## Abstract

We propose and demonstrate a multi-channel tunable optical dispersion compensator (TODC) that consists of an arrayed-waveguide grating (AWG) and liquid crystal on silicon (LCOS). By utilizing the AWG with a large angular dispersion and the LCOS with a flexible phase setting, we can construct a compact and flexible TODC that has a wide tuning range of chromatic dispersion. We confirmed experimentally that the TODC could realize channel-by-channel CD compensation for six WDM channels with a ± 800 ps/nm range and a 3 dB bandwidth of 24 GHz. We believe that the multi-channel operation of this TODC will help to reduce the cost and power consumption of high-speed optical transmission systems.

© 2010 OSA

## 1. Introduction

In recent years, the increase in communication network traffic has led to a demand for higher speed optical transmission and a more agile network. As the bit rate is increased to 40 Gb/s or above, the limitation caused by the chromatic dispersion (CD) of optical fibers becomes much more severe. We should compensate for the waveform distortion caused by the CD by using a dispersion compensator for high-speed medium- or long-haul transmissions.

The CD of the transmission fiber should be adaptively compensated, because it may be subject to changes in the optical wavelength division multiplexing (WDM) network. For example, the CD may change significantly as the temperature around the transmission fibers changes. In addition, the optical path is sometimes reconfigured in ring-type or mesh-type WDM networks, and this changes the CD value that must be compensated. To compensate flexibly for the CD, we must employ a tunable optical dispersion compensator (TODC).

Practical single channel TODCs have already been developed by using a number of technologies including lattice filters [1], fiber-Bragg gratings [2,3] and spectrometer-based TODCs [4–10]. However, as many single-channel TODCs are required as there are WDM channels in the entire system, which results in a high power consumption and large space requirements. It would be very attractive if we were able to realize multi-channel TODCs, because they would reduce the size, power consumption and cost of transmission nodes.

Our goal is to realize multi-channel TODCs. Of the single-channel TODC technologies mentioned above, we believe that a spectrometer-based configuration is the most promising for extension to multiple-channel operation. In general, spectrometer-based TODCs consist of a spatial light modulator (SLM) and a grating device. Several kinds of SLM devices including a micro electro-mechanical system (MEMS) mirror array [8,9] and a liquid crystal on silicon (LCOS) device [10] have been reported for use in spectrometer-based TODCs. These SLMs provide flexible and individual CD settings for multiple channels. Of these SLMs, LCOS appears to be superior to the MEMS mirror array because of its stability and reliability, since it has no mechanical moving parts. In terms of the grating device, it is advantageous to use a highly dispersive grating as a spectrometer such as a virtually imaged phased array (VIPA) [11] or an arrayed-waveguide grating (AWG) [12–14] to obtain a large CD with a spectrometer-based TODC. Although the use of a VIPA makes the device compact, it has a drawback in that the output field profile is exponential, which causes a field mismatch between the incident and reflected optical signals. This results in an insertion loss increase [11]. On the other hand, an AWG has a Gaussian output profile, which minimizes the field mismatch and resulting loss increase. In addition, AWGs have advantages such as design flexibility, mass producibility, and the capacity for dealing with a high channel count.

Thus, we have reported a spectrometer-based TODC using an AWG and an LCOS [15]. In our previous report, we introduce the principle of the TODC, and capital experimental results.

In this paper, we propose and demonstrate a spectrometer-based TODC consisting of an AWG and an LCOS, which can compensate for the CD of multiple WDM signals on a channel-by-channel basis. In the following sections, we first describe the detailed configuration and principle of our TODC, and detail the performance limiting factors of the configuration from a design viewpoint. Then, we evaluate the single channel characteristics experimentally. Finally, we demonstrate the channel-by-channel compensation of six WDM signals with a channel spacing of 100 GHz.

## 2. Principle and design of spectrometer-based TODC using LCOS

#### 2.1 Principle

Figure 1
shows the schematic configuration of our channelized TODC, which consists of an AWG and an LCOS. This configuration is based on that of a spectrometer-type TODC, and the AWG is used as a spectrometer and the LCOS is used as a space phase modulator. We explain the function of this spectrum-type TODC in terms the optical signal flows. An optical signal is fed into the AWG via a circulator and dispersed by the AWG. It is then focused on the LCOS in the *x* direction by the focusing lens. The LCOS, which has a large number of fine pixels, is in the wavelength dispersion axis of the AWG. Thus the LCOS spatially modulates the wavefront of the incoming light, reflects the signal back in the opposite direction, and returns it to the same input waveguide of the AWG. Finally the output signal travels in a lower direction via a circulator.

