The present disclosure relates to waveguide Bragg gratings and more particularly to waveguide Bragg gratings designed by a Layer Peeling/Layer Adding Algorithm.
One of the simplest optical filters is a Fiber Bragg Grating (FBG) which is basically a special optical fiber with periodic index variation in its fiber core region. Periodic index variation is generally effected with photo-sensitive fibers, excimer lasers and phase-masks. When light passes through the FBG with periodic index variation, one particular wavelength band will be blocked and reflected, while all the other wavelengths will not be affected and will be transmitted. The central reflected wavelength λB, which is called Bragg wavelength, can be calculated using the following phase-matching equation:
λB=2neΛ (1.1)
Here, ne is the effective index of the FBG, and Λ is the period for the index variation. Wavelength satisfying the above equation will satisfy constructive interference for reflection. Thus this wavelength will be reflected by the Bragg grating. The degree of reflection and the 3-dB bandwidth of the reflection band, will depend on the number of periods and the amount of effective index variation. The simple FBG reflects only one wavelength and permits all other wavelengths to pass through.
Waveguide Bragg Grating (WBG) is the counterpart of FBG on a planar platform. Compared with FBGs which have the benefits of ultra-low propagation loss and perfect polarization independence, the major advantage of WBGs is their compactness and thus the capabilities of integration with other devices.
There is a common limitation for both FBG and WBG: they can remove just one single wavelength band in the transmission spectrum. For most practical applications, the removal of that particular spectral band may be already sufficient. However, there are still some applications which require the removal of more than just one single wavelength.
The embodiments of the present disclosure provide significant and non-obvious advantages over the prior art by providing a waveguide Bragg grating that is configured as and functions as a complex waveguide Bragg grating that outputs a reconstructed complex optical spectrum from a target input complex optical spectrum.
More particularly, the present disclosure relates to a waveguide Bragg grating that includes a silicon substrate defining a length, a width and a depth, and a silicon dioxide (SiO2) cladding over the silicon substrate and encasing a silicon nitride (Si3Ni4) core extending along the length of the silicon substrate and defining a variable width and thickness wherein the silicon nitride (Si3Ni4) core is configured as and functions as a complex Bragg grating waveguide,
In an aspect of the present disclosure, the thickness of the silicon nitride (Si3Ni4) core ranges from 40-400 nm.
In an aspect of the present disclosure, the thickness of the silicon nitride (Si3Ni4) core is 100 microns (μm).
In an aspect of the present disclosure, the waveguide Bragg grating is designed by: determining a grating profile of the silicon nitride (Si3Ni4) core from a Layer Peeling algorithm and a Layer Adding algorithm; and mapping the grating profile to a 1-layer waveguide structure with varying width dimensions.
In a still further aspect of the present disclosure, the waveguide Bragg grating is further designed by: relating the grating profile to an effective index variation defining a range along the grating and mapping the range of the effective index variation to the 1-layer waveguide structure with varying width dimensions, such that a single specific waveguide width corresponds to a single specific effective index, thereby converting output of the Layer Peeling algorithm and the Layer Addition algorithm to an aperiodic array of widths for the complex waveguide.
In a yet further aspect of the present disclosure, the waveguide Bragg grating is designed by: discretizing the waveguide grating into individual rectangular segments each defining a fixed width and a variable length such that the number of waveguide segments equals to a number of segments in the aperiodic array of widths for the complex waveguide.
In another aspect of the present disclosure, the waveguide Bragg grating is further prepared for electron beam lithography via simulating the aperiodic array of widths via finite difference method (FDM) and eigen mode expansion (EME).
In a still further aspect of the present disclosure, the simulating the aperiodic array of widths via finite difference method (FDM) and eigen mode expansion (EME) includes simulating a three-dimensional array of widths via finite difference method (FDM) and eigen mode expansion (EME).
The present disclosure relates also to a method of designing a waveguide Bragg grating by: waveguide Bragg grating is designed by: determining a grating profile of the silicon nitride (Si3Ni4) core from a Layer Peeling algorithm and a Layer Adding algorithm; and mapping the grating profile to a 1-layer waveguide structure with varying width dimensions.
In a still further aspect of the present disclosure, the waveguide Bragg grating is further designed by: relating the grating profile to an effective index variation defining a range along the grating and mapping the range of the effective index variation to the 1-layer waveguide structure with varying width dimensions, such that a single specific waveguide width corresponds to a single specific effective index, thereby converting output of the Layer Peeling algorithm and the Layer Addition algorithm to an aperiodic array of widths for the complex waveguide.
In a yet further aspect of the present disclosure, the waveguide Bragg grating is designed by: discretizing the waveguide grating into individual rectangular segments each defining a fixed width and a variable length such that the number of waveguide segments equals to a number of segments in the aperiodic array of widths for the complex waveguide.
In another aspect of the present disclosure, the waveguide Bragg grating is further prepared for electron beam lithography via simulating the aperiodic array of widths via finite difference method (FDM) and eigen mode expansion (EME).
In a still further aspect of the present disclosure, the simulating the aperiodic array of widths via finite difference method (FDM) and eigen mode expansion (EME) includes simulating a three-dimensional array of widths via finite difference method (FDM) and eigen mode expansion (EME).
The present disclosure relates also to a method of manufacturing a waveguide Bragg grating that includes providing a silicon wafer thermal SiO2 layer grown on a first surface of the silicon wafer; depositing via using low-pressure chemical vapor deposition (LPCVD) a Si3N4 layer on the thermal SiO2 layer; patterning a profile of the waveguide Bragg grating via electron beam lithography; providing a hard mask on the Si3N4 layer; performing reactive ion etching of the Si3N4 layer where it is not protected by a mask and removing the hard mask; depositing a low-stress SiO2 layer on top of the wafer via one of a silane based plasma-enhanced chemical vapor deposition of SiO2 or a low-stress tetraethoxysilane (TEOS) plasma-enhanced chemical vapor deposition (PECVD) process; and cleaving end-facets to form thereby a complex waveguide Bragg grating.
In another aspect of the present disclosure, the method of manufacturing further includes polishing a second surface of the silicon wafer wherein the second surface is on an opposing side of the first surface of the silicon wafer prior to cleaving end-facets to form thereby a complex waveguide Bragg grating.
In an aspect of the present disclosure, the patterning of a profile of the waveguide Bragg grating via electron beam lithography includes controlling writefield alignment of the profile; and overlapping neighboring writefields with each other to control stitching error.
In an aspect of the present disclosure, the providing a silicon wafer with a thermal SiO2 layer grown on a first surface of the silicon wafer includes providing a silicon wafer with a 3-15 μm thermal SiO2 layer grown on a first surface of the silicon wafer.
In an aspect of the present disclosure, the depositing via using low-pressure chemical vapor deposition (LPCVD) a Si3N4 layer on the thermal SiO2 layer includes depositing via using low-pressure chemical vapor deposition (LPCVD) a 100 nm thick Si3N4 layer on the 3-15 μm thermal SiO2 layer.
In an aspect of the present disclosure, the providing a hard mask on the Si3N4 layer, performing reactive ion etching of the Si3N4 layer and removing the hard mask are performed by providing a chromium hard mask on the Si3N4 layer, performing reactive ion etching of the Si3N4 layer and removing the chromium hard mask.
