1. Introduction

Optical filters have been widely applied to optical communication systems and fiber sensing fields. With the rapid development of optical communication, many techniques have been proposed for optical filters, such as birefringence [1,2], optical-electric thin-films [3], array waveguide gratings [4], ring resonators [5–7], fiber gratings [8,9], Michelson and Mach-Zehnder interferometers [9–11], and Michelson-Gires-Tournois interferometer (MGTI) [12–14].

MGTI, which can be achieved by combining Gires-Tournois etalons (GTE) with traditional Michelson interferometer, have received much attention. They have good passband flatness (flat-top), high extinction ratio, wide passband and stopband bandwidths, which is very close to the rectangular spectrum and quite suitable for DWDM systems [12–14]. For an MGTI with a finite number cavities, passband(or stopband) bandwidth decreases with the increase of extinction ratio. Wider bandwidth can be obtained by increasing the number of cavities, which will increase the complication of MGTI’s structure and difficulty in optimizing parameters [14]. MGTI has severe chromatic dispersion (CD) problems, which is caused by the employing of GTEs. According to the CD characteristics of MGTI, an extra set of cascaded GTEs has been proposed as the tunable dispersion compensators (TDCs) [13,14].

TDCs are important components in high bit-rate optical communication systems. Generally, they are all-pass filters which only have influences on signal’s phase, i.e.,CD. There are various kinds of TDCs, such as virtually imaged phased arrays [18], ring resonators [19,20] and GTEs [13–17]. Ring resonators and GTEs are very familiar and useful TDCs, which have similar phase response and also can be used as the dispersive phase elements in optical interleavers [5–7,12–14]. Generally, the GTEs is proposed consist of bulk-optic thin-film mirrors [13–16] and fiber gratings [12,17]. Ring resonators based on planar lightwave circuits (PLCs) have been widely reported [5–7,19,20]. PLCs is suitable for devices with small size [21]. Obviously, devices based on PLCs are superior to those based on bulk-optic devices and fiber-platform in terms of high density integration. It is a goal of this work to develop the comb filter with perfect spectral performance and low dispersion. The whole model including the comb filter and its TDC we proposed consists of several wide-band 2×2 couplers, which promises an all-waveguide structure and a wide working waveband. For the small size, our model is expected to be integrated on PLCs.

Generally, all the GTEs mentioned above are reflective elements [12–17]. So the reflective GTEs are suitable to be combined with Michelson interferometer to form an MGTI. As the first feature of this paper, we propose a novel multi-cavity transmissive Gires-Tournois etalon (MCT-GTE) composed of cascaded Mach-Zehnder interferometer loops (MZILs). Each MZIL forms a cavity. Mathematically, it has the same dispersive phase as multi-cavity reflective Gires-Tournois etalons (MCR-GTE). Thus MCT-GTE is suitable to be added to each arm of a Mach-Zehnder interferometer (MZI) to form a novel MZI, i.e., Mach-Zehnder Gires-Tournois interferometer (MZGTI). An MZGTI is an interleaver, which has the same spectral characteristics as an MGTI whose merits have been mentioned above.

The basic structure of MZGTI is a 2×2 MZI. The MZI structure is flexible since its input ports and output ports are separate. Moreover, its four ports are equivalent, i.e., its filtering characteristics won’t change no matter which port is used as input port. According to the characteristics, we propose a novel structure to realize super high extinction ratio comb filter (SHERCF) based on cascaded MZGTIs, which is the second feature herein. As has been mentioned before, bandwidth and extinction ratio are incompatible for an MGTI or MZGTI. Increasing the number of cavities (i.e., cascading more MZILs) can yield wider bandwidth and higher extinction ratio, but it also increases the complication of MZGTI’s structure and difficulty in optimizing parameters. The SHERCF we proposed has wide bandwidth when extinction ratio is fairly high (>200dB), which is far better than MGTI and MZGTI. Hence SHERCF is an efficient way to solve the contradiction. However, it loses the other complementary output port.

