MATHEMATICAL MODELING OF RING RESONATOR FILTERS FOR PHOTONIC APPLICATIONS

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MATHEMATICAL MODELING OF RING RESONATOR

FILTERS FOR PHOTONIC APPLICATIONS

Jyoti Kedia1(Assistant professor), Dr. Neena Gupta2(Associate Professor, Member IEEE)

1,2

PEC University of Technology, Sector 12, Chandigarh

Abstract: Optical Filters are crucial passive components for wavelength filtering in WDM systems for photonic networks. Micro-waveguide and fiber based ring resonators are of great interest due to their versatile functionalities and compactness. In this paper the general characteristics of serially coupled ring resonator filters are analyzed and analytically derived the optical transfer function in Z domain as a new concept.

Introduction: Various photonic components, such as add-drop multiplexers and interleavers, are important in wavelength division multiplexed fiber optic networks. With increase in complexity of these systems, the demand for photonic components with smaller footprint, lower cost and low power dissipation has increased. The optical ring resonator is one such photonic component. The layout of an integrated ring resonator channel dropping filter, is described in Fig.1. An integrated optical ring resonator is a single transverse mode waveguide based device formed as a circle. Two couplers enable light to be inserted and extracted from the ring.

These ring resonator based selective bandpass filters can perform functions like channel add-drop, channel selection, demultiplexing and multichannel filtering in dense wavelength division multiplexing

Fig1. Integrated optical ring resonator

(DWDM) systems. High stopband rejection is the most important required characteristic of these functions so that low cross talk between channels, a flat top filter response and low insertion loss can be achieved [1]. Since the single ring resonator filter is insufficiently perceptive for many important applications of DWDM systems, the serially coupled multiple ring resonator filters are required to achieve pass band with a sharper roll off, flatter top and higher out off band rejection around a resonance.

The single ring resonator filters has a simple Lorentzian response. The Lorentzian reponses have a sharp peak whereas filter application require a flat top. Also in DWDM systems high selectivity filters are required

waveguides Integrated

optics microring

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which have wide free spectral range (FSR) to accommodate large channel counts. The FSR of ring resonator is given by

𝐹𝑆𝑅 =_𝑛 𝑐

𝑒𝑓𝑓∗𝐿𝑟𝑖𝑛𝑔 (1)

Lring is the ring circumference, c is velocity of light in vacuum and neff is the effective refractive index of the waveguide. As the equation shows that if we wish to achieve a wide spectral separation between the peaks of the transfer function we must use a ring with smaller diameter. Unfortunately there is a limit for the ring circumference because very small rings have significant bending loss. That’s why, instead of making small rings, we can use multiple rings to achieve better filtering functions.

To design an optical filter, the electromagnetic field equations are used where the fields are solved in the frequency or time domain. To make the solution less tedious, scattering matrix method [2] and the transfer matrix/ chain matrix algebraic, method [2, 3, 4] has been developed for determining the transfer functions in Z domain which is an effective analytical method of signal processing. The optical circuits are considered to be linear and time invariant. There is another approach proposed by [5] called the signal flow graph (SFG) method which is fast and is graphical in nature. This method yet has not been widely employed in analysis of optical circuits but in electrical circuits.

TransferFunction of Ring Resonator

The ring resonator architectures are illustrated in fig 2 and 3 alongwith their signal flow graph. By taking into account the coupling factor ĸi of the ith coupler and the insertion loss

γ for each coupler, the lightpass through the throughput path can be expressed as ci = [(1- γ) (1- ĸi)]1/2 and the light pass through the cross path is given by –jsi = -j[(1- γ) ĸi]1/2. The transmission of light along the ring resonator (the closed path), we can represent it as ξ=xz-1

, where x = exp(-αL/2) is the one round trip loss coefficient, and the z-1 is the Z transform parameter which is defined as z-1 = exp(-jβL) where β=kneff is the propagation constant, k= 2π/λ is the vacuum wave number, neff is the effective refractive index of the waveguide and the circumference of the ring is L= 2πR where R is the radius of the ring.