In another aspect of the present disclosure, the depositing a low-stress SiO2 layer on top of the wafer via one of a silane based plasma-enhanced chemical vapor deposition of SiO2 or a low-stress tetraethoxysilane (TEOS) plasma-enhanced chemical vapor deposition (PECVD) process includes depositing a 3-15 μm low-stress SiO2 layer on top of the wafer via one of a silane based plasma-enhanced chemical vapor deposition of SiO2 or a low-stress tetraethoxysilane (TEOS) plasma-enhanced chemical vapor deposition (PECVD) process;
The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.
The above-mentioned advantages and other advantages will become more apparent from the following detailed description of the various exemplary embodiments of the present disclosure with reference to the drawings wherein:
For the purposes of promoting an understanding of the principles of the present disclosure, reference will now be made to the exemplary embodiments illustrated in the drawings, and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of the present disclosure is thereby intended. Any alterations and further modifications of the inventive features illustrated herein, and any additional applications of the principles of the present disclosure as illustrated herein, which would occur to one skilled in the relevant art and having possession of this disclosure, are to be considered within the scope of the present disclosure.
The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments.
It is to be understood that the method steps described herein need not necessarily be performed in the order as described. Further, words such as “thereafter,” “then,” “next,” etc., are not intended to limit the order of the steps. Such words are simply used to guide the reader through the description of the method steps.
1: Introduction
1.1 Single-Notch Filter vs Multi-Notch Filter
To understand the advantages of the embodiments of the present disclosure, designing and fabricating optical filters is discussed herein It is best to start from one of the simplest optical filters, which is a Fiber Bragg Grating (FBG) [1] [2] [3] [4]. As shown in
λB=2neΛ (1.1)
Here, ne is the effective index of the FBG, and Λ is the period for the index variation.
Wavelength satisfying the above equation will satisfy constructive interference for reflection. Thus this wavelength will be reflected by the Bragg grating. The degree of reflection and the 3-dB bandwidth of the reflection band, will depend on the number of periods and the amount of effective index variation.
As seen in
Waveguide Bragg Grating (WBG) is the counterpart of FBG on a planar platform [5] [6] [7]. Compared with FBGs which have the benefits of ultra-low propagation loss and perfect polarization independence, the major advantage of WBGs is their compactness and thus the capabilities of integration with other devices. The complex waveguide and grating devices according to embodiments of the present disclosure can be fabricated on a small chip of only finger nail size. Also, because fibers and integrated waveguides both have its pros and cons (waveguides are lossy but compact, fibers have low loss but take more space), the interaction between them becomes an important and popular topic.
There is a common limitation for both FBG and WBG: they can remove just one single wavelength band in the transmission spectrum. For most practical applications, the removal of that particular spectral band may be already sufficient. However, there are still some applications which require the removal of more than just one single wavelength. For instance, as shown in
First of all, when some of the target wavelengths to be removed are distributed very closely in the spectral domain, it will be hard to realize these close spectral notches on a physical device. Assume the task is that it is desired to remove both the wavelengths of 1550 nm and 1551 nm with a 3-dB width of only 0.2 nm-0.3 nm.
That means if we are going to use one grating to filter out 1550 nm, and another grating for 1551 nm, assuming an average effective index of 1.500, the periods of these two gratings are
Setting aside whether a period of 516.67 nm can be written accurately or not, just look at the length difference between two periods of the 1st grating and the 2nd grating, which is only 0.33 nm in this example. That is to say, the period of the 2nd grating needs to be only 0.33 nm larger than the period of the 1st grating. In real fabrication, it is impossible to fabricate two separate gratings whose periods are just 0.33 nm different.
The second reason why cascading simple FBGs or WBGs is not desirable for multi-notch filtering applications lies in the total size and the overall loss. Connecting all these gratings together will give a long device overall (especially for FBG), and it will result in very low throughput correspondingly (especially for WBG).
Moreover, there are also potential problems of narrow-band filtering and side-lobes. If the bandwidths of the spectral notches need to be very narrow, say only 0.3 nm for the 3-dB width, then a long grating is required with a large number of periods and very small effective index variations.
Finally, if we want to regenerate a very smooth transmission/reflection spectrum without any obvious spectral side-lobes, then the techniques of apodization should be added to the design of gratings, which increases the complexity of the design and fabrication.
1.2 The Fundamental Problem of Optical Filter Design
It is useful at this point to consider some fundamental and general theory suitable for all the various applications, not just notch filters. Although single-notch and multi-notch filters have been discussed, the filtering function itself need not be limited to one single spectral type. Thus one of the fundamental problems for designing optical filters is whether it is possible to realize any arbitrary filtering function. A traditional optical filter, either a FBG or a WBG, removes only one particular spectral band in its transmission spectrum.
In
1.3 An Important Application in Astrophysics
The absolute and relative intensities are highly time-variable wherein in the astrophysics experiment, these narrow and highly time-variable OH-lines (of order 400 lines) of spectral width of 0.3-0.4 nm have to be removed simultaneously with a large suppression ratio of at least 15 dB.
The signal from these lines completely dominates over that of most astronomical targets, particularly faint distant sources whose emission is red-shifted into the near-infrared window due to cosmological expansion. As a result, astrophysicists have long sought to reduce or eliminate altogether the effects of the Earth's atmosphere by having facilities operating at very high altitudes (e.g., Stratospheric Observatory for Infrared Astronomy) or in space (e.g., Hubble Space Telescope). These facilities are, however, much more expensive than ground-based telescopes of the same sizes.
A potentially cheaper solution is to use ground-based telescopes but eliminate the OH emission lines using complex optical filters. At first thought, thin-film optical filters, fabricated using the optical thin-film coating technology [11], may be a good starting point. However, as will be shown later, to realize a complex filter that removes multiple narrow spectral lines, the filter itself has to be constituted of as many as 200,000 segments/layers, whose index variations have no regularity. This makes it unpractical for thin-film filters of as many as 200,000 segments/layers, whose index variations have no regularity. This makes it unpractical for thin-film filters.
It has been shown that the OH emission lines LOH can be filtered out using an aperiodic Bragg grating implemented on a fiber platform [12]. An aperiodic fiber Bragg grating (FBG) with a total length of 5 cm is capable of removing 100 narrow OH emission lines simultaneously. Such aperiodic FBG devices have been tested on ground-based telescopes and delivered promising results [8] [13] [14] [15].
According to embodiments of the present disclosure, an alternative approach is proposed based on Bragg gratings on planar waveguide platforms for applications requiring integration and compactness. In the astrophysics experiment, a large number (of order 400) of narrow OH lines with spectral width of 0.3-0.4 nm and spectral precision better than ±0.1 nm have to be removed simultaneously with a large suppression ratio of at least 15 dB, as they all contribute to a broad and highly time-variable background light level in the spectrometer. Such a background has to be removed in order to study faint objects in the night sky. The present disclosure presents the theoretical and experimental approaches for realizing such a complex waveguide Bragg grating (CWBG). As discussed below, this type of CWBG may find applications not only in the field of astrophysics, but also potentially in the areas of ultra-fast pulse generation [16] [17] and slow-light [18].
