) dg1 = 105.6 nm, the sum of (du + dg2) is kept constant for (short dashed curves) dg2 = 65.6 nm and (dash-dot-dotted curves) dg2 = 105.6 nm,

respectively. Other parameters are the same as those in Fig. 2 except that (b) (dashed curve)

57.30B and (dash-dotted curve) 56.88B , (c) (short dashed curve) 57.35B and

(dash-dot-dotted curve) 56.86B .

Table 1. Key Parameters of Spectral Response in Fig. 3(a)

Grating thickness (nm) Resonant wavelength (nm) Filter linewidth (nm)

Upper grating Lower grating Upper grating Lower grating

65.6 799.53 799.39 0.05 0.10

85.6 800.00 800.00 0.05 0.05

105.6 800.67 800.72 0.05 0.04

at the central wavelength of 800 nm while keeping the feature of low-reflection sideband, as

shown in Fig. 3(c). Considering the practical applications, we can simultaneously adjust the

upper and lower grating layer to obtain spectral response of the Brewster filter for desiring

linewidth and operating wavelength.

As reported in previous works, multiple channels of the resonant Brewster filter can be

obtained by using multiple resonances [12,13]. For simplification of the fabrication process,

multiple resonances can be achieved by adjusting the thickness of the homogeneous layer due to

the equality of refractive indices of the gratings and the homogeneous layer. We calculate the

reflectance at Brewster angle with varying the thickness of the homogeneous layer, as shown in

Fig. 4. The thicknesses of the grating layers are maintained at the same time. As can be seen,

three resonant peaks (30.0, 355.2 and 681.1 nm) appear almost periodically in the range of

0-800 nm, which correspond to the second stop band of the TM0, TM1 and TM2 waveguide

modes of the filter, respectively [27]. When the thickness of the homogeneous layer is tuned to

these three values, the single, double or triple channels can be obtained in the Brewster filter,

respectively.

The coupling strength between the electromagnetic fields in the gratings and homogeneous

layer can be appropriately affected by the homogeneous layer thickness and lateral alignment

condition between two gratings in this kind of devices [14]. Meanwhile, the tunable range and

ability of the lateral alignment shift along transverse direction, which could be applied to alter

the line shape and linewidth of spectral response of the Brewster filter, is also dependent on the

characteristic of the field inside the homogeneous layer [18]. Below, the tunable range of the

resonant mode location and the characteristic of spectral response of the Brewster filter due to

the lateral alignment shift are studied near the same resonant wavelength of 800 nm for three

different thicknesses of the homogeneous layer. The reflectance of the Brewster filter as the


function of the lateral alignment shift and wavelength for TM0, TM1 and TM2 waveguide modes

are shown in Fig. 5(a), (b) and (c), respectively. For the three waveguide modes, the lateral

alignment condition does not almost cause any shift for the resonant peak but the linewidth and

reflectance peak. The resonant peaks of the TM0 and TM1 modes shift toward short wavelength

as S increases from 0 to 0.5. For example, the wavelength of resonant peak shifts from 800 to

799.12 nm in Fig. 5(a) and from 800 to 799.57 nm in Fig. 5(b). The reason is that the field inside

the Brewster filter is mostly confined in the grating layer due to their relative large thicknesses,

and the lateral alignment shift S can affect the coupling

0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

Refl

ecta

nce

Thickness of the homogeneous layer (nm)

Fig. 4. (Color online) Calculated reflectance as a function of thickness of the homogeneous layer at Brewster angle under TM polarization, other parameters are the same as those in Fig. 2.

(a) (b)

(c) (d)

Fig. 5. (Color online) Calculated reflectance as a function of the lateral alignment shift S and

wavelength at Brewster angle (57.13°) for (a) TM0, (b) TM1 and (c) TM2 guided-modes. (d)

Calculated reflectance peak as a function of the lateral alignment shift S for TM0 (squares), TM1 (circles) and TM2 (up triangles) guided-modes. Other parameters are the same as those in Fig. 2.

strength between the two gratings [18]. With increasing of S from 0 to 0.5, the linewidth of the

TM0 mode increases, however, that of the TM1 mode changes more complicated because of the

comparable thicknesses of two grating layers to the homogeneous layer. The linewidth of the


TM2 guided-mode decreases sharply as S varies from 0 to 0.5, but the location of the resonant

peak is invariable. These characteristics are in good agreement with the coupled-mode theory

[20].

Figure 5(d) shows the variety of reflectance peak around 800 nm as a function of the lateral

alignment shift S for TM0 (squares), TM1 (circles) and TM2 (up triangles) waveguide modes.

For the TM0 and TM1 guided-modes, the electromagnetic field of the excited guided-mode

inside the filter can be greatly affected due to oblique incident angle

0.0

0.5

1.0

0.0

0.5

1.0

796 798 800 802 804

0.0

0.5

1.0

S=0

S=0.25

S=0.5

(a)

Refl

ecta

nce

S=0

S=0.25

S=0.5

(b)

S=0

S=0.25

S=0.5

Wavelength (nm)

(c)

Fig. 6. (Color online) Reflection spectral response of the triple-layer filter for (a) TM0, (b) TM1 and (c) TM2 guided-modes under different alignment conditions: (solid curves) perfect

alignment (S = 0), (dashed curves) quarter-period shifted (S = 0.25), and (dotted curves)

half-period shifted (S = 0.5). Other parameters are the same as those in Fig. 2.