Here the AWG consists of an input waveguide, a slab waveguide, and hundreds of different length channel waveguides. This AWG acts as a transmissive diffraction grating with a large angular dispersion. Figure 1 shows an example of six discrete wavelength optical signals diffracted at different angles by the AWG, and these signals are modulated with the LCOS. We should stress that if we input a continuous light, the light is evenly dispersed on the LCOS. Therefore, there is no wavelength channel boundary on the LCOS.

The LCOS has a large number of fine pixels on the dispersion axis of the AWG. This pixel size is typically around 10 μm, which is much smaller than the channel interval in the *x* axis. Therefore there are typically hundreds of pixels in each channel. As a result we can modulate the wavefront of the incoming light signal very flexibly, as described in more detail later in this section.

### 2.1.1 Chromatic Dispersion value CD

Here we analyze the relationship between the CD and design parameters. When the phase setting distribution in the dispersion axis *x* is *Φ _{SLM}(x)*, the group delay

*τ*is obtained by

*ω*,

*c*and

*λ*are the angular frequency of the signal, the speed of light in a vacuum, and the signal wavelength, respectively. Thus, the CD provided by the system is given by

This equation is generally applied to the spectrometer-type TODC. As seen from the equation, if we set the phase in a quadratic form, we are able to obtain a linear group delay or constant chromatic dispersion. In addition, the equation also tells us that there are two ways to obtain a large CD, namely, by using a highly dispersive spectrometer or by employing a large quadratic coefficient in the spatial phase modulator.

The linear dispersion *dx/dλ* in the first part of Eq. (2) represents the dispersion of the AWG on the LCOS surface, and this can be increased by increasing the diffraction order. This can be given with the AWG design by the equation:

*n*is the group index of the effective index

_{g}*n*of the array waveguide,

_{c}*f*is the focal length of the focusing lens,

*ΔL*is the path length difference between neighboring array waveguides, and

*d*is the array waveguide separation in the output region [16]. The

*ΔL*is not in the bulk grating, and is a characteristic of the AWG, which is given by

*ΔL = λ*, where

_{c}m/n_{c}*m*is the diffraction order. Equation (3) means that the linear dispersion can be increased by

*ΔL*, and that this increase corresponds to higher order diffraction. The

*d*value in an AWG is typically about 10 μm, and

*m*can be above 100, and these parameters can be designed flexibly. On the other hand, in a typical bulk diffraction grating,

*d*is 1 μm, and

*m*is 2. Thus the

*dx/dλ*value of the AWG can be increased by one order compared with that of a typical bulk grating. Therefore, the CD value using the AWG is much larger than that with a bulk-type diffraction grating.

By contrast, in Eq. (2), *d ^{2}/dx^{2}Φ_{SLM}(x)* is the quadratic coefficient of the phase function. As mentioned above, the phase shift setting in a LCOS can be given in each pixel by employing an electrical voltage. As a result, the necessary phase distribution along the dispersion

*x*axis can be set flexibly. Here the phase shift in the LCOS is in the 0 to 2π range. Thus if the necessary phase shift is above 2π, we can give the phase function in the folded every 2π as shown in Fig. 1(b). Therefore, a large CD value can be obtained by combining the AWG and the LCOS. At the same time, we can set different CD values in the different channels, and this enables us to realize channel-by-channel TODC operation.

### 2.1.2 Transmission bandwidth BW

It is known that the spectrometer-based TODC suffers from bandwidth degradation as the CD setting increases [11]. This effect is caused by the phase mismatch between the incident and reflected fields. As mentioned above, the phase function must be quadratic if we provide the device with a finite chromatic dispersion value. Therefore, an incident optical signal that focuses near the edge of the channel is reflected in a different direction of incidence because the wavefront of the reflected electric field tilts according to the phase function slope. On the other hand, a signal that enters near the center of the channel is reflected in the same direction as the incident light. As a result, the coupling between the incident and reflected lights decreases around the edge of the channel, resulting in transmission bandwidth degradation. This effect becomes particularly significant when the quadratic phase function is steep.