1.4 Searching for an Integrated Optical Filter for Any Spectrum
Therefore, the embodiments of the present disclosure relate to designing a single grating which can realize basically any spectral shape. Fortunately, just focusing on the theory and disregarding the difficulty of implementation, there are indeed some very good theoretical models and methods to solve this problem [19] [20]. In general, all these methods are called inverse scattering (IS) algorithms, as they are just the reverse process of finding the spectrum from the effective index profile, which is categorized as forward scattering problems. According to the IS algorithm, one single complex-shaped grating is already enough to remove multiple spectral lines with various depths and bandwidths. The input here is the desired spectral shape, and the output of the theory shows the grating profile or the effective index distribution along the grating. To further comprehend these theories, a comparison between various methods of grating synthesis has been performed in Reference [21].
One objective of the embodiments of the present disclosure is to go from the algorithm to making a real complex grating or complex optical filter.
Such optical filters have already been realized successfully on the fiber platform to remove many spectral lines [12]. However, the size of this aperiodic fiber Bragg gratings (AFBG) is not very compact and the approach does not lead to ease of integration with other integrated photonic devices. Compared with FBG, WBG is implemented on the waveguide platform, and has much smaller footprint and is suitable for dense-integration applications in the future. The fabrication process of WBG is also CMOS-compatible, and thus provides certain advantages, WBG were also written by deep-UV lithography experimentally for random spectral tailoring [22] [23], however the algorithm only assumes low index variation and neglects second-order and higher-order reflections. Moreover, the fabricated WBG is based on a 2-layer waveguide structure, which requires two lithography and two etching steps, giving some difficulty of precise pattern overlap between multiple lithography steps. More lithography steps also indicate larger cost in the fabrication or manufacturing, thus limiting the future potential of mass production and commercialization.
More particularly, the waveguide Bragg grating 150 includes a silicon substrate 152 defining a length L, a width W and a depth D and a silicon dioxide (SiO2) cladding 156 over the silicon substrate 152. The silicon dioxide (SiO2) cladding 156 encases the silicon nitride (Si3Ni4) core 154 and extends along the length L of the silicon substrate 152 and defines a variable width Wc and thickness tc. The silicon nitride (Si3Ni4) core 154 is configured as and functions as a complex waveguide.
The thickness tc of the silicon nitride (Si3Ni4) core 154 may range from from 40-400 nm, depending if the waveguide is weakly or strongly guiding in the vertical X direction.
In one embodiment, the thickness tc of the silicon nitride (Si3Ni4) core 154 is 100 microns (μm).
The Si3N4/SiO2 waveguide material system 150 has compact size, low propagation loss, high coupling efficiency and an ultra-broadband transparency window. It is an elegant structure which scales like a small chip.
As described below in more detail with respect to
2: Layer Peeling & Adding Algorithm
In order to explain the theoretical algorithm for designing the optical filter, some introduction and derivations of the coupled mode theory and the transfer matrix method will be given at first. Then the Layer Peeling/Adding Algorithm will be discussed. Finally, several real filter design examples will be demonstrated to show not only the effectiveness of the algorithm and but also the effective index variations for realizing specific spectra.
2.1 Coupled Mode Theory
Coupled mode theory (CMT) is one of the most successful theories for investigating any type of grating structure. The grating is essentially an optical fiber or waveguide structure with an effective index perturbation. In a medium which has a constant refractive index everywhere, different optical modes are orthogonal to each other in principle. But as long as an effective index perturbation occurs, different modes will start talking to each other. For instance, some optical power of the forward-propagating mode may be transferred to that of the backward-propagating mode, the degree of which depends on the amount of the effective index variation. CMT is presented and described elegantly in a number of papers and books, and some of those references can be found in [24] [25] [26] [27] [28]. It is noted that the scientific notations are slightly different between these references. For simplicity, the notation in the present disclosure follows [24] closely. In this notation, light propagates along only +z (forward) and −z (backward) direction. The x and y are both traverse directions. Also, the implicit time dependence is exp(−iωt), so the forward propagating wave with the propagation constant β will have the form with the term exp[i(βz−ωt)].
To start with, several reasonable assumptions are made here. First of all, in the algorithm it is assumed the waveguide is lossless. In other words, the refractive index is a real number. Secondly, it is assumed that single-mode condition is observed in the entire spectral range, which means no higher order radiating mode occurs. This indicates that only forward-propagating mode and backward-propagating mode should be considered in the theory. It is also the real situation in FBGs and WBGs. Thirdly, the index variation is assumed to be much smaller than the average effective index n0. Let us first recall the four Maxwell's Equations:
where {right arrow over (E)} and {right arrow over (H)} are electric and magnetic field vectors, and {right arrow over (D)} and {right arrow over (B)} are electric and magnetic flux densities, respectively. {right arrow over (J)} is the current density and ρ is the free charge density.
To start with, several reasonable assumptions are made here. First of all, in the Then the derivations indicate the following. The scalar wave equation (deduced directly from the above Maxwell's equations assuming ρ=0 and J=0) tells us that
{∇2+k2n2(x, y, z)}Ε(x, y, z)=0 (2.2)
This can be further written as:
{∇t2+k2n2(x, y, z)+δ2/δz2}Ε2(x, y, z)=0 (2.3)
Here ∇2=δ2/δx2t+δ2/δy2, and k=ω/c is the vacuum wavenumber. Here, n is the overall effective index which includes both the average effective index and the effective index variation. As a comparison, we will now define another parameter no to represent the average effective index, which is a constant for the grating.
Since we are considering the coupling between the forward-propagating mode and the backward-propagating mode, the electric field can be written as
Ε(x, y, z)=b1(z)Ψ(x, y)+b−1(z)Ψ(x, y) (2.4)
The whole electric field Ε(x, y, z) should satisfy (2.3), and Ψ(x, y) should satisfy the wave equation with average index n0 below
{∇t2+k2n02−β2}Ψ(x, y)=0 (2.5)
To reach (2.5), it should be remembered that the electric field in a medium with constant index has a propagation term which is either exp(+iβz) or exp(−iβz).
From (2.3) (2.4) (2.5) we can obtain
Multiplying (2.6) by Ψ and integrating over the whole xy-plane, we can get
where
D
11(z)≈k(n−n0) (2.8)
because n2−n2≈2n0(n−n0), with the third assumption we made before. Now, (2.7) can be decomposed into a set of first order differential equations
If there is no index variation, n=n0, D11=0, then b1(z) ∝ exp(iβz) and b−1(z) ∝ exp(−iβz). Then b1(z) will have no interaction with b−1(z) at all. On the contrary, if there is an index variation, then D11≠0, so the forward and backward modes will start to interact with each other.
For grating 150, the index variation can be represented as
and D11(z) can be rewritten as
where κ(z) is a complex and slowly varying function of z. To further simplify (2.9) and (2.10), the new field amplitude u and v are defined as
Using (2.9) (2.10) (2.12) (2.13) (2.14 the coupled mode equations are obtained
where δ=β−π/Λ is the wavenumber detuning with respect to the central wavelength λ0=2n0Λ, and q(z) is the complex coupling coefficient q(z)=iκ(z)
From the Layer Peeling/Adding algorithm that is described in 2.2 below, we can calculate all the values of q(z), but for a physical optical filter, knowing only q(z) is far from satisfactory. To design a real optical filter, it is most important to know the effective index n(z) along that filter. The important relation between q(z) and n(z) is given below:
Now that we know the coupled-mode equations and the relation between q(z) and n(z). The coupled mode equations derived above are a set of elegant and classic equations. They form the basis for the theory that will be used later.
