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Reconfigurable optical interleaver modules with tunable wavelength transfer matrix function using polymer photonics lightwave circuits

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Abstract

A transparent reconfigurable optical interleaver module composed of cascaded AWGs-based wavelength-channel-selector/interleaver monolithically integrated with multimode interference (MMI) variable optical attenuators (VOAs) and Mach-Zehnder interferometer (MZI) switch arrays was designed and fabricated using polymer photonic lightwave circuits. Highly fluorinated photopolymer and grafting modified organic-inorganic hybrid material were synthesized as the waveguide core and caldding, respectively. Thermo-optic (TO) tunable wavelength transfer matrix (WTM) function of the module can be achieved for optical routing network. The one-chip transmission loss is ~6dB and crosstalk is less than ~25 dB for transverse-magnetic (TM) mode. The crosstalk and extinction ratio of the MMI VOAs were measured as −15.2 dB and 17.5 dB with driving current 8 mA, respectively. The modulation depth of the TO switches is obtained as ~18.2 dB with 2.2 V bias. Proposed novel interleaver module could be well suited for DWDM optical communication systems.

© 2014 Optical Society of America

1. Introduction

Optical interleavers are important wavelength-selective components in dense wavelength-division multiplexed (DWDM) optical fiber communication systems. The technologies were used include Sagnac birefringence-based filter; Gires-Tournois based Michelson interferometers; Mach-Zehnder based interferometers; sampled Bragg gratings; and the arrayed waveguide grating (AWG) router [13]. They are constructed either using optical fibers or planar waveguide circuits (PLC). The fiber-based interleavers suffer from a lack of compactness and ruggedness, whilst the PLC-based interleavers can overcome these shortcomings [4,5]. Specially, monolithic multi-functional integrated photonic waveguide devices can enable the chip compact construction with lower costs and excellent performances [611]. Several material systems [1220], including lithium niobate, silicon-on-insulator (SOI), InP, and polymers, have been used to fabricate the interleaver modules. As a multi-functional material system, polymers exhibit well-controlled refractive indices, highly flexible structures, and large thermo-optic (TO) and electro-optic (EO) coefficients [2125], which can be advantageous to reduce manufacturing costs and realize monolithic integration with functional devices such as lasers and detectors.

In this paper, we propose a novel monolithically integrated module comprised of 4-channel cascaded AWGs-based wavelength-channel-selector/interleaver, 1 × 4 multimode interference (MMI) variable optical attenuators (VOAs) and 4-channel Mach-Zehnder interferometer (MZI) thermo-optic (TO) switch arrays using polymer photonic lightwave circuit. The tunable wavelength transfer matrix (WTM) function of the module can be achieved for optical routing networks. Fluorinated photopolymer and organic-inorganic hybrid materials were synthesized as the waveguide core and cladding, respectively. TO tuning WTM vector properties and parity signal WTM selection switching characteristics were analyzed, simulated and measured. The fabrication process of the device was described. Optimized structural properties of the waveguides and electrode heaters were provided. Through careful design and fabrication of the integrated interleaver modules, the excellent performances of the module were realized.

2. Design and experiments

2.1 Device structure

Novel polymer monolithically multi-functional integrated interleaver module was designed and fabricated. The schematic diagram of the integrated devices is shown in Fig. 1.This chip consists of 4-channel cascaded AWGs-based wavelength-channel-selector/interleaver, 1 × 4 MMI VOAs and 4-channel MZI TO switch arrays. The total size is 30 × 12 mm2. WTM vector tuning and parity signal WTM selection switching characteristics of the cascaded AWG-based inteleaver module can be achieved through TO tuning effect derived from serpentine heaters on arrayed waveguide section. The 1 × 4 VOAs with MMI waveguides and local electrode heaters can adjust each channel optical power of the signal wavelengths. The TO switch arrays with M-Z waveguide structures and arrayed electrode heaters can modulate the optical intensity and response time of multiplexing/demultiplexing signal wavelengths from the first-stage AWG-based wavelength-channel-selector to the last-stage AWG-based interleaver.

 figure: Fig. 1

Fig. 1 Schematic diagram of the monolithically multi-functional integrated interleaver module.

