Multi-band optical bandpass filter with picometer bandwidth in visible spectrum formed by prism pair coupled planar optical waveguide

Jianhua Liu; Li Tao

doi:10.1364/OE.25.012121

1. Introduction

Optical bandpass (OBP) filter is a common device that has been widely used in multiple optical fields such as spectroscopic measurements, laser systems, optical communications, displays, and lighting et al. Generally, commercially available OBP filters are made of stacks of alternating layers of dielectric materials of high and low refractive indices, known as film pairs, with optical thickness of one-fourth of the working wavelength [1]. The stack of film pairs assures the constructive interference in phase at the central wavelength. Due to the vibrational nature of light wave, if the number of film pairs are not sufficient, the transmission leakage in the sideband (or skirt) of this film stack filter is usually unacceptable, and the edges of the pass-band are not very sharp. Therefore in order to obtain a filter of bandwidth in nanometer scale with sideband well depressed, tens or even hundreds of this thin film pairs piling together are usually necessary. However, as the number of the film layers becomes larger, losses also increase due to material imperfection and errors in the film deposition process. For example, the conventional film deposition technology produces porosities in film layers, which cause adsorption of water molecules in the film stacks which would deteriorate the performance of the filter device. In addition, precise control in device fabrication process is also a problem for making films with quarter wavelength thickness. For these reasons, the transmission efficiency typically gets lowered to a range of 50 - 90% [1]. The cost of the device goes much higher with better expected performance.

In the past decades, a new kind of narrow bandwidth optical filter has been extensively studied, which exhibits the so called “optical anomaly,” i.e. 100% transmission efficiency and extremely narrow bandwidth in nano/picometer scale [2–14]. A special constituent component, i.e., one layer of sub wavelength grating, or the so called waveguide grating (WG), is either sandwiched or just cast on top of the planar structure, which provides a phase matching mechanism for the guided mode resonance between the incident and the guided mode of light in the planar structure. Complete exchange of light energy takes place at a special combination of the structural configuration and the incident light properties including the wavelength and the angle of incidence et al. That phase match is intrinsically sensitive to the change of the wavelength or the angle of incidence. Therefore the filter of this WG based structure has a narrow bandwidth. However, it is not easy to precisely control the modulation profile of sub wavelength grating on the refractive index of the grating material, and therefore the measured performances of a real WG based optical filter, either the bandwidth or the transmission efficiency, are usually unsatisfactory with respect to their theoretical predictions [15–17].

Recently, we have proposed a new type of narrow bandwidth optical bandpass filter which is composed of only a few (minimally 3) layers of films that form a planar optical waveguide, and two coupling prisms cladding on both side of the POW [18]. The light transmits from one prism to the other by the bridge of the sandwiched POW under the condition of guided mode resonances (GMR), i.e., a two-step GMR, which makes the transmittance extremely sensitive to the structural configuration, wavelength, and the angle of incident light beam. Light transmission efficiency could well be higher than 99% for each GM, as long as the constituent material losses are not very high [18]. Because of the simple structure with less number of layers, and no requirement for the quarter wavelength thickness, it could be easily implemented as a practical device.

In this paper, with the modified configuration parameters of this prism coupled POW filter, and also with the inclusion of practical material dispersion and extinction for each constituent material, we have obtained simulation results of multiple pass-bands with high transmission efficiencies in the visible spectrum. The most prominent feature of the filter is that bandwidths for most of the pass-bands are in picometer scale. In addition, each pass-band has flat null sideband and symmetric band shape, and can be angularly tuned in practical calibrations.

2. Geometric configuration and theoretical model

The proposed filter in the form of prism pair coupled planar optical waveguide consists of a POW with two prisms cladding to the two sides of it, which is schematically illustrated in its longitudinal cross sectional view in Fig. 1.

Fig. 1 Schematic illustration of the filter structure of a prism pair coupled planar optical waveguide in view of its longitudinal cross section.