2.2 Transfer Matrix Method
Transfer matrix method (TMM) is the typical method used in the forward-scattering problems, as it calculates the transmission/reflection spectrum starting from the effective indices. It is a simple and straightforward approach shown in many books [27] [28]. On the contrary, the Layer Peeling algorithm, which is deduced below, allows us to calculate the effective index from the final spectrum.
Transfer matrix method uses a discretized model, where it discretizes the whole grating into a sufficient number N of small segments. Each segment is so short that it can be treated as having just a constant effective index variation intensity. In other words, the effective index of each Δ segment will vary sinusoidally according to (3.2) with Δn(z) to be constant. Δn(z) of different Δ segment may not be the same, and in our cases it can be very aperiodic. If the overall length of the grating is L, then the length of each segment will be
Δ=L/N (2.19)
Derived from the coupled mode equations before, the electric fields can be written as
Here, light propagates along the z direction, u(z) and v(z) are the forward-propagating and backward-propagating field amplitude at location z, u(z+Δ) and v(z+Δ) are the forward-propagating and backward-propagating field amplitude at location z+Δ. q is the complex coupling coefficient in accordance with the previous section, δ=β−π/Λ is the wavenumber detuning, and γ is defined as γ2=|q|2−δ2. (2.20) calculates u(z+Δ) and v(z+Δ) from the information of u(z) and v(z), given the effective index variation in this Δ segment. Note here that the effective index variation is not shown in (2.20) explicitly, and instead it is contained in q.
From the above equation for a single segment, the overall transfer matrix of the grating is
T=T
N
T
N−1TN−2 . . . T2T1 (2.21)
Notice that in (2.22), v(L)=0 since the light will only go forward at z=L. The transfer matrix T is also wavelength dependent, and the wavelength information is already contained in the γ and δ parameter. Finally, the reflection coefficient can be obtained from T as
r(δ)=−T21/T22 (2.23)
As a brief conclusion of the transfer matrix method, it should be emphasized that its main purpose is to calculate the transmission/reflection spectrum from the effective indices of the gratings/filters for all the wavelengths. It also means the direction of calculation is from the space domain to the frequency domain. It is considered as a forward-scattering method. 2.3 Layer Peeling/Adding Algorithm
Layer Peeling/Adding algorithm actually contains two sub-algorithms which use very similar approaches but totally opposite directions. The Layer Peeling (LP) algorithm receives the target spectrum as the input and output the distribution of the effective indices. The Layer Adding (LA) algorithm receives the effective index distribution as the input, and gives the spectrum as the output. In this sense, Layer Adding and Transfer Matrix Method have the same purpose, although Layer Adding is much faster than the Transfer Matrix Method. The details of LP and LA are discussed below. The present disclosure follows the notation from [19]. Some other good references of LP and LA can also be found in [24] [29] [30].
First of all, starting from (2.20), the transfer matrix Tj of the jth segment is decomposed as the multiplication of two sub-matrices.
This is done by simplifying the transfer matrix of each segment into a propagation matrix TΔ and a reflection matrix Tjρ. In the propagation matrix Tjρ, we only consider the propagation of light, so only the optical phase changes by exp(iδΔ). In the reflection matrix Tρj light from the jth segment sees the effective index of the (j+1)th segment, so some of the forward-propagating power might be reflected. TΔ is obtained from T by letting q→0, and Tj isρ obtained from T by letting q→∞ while holding qΔ to be a constant. ρ is the complex reflection coefficient, and it is related to q in the following equations:
From (2.24) (2.25) (2.26) we can write u(z+Δ) and v(z+Δ) as
It should be noted that the reflectivity is a function of z and δ and it can be written as
Therefore, from (2.29) (2.30) (2.31) we obtain the following two key equations
These two equations above provide the foundation for designing the complex waveguide Bragg grating 150. Eq. (2.32) is the Layer Peeling equation, and Eq. (2.33) is the Layer Adding equation. (2.32) allows us to obtain the reflectivity and the effective index of the (j+1)th segment, if we already have the reflectivity and the effective index of the jth segment. (2.33) does the reverse. It calculates the reconstructed spectrum (for comparison with the original target spectrum) from the calculated effective indices.
A key point to note is the physical meaning of Δ, δ and r(z, δ). As discussed before, Δ is the physical length of each segment used in LP/LA algorithm. δ is the wavelength detuning from the central resonance wavelength. So although the wavelength λ does not appear in (2.32) and (2.33), it has been implicitly represented by the term δ. r(z, δ) is the reflectivity r versus wavelength δ (now we know δ is just another expression of λ) at position z. So r(0, δ) is the original reflectivity spectrum seen from the beginning of the whole grating, r(Δ, δ) is the reflectivity spectrum seen from the beginning of the 1st grating segment, and r(2Δ, δ) is the reflectivity spectrum seen from the beginning of the 2nd grating segment, etc.
A simple analogy is shown below so the readers can understand the LP/LA algorithm better. Suppose there is an onion with a special shape (it may not be perfectly spherical, just like the fact that real applications may not have good-looking simple spectra). The onion is composed of many many layers, and each layer of the onion carries some specific information (thickness, any spot or irregularity for that layer, etc) about that layer, so if we peel this onion layer by layer we could know how the onion is constructed in detail. There is only one rule: to see the jth layer we first have to peel off the (j−1)th layer. So first of all, the outermost layer (the 1st segment) is peeled off, and its information (q, ρ, and neff) is written down (for later reconstruction), revealing the 2nd layer. Then the 2nd layer is peeled off and recorded, and then the 3rd layer, the 4th layer, so on and so forth. The peeling occurs in an iterative and sequential order. Finally, all the layers of the onion are peeled off, and the complete information of the onion is recorded. From these recorded information of layers we are able to reconstruct the same onion perfectly later on. This is exactly how the Layer Peeling works.
In this analogy, the original shape of the onion (without any peeling) is the target reflection spectrum which usually has a special shape, and the information of different layers is the effective indices of different discretized segments in the grating.
Referring to
From the initial target reflectivity spectrum r(z=0, δ), ρ(z=0) is obtained, which corresponds to the coupling coefficient of the 1st segment of the grating. Then using Eqn. 2.32, r(z=Δ, δ) can be calculated, which is essentially the reflectivity spectrum seen from the beginning of the 2nd segment of the grating. Then ρ(z=Δ) is obtained from the value of r(z=Δ, δ). As a next step, r(z=2Δ, δ) and ρ(z=2Δ) are calculated. This procedure can be iterated until information of all those grating segments are found.
LA is just the reverse of LP, and it is used to evaluate the performance of a grating, assuming the effective index profile is obtained already. It is like obtaining the overall shape of a complete onion by assembling all the layers together. At first, we start from the innermost layer (which corresponds to the last segment of the grating). Then the 2nd layer is added on top of the innermost layer. Then the 3rd innermost layer is added. This iterative process comes in a reverse sequence compared to LP. Finally, when all the layers are assembled together, the outer surface of the onion will be presented in a specific shape (which corresponds to the reconstructed reflection spectrum). Hopefully, this is the desired shape that is required to design.