We estimate this bandwidth narrowing analytically, and investigate the limiting factor of the passband. As described in the Appendix, the relation between a chromatic dispersion value *CD* and a 3-dB transmission bandwidth *BW* is expressed by (A5)

*BW*, because

*dx/dλ*can be increased as mentioned above, and

*W*can be suitably designed by designing

_{SLM}*W*. The Gaussian beam spot radius on the LCOS

_{out}*W*can be evaluated from the AWG output light

_{SLM}*W*in the following equation.As shown in Eq. (4), our spectrometer-based TODC requires a small

_{out}*W*if we are to obtain a large

_{SLM}*CD**

*BW*value. With a typical AWG,

*W*is several millimeters long. This means that

_{out}*W*is some tens of microns according to Eq. (5) assuming a focusing length of around 100 mm. This means that an AWG with a wide

_{SLM}*W*is preferable in terms of obtaining a small

_{out}*W*value. If we realize in the normal space optics, we must, for example, install a beam expander (a pair of prisms) to increase the beam width from a typical beam size of around 200 μm. We can easily increase

_{SLM}*W*by increasing the number of arrayed waveguides, and acquire better

_{out}*CD**

*BW*values.

However, if we increase the *W _{out}*, and reduce the

*W*compared with the LCOS pixel pitch

_{SLM}*T*, we can no longer ignore the LCOS pixel discontinuity because we approximate the continuous curvature of the phase function. As a result, ripples appear in the transmission spectrum and in the group delay characteristics. We investigate this effect of LCOS pixel discontinuity on the group delay characteristics numerically in sub-section 2.2.2.

#### 2.2 Design and numerical estimation

In this section, we calculate the dispersion compensation values *CD* and transmission bandwidth *BW* by assuming the experimental parameters described in sub-section 2.2.1. In addition, we evaluate the effect of the diffraction because of the discretely pixelized LCOS cells by calculating the overlap integral of the electric field between the incident and reflected Gaussian beams in sub-section 2.2.2. Based on these calculations, we confirm that this TODC, which consists of an AWG and LCOS, can deal with large dispersion values in a sufficient bandwidth in multi-channel operation.

### 2.2.1 Relationship between chromatic dispersion CD and bandwidth BW

Using Eqs. (2) and (3), we calculated the relationship between chromatic dispersion compensation *CD* and bandwidth *BW* as a parameter of diffraction order *m*. We summarize the parameters we used in Table 1
.

Figure 2
shows the relationship between the chromatic dispersion compensation value *CD* and the transmission bandwidth *BW* as a parameter of the diffraction order *m*. In Fig. 2(a) and 2(b), the ratio of the beam spot sizes on the LCOS *W _{SLM}* and LCOS pixel pitch

*T*(hereafter overlap) are 4 and 8, respectively. Here we also calculate a typical bulk grating as a reference in which the grating pitch is 1 μm and the diffraction order is 2 assuming the same

*W*. As shown in Eq. (3), compared with the bulk grating, the AWG can provide a large

_{SLM}*CD*BW*product because of the large linear dispersion (

*dx/dλ*) originating from high diffraction order m. This

*CD*BW*product can be increased as the spot beam size on the LCOS is decreased.

We also calculate the dependence of the overlap on the bandwidth as a parameter *CD*. Figure 3
shows the results when the diffraction order *m* is 200. As expected with Eq. (4), the bandwidth becomes larger as the spot size *W _{SLM}/T* decreases. If we need the dispersion compensation value

*CD*of the 600 ps/nm bandwidth required for a 40 Gb/s optical signal, we have to design the spot size

*W*to be less than 8. Since

_{SLM}/T*T*is equal to around 5-15 μm, this value can be easily designed in practice.

On the other hand, the Gaussian beam has to be fairly large. This is because the phase discrepancy effect can no longer be ignored as *W _{SLM}/T* decreases. As a result, ripples appear in the transmission and group delay characteristics around the pixel boundary. We discuss this LCOS pixel discrepancy effect numerically in the next section.