MZGTI has the same CD characteristics as MGTI, which is really a challenging problem. According to its CD characteristics, we propose a TDC for MZGTI, which is a set of cascaded ring resonatoros, i.e., multi-cavity ring resonator (MC-RR). Each ring forms a cavity. MC-RR, an all-pass filter, is a generalized multi-cavity transmissive Gires-Tournois etalon. Mathematically, there is only a phase shift of π/2 between MCT-GTEn and MC-RRn, where n is the sum of cavities. Due to this inherent phase shift, MC-RR as the TDC for MZGTI doesn’t need any phase shifters since the compensation bands are exactly located at the passbands of MZGTI provided that their cavity lengths are equal. The compensation band is the band where dispersion is tunable and can be used to compensate MZGTI’s dispersion in passband. In another work we have demonstrated that a set of MC-RRn is an efficient TDC for MGTI. Here, one can see that it is also an efficient TDC for MZGTI.

As the third feature of this paper, we propose the CD compensation scheme for SHERCF. Though we obtain excellent spectral performance, CD problem of SHERCF is much more severe than single MZGTI. In theory, it is several times of single MZGTI. Thus we propose a set of cascaded MC-RRs as the TDC for SHERCF. Further more, we demonstrate its CD compensation ability with the optimized results.

2. Design of super high extinction ratio comb filter (SHERCF)

2.1 Structure of SHERCF

The SHERCF we proposed is shown in Fig. 1(p is an integer). The solid line means that the adjacent two ports are connected and the optical path between them is on. Specially, the solid line with a “x” denotes that the adjacent two ports are disconnected and the corresponding ports are idle. The last element (M) is a mirror with full reflectivity. The simplest mirror is a Sagnac loop mirror composed of a 3dB 2×2 coupler. Each rectangular frame contains a set of mn-MZGTI which is illustrated in Fig. 2. To obtain super high extinction ratio, all the mn-MZGTI in rectangular frames should be identical. MZGTI is an interleaver obtained by combining MZI with MCT-GTE, which is very similar to MGTI [14]. The MCT-GTE we proposed is shown in Fig. 3. We can see that each loop contains two MZIs except the first one. Every MZI is composed of two couplers. So we call it Mach-Zehnder interferometer loop (MZIL). MCT-GTEn consists of n cascaded MZILs. Every MZIL forms a cavity.

 

Fig. 1. Three schematic diagrams of the proposed SHERCF, i.e., p-mn-MZGTI: (a)p is an arbitrary integer; (b)p is an even integer; (c)p is an odd integer

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Fig. 2. Schematic diagram of mn-MZGTI

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Fig. 3. Schematic diagram of MCT-GTEn

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2.2 Principles

2.2.1 MCT-GTEn

Based on the transfer matrix of 2×2 coupler given by Ref [20], let every MZI in Fig. 3 have two arms with equal length. Hence every MZI degenerates into a compound coupler, i.e., every MZI is equivalent to a 2×2 coupler. The transfer matrix of the i-th compound coupler (MZIi) can be written in Eq. (1)

where, j is the imaginary unit, β=n_effk, k=2π/λ, n_eff is the effective refractive index, λ is the wavelength in vacuum. K _i1 and K _i2 are the bar coupling ratios of Coupler i1 and Coupler i2 respectively. _MZi l is the length of each arm of the i-th MZI (coupling region lengths of Coupler i1 and i2 have been included). We refer to _MZi l as the equivalent coupling region length of the i-th compound coupler.

Through recursive analysis as Ref [14], the transmission coefficient _n t of MCT-GTEn in Fig. 3 can be expressed as:

where, r_n=|z_n|. z_n is defined in Eqs. (1)–(2). When n is odd, z_n must be negative. When it is even, z_n must be positive. 2ϕ_n is the dispersive phase of MCT-GTEn. a_n=(1-r_n)/(1+r_n). δ=0.5kL, L=n_eff(l′₁+l _MZ1)=…=n_eff(l_{MZ (n-1)}+l′_n+l_MZn) is the effective cavity length. l′_i, which has been marked in Fig. 3, is the length of the i-th MZIL excluding MZI_i and MZI_(i-1). Specially, ϕ ₀=0. From Eqs. (3)–(4), we infer that the structure proposed in Fig. 3 is really a multi-cavity transmissive Gires-Tournois etalon, i.e., MCT-GTE. It is an all-pass filter.

