In summary, a modified formula for the three-focal-point method is proposed and com-pared with a numerical model of the scalar diffraction theory by a design example in Section 3.1. Simulation results show that the deviation between the two cases becomes serious as the ration E2/E1 approaches unity and the optimal half-separation a can be approximately obtained using the modified formula. The changed passband width can be achieved by changing the ratio E2/E1. The chromatic dispersion characteristic is also considered.
In Section 3.2, the impact of fabrication errors relevant to the phase and amplitude errors on a flat-top planar waveguide demultiplexer is evaluated. Simulation results show that the phase errors caused by the deviations of the positions for the vertices of the grating facets are the main sources of the crosstalk, and the amplitude errors caused by the roundings of the grating corners and the grating side-wall angle offset from the vertical cause additional losses. When the standard deviation of the position errors increases from 0 to 100 nm, the mean value of the crosstalks for 50 samples increases from −36.34 to
−29.78 dB. With a crosstalk criterion of −30 dB in our case, a photomask resolution lower than 40 nm is required.
In Section 3.3, a five-focal-point method is introduced for the optimal design of a flat-top planar waveguide demultiplexer. The multiobjective genetic algorithm is used to optimize the parameters. A larger −1-dB passband width of 30.53 GHz and a lower ripple of 1.09 × 10−4 dB for the case of E3/E1 = 1.0 with a channel spacing of 50 GHz are achieved. It shows that the five-focal-point method gives a near-equivalent ratio of the −1-dB passband width to the channel spacing and a lower ripple Rripple compared with the three-focal-point method based on the Gaussian approximation.
Chapter 4
Planar Waveguide Concave Grating Employing Dielectric Mirrors
In this chapter, a planar waveguide concave grating with dielectric mirrors is proposed to yield a high reflectance and low polarization-dependent loss (PDL). Although metal-ization [24, 41–43] can also be used to increase the reflectance of the grating, a metalized grating usually yields a higher PDL. This is particularly true in a Littrow mount [24].
To solve the problem, we show that a grating with facets employing dielectric mirrors can produce a high reflectance bandwidth covering the entire C-band.
4.1 Transfer-Matrix Analysis
We used the transfer matrix method [44] to design a high-reflectance multi-slot stack.
The fields are continuous across the interfaces of the films. For the dielectric mirror design, a series of the air slots and the high-index stacks with the quarter-wave widths yield a desired high reflectance. The fields at two outmost boundaries can be expressed in matrix form as where Ea and Ba denote the E- and B-fields at the input boundary, Eb and Bb denote the E- and B-fields at the output boundary. The overall transfer matrix M is the product
of all individual 2 × 2 transfer matrices. The individual 2 × 2 transfer matrix Mj of the jth air slot or high-index stack can be expressed as [44]
Mj =
where δj denotes the phase difference due to one travel of the jth air slot or high-index stack and can be expressed as
δj = k0∆j = 2π λ0
njwjcos θj. (4.4)
For TE modes, the propagation constant κj can be expressed as
κj = nj√ǫ0µ0cos θj; (4.5)
for TM modes, the propagation constant κj can be expressed as
κj = nj√ǫ0µ0/ cos θj, (4.6)
where k0 denotes the free space wave number, nj, wj, and θj denote the refractive index, width, and refractive angle of the jth air slot or high-index stack, respectively. The transmittance T and reflectance R can be obtained as
T =
where κa and κb denote the propagation constants for the input and output mediums.
The width of the jth air slot or high-index stack, needed for the odd multiple mquar of quarter-wavelength, can be derived from Eq. (4.4) as
wj = ∆j
njcos θj = mquar· λ40
njcos θj . (4.9)
The side view of the slab waveguide structure with the proposed planar waveguide concave grating is shown in Fig. 4-1. The etched trenches form the air slots and the
Figure 4-1: Side view of the etched trenches for a dielectric mirror.
non-etched parts form the high-index stack for the dielectric mirror design. The slab waveguide consists of a 6-µm SiON core layer with the upper 6-µm and the lower 10-µm SiO2 cladding layers grown on the silicon substrate. The refractive indices for the core layer, cladding layer, and the silicon substrate are 1.456, 1.450, and 3.476, respectively.