The importance of LP/LA is that by using the LP/LA algorithm, any types of onions (optical filters) can be designed, analyzed and reconstructed.
One more property of the LP/LA algorithm is that (2.32) only gives the distribution of effective index of the grating, but it does not indicate the exact material system and platform upon which the grating will be implemented. Therefore, the LP/LA algorithm introduced here can be applied to all the optical filter structures and platforms. In practice, this algorithm can be applied for both FBG and WBG platforms.
2.4 Discrete Fourier Transform & Target Spectrum Preparation
In the previous section, we derived equations for Layer Peeling and Layer Adding algorithms. And it should be clear now that q(z), ρ(z) and neff(z)−n0(z) actually all indicate the same thing: the effective index variation. The current question is: what is the relation between the effective index neff(z) of a certain segment and the term r(z, δ), the reflectivity spectrum seen from the beginning of that segment?
In order to answer this question, let's write the equation of the inverse Discrete Fourier Transform (DFT) in the following way:
(2.34) is the inverse DFT equation which gives the impulse response h(j), starting from the discretized reflectivity spectrum. M is the number of spectral points in the algorithm. Let us recall that r(z, δ) is the reflectivity spectrum seen from the beginning of the location z versus wavelength δ. And it is important to notice that the effective index of the segment in the location z corresponds only to the first element of the impulse response h(0). The reason is that if we send an impulse to the grating, h(0) will be affected by the 1st segment of the grating only, since light at that time does not have enough time to travel to the other (2nd) segments. This rule applies for all the layers, so ρ(z)=h(0). Therefore
Also, in order for the DFT to work correctly, there is a relation between Δ in the time domain and the overall spectral detuning range δw in the frequency domain, which is shown below in (2.36). A whole step-by-step process of the algorithm and all these parameters will be discussed in the next section.
There is one last problem before we can run the algorithm correctly: the target spectrum needs some special treatment [19] [20]. If we take the inverse DFT to the original target spectrum (the default reflectivity spectrum is only a list of real numbers, without any imaginary part), the impulse response will probably have components for t<0. Such an impulse response does not exist in real filters and gratings. For real gratings, the impulse function h(t) always starts from t=0. Otherwise if h(t<0)≠0, it simply indicates that even before the impulse sees the grating, it already has some type of response from that grating!! That is not possible physically. In a physical world, we can never know what's inside a mysterious treasure box unless we first check its contents. No matter what detection methods we use (hands, eyes, or even X-rays), we have to first check it somehow. For a physical optical filter, the impulse response at time t<0 should always be zero.
Due to this consideration, the reflectivity spectrum used for the algorithm needs to be treated in the following way. An apodizing window (such as a Hanning function) is used to force the impulse function to become zero beyond a certain limit. Then the whole impulse response is shifted so that the first non-zero element starts at t=0. This new impulse response is then converted back to the new reflectivity spectrum using DFT. Finally, the new reflectivity spectrum can be used as a valid input to the Layer Peeling/Adding algorithm, so we can start our calculation and design.
2.5 Step-by-Step Guide of Layer Peeling/Adding Algorithm
The steps of the Layer Peeling/Adding algorithm are summarized below:
2.6 Design Examples of Layer Peeling/Adding Algorithm
Generally speaking, LP/LA algorithm can be applied to the synthesis of any random spectrum. As a comparison, traditional design methods can only regenerate spectrum of very limited shapes. To prove the capability of the LP/LA algorithm, two design examples are demonstrated below, showing the reliability and accuracy of the proposed LP/LA algorithm. For each design, we always start with a target transmission/reflection spectrum, then LP algorithm would tell us the effective index variation. Then the LA algorithm or the Transfer Matrix Method will show us the reconstructed spectrum. If the parameters of the grating are chosen appropriately, then the reconstructed spectrum should look very similar to the original target spectrum. The calculated effective index usually appears to have a strange shape and appears to be very aperiodic, and seen from naked eyes there seems to be no regularity at all. The advantages of the approach is that, with such seemingly “weird” effective index variations, different wavelengths will respond exactly how we want them to respond.
In the first example, a single grating G1 is designed whose filtering function has a special shape, with a rectangle, a triangle and an arc in its transmission spectrum.
2.6.2 Design Example 2: A Multi-Notch Filter with 150 Lines
More particularly, in this example, a very complex grating G2 is to be synthesized, which has 150 randomly-distributed narrow notches in the transmission spectrum. To make it more random and arbitrary, each of the notches will have its own suppression ratio. Again, the reconstructed grating RCS2 regenerates the original target spectrum TS2 successfully. LP/LA algorithm still works perfectly in this case.
From these two examples, the power and potential of our LP/LA algorithm is fully demonstrated. In the theory which is assumed to be lossless, the target spectrum can be either a reflection spectrum or a transmission spectrum, since they just add up to unity. In reality however, if a WBG is fabricated, some of the incoming light may experience the scattering loss or the radiation loss, which can be caused by the stepped waveguide widths due to the discretization process. Therefore, the detailed shape of the grating (e.g. the profile of the widths of the WBG) has to be determined as explained below in section 3. The information of the effective index variation obtained from the LP/LA is just a start for designing the real grating device, a 3-dimensional structure that can be fabricated and tested experimentally.
From the previous section 2, it has been proven that by using the LP/LA algorithm, a grating profile can be calculated for any random spectrum. The output of the LP/LA algorithm gives the coupling coefficient q(z) (also sometimes called the grating profile) of the grating directly, but that does not indicate anything about the detailed structure of the grating. To realize a physical grating, we need to obtain its effective index variation. In this section, a design procedure is described so a real CWBG can be synthesized.
The core part of designing a CWBG is to find an appropriate waveguide structure so that the grating profile q(z) can be realized by fabrication. The amplitude of q(z) reflects the change of the effective index at location z, which also denotes the strength of reflection at that point. q(z) and the change of the effective index are related by the following equation:
while the effective index of the waveguide at position z can be written as
Here, n0 is the average effective index, and the real effective index is varying in a sinusoidal way, controlled by the amplitude Δn(z), the period Λ and the phase θ(z). For a practical CWBG, q(z) will vary aperiodically along the z direction of the grating. To analyze the real change of the effective index, let us assume that the maximum amplitude of q(z) is about 40 cm−1 [8]. As a consequence, Δn(z) will also vary according to q(z). At the wavelength of 1550 nm, the maximum value of Δn(z) is about
Therefore, the difference between the maximum effective index and the minimum effective index needs to be about 0.002×2=0.004, which is the range over which n(z) can vary. Considering the limitation of practical e-beam lithography, it is reasonable at first to assume that 20 discrete waveguide widths are written in order to realize a CWBG, whose effective indices cover this region of about 0.004. Therefore, at first thought we can roughly divide 0.004 by 20, which means that the real effective indices need to vary as navg−0.002, navg−0.0018, navg−0.0016, . . . , navg, . . . , navg+0.002. The number 20 is just an assumption in the beginning, and if a more precise CWBG is required in the future, 100 or even more discrete waveguide widths may be necessary. For a FBG which is photosensitive, the effective index variation can be realized by adjusting the intensity of the interferometer, or the period/phase of the phase mask. On the other hand, for a WBG with a planar structure, the effective index variation is made possible by changing some parameters of the waveguide, for example, by varying the width of the waveguide core in different positions of the grating. Obviously, if the width of the waveguide can only change by a very small range, for instance if the width can only vary from 1.000 μm to 1.001 μm, then such a narrow waveguide width step will be impossible even using the state-of-the-art e-beam lithography (EBL). On the other hand, if the width is made to change too much, say from 0.5 μm to 1.5 μm, then the scattering loss caused from the many periods of the CWBG will be too large and will affect the overall transmission severely. A delicate balance needs to be established here.