A real MCT-GTE has the same dispersive phase [Eq. (4)] as an MCR-GTE consists of bulk-optic mirrors discussed in Ref [14]. We should emphasize that the structure shown in Fig. 3 can be a real MCT-GTE only when z_i is chosen properly. As has been mentioned above, z_i must be negative when i is odd and it must be positive when i is even. The sign of z_i is closely related to K_i1 and K_i2 [see Eq. (2)]. If all the z_i is positive, the structure in Fig. 3 will degenerate into a set of MC-RR which will be discussed detailedly in Section 3.1. However, MC-RR is not a real MCT-GTE since there is a phase shift of π/2 between them. Therefore, the compound couplers are necessary to be used in the MCT-GTE.

2.2.2 mn-MZGTI

Based on Eqs. (3)–(4), we use 2ϕ_m and 2θ_n to denote the dispersive phases of MCT-GTEm(on Arm a) and MCT-GTEn(on Arm b) respectively. When both m and n are odd or even, the normalized output intensity of the mn-MZGTI in Fig. 2 can be written in Eq. (5). But when they have different parity, the expressions for I_bar and I_cross should be exchanged. The two output ports (i.e., I_bar and I_cross) are complementary, which make MZGTI be an interleaver that is suitable for DWDM systems.

Effective cavity lengths of MCT-GTEs in two arms must be equal. The phase matching condition of MZGTI is ΔL=0.5L, and ΔL=n_eff[(L_b+l^b_MZn)-(L_a+l^a_MZm)], which is the same as MGTI [13]. L_a and L_b are the length of the two arms of MZGTI excluding MCT-GTE respectively.(’a’ and ’b’means Arm a and Arm b respectively).

Assuming that the refractive index (n_eff) does not change with wavelength, then the dispersive phase for each output port of mn-MZGTI is

The output dispersive phase of mn-MZGTI is the average of MCT-GTEm and MCT-GTEn. Notice that MZGTI in Fig. 2 has four ports. Unlike MGTI [14], MZGTI is a transmissive device. Its input ports and output ports are separate. Hence the output signal won’t appear in the input port, i.e., the input signal won’t be reflected. In fact, its four ports are equivalent, i.e., every port can be defined as input port(E_i). If so, the bar output port (E_bar) and cross output port(E_cross) should be rearranged according to the new input port(E_i). But no matter which port is used as input port, expressions for I_bar and I_cross remain the same as Eq. (5), which means that the filtering characteristics are unrelated to the input port chosen. So are the dispersion characteristics. These merits make MZGTI a flexible structure, which are pivotal for SHERCF.

2.2.3 SHERCF

In Fig. 1, the optical path is controlled strictly. In Fig. 1(a), the incident light is confined to transfer in the bar direction of mn-MZGTI. p can be an arbitrary integer. To make sure that the incident light is confined to transfer in the cross direction, we give two structures which have been illustrated in Fig. 1(b) and Fig. 1(c). They correspond to the situations when p is even and odd respectively. When the light transfers to the mirror(M), it’s reflected by the mirror and returns to the circulator along the same optical path(bar or cross direction). At last, the output light is obtained at the output port of the optical circulator. Hence every mn-MZGTI can be used twice.

The normalized output intensity and the dispersive phase of SHERCF (i.e., p-mn-MZGTI) in Fig. 1 can be expressed as

As has been mentioned above, structures proposed in Fig. 1 can make every mn-MZGTI be used twice, which is the merit of the 2×2 MZI structure. Thus the exponent in Eq. (7) is 2p when we only use p cascaded mn-MZGTIs, which means that the normalized output intensity in dB is 2p times that of single mn-MZGTI. It is the very key for super-high extinction ratio. Here we infer that performance of SHERCF is determined by single mn-MZGTI, i.e. its basic cell. The dispersive phase of SHERCF is also 2p times that of single mn-MZGTI, which suggests that the CD of SHERCF is 2p times that of single mn-MZGTI.