There are basically three essential paramteres describing the behavior of a MRR filter:

1. the -3DB bandwidth or Full Width at Half maximum (FWHM)

2. the on-off ratio and

3. the shape factor.

The on-of ratio for the throughput and drop port, which is the ratio of the on-resonance intensity to the off-resonance intensity, is given by:

𝑜𝑛 − 𝑜𝑓𝑓 𝑟𝑎𝑡𝑖𝑜 = 𝑇max ⁡(𝑡𝑕 𝑟𝑜𝑢𝑔 𝑕 𝑝𝑢𝑡 𝑝𝑜𝑟𝑡 )

𝑇min ⁡(𝑑𝑟𝑜𝑝 𝑝𝑜𝑟𝑡 ) (2)

And

𝑆𝑕𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 = _{−10 𝑑𝐵 𝑏𝑎𝑛𝑑𝑤𝑖𝑑𝑡 𝑕}−1 𝑑𝐵 𝑏𝑎𝑛𝑑𝑤𝑖𝑑𝑡 𝑕 (3)

The ideal response shape is a rectangular filter function with the shape factor of unity.

Mason’s rule for optical circuits

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along the ring direction) taking one node only once. A loop is a forward path that begins and ends on same node. The loop gain or path gain is the product of all the links along that loop or path respectively. If there is no node in common between two loops, they are said to be non-touching. The mason’s rule states that the transfer function or input-output transmittance relationship from node E1(z) to node En(z) in a signal flow graph is given by

𝐻 =1_∆ 𝑛𝑖=1𝑇𝑖∆𝑖 (4)

Where H is the network function relating an input and an output port, Ti is the gain of the i-th forward path from an input to an output port, and n is the total number of forward paths from an input to an output. The signal flow graph determinant is given by

∆= 1 − 𝑇𝑖 𝑖+ 𝑇𝑖,𝑗 𝑖𝑇𝑗 − 𝑖,𝑗 ,𝑘𝑇𝑖𝑇𝑗𝑇𝑘+. . . (5)

Where Ti is the transmittance gain of the i-th loop. In the above equation, the products of non-touching loops are only included. The symbol Δi in (4) is the determinant Δ after all loops which touch the Ti path at any node have been eliminated.

Transfer functions of single and double ring resonator

The architecture and signal flow graph for single ring and double ring resonators is shown in fig 2 and 3 respectively.

The transfer function 𝑬_𝑬𝟑(𝒛)

𝟏(𝒛)

There is one individual loop gain of the signal flow graph (SFG) which is given by

𝐿1₁ _{= 𝐶}

1𝐶2𝜉 (6)

For throughput port the forward path transmittances from node 1 to node 3 and its determinant (corresponding to non touching loop) is given by

𝑇_1𝑡1 _{= −𝐶} 2𝑠12𝜉

∆₁= 1 (7)

𝑇_2𝑡1 _{= 𝐶} 1

∆₂= 1 − 𝐿1₁ ₌_1-_𝐶

1𝐶2𝜉 (8)

From Mason’s rule (5), the determinant of SFG of fig 2b is given by

∆= 1 − 𝐿1₁ ₌_1-_𝐶

1𝐶2𝜉 (9)

By substituting (7)- (9) into (4), the transfer function for the throughput port in fig 2b is given by

𝐸3(𝑧)

𝐸1(𝑧)= 𝐻𝑡

1 ₌ 𝑐1−𝐶2𝜉

1−𝐶1𝐶2𝜉 (10)

The transfer function 𝑬_𝑬𝟐(𝒛)

𝟏(𝒛)

There is only one forward path transmittance from node 1 to node 2 for the drop port which also touches the loop L11 given by (6). So

𝑇_1𝑑1 _{= −𝑆} 1𝑆2 𝜉

∆1= 1 (11)

Substituting (9), (11) into (4), the transfer function for the drop port can be given as

𝐸₂(𝑧)

𝐸₁(𝑧)= 𝐻𝑑1 = −

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Fig 2a. Architecture of SRR add/drop filter

Fig 2b. Z-transform diagram (SFG)

Transfer function of a double ring resonator

SFG of a serially coupled double ring resonator is shown in fig 3b. The input node is E1(z), E3(z) is the throughput node and E12(z) is the drop node.