### 2.2.2 LCOS discrepancy effect

To investigate the LCOS discrepancy effect, we numerically integrated the electric field of the incoming light *φ*(*x*) and the field reflected by the LCOS *ϕ*(*x*). The electric field of the incoming light *φ*(*x*) and the field spatially modulated and reflected by the LCOS *ϕ*(*x*) are expressed as

*A*,

*x, ω*and

*θ*are the electric field intensity, the position on the LCOS in the

_{SLM}(x)*x*direction, the beam spot size on the LCOS surface, and the phase distribution (hereafter phase function) of the LCOS, respectively. The phase of the output signal

*Φ*(

*x*) and the group delay

*τ*are obtained by

*τ*(

*λ*) in Eq. (9) instead of Eq. (1).

Figure 4
shows the calculated dependence of the group delay ripple (GDR) and the 3 dB bandwidth when we change the beam spot size. In this figure, the horizontal axis indicates the number of pixels in the beam spot size as shown in the inset of Fig. 4. The plot is for a case where we set the CD value at 600 ps/nm. As shown in Fig. 4, the GDR decreases greatly as *W _{SLM}* increases, and falls to a sufficiently small value of less than 2 ps by the time the beam width reaches twice the pixel size. This result indicates that if we can adopt a

*W*/

_{SLM}*T*of more than 4, the LCOS discrepancy effect becomes negligible.

As regards the bandwidth, the plot shows that it becomes narrower as the beam width increases. Therefore, in this instance, we fixed the beam width at 8 times the pixel period. This value can secure a small ripple of less than 1 ps with a bandwidth of about 20 GHz. It should be noted that, to confirm the principle of our TODC, it does not matter if we set the spot size at 40 μm, because the parameter provides a CD of 617 ps/nm and a GDR of less than ± 2 ps.

As summarized above, we investigate numerically the relationship between the design parameters and the characteristics achieved with the AWG-LCOS multi-channel TODC. A fairly large *CD*BW* value can be realized when we utilize the higher order diffraction of the AWG. Typically, we could attain a *CD* of 800 ps/nm in a 40 GHz bandwidth in a multi-channel TODC, when we used a diffraction order of 200 and an overlap of 4 pixels of the LCOS. The LCOS discrepancy effect is less than ± 2 ps when the overlap is more than 2 pixels.

## 3. Experimental results and discussions for channelized TODC

To demonstrate the feasibility of the multi-channel TODC described in Section 2, we measured the AWG-LCOS system shown in Fig. 1. In this experiment, we used an easily available AWG with a free spectral range (FSR) of 1000 GHz and a center wavelength of 1587 nm. Although this AWG is not optimized sufficiently for maximizing the *CD*BW* value, it is adequate for investigating the validity of our design described in the previous section. The focal length of the focusing lens was set at 100 mm. The beam waist was set at eight times the pixel size. The pixel number was set at 256, which corresponds to a WDM channel with a bandwidth of 0.84 nm. These parameters are based on the discussion in section 2. Multi-channel operation (6 channels) can be achieved by using 1536 LCOS pixels. For the phase setting in the LCOS, we applied voltage to these pixels with complementary metal oxide semiconductor (CMOS) circuits through dozens of electrical I/O interfaces. For our purpose, this voltage is in the 0 to 5 V range and is capable of fine resolution.

In Sec. 3.1, the relationship between chromatic dispersion *CD* and 3 dB bandwidth *BW* is examined for ch. 3 to confirm the detailed correspondence between the design in Sec. 2 and the results with this setup. In addition, it is shown that channel-by-channel operation has been achieved according to the design in Sec. 3.2. We clarified that ripples that appeared in the transmittance and group delay spectra when we analyzed these results. The analysis of the cause of the ripples and a corrective strategy are discussed in Sec. 3.3. Moreover, Sec. 3.4 demonstrates the ability to prescribe third-order dispersion compensation. We confirmed experimentally that a feature of our TODC is that the optical phase is programmable.

#### 3.1 Single channel characteristics

First, we evaluated the single channel performance of our TODC and Fig. 5 shows its measured single channel characteristics. As seen in Fig. 5, we obtained both positive and negative dispersions of ± 800 ps/nm with a 3 dB bandwidth of 24 GHz. Figure 6 shows the relationship between the 3 dB bandwidth and the CD setting. As shown in Fig. 6, the CD can be tuned continuously to the quadratic coefficient of the phase function. This represents a fundamental limitation, or figure of merit (FOM), which is that the CD setting is inversely proportional to the bandwidth (Eq. (4)). It is possible to increase the bandwidth by making the beam waist small enough for the phase mismatch to be less significant, as explained in sub-section 2.2.2. As shown in Fig. 6, the measured results agree with our calculations. Overall the value is slightly lower than the calculated result. This is because the 3 dB bandwidth is degraded by ripples caused by the finite fill factor of the LCOS, as mentioned in Sec. 3.4.