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The POW is composed of at least three layers named as substrate, guiding, and cladding of the POW as indicated by ‘s’, ‘g’, and ‘c’ respectively in Fig. 1. To act as an optical waveguide, the refractive index of the guiding layer, represented by ‘n_g’, should be larger than that of the substrate, ‘n_s’, and the cladding, ‘n_c’. The two coupling prisms, as the entrance prism and the exit one as abbreviated by ‘en’ and ‘ex’ in Fig. 1, which can be made of a cylindrical bulk material cut at a desired angle along its longitudinal axis, are attached to the substrate and the cladding sides of the POW. Their refractive indices represented by ‘n_en’ and ‘n_ex’ respectively, are larger than that of the substrate and the cladding layers. A beam of light, as indicated as ‘LB’ in Fig. 1, passes the filter along its longitudinal axis. A coordinate system is built in the figure, with the X axis along the normal and the Z axis along the guiding direction of the POW, respectively. The origin point of the X axis is set at the interface between the en prism and the substrate of the POW. The Y axis is normal to the paper plane, which is not shown. The angle θ inside of the en prism between the X axis and the light beam is the coupling angle, which has to be greater than the critical angle, defined as θ_c = arcsin(n_s/n_en), at that interface.

To characterize the light transmission through this kind of filter, a multi-layer model is implemented [18,19], i.e., assume that a beam of plane wave of TE polarization, whose electric field vector is perpendicular to the XZ plane of Fig. 1, is incident onto the filter. Neglecting the reflection loss at the prism surfaces, the light beam encounters multiple reflections and transmissions at each interface of the POW. The wave vector inside of the filter can be decomposed into two orthogonal components, one is along the guiding (the Z) direction, and the other one is along X direction, which is the direction of stratification of the POW layers. The effective wave propagation vector along the Z direction is an unknown constant β, depending on the coupling angle, while the wave component in the X direction can be modelled as a summation of the forward and backward parts of the light wave, whose propagation constants are of -λ_i and λ_i respectively, where i is the layer index of the POW. Each part of the wave is the accumulation of those multiple reflected and transmitted sub wavelets, respectively. The electric field of the optical wave along the direction of the filer can be expressed as the following set of equations [18]:

E_{e n} = \begin{matrix} (A_{e n} e^{- λ_{e n} x} + B_{e n} e^{λ_{e n} x}) e^{- j β z} & x \leq 0 \end{matrix}

\begin{array}{l} E_{i} = (A_{i} e^{- λ_{i} (x - p_{i - 1})} + B_{i} e^{λ_{i} (x - p_{i})}) e^{- j β z} \\ \begin{matrix} p_{i} \geq x \geq p_{i - 1}, i = 1... N \end{matrix} \end{array}

E_{e x} = \begin{matrix} (A_{e x} e^{- λ_{e x} (x - p_{N})}) e^{- j β z} & \begin{matrix} x \geq p_{N} \end{matrix} \end{matrix}

where

A_{j}

,

B_{j}

(j = 1…N) represent the coefficients of the forward and backward propagating eigen waves, respectively. The eigenvalue is defined as

λ_{j} = \sqrt{β^{2} - k_{0}^{2} n_{j}^{2}}

, where

β = k_{0} n_{e f f}

is the effective wave vector along the Z direction.

k_{0} = 2 π / λ_{w}

is the wave vector with wavelength of λ_w in vacuum. The effective refractive index is defined as

n_{e f f} = n_{e n} \sin θ_{}

.

p_{i} = \sum_{j = 1}^{i} d_{j}

are the positions of the interfaces in the filter structure, where d_j indicates the thickness of each layer in the POW.