The transfer function 𝑬_𝑬𝟑(𝒛)

𝟏(𝒛)

There are three individual loop gains of the SFG whose gains are given by

𝐿₁2 _{= 𝐶}

1𝐶2𝜉 (12)

𝐿2₂ _{= 𝐶}

2𝐶3𝜉 (13)

𝐿2₂ _{= 𝐶}

1 𝜉 −𝑗𝑆2 𝜉𝐶3 𝜉 −𝑗𝑆2 𝜉 =

𝐶₁𝐶₃𝜉2_𝑆

22 (14)

Fig 3a. Architecture of DRR add/drop filter

Fig 3b. Z-transform diagram (SFG) 10 C3 12

𝜉 𝜉

8 C2 6

9 C3 11

-jS3

1 C1 3

2 C1 4

7 C2 5

𝜉 𝜉

-jS1

-jS2

E1(z) E3(z)

E12(z)

Drop port, E 12

Input port, E1 Throughput port, E3

K1

K2

K3

E1(z) E3(z)

E2(z)

𝜉 𝜉

1 C1 3

8 C2 6

2 C1 4

7 C2 5 -jS1

-jS2

Input port, E1 Throughput port, E3

Drop port, E2

K1

K2

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There is one possible product of transmittance of two non-touching loops, resulting from separation of the loop L12 and L22, given by

𝐿₁₂2 _{= 𝐶}

1𝐶3𝐶22𝜉2 (15)

The forward path transmittance from node 1 to node 3 for the throughput port and its determinant which corresponds to the non-touching loop can be denoted as

𝑇_1𝑡2 _{= −𝐶}

2𝑆12𝜉

∆1= 1 − 𝐿22 = 1 − 𝐶2𝐶3𝜉 (16)

𝑇_2𝑡2 _{= 𝐶}

3𝑆12𝑆22𝜉2

∆₂= 1 (17)

𝑇_3𝑡2 _{= 𝐶} 1

∆₃= 1 − 𝐿₁2_{+ 𝐿} 2

2 _{+ 𝐿}

3 2_{+ 𝐿}

12

2 ₌

= 1 − 𝐶1𝐶2𝜉 − 𝐶2𝐶3𝜉 + 𝐶1𝐶3𝜉2𝑆22+

𝐶₁𝐶₃𝐶₂2_𝜉2 ₍₁₈₎

From (5) and by using the relation S22+C22=1, the determinant of the SFG from the Mason’s rule is given by

∆= 1 − 𝐿₁2_{+ 𝐿} 2

2 _{+ 𝐿}

3 2_{+ 𝐿}

12

2 ₌

= 1 − 𝐶1𝐶2𝜉 − 𝐶2𝐶3𝜉 + 𝐶1𝐶3𝜉2 (19)

Substituting (16)-(19) into (4) the transfer function for throughput port is given by

𝐸3(𝑧)

𝐸1(𝑧)= 𝐻𝑡

2₌ 𝐶1− 𝐶2𝜉 − C1𝐶2𝐶3𝜉 + 𝐶3𝜉2

1 − 𝐶1𝐶2𝜉 − 𝐶2𝐶3𝜉 + 𝐶1𝐶3𝜉2

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The transfer function 𝑬_𝑬𝟏𝟐(𝒛)

𝟏(𝒛)

There is only one forward path transmittance from node 1 to 12 for the drop port and since all loops touch this forward path,

𝑇_1𝑑2 _{= −𝑗𝑆}

1𝑆2𝑆3𝜉

∆₁= 1 (21)

Substituting (19) and (21) into (4) we get the transfer function at the drop port as