The effective indices of TE0 and TM0 modes are 1.45393 and 1.45392 with the negligi-ble propagation losses, which is due to the leakages to the silicon substrate. The slab waveguide supports the fundamental mode for both polarizations. Fig. 4-2 shows the top view of a planar waveguide concave grating with dielectric mirrors. The entire C-band, ranging from 1528.77 to 1560.61 nm, includes 81 wavelength channels with a channel spacing of 0.4 nm according to ITU grids [1]. The design wavelength is chosen as 1544.69 nm. In the planar waveguide concave grating design, the blaze angle of the grating facet is chosen to make the incident angle of the light to be equal to the reflective angle of the light relative to the facet normal. In a Littrow mount, this leads to the light incident upon the grating facet at a normal angle. From Eqs. (4.5) and (4.6), the reflectances R for the TE and TM modes are the same at normal incidence while they behave differently for oblique incidence.
In a Littrow mount with the air and high effective indices of 1 and 1.456, respectively, and mquarof 11, the widths wland whof the air slot and high-index stack can be obtained from Eq. (4.9) to be 4.25 µm and 2.92 µm, respectively. Simulation results show that the bandwidth of the high-reflectance region increases with the number Nslot of the air slots increases as shown in Fig. 4-3. Here, the high-reflectance region is defined as the region
Figure 4-2: Top view of a planar waveguide concave grating with dielectric mirrors.
0 5 10 15 20
0 5 10 15 20 25 30 35
Bandwidth(nm)
Nslot
Figure 4-3: Bandwidth of the high-reflectance region versus Nslot in a Littrow mount with mquar of 11 at a design wavelength λ0 of 1544.69 nm.
1.50 1.52 1.54 1.56 1.58 1.60 0
20 40 60 80 100
TE and TM modes
R(%)
Wavelength ( m)
Figure 4-4: Reflectances R for the TE and TM modes versus the corresponding wave-length in a Littrow mount with mquarof 11 and Nslotof 15 for the transfer-matrix analysis.
Table 4.1: Bandwidth results of the high-reflectance region in a Littrow mount with Nslot
of 15 and λ0 of 1544.69 nm
mquar wl (µm) wh (µm) Range (nm) Bandwidth (nm)
1 0.39 0.27 1377.26 – 1758.26 381.00
3 1.16 0.80 1484.79 – 1610.14 125.35
5 1.93 1.33 1508.28 – 1583.41 75.13
7 2.70 1.86 1518.58 – 1572.22 53.64
9 3.48 2.39 1524.36 – 1566.07 41.71
11 4.25 2.92 1528.06 – 1562.19 34.13
13 5.02 3.45 1530.63 – 1559.51 28.88
15 5.79 3.98 1532.52 – 1557.55 25.03
17 6.56 4.52 1533.97 – 1556.05 22.08
19 7.34 5.05 1535.12 – 1554.87 19.75
of which the reflectance is higher than 95 %. When the number Nslot of the air slots is below 5, the reflectances of the wavelengths in the C-band are all below 95 %, therefore the bandwidth of the high-reflectance region does not exist according to this definition.
For further analysis, the number Nslot is chosen as 15 and the calculated bandwidth of the high-reflectance region versus mquarin a Littrow mount are summarized in Table 4.1.
By choosing mquar = 11 and Nslot = 15, the reflectances R for the TE and TM modes in a Littrow mount are shown in Fig. 4-4. Notice that the diffraction efficiency of a conventional metalized grating is polarization dependent, originating from the induced surface current of the metal. The proposed grating employing a dielectric mirror with a dielectric-air interface therefore would be able to mitigate this undesired loss.