2.3 Spectrum characteristics

From Eqs. (4)–(6), we find that mn-MZGTI has the same mathematical expressions as MGTI [14]. So MZGTI and MGTI have the same characteristics, such as good passband flatness, high extinction ratio, wide passband and stopband bandwidths [12–14]. The passband bandwidth herein is the width within which the maximum attenuation of normalized intensity is ripple in passband [see Fig. 4(a)]. And the stopband bandwidth herein is defined as the width within which the minimum attenuation is extinction ratio in stopband [see Fig. 4(b)]. The stopband bandwidth is equal to passband bandwidth. In theory, mn-MZGTI with larger m and n has wider bandwidth when the extinction ratio is fixed at a certain value, which means that performance of such kind of interleaver can be enhanced by increasing the number of cavities [14]. Thus performance of 22-MZGTI is better than 21-11- and 10-MZGTI. In Section 2.2.3, one gets that performance of SHERCF is determined by its basic cell. Hence we use 22-MZGTI as the basic cell of SHERCF here.

Based on Eqs. (3)–(5) (7), let L be 6mm and the basic cell in every rectangular frame of Fig. 1(a) be 22-MZGTI. Parameters ri (defined in Section 2.2.1, i=1-2) for MCT-GTEs in 22-MZGTI are listed in Table 1. Let p be 1 and 2 respectively, then we obtain two SHERCFs, i.e., 1-22-MZGTI and 2-22-MZGTI. The normalized intensity for 22-MZGTI and the two SHERCFs are shown in Fig. 4. To compare their performance detailedly, we list some indices for 22-MZGTI and the two SHERCFs in Table 2. From Fig. 4 and Table 2, one can see that 22-MZGTI has good passband flatness and wide bandwidth, which is the inherent advantage of MZGTI. If we need larger extinction ratio under the same bandwidth, the only way is to increase MZILs in MZGTI, which will increase the complication of MZGTI’s structure and difficulty in optimizing parameters [14]. SHERCF has a super high extinction ratio that is 2p times that of single MZGTI and keeps the same bandwidth as single MZGTI, which is far better than single MZGTI. So SHERCF is an effective way to solve the contradiction between bandwidth and extinction ratio. The ripple in passband is multiplied as well. Because the ripple of 22-MZGTI is fairly small, passband flatness of SHERCF is also good. So performance of SHERCF is determined by its basic cell. However, because the optical path is strictly controlled, SHERCF lose the other complementary output port.

From Eqs. (3)–(5) (7), we infer that passband centers(I_oa=1) and stopband centers(I_oa=0)are determined by δ=π+2qπ and δ=2qπ respectively(q is an integer). The cases for I_ob and I_oc are contrary to I_oa. All the wavelengths labeled along the horizontal coordinate axis in Fig. 4 are passband centers and stopband centers. SHERCF has the same spectral periodicity as its basic cell, i.e., mn-MZGTI. Spectrum spacing is the wavelength distance between two adjacent passband centers. Generally q is a large integer, so we use this approximate formula Δλ=λ ² ₀/ΔL to calculate it, where λ ₀ is the center wavelength. Δλ is approximately 0.8nm in Fig. 4.

Table 1. parameters r_i for 22-MZGTI

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Fig. 4. the normalized intensity of 22-MZGTI and the two SHERCFs, i.e., 1-22-MZGTI and 2-22-MZGTI: (a) detailed passband, and (b) two periods

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Table 2. Indices for 22-MZGTI and the two SHERCFs,i.e.,1-22-MZGTI and 2-22-MZGTI

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2.4 Chromatic dispersion of SHERCF

The group delay (GD) and the chromatic dispersion(CD) are defined by GD=-dΦ/dω and CD=dGD/dλ respectively [13], where Φ is phase. From Eq. (3), the dispersive phase of MCT-GTEn is Φ=2ϕ_n. Then we obtain the GD and CD for MCT-GTEn which also have recursive characteristics as phase.

with

where τ=L/c, c is the light velocity in vacuum. Specially, GD ₀=0.

with

where h=πL/λ ².