𝐸12(𝑧)

𝐸₁(𝑧) = 𝐻𝑑2

= 𝑗S1𝑆2𝑆3𝜉

1 − 𝐶₁𝐶₂𝜉 − 𝐶₂𝐶₃𝜉 + 𝐶₁𝐶₃𝜉2

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The transfer function 𝑬_𝑬𝟏𝟐(𝒛)

𝟏𝟎(𝒛)

The loop gains are same as that of (12)-(15) for calculating this transfer function. The forward path transmittances from node 10 to 12 and its determinant can be denoted as

𝑇_1𝑡2 _{= −𝐶}

2𝑆32𝜉

∆₁= 1 − 𝐿2₂ _{= 1 − 𝐶}

2𝐶1𝜉 (23)

𝑇_2𝑡2 _{= 𝐶}

1𝑆32𝑆22𝜉2

∆₂= 1 (24)

𝑇_3𝑡2 _{= 𝐶} 3

∆3= 1 − 𝐿21+ 𝐿22 + 𝐿23 + 𝐿212 = ∆

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Substituting (23)-(25) into (4) the transfer function is given by

𝐸12(𝑧)

𝐸₁₀(𝑧)=

𝐶3− 𝐶2𝜉 − C1𝐶2𝐶3𝜉 + 𝐶1𝜉2

1 − 𝐶₁𝐶₂𝜉 − 𝐶₂𝐶₃𝜉 + 𝐶₁𝐶₃𝜉2

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The transfer function _𝑬𝑬𝟑(𝒛)

𝟏𝟎(𝒛) is same as that of

𝑬𝟏𝟐(𝒛)

𝑬𝟏(𝒛). The obtained results for each transfer

function of double ring resonator add/drop filter can be expressed in matrix form. This relates the input ports to the output ports and is called scattering matrix which is given by

𝐸₃(𝑧)

𝐸12(𝑧) = 𝑆𝑅𝑅(𝑧)

𝐸₁(𝑧) 𝐸10(𝑧)

𝑆_𝑅𝑅 𝑧 = 1 ∆

𝑋₁ 𝑗S₁𝑆₂𝑆₃𝜉 𝑗S₁𝑆₂𝑆₃𝜉 𝑋₂

Where Δ=1 − 𝐶1𝐶2𝜉 − 𝐶2𝐶3𝜉 + 𝐶1𝐶3𝜉2

X1=𝐶1− 𝐶2𝜉 − C1𝐶2𝐶3𝜉 + 𝐶3𝜉2

X2=𝐶3− 𝐶2𝜉 − C1𝐶2𝐶3𝜉 + 𝐶1𝜉2

Using these equations, it is possible to design a double ring resonator with box like filter response shape. The couplers for SRR may be assumed symmetrical i.e. k1=k2 with internal losses fully compensated (α=0).

With full compensation of rings for DRR the outer couplers may be taken symmetrical with coupling coefficients k1=k3 and a smaller value of coupler k2 at the center.

Conclusion

The transfer function of single ring resonator and serially coupled double ring resonator employing graphical approach has been presented. The approach uses z transform

which is very easy to implement and simulate using MATLAB.

References

[1] Melloni A, Martinelli M, “Synthesis of direct coupled resonators bandpass filters for WDM systems, IEEE journal of Lightwave technology, 20 (2), 2002, pp. 296-303.

[2] Schwelb O., “Generalized analysis for a class of linear interferometric networks. I. Analysis”, IEEE transaction on Microwave theory and techniques, 46 (10), 1998, pp. 1399-1408.

[3] Capmany J, Muriel M. A., “A new transfer matrix for the analysis of fiber ring resonators: compound coupled structures for FDMA Demultiplexing”, IEEE Journal of Lightwave Technology, 8(12), 1990, pp. 1904-1919.

[4] Moslehi B, Goodman J. W., Tur M., Shaw H.J., Fiber optic lattice signal Processing”, Proceedings of the IEEE, 72(7), 1984, pp. 909-930.