4.2 2D Waveguide Analysis
In the slab waveguide region, the electromagnetic field are independent of the y-axis and only the transverse wave distributed along the z-axis is considered. To further evaluate the effect of etching depth on the reflectance and the bandwidth of the high-reflectance region, we further analyze the 2D waveguide with FullWAVE software from R-Soft. The waveguide structure of the etched trenches for one stack is coded into the software and the software layout is shown in Fig. 4-5. A Gaussian filed is chosen as the launch field at the input end. The light propagates through the slab waveguide to the dielectric mirror with a distance of 100 µm, is partially reflected by the dielectric mirror, propagates through the slab waveguide to the time monitor, and is detected by the time monitor with a width of 10 µm. For the accuracy of the simulation results, the gird size is chosen as 0.06 µm.
Fig. 4-6 shows the reflectances R for the TE and TM modes versus the etched depth Detch in a Littrow mount with mquar = 11. We can see that almost no light is reflected by the mirror until the trench is etched to the core layer. Then the reflectance increases sharply and then saturates when the trench reaches to the lower cladding layer. The reflectance maintain stable when the etched depth is higher than 15 µm. Therefore, in our analysis the etched depth is chosen as 15 µm. Most of light propagates in the core layer so the reflectance is low until the trench is etched through the core layer to reach the lower cladding layer.
Fig. 4-7 shows the reflectances R for the TE and TM modes versus mquar in a Littrow
Figure 4-5: The software layout of the waveguide structure of the etched trenches for one stack.
0 5 10 15 20 25
0 20 40 60 80 100
TE mode
TM mode
R(%)
D ( m)
Figure 4-6: Reflectance R versus Detch in a Littrow mount with mquar of 11 and Nslot of 15 at a design wavelength λ0 of 1544.69 nm.
0 5 10 15 20 -1.0
-0.5 0.0
mquar = 11
R(dB)
mquar
TE mode
TM mode
Figure 4-7: Reflectance R versus mquar in a Littrow mount with Nslot of 15 and Detch of 15 µm at a design wavelength λ0 of 1544.69 nm.
1.50 1.52 1.54 1.56 1.58 1.60
0 20 40 60 80 100
R(%)
Wavelength ( m)
TE mode
TM mode
Figure 4-8: Reflectance R versus the corresponding wavelength in a Littrow mount with mquar of 11, Nslot of 15, and Detch of 15 µm for the 2D waveguide analysis.
mount with Nslot of 15, Detch of 15 µm, and λ0 of 1544.69 nm. Simulation results show that the reflectances for both modes decrease as mquar increases. The loss mainly comes from the scattering of light to the air when passing through the air slot. So when the width of the air slot increases with mquar, it leads to a deterioration of the reflectance.
The PDL, defined as the difference between the diffraction efficiencies for the TE and TM modes, is 0.05 dB with mquar of 11, Nslot of 15, and Detch of 15 µm. Fig. 4-8 shows the reflectances R for the TE and TM modes versus the corresponding wavelength in a Littrow mount with mquar of 11, Nslot of 15, and Detch of 15 µm. Compared with the results obtained from the transfer-matrix method, the reflectance and the bandwidth of the high-reflectance region both decrease and the reflectances for the TE and TM modes are slightly different in a Littrow mount. The reason is that the incident light used in the transfer matrix analysis is the plane wave while optical beam profile used in the 2D waveguide analysis has a finite beam waist, which turns out to have a large influence on the reflectance.
For a conventional design, the metal coating on the shaded facet is the main source of the PDL especially in a Littrow mount [24]. The induced surface current does not exist in a dielectric-air interface and the dielectric mirror mitigates the effect of the PDL.
However, the grating nonideality due to the width variation ∆w of the air slot and high-index stack during the fabrication process can significantly decrease the reflectance. The perturbed width w′l and wh′ of the air slot and high-index stack can be expressed as
wl′ = wl+ ∆w, (4.10)
wh′ = wh− ∆w. (4.11)
Fig. 4-9 shows that the additional loss increases as the width variation increases. Simu-lation results show that when the width variation ∆w is below ±0.1 µm the losses can be kept below 0.04 dB for both modes so the width variation below 0.1 µm is required to yield a dielectric mirror with high reflectance.