From Eq. (6), we get that the output dispersive phase of mn-MZGTI is the average of MCT-GTEm and MCT-GTEn. So are the group delay and chromatic dispersion. In other words, chromatic dispersion of mn-MZGTI is caused by the two sets of MCT-GTEs.

From Eqs. (9)–(12), it’s easy to find that GD and CD are proportional to L and L² respectively, where L is the effective cavity length. Both GD and CD have periodic response. Their periods are half the spectrum spacing. Moreover, by easy analysis, we infer that the passband and stopband centers are all zero dispersion points. So are the wavelength points determined by δ=π/2+yπ (y is an integer).

From Eq. (8), we get that both the GD and CD of SHERCF are 2p times those of single mn-MZGTI. Their CD curves intersect at the zero dispersion points. According to Table 1 and Eqs. (9)–(12), we plot the CD curves for 22-MZGTI and the two SHERCFs (i.e.,1-22-MZGTI and 2-22-MZGTI) in Fig. 5. One can see that CD in the region near passband centers and stopband centers is flattest. The worst dispersion region lies near the zero disperson points (i.e.,1549.7869nm and 1550.1873nm) determined by δ=π/2+yπ where exists the sharp edge of the spectrum. Generally speaking, CD of 22-MZGTI is bad, which is the inherent defect of interleavers based on resonators. Though extinction ratio of SHERCF multiplies, its CD multiplies as well. Hence CD problem of SHERCF is much more severe than single mn-MZGTI. The more cascaded MZGTIs are, the worse the CD is.

 

Fig. 5. CD curves for 22-MZGTI and the two SHERCFs, i.e.,1-22-MZGTI and 2-22-MZGTI

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3. Dispersion compensation

From Section 2.4, CD problem of mn-MZGTI is severe. However, CD of SHERCF is 2p times that of single mn-MZGTI, which is much more severe. According to the CD characteristics of mn-MZGTI, we propose a set of cascaded ring resonators as its TDC, i.e., multi-cavity ring resonator (MC-RR). A set of MC-RR is essentially a generalized multi-cavity transmissive Gires-Tournois etalon. We mark it MC-RRu, where u is the sum of cavities. MC-RRu illustrated in Fig. 6 is an all-pass filter as well.

 

Fig. 6. Schematic diagram of the proposed TDC,i.e., MC-RRu 3.1 MC-RRu

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By recursive analysis, the transmission coefficient of MC-RRu can be expressed as

where l_cu is the length of the coupling region of the u-th coupler. $r_{\mathrm{cu}}=\sqrt{K_{\mathrm{cu}}},K_{\mathrm{cu}}$ is the bar coupling ratio of the u-th coupler. f_u=(1-r_cu)/(1+r_cu), δ_c=0.5kL_c, and L_c=n_eff(l _c1+l′₁)=…=n_eff(l_{c (u-1)}+l′_u+l_cu). L_c is the effective cavity length of each ring resonator. Specially ϕ _co=0. (the extra subscript character ‘c’ means compensator)

Comparing Eqs. (3)–(4) with Eqs. (13)–(14), there is only a phase shift of π/2 between MCT-GTE and MC-RR. So we infer that a set of MC-RR is a set of generalized multi-cavity transmissive Gires-Tournois etalon. Each loop forms a cavity.

3.2 Chromatic dispersion of MC-RRu

From Eq. (13), we obtain the dispersive phase of MC-RRu is Φ_c=2ϕ_cu. Then the recursive formulas for group delay (GD_c) and chromatic dispersion (CD_c) of MC-RRu are

with

where τ_c=L_c/c, c is the light velocity in vacuum. Specially, GD _c0=0.

with

where h_c=πL_c/λ ².

Comparing Eqs. (9)–(12) with Eqs. (15)–(18), the recursive mathematical forms of GD and CD for MCT-GTEn and MC-RRu are identical. Differences only exist in c_cu and g_cu, which is due to the phase shift of π/2. If Lc is equal to L, MC-RRu has the same CD periodicity as mn-MZGTI and SHERCF. By easy analysis, all the zero dispersion points of MCT-GTEn (mn-MZGTI or SHERCF) mentioned in Secton 2.4 are also zero dispersion points of MC-RRu. The CD property of MC-RRu is closely related to parameters r_ci (i=1-u) chosen. So the CD curve can be tailored by adjusting r_ci. More details about the CD compensation ability of cascaded ring resonators have been described in another work. Generally speaking, the more parameters r_ci are, the stronger the tunability becomes. And then, the ability to compensate CD will be stronger.