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The integrated chip comprises a first-stage AWG-based selector with 100 GHz channel spacing and a last-stage AWG-based interleaver with 200 GHz channel spacing. The metal serpentine electrode heaters are set on the arrayed waveguides region of the cascaded AWGs. The length difference between the adjacent electrodes is ΔLe. The adjacent electrodes are connected to each other end to end. The functional AWG device is based on the grating equation [26]

Δϕ2πm2πnsd/λ=θ
where ns is the effective index of the slab region, m is the diffraction order of the array, d is the arrayed waveguides separation and λ is the wavelength of the incident beam. Equation (1) shows that dispersion angle θ is resulting from a phase difference between adjacent waveguides ΔΦ. However, if the temperature of the waveguides shifts jΔT (j = 0, ± 1, ± 2, …) owing to the electrodes, the refractive index of the arrayed waveguides with change jΔnc, ΔΦ is determined by two compositions
Δϕ=2πλ(ncΔL+jΔncΔLe)
The relations can be written as
jΔxjΔnc=RΔLnsd
where Δx is the output waveguides separation, and R is the focal length. The variation of the focal position x will depend on the index migration Δnc. When the thermal shift from (T0-jΔT) to (T0 + jΔT), the beam will export from channel –j to channel j. Then the wavelength-channel-selective function is realized.

2.2 Analysis and simulation of wavelength transfer matrix

To ensure the low-loss single-mode polymer optical waveguide for planar lightwave circuits (PLCs), Negative-type fluorinated photoresist and organic-inorganic grafting PMMA were used as the waveguide core and cladding, respectively. Highly fluorinated polystyrene derivates (FPSDs) [27] were synthesized by copolymerization of 2,3,4,5,6-pentafluorostyrene (PFS) and fluorinated styrene derivate monomer (FSDM). The fluorinated polymers were doped into fluorinated bis-phenol-A novolac resin (FSU-8) using diphenyl iodonium salt as a photoacid generator (PAG). The refractive index and crosslinking density of the negative-type fluorinated photopolymer can be tuned and controlled by monitoring the feed ratio of comonomers. The SiO2-TiO2 network grafting PMMA material [28] offers several advantages such as low birefringence, good thermal stability and low wavelength dispersion. The refractive index of the sol-gels can be adjusted by monitoring the composition of TiO2 in hybrid materials. The refractive indices (n) of the polymeric core and cladding materials measured with an M-2000UI variable angle incidence spectroscopic ellipsometer are 1.571 and 1.560 at 1550-nm wavelength, respectively. The relative refractive index difference between the core and the cladding is about Δ = (n1-n2)/n1 = 0.7%. The design parameters of the cascaded AWGs are given in Table 1.

Tables Icon

Table 1. Design Parameters of the Polymer Cascaded AWGs

Cyclical free spectral region and wavelength assignment characteristics of the cascaded AWGs are designed for variable WTM functions [29]. The relations can be given as

λi+N=λi+FSR=λi+NΔλ
λij=λ=λ0+(i+j)Δλ
when four nodes A-D are connected to input channel ports −1~2 of the first-stage AWG-based selector, each with λ1-λ4, λ1 is defined as center wavelength 1550 nm, wavelength spacingΔλ is 0.8 nm. Input and output variable WTMs of the first-stage AWG-based selector are indicated in Fig. 2(a) and 2(b). The output wavelengths of the first-stage AWG-based selector port M are given as the product of unit matrix, wavelength transfer matrix F, and input wavelength matrix N as
M=IFN
M=IFN=[1000010000100001][δ3δ4δ1δ2δ4δ1δ2δ3δ1δ2δ3δ4δ2δ3δ4δ1][A1+A2+A3+A4B1+B2+B3+B4C1+C2+C3+C4D1+D2+D3+D4]=[A3+B4+C1+D2A4+B1+C2+D3A1+B2+C3+D4A2+B3+C4+D1]
Where Χk = λk (Χ = A, B, C, or D), δk·λk = λk, and δk·λl = 0 (kl). Demultiplexing output wavelength transmission spectral as each column vector of output wavelength matrix is shown in Fig. 3(a)-3(b) and Fig. 4(a)-4(b), respectively.

 figure: Fig. 2

Fig. 2 The operating principle and schematic configuration of the first-stage AWG selector (a) input WTM N and output WTM M; (b) input WTM N and variable WTM M’ with TO tuning effect.

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 figure: Fig. 3

Fig. 3 Output wavelength transmission spectral of output wavelength matrix M (a) as the first column vector and (b) as the second column vector.

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 figure: Fig. 4

Fig. 4 Output wavelength transmission spectral of output wavelength matrix M (a) as the third column vector and (b) as the fourth column vector.