With the continuity conditions for the TE wave at each interface (p_i), i.e. the electric field and its derivative in the X direction are continuous [20], and with an arbitrarily assumed field amplitude A_en at the en-substrate interface, the field amplitude reflection r and transmission coefficient t in the en and ex prism region can be defined as $r = B_{e n} / A_{e n}$ and $t = A_{e x} / A_{e n}$ , respectively. Analytical expressions for r and t can be derived as functions of the eigen values of all spatial regions of the filter and layer thicknesses in the POW:

r = \frac{λ_{e n} M_{11} + M_{21} - λ_{e n} λ_{e x} M_{12} - λ_{e x} M_{22}}{λ_{e n} M_{11} - M_{21} - λ_{e n} λ_{e x} M_{12} + λ_{e x} M_{22}}

t = \frac{2 λ_{e n} \prod_{i = 1}^{N} (2 λ_{i} e^{- λ_{i} d_{i}})}{λ_{e n} M_{11} - M_{21} - λ_{e n} λ_{e x} M_{12} + λ_{e x} M_{22}}

where

(\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) = \prod_{i = 1}^{N} (\begin{matrix} λ_{i} (1 + e^{- 2 λ_{i} d_{i}}) & - (1 - e^{- 2 λ_{i} d_{i}}) \\ - λ_{i}^{2} (1 - e^{- 2 λ_{i} d_{i}}) & λ_{i} (1 + e^{- 2 λ_{i} d_{i}}) \end{matrix})

, and N is the total number of layers in the POW.

The intensity reflectance and transmittance in the en and ex prism regions, can then be obtained as $R = {| r |}^{2}$ and $T = (λ_{e x} / λ_{e n}) {| t |}_{}^{2}$ [18,21], respectively, and by means of which, the numerical simulations on R and T, with material dispersions as well as the material absorptions in all of the constituent components, can then be conducted. Since we are concerned more about the transmittance of filter as a bandpass device, in the following section the transmission losses at the two surfaces of the filter were neglected as with appropriate anti-reflection (AR) coatings. It is also convenient to separate the efficiency of the light transmission through the filter into two parts; one is the efficiency in the two cladding prisms, which can be estimated as a transmission reduction of the light through a whole cylindrical bulk, as long as the length of the cylinder is determined. And the other part is the transmittance T of the light traversing across the POW as derived above. Therefore the following results of the transmittance refer to the later part of the efficiency, after which the first part is addressed.

3. Numerical simulations and results

With the expressions for the R and T derived from Eq. (4) and (5), the light transmission upon the POW in the filter as a function of the coupling angle θ and wavelength in the visible spectrum are calculated. The simulations are decomposed into two steps. The first one is the searching for guided modes (GMs), as a function of the coupling angle starting from the critical angle θ_c to the grazing one (90°). Once the transmittance T reaches a peak value, or the reflectance reaches a minimum, a GM is found, and the corresponding incident angle is called the resonance angle θ_r. With all GMs found in that way, the second step is that, at each resonance angle, the transmittance is calculated in the visible spectrum ranging from 0.4 to 0.8μm at a certain wavelength resolution which depends on the bandwidth. Once a high value of T is reached, a pass-band is then found. During the process of the second step, the material dispersion and the extinction coefficient dispersion of all of the constituent materials of the POW are taken into considerations in the filter. The dispersion parameters are adopted from a public data source [22].

To make the simulated results feasible, we have chosen some common materials that are available from the market. SCHOTT flint glasses for example, their dispersion parameters as well as constants in Sellmeier’s formula, are also accessible publically [22]. Table 1 lists the proposed design for such a filter, as depicted in Fig. 1, and the material parameters at wavelength of 0.6328μm. SCHOTT flint glass with product numbers N-BAF10 and N-SF4 are selected as the two coupling prisms and the guiding layer, respectively. A fused silica glass is symmetrically selected as the substrate and the cladding layers of the POW. Although the explicit extinction coefficients for the materials we chose are not available, the values of the extinction coefficients at a certain given wavelength could be retrieved from the public source [22], except for the fused silica. A moderate value of 1.0 × 10⁻⁸ for fused silica without dispersion is then adopted in the simulations. We feed a series of discrete wavelengths to get the corresponding values of the extinction coefficients (κ) for the constituent materials, and then made smooth connections to those discrete points by means of numerical interpolation with Akima algorithm [23].