The insertion loss of the TODC was 6.4 dB. This loss is composed of the 1.5 dB coupling loss between the optical fiber and the AWG, the 2.1 dB loss of the circulator, the 0.5 dB loss of the lens without AR coating, the 0.5 dB loss at the LCOS, and the 1.8 dB loss of AWG diffraction and free space optics. Of these optical loss factors, the loss caused by AWG diffraction is comparatively large. This is because there is a field mismatch between the arrayed waveguides and slab waveguide at the contact part [17]. Therefore, it will be possible to improve the insertion loss by compensating for this mode mismatch by employing vertically tapered waveguides. We expect this loss can be reduced by around 5 dB based on this loss factor analysis. The insertion loss of our TODC is relatively small compared with that of other spectrometer-based TODCs. This reflects the excellent optical characteristics of the AWG.

#### 3.2 Channel-by-channel operation

We demonstrate the channel-by-channel operation of six channels in the TODC. As explained in the previous section, our setup utilizes an AWG with a 1000-GHz FSR and an LCOS with 2560 pixels. Accordingly, we can perform a 10-channel operation because the channel spacing is 100 GHz. However, the losses of the two channels around the FSR edge are fairly large because of the cyclic properties of the AWG transmission spectra. Figure 7(a) shows the measured transmission spectrum when the CD value setting is 0 ps/nm. The frequency range with loss uniformity of 3 dB was about 600 GHz. Thus, we realized the independent operation of six channels in a 600 GHz frequency range of with a flat transmission spectrum.

Figure 7(b) shows the channel-by-channel operation of the TODC for six WDM channels. We obtained CD values of 83, 402, −479, 0, −782 and 766 ps/nm for channels with frequencies ranging from 191.7 to 192.3 THz. The plots that fall outside the 3 dB bandwidth are omitted in each channel. We obtained both positive and negative dispersions of ± 800 ps/nm in each channel. A narrow frequency range of 600 GHz (six channels) is sufficient for the FSR of the designed AWG, so the same transmittance and group delay characteristics were obtained in each channel. It can be clearly seen that the six channels are provided with arbitrary CDs independently.

As described above, this LCOS can accommodate ten channels with a 100 GHz spacing, thus we can increase the TODC channel number to ten if we increase the AWG’s FSR and widen the flat transmission spectrum range to more than 1000 GHz without any degradation in the characteristics of the individual channels.

However, if we increase the channel number above ten, we must decrease the number of pixels per channel of the LCOS. This means that we must increase the linear dispersion on the LCOS. As described in detail in the Appendix, the FOM of our TODC is in inverse proportion to *W _{SLM}*, and proportional to the linear dispersion on the LCOS. Therefore, to preserve the same characteristics in each channel, we must reduce

*W*. However, it should be noted that the overlap,

_{SLM}*W*, should be two or more pixels, as described in sub-section 2.2.2. Therefore, we can expand up to 20 channels with this method. If we want to design a TODC with 20 or more channels, we have to change the number of pixels and the pixel gap of the LCOS.

_{SLM}/TIt appears to be difficult to increase the number of channels to 40 or more with the principle that we described in Sec. 2 without degrading the FOM of the TODC. With a spectrometer-based TODC using a bulk grating, it has been proposed that the trade-off between the bandwidth and the dispersion value be eliminated, namely that the FOM be greatly increased [9]. Furthermore, we have reported a 40-wavelength channelized TODC [18], which has a CD tuning range of ± 400 ps/nm with a 3 dB bandwidth of 40 GHz, which is 1.8 times wider than that obtained with the conventional configuration. This TODC is achieved by using a configuration that employs two AWGs and an LCOS for a channelized TODC. Alternatively, a TODC which has a configuration consisting of a cyclic frequency AWG, a bulk grating and an LCOS realized 50-wavelength channel-by-channel operation. The AWG supplies a large dispersion setting, while the bulk grating provides operation over a wide wavelength range [19,20]. The configuration simultaneously provides excellent TODC characteristics, large channel number operation and optics alignment with a relatively large tolerance.