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The temperature distribution of the heating structure of the serpentine electrode heaters on the casdaded AWGs with wavelength-channel-selected function could be written as

T(x,y)=PπKhLWtanh1[sinh(πy2Ls)cosh(πτ2Ls)]dτ
whereL=k=1MLk=ΔLek=1Mk, Lk is the length of the kth electrode, W is the width of the electrodes, Ls is the waveguide thickness which include the core and the cladding thickness, and Kh is the thermal conductivity of the core and cladding materials. The thermal conductivity Kh of fluorinated photopolymer core and sol-gel cladding was 0.35 and 0.18 Wm−1K−1, respectively. Ls is measured as 22-μm and the width electrode W is obtained as 30-μm. The 8-μm thickness top cladding is enough to reduce metal absorption caused by electrodes.

When the temperature changing ΔT1 from the serpentine electrode heaters is defined as 15.6 K, row vector of the output WTM can be adjusted between adjacent channels. Heat-driven power of electrodes is about 9.5 mW/channel based on the three-layer active region’s temperature distributions by Fourier transform method [30,31]. The output matrix M can be transformed into matrix M’, shown as

M'=I'FN=[0100001000011000][δ3δ4δ1δ2δ4δ1δ2δ3δ1δ2δ3δ4δ2δ3δ4δ1][A1+A2+A3+A4B1+B2+B3+B4C1+C2+C3+C4D1+D2+D3+D4]=[A4+B1+C2+D3A1+B2+C3+D4A2+B3+C4+D1A3+B4+C1+D2]
when the amount of temperature changing ΔT = 4NΔT1 (N = 0,1,2,3,…), row vectors of matrix M can remain unchanged. The WTM M are connected to input channel ports −1~2 of the last-stage AWG-based interleaver. Input and output variable WTMs of the last-stage AWG interleaver are indicated in Fig. 5(a) and 5(b). The output odd wavelengths Oodd of the last-stage AWG-based interleaver port are given as the product of wavelength transfer matrixFodd' and input wavelength matrix M as
Oodd=Fodd'M=[00δ1δ30δ1δ30δ1δ300δ300δ1][A3+B4+C1+D2A4+B1+C2+D3A1+B2+C3+D4A2+B3+C4+D1]=[0+0+A1+B30+B1+C3+0C1+D3+0+0A3+0+0+D1]
when temperature changing ΔT2 from the serpentine electrode heaters is 7.8 K, the output odd WTM Oodd can be transformed into the output even WTM Oeven by the product of wavelength transfer matrixFevenand input wavelength matrix M as
Oeven=Feven'M=[00δ2δ40δ2δ40δ2δ400δ400δ2][A3+B4+C1+D2A4+B1+C2+D3A1+B2+C3+D4A2+B3+C4+D1]=[0+0+B2+C40+C2+D4+0D2+A4+0+0B4+0+0+A2]
Fig. 6(a)-6(b) and Fig. 7(a)-7(b) show the TO tuning conversion relationships between WTM Oodd and Oeven corresponding to different channels, respectively. Heat-driven power of electrodes is about 4.5 mW/channel. When the amount of temperature changing ΔT = 2NΔT2 (N = 0,1,2,3,…), the output odd WTM Oodd can remain unchanged. The TO tuning WTM function is efficient and practical for designing for complicated wavelength routing networks.

 figure: Fig. 5

Fig. 5 The operating principle and schematic configuration of the last-stage AWG interleaver (a) input WTM M and output odd WTM Oodd; (b) input WTM M and even WTM Oeven with TO tuning effect.

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 figure: Fig. 6

Fig. 6 The TO tuning conversion relationships between WTM Oodd and Oeven (a) as the first column vector; (b) as the second column vector.

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 figure: Fig. 7

Fig. 7 The TO tuning conversion relationships between WTM Oodd and Oeven (a) as the third column vector; (b) as the fourth column vector.

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By the finite-different beam propagation method (FD-BPM) [32], the parameters of the 1 × 4 MMI VOAs were designed and optimized. MMI length, MMI width, and waveguide width can be defined as 2146 μm, 50 μm and 4 μm, respectively. Each arm is separated by a lateral distance of 13.8 μm, to achieve equal output optical power of each channel. By TO tuning effect of heaters on MMI region, output optical power from different channels can be adjusted. Extinction ratio of more than 20 dB can be obtained for each port of the VOAs. Cosine S bend arrayed waveguide section can achieve the smooth connection between output channel pitch 13.8 μm of the VOAs and input channel pitch 100 μm of AWG. For the switch arrays, the maximum power consumption of a single switch is less than 12 mW. A three-dimensional finite-difference beam propagation method (BeamPROP, Rsoft Co.) was used to numerically calculate the optical switch properties. The temperature field is simulated under the experimental condition with operating a phase difference of π. The result shows that there is a temperature of 3.6 K between the two phase arms under the operating π phase difference condition. The thermo-optic effect on other switch arrays can also be realized.