It should be noted that the MC-RR can take place of MCT-GTE in Fig. 2 [7]. If so, a phase-shifter are required [6,7]. From the analysis above, there is a π/2 -phase-shift between MC-RR and MCT-GTE. So, a π/2 -phase-shifter should be added to either arm of the MZI [6]. An extra length of 0/(4) eff λ n is usually used to serve as the π/2 -phase-shifter. Then the phase matching condition (see Section 2.2.2) must be modified as ΔL=0.5L±λ ₀/4, where λ ₀ is the center-wavelength of the interleaver’s working waveband. However, only the center-wavelength (i.e., λ ₀) can obtain an accurate phase shift of π/2. For the wavelength away from λ ₀, the amount of phase shift will deviate from π/2. Therefore, the interleaver’s spectral performance away from the center-wavelength will be degraded since the modified phase matching condition is relative to the center-wavelength, which means that spectral performance will be inhomogenous along with wavelength. MCT-GTE can solve this problem since MZGTI doesn’t need any phase-shifters.

In Ref [14], a set of MCR-GTE is proposed as the TDC for MGTI. However, extra phase shifters should be added to every cavity of MCR-GTE to shift its compensation band to the passband of MGTI. The compensation band is the band where dispersion is tunable and can be used to compensate MGTI’s dispersion in passband. Thus if an extra set of MCT-GTE is used as the TDC for MZGTI, extra phase shifters are required as well. The merit of MC-RR as the TDC for MZGTI and SHERCF is that it doesn’t need any phase shifters. Due to the inherent π/2 -phase-shift of MC-RR mentioned above, the compensation band of MC-RR is exactly located at the passband of MZGTI if only all the effective cavity lengths are equal (Lc=L).

3.3 Dispersion compensation for MZGTI and SHERCF

According to the CD characteristics of SHERCF described in Section 2.4, the CD compensator proposed for the SHERCF is illustrated in Fig. 7, i.e., 2p-MC-RRu. Each rectangular frame contains a set of MC-RRu. And all the basic cells (i.e., MC-RRu) are identical. p is an integer, which is defined in Eqs. (7)–(8). The total CD of 2p-MC-RRu is 2p times that of single MC-RRu. MC-RRu is a transmissive and reciprocal element, so we give another structure of CD compensator for SHERCF, which is shown in Fig. 8. There is a mirror with full reflectivity composed of a 3dB coupler. Every basic cell is used twice, so this structure is superior to that shown in Fig. 7 in terms of saving the basic cells. Thus p-MC-RRu in Fig. 8 is equivalent to 2p-MC-RRu.

 

Fig. 7. Schematic diagram of the CD compensator for SHERCF,i.e., 2p-MC-RRu

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Fig. 8. Saving type of CD compensator for SHERCF,i.e.,2p-MC-RRu

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Through adjusting parameters r_ci (i=1-u), the compensation effect will be the best if the CD compensator has exactly the same dispersion and dispersion slope as that of SHERCF in passband, but with opposite sign respectively. In this case, CD of SHERCF can be compensated to zero. Actually, only in some points can the CD be compensated to zero. We only use single MC-RRu as a TDC for mn-MZGTI. But we use a set of 2p cascaded MC-RRs (i.e., 2p-MC-RRu) as a CD compensator for SHERCF (i.e., p-mn-MZGTI). Then what’s the optimized goal? Using MCT-GTEu (or MCR-GTEu) as a TDC, the compensated CD of single mn-MZGTI must be a curve with ripples, which is the same as the resultant CD of MGTI compensated by a set of multi-cavity Gires-Tournois etalons discussed in Ref [14]. So we also refer to the status when the sum of peaks or troughs in ripple region is equal to the number of ring resonators (i.e.,u) as the optimized goal for single mn-MZGTI. Likewise, we refer to the same status when the sum of peaks or troughs is equal to u as the optimized goal for the resultant CD of p-mn-MZGTI compensated by 2p-MC-RRu.