In [7], a comparison between a ridge waveguide (2-layer structure) and a strip waveguide (1-layer structure) was made on the SOI platform. A ridge waveguide cross-section was determined to be preferable, since the change of the waveguide width was large enough to be written by lithography. Ref. [22] [23] also utilize this kind of ridge waveguide for implementing a weak WBG.
According to embodiments of the present disclosure, a 1-layer strip waveguide of Si3N4/SiO2 is actually more appropriate, which also has the additional advantage of only one lithography step and one etching step.
To explain in more detail, some simulation is done and the results are shown in
More particularly,
Suppose now that a Si3N4 waveguide thickness of 200 nm or 300 nm is used to fabricate the WBG 150 of
In order to get an effective index change of 0.004, the width can only vary by ˜50 nm and ˜20 nm, respectively. If we simply divide this number by 20, this gives a width step of 2.5 nm and 1 nm, respectively. This means that in order to fabricate this CWBG, a series of waveguide widths such as 1 μm, 1.001 μm, 1.002 μm, . . . , etc., need to be patterned clearly. It is impossible to write such small steps of the waveguide widths, even with the current state-of-the-art EBL system. On the other hand, if the thickness of the Si3N4 waveguide is only 50 nm, then even when the waveguide width varies from 0.4 μm to 2 μm, the change of the effective index is only about 0.002. As a consequence, large scattering losses will occur if the thickness of the WBG is only 50 nm.
Fortunately, if the thickness of the Si3N4 waveguide is 100 nm, then, when the waveguide width changes from 0.4 μm to 0.9 μm, the effective index is changed by about 0.004. In this case, it is possible to write various waveguide widths varying between 0.4 μm and 0.9 μm using EBL with enough resolution, and the scattering loss may also be tolerable in practice. Moreover, as described below, it is shown that such an Si3N4/SiO2 grating is inherently compatible with the high coupling efficiency of a Si3N4/SiO2 waveguide coupler, since 100 nm thick, 500 nm wide Si3N4 waveguide will give a coupling efficiency of 84% to SMF28 fiber, 100 nm thick, 700 nm wide Si3N4 waveguide will give a coupling efficiency of 92% to SM1500G80 fiber, and 100 nm thick, 900 nm wide Si3N4 waveguide will give a coupling efficiency of 96% to UHNA3 fiber. The required waveguide widths would then need to be within the range of 0.4-0.9 μm in order to realize the required range of effective index variation. Therefore, the high-efficiency, easy-to-fabricate Si3N4 waveguide couplers as concurrently disclosed in U.S. provisional applications 62/360,814 and 62/530,441 that are incorporated herein by reference, and to which priority is claimed, would be completely compatible with the fabrication of the CWBG 150 here.
The procedure to design the real CWBG is now understood: from the LP/LA algorithm, the grating profile q(z) or ρ(z) is obtained initially, which can be related to the effective index variation n(z) along the grating. This range of the effective index variation is then mapped to a 1-layer waveguide structure with varying widths. It means one specific waveguide width will correspond to one specific effective index. In reality, the mapping is realized by a Matlab script with a loop.
As illustrated in
For a real CWBG which is about 2 cm-3 cm in length, it is generally discretized into 200,000-300,000 small individual segments. Each segment is a tiny rectangle, with a length of 100 nm and a fixed width obtained from the mapping process before. The number of waveguide segments equals to the numbers in the grating's effective index array. In other words, if there is 200,000 points in the effective index profile, there will also be 200,000 small segments which constitute the whole CWBG.
So far, it is possible to obtain the widths of the grating, and now a complete GDSII file is required for the subsequent EBL. Since these 200,000 segments are not periodic at all, another Matlab script is written to remotely control the assembling of CWBG in FIMMWAVE/FIMMPROP. The advantage is that, after the grating assembling, as shown in
More particularly,
Thus, the process of effective index mapping, grating assembling and 3D simulation is shown in
In contrast to the prior art, to implement the embodiments of the present disclosure, a complex waveguide grating was constructed with 200,000 aperiodic segments. Although FIMMWAVE/FIMMPROP is reliable software, it is time-consuming to assemble all the 200,000 individual segments in FIMMWAVE/FIMMPROP. The embodiments of the present disclosure are the first ever simulation of such a complex waveguide grating with so many aperiodic segments. In the 3D simulation, FIMMWAVE/FIMMPROP only recognizes that the waveguide is a complex waveguide with varying width, and there are no accommodations in the software to address the configuration wherein the width is arranged and assembled in such an unusual way. The 3D simulation requires from a few days to a few weeks 24/7 unceasingly to execute, as FIMMWAVE/FIMMPROP has to calculate the overlap integral for 199,999 (200,000−1) interfaces one by one. And each run is only for a single wavelength. In the spectrum, there may be 1000 or even more spectral points, so the overall time is just 1000 times that. A future powerful upgrade of FIMMWAVE/FIMMPROP to support such intensive calculations would yield much faster simulation results.
The specific method steps for the method 340 of designing the complex waveguide Bragg grating 150 are illustrated in
The method includes step 3406 of relating the grating profile to an effective index variation defining a range along the grating and, in
In
In
In one embodiment, the step 3412 of simulating the aperiodic array of widths via finite difference method (FDM) and eigen mode expansion (EME) includes step 3414 of simulating a three-dimensional array of widths via finite difference method (FDM) and eigen mode expansion (EME).
The entire nano-fabrication process steps are listed in Appendix A of U.S. provisional application 62/530,441 that is incorporated herein by reference in its entirety.
More particularly, still referring to
In step 420, via low-pressure chemical vapor deposition (LPCVD) 415, the method includes depositing Si3N4 layer 154, having a thickness in one embodiment of 100 nm, on first surface 156′ of the 3-15 μm thermal SiO2 layer 156.
In step 430, via electron beam lithography (EBL) 425, the method includes patterning a profile of the waveguide Bragg grating 150 in a layer of poly(methylmethacrylate) PMMA 158 disposed on first surface 154′ of the Si3N4 layer 154.
In step 440, in conjunction with E-beam deposition 435, the method include providing a chromium hard mask 160 on first surface 154′ of the Si3N4 layer 154 and etching the Si3N4 layer 154 Other masks can be used for etching a pattern, apart from chromium. For instance, other metals apparent to those skilled in the art can be used. Chromium is just one option.