#### 3.3 Cause of ripples in the group delay and the transmission spectra

As shown in Fig. 5, the group delay and transmittance spectra have ripples. These ripples have adverse effects on the group delay and transmission characteristics. Therefore, we investigate the causes of these ripples, and discuss the possibility of reducing them. We demonstrated that the phase setting of 2π-folding was not the cause of the ripples under our experimental conditions.

Figures 8(a) and 8(b) also show the measured transmission and group delay spectra, and the corresponding phase function, respectively, when the dispersion value is 792 ps/nm in Fig. 5. The frequency range in this graph corresponds to 100 GHz of 1 channel. We also show calculated values as described in sub-section 2.2.2. An important LCOS parameter is the fill factor, because a reduction in the fill factor induces noise generation as a result of the diffraction effect. The fill factor is defined by

where*T*and

*S*are the pixel pitch of the LCOS and the space between adjacent pixels, respectively.

When we investigated the influence of the fill factor, in our calculation we assumed that the fill factor was 95%, the reflectivity of the LCOS pixels and gaps between the pixels was 100%, and the phase shifts on the edge of the pixels were steep.

In Fig. 8(a), the measured and calculated spectra tend to agree qualitatively. Here we presume that the ripples appear for two reasons. One is the interference of the reflected light from the gaps between pixels as discussed in sub-section 2.2.2. The other is a phase setting error around the 2π phase step of the phase function as shown in Fig. 8(b). The ripples caused by the former should appear at point A, and those caused by the latter should appear at point B. In Fig. 8, it is difficult to identify the cause of the ripples.

To specify the ripples caused by the phase setting error, we performed a reference experiment. In this experiment, we limited ourselves to a discussion of the GDR, and a similar discussion is possible for transmittance ripple. If the phase function is folded accurately at a phase of 2π, the GDR does not appear. However, if the phase is detuned from 2π, the GDR emerges, because the phase function is not smooth. Therefore, we measured the transmission ripples and GDR when the phase distribution of LCOS was set at a step phase function in the experimental setup in Sec. 3.1, shown in Figs. 9(a) and 9(b). Figure 9(c) shows the calculated and measured GDR dependence on the phase detuning from 2π. When the phase function is set to an accuracy of ± 0.05 rad of 2π at the reset point, the GDR is minimized to less than ± 1 ps. Here our phase setting accuracy is within 1%. Therefore, the ripples observed in the previous experiment are not caused by the phase setting errors of the 2π-folding.

Thus, we presume that the ripples are caused by the interference of the reflected light from the gaps between pixels. As shown in Fig. 8(b), we can see that the ripples appear at frequencies with phase settings of around π.

Here, we assume that the reflected light from the gap is not modulated by any pixels, and thus has a phase shift of 0 compared with the signal light.

As shown in the inset of Fig. 1, the cross-sectional view the LCOS consists of a phase-shifting layer of liquid crystal placed directly on a CMOS integrated circuit. The LCOS is an electrically programmable device with a fine pixel pattern that modulates the phase of the incident light. Although a cutting edge CMOS fabrication process realizes an extremely fine pattern, it is impossible to null the space width between pixels. Accordingly, the LCOS forms a phase grating between the effective pixels and ineffective pixel spaces. Thus the diffraction efficiency becomes less than unity and this effect degrades the insertion loss because of the adverse effect of the optical coupling as shown in Eq. (A4) in the Appendix. Moreover, since the phase function is quadratic in our setup, the diffraction efficiency varies as the signal frequency changes. The effect becomes significant where the LCOS functions like a zero/π phased grating, in which case the diffraction efficiency becomes the worst.

Therefore, the reflection interferes with the signal light in an opposite phase. As a result, the transmittance and group delay are distorted by the ripples. The ripples have to be suppressed to obtain good transmission system performance. Since the LCOS has gaps between the pixels in practice, the phase and reflectance around these gaps deviated periodically from the set values. In this section, we roughly estimate this diffraction effect in the worst case using a simple model. Because it is not easy to calculate the electric field distribution in an LCOS, we simply investigated the influence of the diffraction of an LCOS by changing the reflectance in the gaps as a parameter. Figure 10 shows the calculated transmission and group delay characteristics with reflectance in gaps of 0, 50 and 100% on the assumption that the light incident on a space is reflected by the CMOS surface and is phase-shifted by zero. As shown in Fig. 10, the frequencies at which the transmission intensity drops to a lower value appear regardless of reflectance. For instance, the typical frequencies that are nearest the center frequency are ± 12.8 GHz. The assumption in our calculation takes the worst case into account, because the signal reflected from the space interferes with the signal. Other than the case of 0%, ripples appear for both transmission and group delay at specific frequencies, where the phase functions are set around π. This reflects the fact that the diffraction efficiency is minimized when the phased grating is zero/π. In addition, the ripples become steeper as the reflectance approaches 100%, because the signal light and the unwanted light from the space have the same intensity, so that the visibility of the interference reaches maximum.