2.3 Fabrication procedure

The fabrication process is shown as Fig. 8.It shows that the organic - inorganic hybrid thin film of 10-μm thickness was formed as the cladding layer by spin coating on Si substrate, and the wafer was done by thermal annealing at 125 °C for 1 h to cross-link the polymer as the bottom layer. The layer thickness of 8-μm is sufficient to reduce the optical leakage into the substrate. A 4-μm thickness fluorinated photopolymer was spin-coated on the bottom cladding as waveguide layer, and then pre-baked at 65 °C for 10 min and 90 °C for 20 min to remove any traces of the solvent. The pattern exposure was performed at a wavelength of 365 nm using the 350 mW Hg lamp power through a contact chromium mask. The exposure time was 180 s. After post-baking, the resist was developed in propylene glycol-monomethyl ether-acetate (PGMEA) for 40 s, rinsed in isopropyl alcohol and then deionized water, and blown dry to form the channel waveguides. After that, it is very important to curing-bake the wafer at 150 °C for 30 min so that the adhesion between polymeric waveguides and bottom cladding layer can be enhanced well. A 8-μm-thick organic - inorganic hybrid film was spin-coated as the upper cladding layer to further reduce the optical leakage from waveguides into the metal film. Finally, the aluminum electrode heaters were patterned by photolithography and wet etching.

 figure: Fig. 8

Fig. 8 Fabrication process for UV defined waveguide and electrode heater structure.

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Scanning electron microscope (SEM) micrographs of the waveguides are shown in Fig. 9(a), 9(b), 9(c) and 9(d). They indicate the cross section of the waveguide, the top views of Y branchs and output channel arrays by SEM. It shows that the ridge-wall is smooth and almost vertical. It depicts that the process enables precise control of the core size.

 figure: Fig. 9

Fig. 9 SEM photographs of (a) input and (b) transmission segment patterns of cross-sectional waveguides; (c) Y branchs and (d) output channel arrays.

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Figure 10(a) gives structural patterns of electrode heaters from the 1 × 4 MMI VOAs by microscope ( × 500). The value of the resistance is about 150 Ω. Figure 10(b) gives interactional segments patterns of the serpentine electrode heaters on cascaded AWGs by microscope ( × 500). The measured total resistance was 800 Ω. Figure 10(c) gives structural patterns of the electrode heaters from switch arrays by microscope ( × 500). The value of the resistance is about 200 Ω. They show that the parameters designed of the serpentine electrode heaters can be realized very well.

 figure: Fig. 10

Fig. 10 The surface profiles of (a) MMI-VOA; (b) serpentine and (c) switch-arrayed electrode heaters ( × 500).

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2.4 Results and discussion

The propagation loss of a 4-μm-wide straight waveguide, measured by the cutback method at 1550 nm, was found to be 0.5 dB/cm. Schematic photographs of the proposed polymer 4-channel integrated interleaver module measured were shown as Fig. 11(a).Figure 11(b) gives the near-field patterns of the device. Signal light from a wide-band erbium-doped optical fiber amplifier (EDFA) was butt-coupled into the input waveguide through standard single-mode fiber. The signals from the output waveguides were magnified ( × 60) by lens and received by the CCD camera. The output channel spacing is 1.595 nm/channel, the fiber–fiber insertion loss at each channel is from 5.55 dB to 6.86 dB, and the crosstalk of the 4 channels is about −25 dB.

 figure: Fig. 11

Fig. 11 (a) Schematic photographs of the proposed polymer 4-channel integrated interleaver module measured. (b) Near-field guide-mode patterns of the device with signal light from a wide-band EDFA.