In step 450, in conjunction with reactive ion etching 445, the method includes removing the chromium hard mask 160 and performing reactive ion etching of the Si3N4 layer 154 to yield an Si3N4 layer 1540 on the thermal SiO2 layer 156 having a width dimension equal to the width dimension of the chromium hard mask 160, wherein the width dimension is reduced as compared to the width dimension of the thermal SiO2 layer 156 prior to reactive ion etching 445.
In step 460, in conjunction with a silane based plasma-enhanced chemical vapor deposition of SiO2 process 455a or low-stress tetraethoxysilane (TEOS) plasma-enhanced chemical vapor deposition (PECVD) process 455b, the method includes depositing 3-15 μm low-stress SiO2 layer 1562 on top of the wafer via the silane based plasma-enhanced chemical vapor deposition of SiO2 process 455a or the low-stress tetraethoxysilane (TEOS) plasma-enhanced chemical vapor deposition (PECVD) process 455b to encase the Si3N4 layer 1540 in SiO2.
In one embodiment, prior to method step 480, wherein the method includes as step 475 cleaving end-facets 158a and 158b (see
Step 480 includes yielding the complex waveguide Bragg grating 150 by step 475 of cleaving end-facet to form thereby complex waveguide Bragg grating 150.
It should be noted that polishing the second surface 152″ to reduce the thickness of the Si substrate 152 along the y-z plane is realized by polishing the Si layer from the bottom or second surface 152″. It is easier to put a cleave mark in upper surface 1562′ of SiO2 layer 1562 and then separate (cleave) the waveguide Bragg grating 150 by applying a force down from the top toward the bottom of the sample (force along the x-y plane) to result in facets 158a and 158b on the near and far ends in the z-direction and which extend across the y-direction of waveguide Bragg grating 150. If the waveguide Bragg grating 150 is thicker, it is more difficult to break the sample and the surface of the cleaved facet (in the x-y plane) might not be smooth anymore because the cleavage does not occur along a crystal cleavage plane. The Si causes the cleave to occur along a cleavage plane because of the Si crystalline structure. The SiO2 and the Si3N4 are not crystals. The Si layer 152 is much thicker than either the SiO2 or the Si3N4 layers.
In
It should be noted that while polishing the second surface 152″ assists in cleaving the sample, strictly speaking, the polishing is not necessary. Having a properly cleaved facet allows for optically coupling light into the waveguide 150 from a fiber.
Thus, the waveguide facets 158a and 158b are on the edges of the wafer, at normal (90 degrees) angle from the surface of the wafer. Alternatively, light can also be coupled in the waveguide using an on-chip grating that allows coupling of light at, or near, normal incidence to the wafer surface.
As defined herein, a film with stress level less than of order 100 MPa is a low stress film.
A SiO2 film grown using silane by PECVD would have losses of order few dB/cm. The stress in the film would be of order 3 GPa. By reducing the stress, the number of microcracks and the amount of light scattering in the waveguide are reduced, which translate to a lower propagation loss (<1 dB/cm), which would be desirable but not required for the basic demonstration of the complex waveguide Bragg grating 150 designed and constructed according to the present disclosure. An approach for reducing the stress in the complex waveguide Bragg grating 150 to a level less than 100 MPa is to perform TEOS based SiO2 deposition using PECVD.
Other deposition techniques apart from PECVD and other materials apparent to those skilled in the art are also contemplated.
In addition, a thermal anneal step on the sample at temperatures of order 1000-1200° C. for 2-4 hours might be beneficial for reducing absorption due to OH, Si—H, and N—H bonds in the spectral region between 1300-1600 nm from a few dB/cm to appreciably less than <1 dB/cm.
Since the method 400 involves nano-fabrication, it is important to show some Scanning Electron Microscope (SEM) figures of our real waveguide patterns produced by the method. For the presently disclosed CWBGs 150 which have continuously-varying waveguide widths and small theoretical feature size of less than 10 nm, a stable EBL with high resolution is highly beneficial.
One of the major issues with current Raith EBL systems involves human factors in operating the machine. The reason is that all the adjustments about focus, stigma and aperture are judged and optimized manually by the machine operator. These adjustments (especially stigma adjustments) can be somewhat difficult for those not having a great deal of SEM experience. It requires practice and real sample processing to learn and become familiarized.
Another small but really useful procedure is the Laser Height Sensing (LHS). LHS is an active focus-adjustment technique for large patterns with large dimensions and many writefields, especially centimeter-sized waveguides and waveguide gratings according to the present disclosure.
More particularly,
Although LHS is optional for Raith EBL system employed to perform the fabrication process for the complex waveguide Bragg gratings according to the present disclosure, it is recommended to use this function if the pattern has dimensions larger than a few millimeters. It is worth noting that on some of the newer EBL systems, such as those operating at 100 kV, LHS becomes a necessary step.
In
In
Stitching error is also a very common problem in e-beam lithography, and it can have several causes. The beam will jump between neighboring writefields, and this can cause the two writefields to be stitched with an offset, as shown in
More particularly, while the stitching is defined by crests spaced apart by 1.1 μm and troughs spaced apart by 1.4 μm, a stitch error 450 may occur between two writefields where the crests do not align and/or the troughs do not align.
On the Raith machine utilized to perform the nanofabrication process for the complex waveguide Bragg gratings of the present disclosure, there is a Fixed Beam Moving Stage (FBMS) configuration which can eliminate the stitching error. The reason why FBMS can eliminate the stitching error is actually quite simple: FBMS commands the beam to go from the beginning to the end without any jumps. A beam jump from one writefield to the next writefield causes the stitching error. When using the FBMS feature, the whole big pattern is no longer separated into small writefields. Instead, the beam will move continuously from one end of the waveguide to the other end with no jump.
However, FBMS handles only simple patterns, such as straight waveguides and circles. FBMS cannot write a CWBG with 200,000 individual segments, each with its own width. Secondly, FBMS does not allow the Laser Height Sensing, which is the feature of active focus-adjustment. The consequence is that using FBMS, the pattern will be perfectly focused only in one location, and will be more or less defocused in other positions.
Therefore, it is not recommended to use FBMS (unless stitching error is a very severe problem), because it cannot write complex waveguide gratings with constantly-varying widths, and it does not support LHS for active focus-adjustment.
To overcome the stitching error, a layer of conducting polymer (such as aqua-save) on top of the sample surface and a careful execution of the EBL process will greatly avoid stitching errors.
Finally, two SEM figures of an actual 47-notch grating are shown in
Referring to
That is,
In
In
In
In
As described above with respect to
TABLE 4.1 summarizes and compares the positions of the experimental transmission notches of the CWBG with its theoretical values. TABLE 4.2 lists the major parameters and experimental performance of the CWBG device. All the spectral lines are suppressed by at least 15 dB, while some of the deepest notches reach as much as 33.6 dB. The variance in the suppression ratios is due to the limited resolution of the current fabrication process. As explained above, the designed CWBG 150 has a width that varies between 800 nm and 1.6 μm, corresponding to the range of effective index variation (±0.0064). In order to further map each discretized width to the varying effective index precisely, a small width step is desirable. In this case, the actual waveguide widths are designed to be 800 nm, 808 nm, 816 nm, . . . , 1592 nm, 1600 nm, with a step of 8 nm. In other words, the continuously-varying effective index of ±0.0064 is sampled into (1600−800)/8+1=101 discretized values. Smaller steps (e.g. 4 nm or 2 nm) do not improve the CWBG performance in our simulations. This 8 nm width step can be considered ideal for CWBG 150 design, but in practice, such a small step cannot easily be patterned using a state-of-the-art e-beam lithography, because this length resolution stretches the capability of the instrument. This is why the suppression ratios in our experiment is not exactly the same for all the 20 notches.