Figure 11 shows the GDR and transmittance ripples on the reflectance of the LCOS where the relative frequency is 12.8 GHz in Fig. 10. Both ripples become small, because the diffraction of the gaps is suppressed by improving the reflectance. To fully meet the tight GDR tolerances required by 40 Gbps systems, the TODC should have a low GDR within ± 5 ps. If we build a TODC in accordance with this simulation condition, the LCOS reflectance must be less than 50%.

There are two ways to suppress these unwanted reflections: one is to transmit the light to the rear of the SLM, and the other is to fill the gaps with an absorbent material. The former can be realized relatively easily by replacing the CMOS-based LCOS with a liquid crystal phase modulator array whose substrates and electrodes are made of glass and ITO, respectively. In Fig. 10, no ripples appear in the intensity and group delay, which has a reflectance of 0%; therefore, the bandwidth is increased from 25.6 to 32.6 GHz, and the GDR is eliminated completely. The diffraction-suppressed SLM causes an additional insertion loss, because it eliminates the optical power that is launched into the gaps. However, if we assume a reflectance of 0%, the loss is about 0.5 dB.

#### 3.4 High order dispersion

In a high bit-rate system, the transmission quality is very sensitive to chromatic dispersion. Techniques to compensate precisely for the dispersion slope are becoming increasingly important for such advanced high bit rate WDM systems.

Since the LCOS has very fine pixels, and the optical phase in each pixel can be set flexibly, this provides very good controllability. Thus we can realize various dispersion characteristics. Figure 12 is an example of a higher order dispersion setting. If we set a cubic function as the phase function, the group delay is quadratic. It is also possible to realize much higher order dispersion. This higher order dispersion compensation will be important for transmission at above 40 Gb/s in the next generation networks. A TODC that utilizes LCOS is advantageous from this viewpoint.

## 4. Conclusion

We proposed a spectrometer-based TODC consisting of an AWG and an LCOS. We described the principle of the TODC and investigated the factors that limited its performance, which were caused by the limitation of the LCOS. We also experimentally demonstrated a TODC that is capable of setting arbitrary CD values for six WDM channels independently. The TODC exhibited a maximum CD of ± 800 ps/nm with a 3-dB transmission bandwidth of 24 GHz.

The combination of an AWG and an LCOS provides us with a large CD setting and multiple channel-by-channel operation. The TODC is a promising candidate for use in future ring/mesh networks.

## 5. Appendix

We estimate the bandwidth narrowing analytically, and investigate the limiting factor of the passband.

Appendix: If we assume the phase function has a quadratic form

Figure 13 is a schematic diagram of a quadratic phase function and the concomitant angular displacement of a beam. If the phase shift given by the LCOS changes sufficiently slowly over the beam waist of the signal on the LCOS surface, the wavefront tilt can be represented by a first derivative of the phase function as

_{2}and

*x*are the quadratic coefficient of the phase function and the geometric position on the LCOS. The position

*x*is related to a specific signal wavelength bywhere

*β*is the linear dispersion of the spectrometer. Now we consider the coupling efficiency between two optical fields that are the incident into and reflected out of the LCOS. The signal from the device has to trace the same path of incidence in order to be output from the device. A simple equation can be applied to the coupling efficiency, because the fields are both Gaussian. The coupling efficiency is expressed by

*W*is the spot size of the beam on the LCOS plane. Here, if we assume the signal has a narrow wavelength range, the wavelength range is approximately proportional to the frequency range. In addition, according to Eq. (2), the chromatic dispersion is proportional to the quadratic coefficient of the phase function.

_{SLM}Finally, substituting Eqs. (A2), (A3) and (2) in Sec. 2.1, into Eq. (A4), the relation between the tuning range of the chromatic dispersion *CD* and the 3-dB transmission bandwidth *BW* is expressed by

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