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Figure 12(a) and 12(b) show the actual spectral response of the first-stage AWG-based selector with TO tuning effect from the output channel#0. It is indicated that when temperature changing ΔT of the serpentine electrode heaters is 17 K, row-vector adjustment of the output WTM can be achieved. The driving voltage is applied as 7.5 V and Heat-driven power of electrodes is about 10.5 mW/channel. Figure 12(c) and 12(d) show the actual spectral response of the last-stage AWG-based interleaver with TO tuning effect from the output channel#0. It is given that when temperature changing ΔT of the serpentine electrode heaters is 10 K, the output odd signal wavelengths (λ1 and λ3) can be transformed into the output even singal wavelengths (λ2 and λ4). Parity signal row-vector adjustment of the output WTM can be also obtained. The driving voltage is applied as 4.5 V and Heat-driven power of electrodes is about 6.5 mW/channel. Wavelength selective switching characteristics of the integrated interleaver module with variable WTM function are realized.

 figure: Fig. 12

Fig. 12 The actual output spectral response of the first-stage AWG-based selector from the output channel#0 with (a) ΔT = 0 and (b) ΔT = 17 K ; The actual output spectral response of the Last-stage AWG-based interleaver from the output channel#0 with (a) ΔT = 0 and (b) ΔT = 10 K.

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The operating performances of the 1 × 4 MMI VOAs and 4-channel MZI TO switch arrays were obtained. In the range of signal wavelengths, the crosstalk and extinction ratio of the MMI VOAs were measured as −15.2 dB and 17.5 dB, respectively. The value of minimum driving current was required as 8 mA. For the MZI TO switch arrays, Fig. 13(a) shows that the thermo-optic switching response observed by applying square–wave voltage at a frequency of 100 Hz. It can be noted that the rise and fall times were 330-μs and 510-μs, respectively. Figure 13(b) gives that channel output intensity versus power consumption of the optical switch at 1550 nm for TM mode. The extinction ratio of the TO switch was measured about ~18.2 dB with 2.2 V bias. The applied electric power as the switching power is actually 8.6 mW.

 figure: Fig. 13

Fig. 13 Performances of MZI TO switch arrays. (a) TO switch responses measured by applying square-wave voltage at frequency of 100 Hz. (b) Actual channel output versus power consumption of optical switch at 1550 nm for TM mode.

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3. Conclusions

In summary, a transparent interleaver module composed of cascaded AWGs-based wavelength-channel-selector/interleaver monolithically integrated with VOAs and switch arrays is successfully designed and fabricated using polymer photonic lightwave circuit technology. Excellent TO tuning WTM vector properties and parity signal WTM selection switching characteristics were obtained. The preferable structural profiles of waveguide and electrode were obtained by the pictures of SEM and microscope. These features were advantageous to optimize producing process and enhance optical performances of polymer. The one-chip transmission loss is ~6 dB and crosstalk is less than ~25 dB for transverse-magnetic (TM) mode. The crosstalk and extinction ratio of the MMI VOAs were measured as −15.2 dB and 17.5 dB with driving current 8 mA, respectively. The modulation depth of the TO switches is obtained as ~18.2 dB with 2.2 V bias. The maximum power consumption of a single switch is less than 9 mW. The monlithic multi-functional integrated interleaver module improves performances of the device, greatly simplifies the assembly and represents significant cost saving in package. The technique is very useful for efficient DWDM optical communication systems.

Acknowledgments

The authors gratefully acknowledge financial support from National Natural Science Foundation of China (No. 61107019, 61177027, 61275033, 61205032, 61261130586), Ph.D. Programs Foundation of Ministry of Education of China (No. 20110061120054), China Postdoctoral Science Foundation (No. 2011M500597, 2012M510900), China Postdoctoral Science special Foundation (No.2012T50277), Program for Special Funds of Basic Science & Technology of Jilin University (No. 201103071, 201100253), Science and Technology Development Plan of Jilin Province (No. 20130522151JH, 20140519006JH).

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Figures (13)