4.3 Summary
In summary, a planar waveguide concave grating employing dielectric mirrors is proposed to mitigate the PDL which comes from the induced surface current of the metal, especially
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0
5 10 15 20
Loss(dB)
w ( m)
TE mode
TM mode
Figure 4-9: Loss due to the width variation ∆w in a Littrow mount with mquar of 11, Nslot of 15, and Detch of 15 µm at a design wavelength λ0 of 1544.69 nm.
in a Littrow mount. The transfer matrix method is used to obtain the reflectance R and the number Nslot of the air slots is chosen as 15. Simulation results show that the reflectance is high for a wide range of the wavelength. The FullWAVE software from R-Soft is used to analyze the reflectance R for the 2D waveguide with etched air slots.
Simulation results show that a PDL of 0.05 dB can be achieved with mquar of 11, Nslot of 15, and Detch of 15 µm. The impact of the fabrication error of etched air slots is also taken into account. To yield a proposed grating with high reflectance, the width variation ∆w should be below ±0.1 µm.
Chapter 5
16-Channel Optical Add-Drop Multiplexer
An optical add-drop multiplexer (OADM), which is capable of transmitting and drop-ping the wavelength signals selectively, play a crucial role in fiber-to-the-home (FTTH) systems. In a conventional design based on the planar waveguide devices, an OADM em-ploys AWGs [2, 25]. However, AWGs encounter the inherent limits due to the lower free spectral range (FSR) and larger die size [10]. In this chapter, a novel OADM employing a planar waveguide concave grating is proposed. The arrangements of the components and the design considerations of the concave grating will be discussed in detail. The transmission characteristics are introduced for our analysis. A design example is used to analyze the spectral characteristics of the devices. The bending loss versus the radius of the curvature in a 90◦ arc channel waveguide for the fundamental TE mode (TE0) is also simulated.
5.1 Structure and Design Considerations
The schematic configuration of the 16-channel OADM, which consists of one planar waveguide concave grating, two input waveguides, two output waveguides, coupled waveg-uides, and sixteen sets of 2 × 2 switches is shown in Fig. 5-1. 16 wavelength signals, λ1, λ2,..., and λ16, are coupled out from the output end of input port 1 to the slab waveguide, while 16 waveguide signals, λ′1, λ′2,..., and λ′16, with the same channel spacing are coupled out from the output end of input port 2 to the slab waveguide at the same time. The
Figure 5-1: Schematic configuration of the 16-channel optical add-drop multiplexer sys-tem.
structures of the channel and slab waveguides are shown in Fig. 5-2. After propagating through the slab waveguide to the concave grating, they are diffracted by the concave grating, focused to the position along the imaging curve, and then coupled into the input ends of the different coupled waveguides according to the corresponding wavelengths as shown in Fig. 5-3. After transmitting and dropping the signals selectively by the 2 × 2 switches [2, 25], the signals are coupled out from the output ends of the coupled waveg-uides to the slab waveguide. They are diffracted again by the concave grating, focused to the position along the imaging curve, and then coupled into the input ends of output port 1 and port 2. To perform the functions of the demultiplexing from the input prot and the multiplexing to the output port, a concave grating with specifically designed tilt angles of two sides of the grating facets is proposed.
To design the concave grating, the structures of the channel and slab waveguides are chosen as shown in Fig. 5-2. By using the transfer-matrix method [29], the effective index neff in the slab waveguide can be solved. After the effective index neff in the slab waveguide is obtained, we choose the design wavelength λ0 in vacuum, the grating period d along the grating chord, the incident angle for the input port with respect to the y-axis at the grating pole, and the diffraction order m. Based on the diffraction formula of the plane grating from Eq. (3.17), the diffraction angle β(f0) with respect to the y-axis at
Figure 5-2: Side views of the single-mode channel waveguide (left) and the slab waveguide (right).