Table 1. Optimized parameters of the constituent materials of a multi-mode prism pair coupled POW filter

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The structural configuration in this work is symmetric, i.e. the two coupling prisms are of the same material, and so do the substrate and cladding layers of the POW, as listed in Table 1. Figure 2 illustrates the dispersions of the refractive indices (n_s) and the extinction coefficients (κ_s) of the three constituent materials in the whole visible spectrum. It is shown that the change rates for the refractive indices are smooth in the whole spectral range, while that for the extinction coefficients are more variable especially in the blue part of the spectrum. The pass-band spectra of the filter are then calculated with these interpolated values at every interested wavelength.

Fig. 2 Dispersions of the refractive indices (a), and extinction coefficients (b), of the constituent materials of the bandpass filter. Refer to the context for the detailed descriptions of the roles these materials played in the filter.

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The central wavelength λ_c of our filter is designated as 0.6328μm, for the sake of the availability of the light source for future experiments. Figure 3 illustrates the spatial profiles of refractive index and the extinction coefficients of the constituent materials in the filter structure at the central wavelength λ_c. In searching for the guided modes (GMs) in the structure, the coupling angle varied while the wavelength of the light was set fixed at λ_c. Figure 4 illustrates the guided mode resonances in terms of the transmittance and reflectance in the filter as functions of the coupling angle and the wavelength, respectively. In Fig. 4(a) at λ_c, there are three GMRs sustainable in the filter in the range of the searching angles, 65.554°, 73.802°, 88.333°, corresponding to the 2nd, 1st, and the 0th order of the GMs, respectively. Because of the existence of material losses in the constituent layers, the transmittance of the 0th order GM is lower than that of the other two GMs, but it is still over 93%, which is acceptable in terms of a practical device. Transmittances for each pass-band are listed in Table 2.

Fig. 3 Spatial profiles of the refractive index (a), and extinction coefficient (b), along the X direction in the filter structure as in Fig. 1 at the central wavelength of λ_c = 0.6328μm.

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Fig. 4 Guided mode resonances for the transmittance and reflectance in the filter as functions of (a), the coupling angle at the central wavelength λ_c = 0.6328μm, and (b), the wavelength at the fixed incident angle of 65.554°, i.e. the 2nd order GM.

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Table 2. Pass-band parameters for all guided modes

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In searching for the pass-band at each GM, the incident angle of the light beam was set at the corresponding resonance angles, and the wavelength was scanned from 0.4μm to 0.8μm at a certain precision step depending on the bandwidth of the pass-band. Figure 4(b) illustrates the pass-band for the 2nd order GM as an example, i.e., at the coupling angle of 65.554°. Similar results were also obtainable for the 1st and the 0th order GMs except with different numbers of pass-bands. It is shown in Fig. 4(b) that for this 2nd order GM, high transmittance for each pass-band can be obtainable even though there exist material losses, especially for the violet band that is centered at λ_c = 0.4026μm.

Figure 5 illustrates the pass-bands in terms of the transmittance and reflectance of the filter for the violet band of the 2nd order GM and the red band of the 0th order GM. The bandwidths are 26.30pm and 9.29pm respectively. It is clear that the pass-band of this kind of filter is very sharp in band edges and symmetric in band shape, with complete null leakage in side band. All of the parameters of the pass-bands for the three GMs, i.e. the central wavelength λ_c, bandwidth Δλ, and the corresponding peak transmittance, are listed in Table 2.

Fig. 5 Pass-bands in terms of the transmittance and reflectance of the filter for (a), the violet band of the 2nd order GM, and (b), the red band of the 0th order GM, respectively. The bandwidths are 26.30pm and 9.29pm respectively.

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Peak transmittances and bandwidths of all of the pass-bands in the filter are illustrated in Fig. 6. It is clear from Fig. 6(a) that the transmittance of the pass-band is higher with increasing GM order. Especially for the 2nd order GM, all of the pass-bands have high transmittance even though material loss is relatively high in the violet band. For the 0th order GM, the material loss plays a vital part for the shorter wavelength bands, while the red band survives. On the other hand, from Fig. 6(b), it is clear that the higher the GM order, the wider the bandwidth is. And the bandwidth becomes wider in approximately exponential way for a certain order GM. Fortunately however the bandwidths for most of the pass-bands in the visible spectrum are in picometer scale except for the red band of the 2nd order GM. If the material losses could be reduced, bandwidths of the shorter wavelength bands of the 0th order GM could be in the sub-picometer regime.