Fig. 1
Fig. 1 Schematic diagram of the monolithically multi-functional integrated interleaver module.
Fig. 2
Fig. 2 The operating principle and schematic configuration of the first-stage AWG selector (a) input WTM N and output WTM M; (b) input WTM N and variable WTM M’ with TO tuning effect.
Fig. 3
Fig. 3 Output wavelength transmission spectral of output wavelength matrix M (a) as the first column vector and (b) as the second column vector.
Fig. 4
Fig. 4 Output wavelength transmission spectral of output wavelength matrix M (a) as the third column vector and (b) as the fourth column vector.
Fig. 5
Fig. 5 The operating principle and schematic configuration of the last-stage AWG interleaver (a) input WTM M and output odd WTM Oodd; (b) input WTM M and even WTM Oeven with TO tuning effect.
Fig. 6
Fig. 6 The TO tuning conversion relationships between WTM Oodd and Oeven (a) as the first column vector; (b) as the second column vector.
Fig. 7
Fig. 7 The TO tuning conversion relationships between WTM Oodd and Oeven (a) as the third column vector; (b) as the fourth column vector.
Fig. 8
Fig. 8 Fabrication process for UV defined waveguide and electrode heater structure.
Fig. 9
Fig. 9 SEM photographs of (a) input and (b) transmission segment patterns of cross-sectional waveguides; (c) Y branchs and (d) output channel arrays.
Fig. 10
Fig. 10 The surface profiles of (a) MMI-VOA; (b) serpentine and (c) switch-arrayed electrode heaters ( × 500).
Fig. 11
Fig. 11 (a) Schematic photographs of the proposed polymer 4-channel integrated interleaver module measured. (b) Near-field guide-mode patterns of the device with signal light from a wide-band EDFA.
Fig. 12
Fig. 12 The actual output spectral response of the first-stage AWG-based selector from the output channel#0 with (a) ΔT = 0 and (b) ΔT = 17 K ; The actual output spectral response of the Last-stage AWG-based interleaver from the output channel#0 with (a) ΔT = 0 and (b) ΔT = 10 K.
Fig. 13
Fig. 13 Performances of MZI TO switch arrays. (a) TO switch responses measured by applying square-wave voltage at frequency of 100 Hz. (b) Actual channel output versus power consumption of optical switch at 1550 nm for TM mode.

Tables (1)

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Table 1 Design Parameters of the Polymer Cascaded AWGs

Equations (11)

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Δϕ2πm 2π n s d/λ =θ
Δϕ= 2π λ ( n c ΔL+jΔ n c Δ L e )
jΔx jΔ n c = RΔL n s d
λ i + N = λ i + F S R = λ i + N Δ λ
λ i j = λ = λ 0 + ( i + j ) Δ λ
M = I F N
M = I F N = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] [ δ 3 δ 4 δ 1 δ 2 δ 4 δ 1 δ 2 δ 3 δ 1 δ 2 δ 3 δ 4 δ 2 δ 3 δ 4 δ 1 ] [ A 1 + A 2 + A 3 + A 4 B 1 + B 2 + B 3 + B 4 C 1 + C 2 + C 3 + C 4 D 1 + D 2 + D 3 + D 4 ] = [ A 3 + B 4 + C 1 + D 2 A 4 + B 1 + C 2 + D 3 A 1 + B 2 + C 3 + D 4 A 2 + B 3 + C 4 + D 1 ]
T(x,y)= P π K h LW tan h 1 [ sinh( πy 2 L s ) cosh( πτ 2 L s ) ]dτ
M ' = I ' F N = [ 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 ] [ δ 3 δ 4 δ 1 δ 2 δ 4 δ 1 δ 2 δ 3 δ 1 δ 2 δ 3 δ 4 δ 2 δ 3 δ 4 δ 1 ] [ A 1 + A 2 + A 3 + A 4 B 1 + B 2 + B 3 + B 4 C 1 + C 2 + C 3 + C 4 D 1 + D 2 + D 3 + D 4 ] = [ A 4 + B 1 + C 2 + D 3 A 1 + B 2 + C 3 + D 4 A 2 + B 3 + C 4 + D 1 A 3 + B 4 + C 1 + D 2 ]
O o d d = F o d d ' M = [ 0 0 δ 1 δ 3 0 δ 1 δ 3 0 δ 1 δ 3 0 0 δ 3 0 0 δ 1 ] [ A 3 + B 4 + C 1 + D 2 A 4 + B 1 + C 2 + D 3 A 1 + B 2 + C 3 + D 4 A 2 + B 3 + C 4 + D 1 ] = [ 0 + 0 + A 1 + B 3 0 + B 1 + C 3 + 0 C 1 + D 3 + 0 + 0 A 3 + 0 + 0 + D 1 ]
O e v e n = F e v e n ' M = [ 0 0 δ 2 δ 4 0 δ 2 δ 4 0 δ 2 δ 4 0 0 δ 4 0 0 δ 2 ] [ A 3 + B 4 + C 1 + D 2 A 4 + B 1 + C 2 + D 3 A 1 + B 2 + C 3 + D 4 A 2 + B 3 + C 4 + D 1 ] = [ 0 + 0 + B 2 + C 4 0 + C 2 + D 4 + 0 D 2 + A 4 + 0 + 0 B 4 + 0 + 0 + A 2 ]
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