In Fig. 9, we use MC-RR4 and 2p-MC-RR₄ to compensate the CD of 22-MZGTI and SHERCF (p-22-MZGTI) respectively. In Fig. 9(a), CD curves of the same line style are the CD curves of the filter(22-MZGTI or p-22-MZGTI) and its corresponding CD compensator(MC-RR4 or 2p-MC-RR4) respectively. A compensated CD curve in Fig. 9(b)–9(c) is obtained by adding the two CD curves of the same line style in Fig. 9(a). In Fig. 9(c), there are four peaks and four troughs in the ripple region, i.e., quasi-flat dispersion region. One can see that the compensation effect of 22-MZGTI is the best. For p-22-MZGTI, the quasi-flat dispersion region gets narrower with the increase of p.

CD ripple bandwidth is the width of the area covered by the dispersion ripple, i.e., quasi-flat dispersion region [see Fig. 9(c)]. We define the ratio of CD ripple bandwidth to passband bandwidth (see Section 2.3) as the bandwidth ratio. With dispersion ripple fixed at ± 1ps/nm, Table 3 lists the bandwidth ratios of compensated filters using MC-RRu as the basic cell of CD compensator, i.e., w-MC-RRu. Specially, 1-MC-RRu is the TDC for 22-MZGTI. For SHERCF (i.e., p-22-MZGTI), w is equal to 2p. Table 3 shows that the larger u is, the better compensation effect will be. In other words, MC-RRu with more cascaded ring resonators has stronger CD compensation ability. And we also see that the compensation effect becomes worse with the increase of p.

 

Fig. 9. (a). CD curves of 22-MZGTI,the two SHERCFs and their corresponding CD compensators when the dispersion ripple is fixed at ± 1ps/nm ; (b) compensated CD curves of 22-MZGTI and the two SHERCFs; (c)Details of the quasi-flat dispersion region in (b).

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Table 3. Bandwidth ratios for 22-MZGTI and the two SHERCFs compensated by different compensators (dispersion ripple=± 1ps/nm)

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Figure 10 shows bandwidth ratio versus dispersion ripple for 22-MZGTI (w=1) and the two SHERCFs(w=2 and 4) using w-MC-RR4. One can see that the bandwidth ratio increases with the increase of the permissible dispersion ripple. For example, when the permissible dispersion ripple is ±12ps/nm, the bandwidth ratios for the three filters (i.e., 22-MZGTI, 1-22-MZGTI and 2-22-MZGTI) reach 78.5%, 74.0%, 69.7% respectively. Figure 11 shows the optimized parameters r_ci (i=1-4) versus dispersion ripple for w-MC-RR4. ci r increase with the increase of dispersion ripple and parameters follow the order of r _c1≫r _c2≫r _c3≫r _c4. Parameters should be smaller for w-MC-RR₄ with larger w when dispersion ripple is fixed at a certain value.

 

Fig. 10. Bandwidth ratio versus dispersion ripple for 22-MZGTI(w=1) and the two SHERCFs(w=2 and 4) using w-MC-RR₄

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Fig. 11. Optimized parameters _ci r versus dispersion ripple for w-MC-RR₄(w=1,2 or 4)

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4. Discussion

From Sections 2–3, one can see that the comb filters we proposed have perfect spectral performance and low dispersion which are composed of several wide-band 2×2 couplers. For the small size, the whole model is expected to be integrated on PLCs. The analysis above is under ideal conditions, i.e., the lengths mentioned are all accurately matched, all the components are lossless, and all the the coupling coefficients of the couplers are precisely controlled. In fact, the conditions are difficult to satisfy.