It is worth noting that others have approached this same problem from the point of view of volume holography, using a 2-layer SOI waveguide structure fabricated by deep-UV lithography [22, 23]. Large differences between the theory and the experimental realization are observed. The theory assumes small index variation so the second-order reflections are neglected, which is good enough for weak gratings, but not suitable for gratings with both deep and narrow notches. By comparison, the LP/LA algorithm utilized herein according to the present disclosure makes no such assumptions, and can be applied to any arbitrary spectrum. Moreover, we experimentally demonstrate that one can design a CWBG with a simpler one-layer waveguide structure, fewer fabrication steps, deeper and narrower notches, and better spectral precision. This makes CWBG promising for various applications, especially in astrophysical observations.
Based on the first generation of CWBG, we designed and fabricated the second generation of CWBG, which has 47 notches between 1510 nm and 1610 nm.
It is noted that in
The major parameters of this CWBG device are listed in TABLE 4.5.
In the previous section, although a powerful CWBG has been experimentally fabricated to remove 20 prescribed spectral dips, there is still a critical problem which remains unanswered: what if those spectral notches are not exactly in their desired locations in the spectrum? How about those dips are all shifted to the left or to the right in the spectrum? This can be a serious problem, as the theoretical index used in the simulation may not be exactly the same as the real index of the CWBG materials. As a very simple example, if we want to design a spectral dip which is centered at 1550.00 nm, in practice the notch may be centered at 1546.55 nm.
To solve this problem, a method which can fully adjust the positions of the spectral notches is found. The basic principle behind this method is this: if the real index of the CWBG is larger than the theoretical index of the CWBG used in the design, then the spectral locations of the real notches will be on the longer-wavelength side of the original desired positions, i.e. they will be red-shifted in practice. So, if we want the real notches to be in the right place, we need to increase the theoretical index correspondingly.
Experimental verification of this idea has been performed, and it proves to be a successful solution. Two CWBGs (CWBG 1 and CWBG 2) are fabricated both with 20 spectral notches. Every parameter is set to be equal for these two CWBGs, but the theoretical index of CWBG1 is set to be 0.01 smaller than the index of CWBG 2. As the theoretical index of CWBG 2 is higher than that of CWBG 1, the notches of CWBG 2 should be shifted 10 nm left with respect to CWBG 1. This theoretical prediction is fully verified by the experimental results, as shown in
Therefore, the spectrum of the CWBG can be fully tuned, just by changing a parameter in the algorithm. The fabrication process will always remain the same and straightforward. Using the same principle, separations of the neighboring notches can also be tuned precisely, also simply by adjusting the designed notch position accordingly.
If the spacing between the lines is well-adjusted but, if the whole spectrum is evenly shifted from the desired spectral position, the whole spectrum can be shifted back to the right position by placing the filter chip on a temperature controller. By adjusting the temperature of the TE cooler, the whole spectrum can be shifted by increasing or decreasing the chip temperature.
As shown in
To solve this problem of absorption at 1500 nm, a thermal annealing process at the temperature of 1200° C. was performed. The spectrum of the 47-notch grating sample was measured before and after 0.5 hour at 1200° C. in
There is also another absorption dip at around the wavelength of 1.4 μm (not shown), also caused by H in SiO2 and Si3N4 layers. It is contemplated that that absorption dip may be removed using the annealing process.
Although the current experimental results are highly advantageous and non-obvious over the prior art, there are still many improvements which can be investigated further or performed in the future.
The present disclosure has demonstrated a complex waveguide Bragg grating CWBG 150 both theoretically and experimentally. The CWBG is able to remove multiple randomly-distributed wavelengths in the spectrum. It is fabricated using silica-on-silicon technology with the assistance of e-beam lithography. The whole process is designed to be very straightforward and easy to follow, but it produces one of the most powerful integrated optical filters.
To realize such a CWBG, a LP/LA algorithm is used at first for calculating the grating profile, or the effective index variation along the grating. Then the effective index is converted into a detailed shape/structure of a waveguide grating. The thickness of the waveguide grating is a constant, but the width varies in an aperiodic way. With the help of Matlab script control, about 200,000 small segments are assembled together to form the final shape of CWBG, whose overall length is about 2 cm. After this, a 3D simulation is performed using FIMMWAVE/FIMMPROP, to check if the reconstructed spectrum agrees with the target spectrum or not. If the results are acceptable, the profile of the CWBG is exported from FIMMWAVE/FIMMPROP to a GDSII file for e-beam lithography directly. High coupling fiber-to-waveguide couplers are also added to both sides of the CWBG. Finally, the spectrum of the CWBG is measured with a broadband lightsource, a fiber rotator and an Optical Spectral Analyzer (OSA). Full tunablility of the CWBG is also demonstrated, so the spectral locations of the real dips can appear precisely at their desired positions in the spectrum.
The future improvements will involve realization of 50 and 100 random spectral notches, better accuracy (±0.1 nm) of the spectral position for each notch, and further reduction of the propagation loss. Integration of CWBG with AWG and integrated waveguide lanterns will also be realized.
While several embodiments and methodologies of the present disclosure have been described and shown in the drawings, it is not intended that the present disclosure be limited thereto, as it is intended that the present disclosure be as broad in scope as the art will allow and that the specification be read likewise. Therefore, the above description should not be construed as limiting, but merely as exemplifications of particular embodiments and methodologies. Those skilled in the art will envision other modifications within the scope of the claims appended hereto.
Matlab programs used for Layer Peeling/Adding algorithm are disclosed herein. In this example, the detailed steps on designing a 20-notch filter are shown herein.
The following documents listed in this Bibliography are incorporated herein by reference in their entirety:
Publications
The following documents in this listing of Publications are incorporated herein by reference in their entirety:
Journal Papers
Conference Papers
This application claims the benefit of, and priority to, U.S. Provisional Patent Application No. 62/360,811 filed on Jul. 11, 2016, entitled “Generation of Arbitrary Optical Filtering Function Using Complex Bragg Gratings”, by Tiecheng Zhu et al., U.S. Provisional Patent Application No. 62/360,814 filed on Jul. 11, 2016, entitled “High Coupling Efficiency Between a Single Mode Optical Fiber and an On-Chip Planar Single Mode Optical Waveguide:” by Tiecheng Zhu et al., and U.S. Provisional Patent Application No. 62/530,441 filed on Jul. 10, 2017, entitled “Layer Peeling/Adding Algorithm and Complex Waveguide Bragg Grating For Any Spectrum Regeneration and Fiber-to-Waveguide Coupler with Ultra-High Coupling Efficiency”, by Tiecheng Zhu, the entire contents of each of which are incorporated by reference herein.
Number | Date | Country | |
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62360811 | Jul 2016 | US | |
62360814 | Jul 2016 | US | |
62530441 | Jul 2017 | US |