Fig. 6 Peak transmittances (a) and bandwidths (b) of different pass-bands of the three guided modes in the proposed filter.

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A small change of the coupling angle on the side of the filter can cause slight changes in the major parameters of a pass-band. Figure 7 shows as an example of the angular tuning on the violet pass-band of the 2nd order GM. The step of the angular variation is 0.025°. Figure 7(a) illustrates the blue shift of the pass-band as the coupling angle increases. Figure 7(b) shows the changes of the major parameters, including the peak transmittance, the bandwidth, and the central wavelength, of the pass-band with angular tuning. It is shown that, in general, with the increase of the coupling angle, those major parameters decrease only slightly.

Fig. 7 (a) Angular tuning on the violet pass-band of the 2nd order GM about the resonance angle θ_r = 65.554° with an angular step of 0.025deg. (b) Tuning effect on the major parameters, i.e. the peak transmittance, bandwidth, and the central wavelength of the pass-band of (a).

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4. Discussions

According to the results above, especially from Fig. 1 and Fig. 6, it is clear that with a structure as simple as a 3-layer POW together with two cladding prisms, a high efficiency multi-band optical bandpass filter can be formed. It is structurally simple compared to those interference filters that are made of large numbers of film stacks. The bandwidths of the pass-bands can be achieved in pico-meter scale compared to our previous work with nanometer bandwidths [18]. The bandwidth compression can be accomplished by simply increasing the refractive index of the guiding layer. Therefore by changing the material of the guiding layer, filter devices of different bandwidth from nano to pico-meters can easily be made. Meanwhile from Table 2, it is clear that even though with the practical material dispersions and losses, the efficiencies of those pass-bands, especially for the higher order GMs, are still higher than that of those conventional filters.

Simplicity is also apparent in comparison with the extensively investigated waveguide grating (WG) based optical filters, in which a grating of sub wavelength pitch is deposited within or on top of the filter structure. Since the real profile of the sub wavelength grating is usually not easy to be fully matched to the designed one, the performance of this WG based bandpass optical filter is therefore deteriorated, usually in bandwidth [15–17]. Our proposed filter structure consists of only planar layers, which are practically easy in fabrication. Another advantage of our filter is that the pass-band is highly symmetric with null sideband. For those WG based filters, the pass-band is usually asymmetric in band shape, with high skirt extending widely into the sideband [4]. In addition, large aperture devices, which are desirable in high precision spectroscopic measurements, are easier to be implemented in our proposed form, whereas large sized WGs are not feasible with common facilities.

We should note that the feature of multiple bands in the simulated filter is not unexpected because of multiple guided modes sustainable in the POW, which can also be found in WG based filters [5]. Their common foundation is the planar optical waveguide that supports multiple guided modes. The two-step resonance in our structure makes it more sensitive to the GM coupling condition, so that the bandwidths become more compressed [24–26].

It needs to be pointed out that the transmittances in the simulations above are for the POW layers only. The overall transmission efficiency of the filter needs to include the energy loss in the bodies of the two prisms. That part of the efficiency reduction depends not only on the extinction coefficient of the prism material, but also on the beam path length inside of the two prisms. Therefore an acceptably quasi-empirical level of the loss in the prism cylinder is required for a specific situation, 1% for example, which is comparable to the surface losses of the prism coatings. Once the level of the prism loss is designated, the length of the prism cylinder can be determined according to the extinction coefficient at the central wavelength due to the extinction dispersion. Thereafter with the given GM order, which corresponds to the cutting angle of the prism cylinder, the minimal geometrical configuration of the filter, in terms of length by diameter, can then be settled. For example, let us assume that the violet band of the 2nd order GM is to be used in a filter, given a 1% prism loss, then the minimal filter size can be determined as 4.34 × 1.97 mm² in length by diameter. In this case the peak efficiency of the filter is 93.89%, with 1% reduction from the value (94.84%) listed in Table 2. Another example is the size of the filter of the red band of the 0th order GM, which is 53.59 × 1.56 mm², and its peak efficiency is reduced to 92.50%, which is 1% lower than 93.43% of the tabulated value in Table 2.