Figure 5-3: Schematic figure of the light propagating in the slab waveguide and being diffracted by the concave grating.
the grating pole can be obtained. As shown in Fig. 5-3, the distance r1,0 from the output end of the input port to the grating pole and the distance r2,0(f0) from the grating pole to the input end of the design coupled waveguide for the design wavelength λ0 with a corresponding frequency f0 are chosen. If the grating pole is chosen at the origin of the coordinates, the positions of the output end of the input port and the input end of the coupled waveguide can be obtained as (a1, b1) = (r1,0· sin α, r1,0· cos α) and (a2, b2) = (r2,0(f0) · sin β(f0), r2,0(f0) · cos β(f0)). The design of a planar waveguide concave grating here is based on the recursive definition of facet positions due to its superior spectral characteristics [32, 34]. After determining the x-axis coordinate position of the vertex of the ith groove as xi = i · d, the free -aberration y-axis coordinate position yi of the vertex of the ith groove for a design wavelength of λ0 can be obtained from the solution of the root for Eq.(3.6).
The optimum tilt angle θi of the ith grating facet with respect to the x-axis is deter-mined by Eq. (3.7). The optimum tilt angle θi of the ith grating facet can be obtained as
The incident angle α′ (= −α) for the input end of the output port with respect to the y-axis at the grating pole, the diffraction order m′ (= −m), and the diffraction angle β′(f0) (= −β(f0)) for the output end of the coupled waveguide with respect to the y-axis at the grating pole are determined. The distance r1′,0 (= r1,0) from the grating pole to the input end of the output port and the distance r′2,0(f0) (= r2,0(f0)) from the output end of the coupled waveguide to the grating pole are chosen. The positions of the input end of the output port and the output end of the coupled waveguide can be obtained as (−a1, b1) and (−a2, b2). The optimum tilt angle ϕi of the other side for the ith grating facet can be obtained as
ϕi = tan−1 −
The focal equation for the concave grating can be obtained from Eq. (3.8). α, β(f ), r1,0, and r2,0(f ) in Eq. (3.8) for the waves from the input port coupled to the corresponding coupled waveguides can be replaced with α′, β′(f ), r′1,0, and r2′,0(f ) for the waves from the corresponding coupled waveguides coupled to the output port. The numerical model
of the scalar diffraction theory for the concave grating in the planar waveguide can be obtained from Eq. (3.9).
At first, the signal at a frequency of f is coupled out from the output end of input port 2 (port 1) to the slab waveguide, diffracted by the concave grating, focused to the position along the imaging curve, and coupled into the input end of the jth coupled waveguide. The spectral response for the wave from input port 2 (port 1) to the jth coupled waveguide is denoted by I2,j(f ) (I1,j(f )) and can be obtained from the overlap integral of the imaging field with the mode field of the jth coupled waveguide. When the 2 × 2 switch is turned off, it is in the cross state [25]. After the signal passes through the switch, the signal is coupled out from the output end of the jth coupled waveguide to the slab waveguide. The transmission efficiency of the light transmitted across the jth coupled waveguide, switch, and jth coupled waveguide is denoted by ηj,2→1′ (ηj,1→2′).
Then the signal is diffracted again by the concave grating, focused to the position along the imaging curve, and coupled into the input end of output port 1 (port 2). The spectral response for the wave from the jth coupled waveguide to output port 1 (port 2) is denoted by Ij,1′(f ) (Ij,2′(f )) and can be obtained from the overlap integral of the imaging field with the mode field of output port 1 (port 2). We use the transmission characteristics T1′(f ) and T2′(f ) to denote the intensity of the light with a frequency of f detected at output port 1 and port 2. When all the switches are off, the transmission characteristics,
Then the signal is diffracted again by the concave grating, focused to the position along the imaging curve, and coupled into the input end of output port 1 (port 2). The spectral response for the wave from the jth coupled waveguide to output port 1 (port 2) is denoted by Ij,1′(f ) (Ij,2′(f )) and can be obtained from the overlap integral of the imaging field with the mode field of output port 1 (port 2). We use the transmission characteristics T1′(f ) and T2′(f ) to denote the intensity of the light with a frequency of f detected at output port 1 and port 2. When all the switches are off, the transmission characteristics,