Length matching conditions are pivotal for comb filters based on resonators such as GTEs [12–14] and ring resonators [5–7]. It requires that all the effective cavity lengths of MCT-GTEm, MCT-GTEn and MC-RRu should be accurately equal (L=Lc), which means that the phase delay of every cavity (MZIL or ring resonator) should be accurately equal. Moreover, the phase matching condition[see Section 2.2.2] should be satisfied(Δ_L=0.5L), which means that the phase delay difference between the two arms of MZGTI should be half the phase delay of single cavity. Hence the length matching is essentially phase matching. If the real length deviation is δ′_L, the effective length mismatch amount is δ_L=n_eff δ′L. Then the phase mismatch amount (δϕ) is 2πδ_L/π. Performance of MZGTI is very sensitive to phase mismatch. For the filter working in C-band (λ=1550nm), if δϕ is restricted within 0.087rad (5°), then _L δ should be limited at least within 21.5nm. It’s very severe that the effective length should be accurate to nanometers, which is mainly due to the wavelength of C-band is very short. If λ is10µm, _Lδ can be larger, i.e., 138.5nm δ_L<138.5nm. The length mismatch decreases the extinction ratio, destroys the spectral symmetry and degrades the passband flatness [6].

It’s inevitable that the cavities suffer from several kinds of loss, such as material, bending, and waveguide imperfection. The cavity loss decreases the extinction ratio and degrades the passband flatness [5]. If the coupling coefficients are inaccurate, the ripples in passband and stopband will be irregular, which will decrease the extinction ratio and degrade the passband flatness. Hence the coupling coefficients need to be optimized [14]. Besides the bad influence to spectral performance, all the degradation factors also weaken the dispersion compensation effect, i.e., the CD curve in quasi-flat dispersion region is no longer a regular wavy curve and dispersion-compensated bandwidth in passband narrows.

Among all the degradation factors, the length mismatch (δ_L) is the most important and difficult one which should be overcomed firstly. It can be efficiently overcomed by means of the waveguide temperature control, i.e., inserting a heater whose refractive index can be adjusted [6,22]. Based on such technique, the effective length mismatch amount (δ_L) can be controlled within 10nm [22].

5. Conclusion

We have proposed and theoretically studied the novel MZGTI and super high extinction ratio comb filter (SHERCF) based on cascaded MZGTIs. The MZGTI which is the basic cell of the SHERCF is obtained by combining MZI and MCT-GTE composed of cascaded MZILs. It has the same spectral characteristics as MGTI, such as good passband flatness, high extinction ratio, wide passband bandwidth and stopband bandwidth, which are the inherent merits of interleavers based on GTEs. Through choosing proper parameters of MZGTI, we get an SHERCF with a spectral spacing of 0.8nm, a ripple in passband of 1.35×10^-4dB and an extinction ratio of 204dB by cascading only two MZGTIs. Both the passband bandwidth and stopband bandwidth are 0.329nm. The performance of SHERCF is determined by its basic cell. In theory, ripple and extinction ratio (dB) of SHERCF are several times of its basic cell, but the passband bandwidth and stopband bandwidth remain the same as its basic cell. So the more cascaded MZGTI are, the higher extinction ratio will be achieved.

Like MGTI, MZGTI has severe CD problem, which is the inherent defect of interleavers based on GTEs. A set of MC-RR composed of cascaded ring resonators is proposed as a TDC for MZGTI. Mathematically, there is only a phase shift of π/2 between MCT-GTE and MC-RR. Due to this inherent phase shift, MC-RR as the TDC for MZGTI doesn’t need any phase shifters, which is better than MCR-GTE (or MCT-GTE) [14]. Though the spectral performance of SHERCF is far better than its basic cell (MZGTI), its CD is several times of its basic cell. We propose a set of cascaded MC-RRs as the CD compensator for SHERCF. Moreover, we demonstrate that it is an efficient TDC for SHERCF with the optimized results. We find that the compensated bandwidth gets wider with the increase of permissible dispersion ripple and ring resonators in single MC-RR. However, it gets narrower with the increase of cascaded MZGTIs.

Brief discussions are given on the factors which degrade the performance of the filters, i.e., length mismatch, loss and coupling coefficient inaccuracy. It’s severe that the effective length mismatch amount should be restricted to several nanometers, e.g., 21nm. Hence, the length mismatch is the most important and difficult one which should be overcame firstly when the filters are implemented.