The tuning capability of the pass-bands, as illustrated in Fig. 7, demonstrates the flexibility of the filter in practical usage. If there are minor drifts of the structural parameters, such as the refractive index and/or the thickness of any of the layers, the filter can still be usable with slight adjustment of the angular alignment of the filter on site. This property can make the filter versatile in many applications such as in laser system, ultra-precision spectroscopic measurements, optical communications, as well as in sensing.

5. Conclusion

With the adoption of practical material dispersions in refractive index and extinction, we have numerically simulated a new type of optical bandpass filter that is able to have multiple bands in the visible spectrum with high efficiencies and extremely narrow bandwidths in picometer scale. This new type of filter is composed of a multi-mode planar optical waveguide that is sandwiched between two cladding prisms. The number of the pass-bands depends on the guided modes sustainable in the POW of the filter, as well as the loss level in the constituent materials. Due to its simple structure of only a few layers (three in minimum) of planar films, it is easier in device fabrication compared to the conventional interference filters and the newly WG structures. Advantages over the WG filters are also apparent, such as highly symmetric pass-band shape and null leakage in sideband. In addition, it is easier to make large aperture devices which are desirable in high precision spectroscopic measurements. Therefore potential applications of this new type of optical filter can be found in fields of laser system, optical communication, spectroscopy, and sensing et al.

Funding

National Natural Science Foundation of China (NSFC) (61575047).

References and links

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Constituent	Refractive index	Extinction coefficient κ ( × 10⁻⁸)	Thickness (μm)	Product number
En prism	1.6671	4.344		N-BAF10
Substrate	1.4570	1.0	ds = 0.5	Fused silica
Guiding	1.7496	1.889	dg = 1.56	N-SF4
Cladding	1.4570	1.0	dc = 0.5	Fused glass
Ex prism	1.6671	4.344		N-BAF10

GM order	θ_r (deg.)	band	T_p (%)	λc (μm)	Δλ (pm)
2	65.554	v.*	94.84	0.4026	26.30
		b.	99.48	0.4525	101.6
		g.	99.95	0.5244	417.0
		r.	99.98	0.6328	1799
1	73.802	b.	78.82	0.4350	3.72
		g.	98.45	0.5100	27.0
		r.	99.86	0.6328	265.0
0	88.333	v.	0.140	0.4180	0.73
		b.	24.63	0.4932	0.78
		r.	93.43	0.6328	9.29

Constituent	Refractive index	Extinction coefficient κ ( × 10⁻⁸)	Thickness (μm)	Product number
En prism	1.6671	4.344		N-BAF10
Substrate	1.4570	1.0	ds = 0.5	Fused silica
Guiding	1.7496	1.889	dg = 1.56	N-SF4
Cladding	1.4570	1.0	dc = 0.5	Fused glass
Ex prism	1.6671	4.344		N-BAF10

GM order	θ_r (deg.)	band	T_p (%)	λc (μm)	Δλ (pm)
2	65.554	v.*	94.84	0.4026	26.30
		b.	99.48	0.4525	101.6
		g.	99.95	0.5244	417.0
		r.	99.98	0.6328	1799
1	73.802	b.	78.82	0.4350	3.72
		g.	98.45	0.5100	27.0
		r.	99.86	0.6328	265.0
0	88.333	v.	0.140	0.4180	0.73
		b.	24.63	0.4932	0.78
		r.	93.43	0.6328	9.29

Multi-band optical bandpass filter with picometer bandwidth in visible spectrum formed by prism pair coupled planar optical waveguide

Abstract

1. Introduction

2. Geometric configuration and theoretical model

3. Numerical simulations and results

4. Discussions

5. Conclusion

Funding

References and links

Cited By

Figures (7)

Tables (2)

Equations (5)

Optics Express