0.0

0.2

0.4

0.6

0.8

1.0

799.6 799.8 800.0 800.2 800.4

0.0

0.2

0.4

0.6

0.8

1.0

(a)

Refl

ecta

nce

S=0.0

S=0.25

S=0.5

(b)

Wavelength (nm)

S=0.0

S=0.25

S=0.5

Fig. 7. (Color online) Reflection spectral response of the triple-layer Brewster filter for (a) F =

0.1 and (b) F = 0.9 under different alignment conditions: (solid curves) perfect alignment (S = 0),

(dashed curves) quarter-period shifted (S = 0.25), and (dotted curves) half-period shifted (S = 0.5). The other parameters are the same as those in Fig. 2 except the period of the grating: (a)

342.10 nm and (b) 321.71 nm .

when the two gratings are not perfectly aligned. Thus, as the lateral alignment shift S changes,

the reflectance peak varies in the range of 0.2-1 for TM0 mode and 0.75-1 for TM1 mode,

respectively. For the TM2 waveguide mode, the field distribution in the homogeneous layer

reduced the difference between the diffractive characteristics of the two gratings when the

lateral alignment S varies and the condition of phase matching is satisfied. Thus, the spectral

reflectance peak of the TM2 waveguide mode in Fig. 5(d) is almost unchanged. This


configuration can be considered as a perfect GMR filter at Brewster incident angle for practical

applications of narrowing the linewidth of reflective spectrum. The spectral responses under

particular alignment conditions (S = 0, 0.25 and 0.5) TM0, TM1 and TM2 waveguide modes are

shown in Fig. 6(a), (b) and (c), respectively. As can be seen, by adjusting the lateral alignment

condition, TM0 and TM2 guided-modes can be respectively used for expanding or narrowing

the linewidth of the spectral response of the Brewster filter with symmetric line shape and low

sideband features maintained.

Since the field inside the GMR filter is mostly confined in the medium with high refractive

index [19], therefore, the tunable range of the resonant mode location and the characteristic of

spectral response of the Brewster filter due to the lateral alignment shift could be affected by the

filling factors of the gratings. Figures 7(a) and (b) show the spectrums of the Brewster filter

with a grating filling factor of F = 0.1 and F = 0.9, respectively, which correspond to two

modulated grating structures. For locating the resonant wavelength at 800 nm, the periods of the

gratings are adjusted to 342.10 nm and 321.71 nm in Fig. 7(a) and (b), respectively. The other

parameters are the same as those in Fig. 5(a). In Fig. 7(a), results show that the GMR effect does

not vanish under the perfect alignment condition (S = 0). The range of the spectral linewidth is

still tunable as S varies, since the field in the gratings is extremely confined in the narrow high

refractive index region. In Fig. 7(a), the effective refractive indices of the identical gratings are

close to that of the substrate. Therefore, the location of the resonant peak cannot reach the

minimum value of the tunable range when S = 0.5. When the filling factor is set to 0.9, a strong

coupling strength between the fields in the gratings is kept. Therefore, the linewidth of the

spectral response is almost unchanged as varying of S from 0 to 0.5, which is shown in Fig.

7(b).

4. Conclusion

In summary, a triple-layer GMR Brewster filter can be fabricated by selecting a homogeneous

layer with refractive index equal to two identical gratings with filling factors of 0.5. It is shown

that the linewidth of the spectrum can be significantly changed by altering the thickness of the

lower grating layer, but the upper grating layer mainly affects the resonant mode location and

sideband levels. Higher order of guided-mode can be excited when the thickness of

homogeneous layer increase and different line shape of spectral response can be obtained by

selecting different homogeneous layer thicknesses. The tunable range and ability of the lateral

alignment shift along the transverse direction is also dependent on the thickness of the

homogeneous layer. Different dependence of the linewidth and reflectance peak on the lateral

alignment shift S can be obtained at TM0 and TM2 waveguide modes, the later case can be used

as a perfect GMR filter with the reflectance peak unchanged as S varies. For practical

applications, the triple-layer Brewster filter can still be used to obtain extremely narrow

linewidth of spectral response when the filling factor is set to 0.1. Meanwhile, the resonant

mode location has a certain tunable range when the filling factor is set to 0.9.

Acknowledgments

This work is supported by the Chinese National Key Basic Research Special Fund (grant

2011CB922003), the Natural Science Foundation of China (grant 60678025 and 61008002),

111 Project (grant B07013), the Specialized Research Fund for the Doctoral Program of Higher

Education (grant 20100031120005), and the Fundamental Research Funds for the Central

Universities.


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Triple-layer guided-mode resonance Brewster filter ... · A triple-layer guided-mode resonance Brewster filter consisting of a homogeneous layer and two identical gratings with their