Similarities Between Birefringent Gires–TournoisInterferometer and Double Ring Assisted Mach–

Zehnder Electrooptic ModulatorJosé Edimar Barbosa Oliveira1, Bráulio Fernando R Sakamoto2 and William dos Santos Fegadolli3

Departamento de Microondas e Optoeletrônica – Divisão de Engenharia Eletrônica, Instituto Tecnológico de Aeronáutica-ITAPraça Marechal Eduardo Gomes, 50 – Vila das Acácias – São José dos Campos, São Paulo (SP) – Brazil. ZIP: 12228-900.

1edimar@ita.br2sakamoto@ita.br3fegadoli@ita.br

Abstract ⎯ Recently, Gires –Tournois’s interferometer (GTI)relying on electrooptic materials have been attracting greatinterest because they may enable high linear dynamic rangeoptical modulators. Therefore GTI’s may play a major role onthe subject of signal processing in the field of microwavephotonics. This paper addresses the modeling of a novelbirefringent GTI modulator set – up as a double ring assistedMach–Zehnder modulator (RAMZI). The modeling takes intoaccount the front surface mirror reflectance and reveals thatGTI based modulators may yield a dynamic range as high asthat achieved with RAMZI, with requires a rather morecomplex fabrication technology. Results of numerical simulations obtained with a GTI configuration which relies on LithiumNiobate (LiNbO3) and enables a push–pull operation with asingle voltage source are presented.

Index Terms ⎯ Dynamic Range, Electrooptic Modulators, Gires–Tournois Interferometer, LiNbO3, Microwave Photonics.

I – INTRODUCTION

Optical modulators with high linear dynamic range play asignificant role on the subject of microwave photonics signalprocessing. In the past few years there have been numerousproposals made to improve the linearity of electroopticmodulators based on either the Mach–Zehnder interferometer(MZI) or on the so called Ring–Assisted Mach–Zehnderscheme (RAMZI), in which one or two ring resonators arecoupled to the MZ arms [1], [2] and [3]. However, veryrecently, an innovative modulator schemes that combines thesuperlinear characteristics of Gires–Tournois Interferometer(GTI) with the sublinear characteristics of the Michelsoninterferometer to cancel the third order nonlinearity, wasproposed [3].

This paper deals with the modeling of optical modulatorsbased on Birefringent Gires–Tournois interferometer (BGTI),which consists of a Lithium Niobate (LiNbO3) Pockel’s cellthat enables a push–pull scheme operation with a singlevoltage source. It is shown that such BGTI’s yields a Spur–Free Dynamic Range (SFDR) as high as that of RAMZImodulator which comprises two identical rings coupled oneach arm and operating according to a push–pull schemeoperation. Furthermore, the properties of the BGTI areinvestigated by the following parameters change: front surface

mirror reflectance, interferometer length and Pockel’s cellhalf–wave voltage.

II – MZI ELECTROOPTIC MODULATORS PRINCIPLESThe schematic representation of an integrated optic modulator

based on a Mach– Zehnder interferometer (MZI), havingLiNbO3 substrate, is shown in Figure 1.

Fig. 1: The schematic representation of an integrated optic modulator usingMach–Zehnder interferometer (MZI).

Based on Fig. 1 one may describes the MZI as follows: first, an incoming input optical power, here after called inputoptical signal, is evenly divided by the input Y-junction;second, the two optical beam, in such way generated, travelsthrough the MZI’s arms, which are labeled by � and �);third, as the two signals propagates along the MZI’s arms theirphase are independently controlled by the microwavemodulating fields - such field have spatial and temporalpattern which depend on quite a few factors, such as: themicrowave source temporal waveform, the electrodesgeometry and their relative position with respect to the MZIarms; Finally, the two optical signals are recombined at theexit Y - junction and then an intensity-modulated opticalsignal is obtained.

Under the previously stated conditions and assuming apush–pull type operation, one obtains following expression for the optical intensity at the exit of the MZI:

( ) ( ) ( )2 sincosout inB m m mo o V v tI I

Vπ

π ω φ⎧ ⎫+ +⎡ ⎤= ⎨ ⎬⎣ ⎦⎩ ⎭

(1)

978-2-87487-002-6 © 2007 EuMA October 2007, Munich Germany

Proceedings of the 2nd European Microwave Integrated Circuits Conference

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where ( )inoI is the optical input intensity, BV and Vπ are,

respectively, the DC bias voltage and the half-wave voltage of phase modulators that constitutes the MZM arms; mω and mφare, respectively, the frequency and phase of the modulatorwaveform.

Equation (1) reveals that the output intensity exhibits anon–linear dependence with respect to the amplitude of themodulation signal, as it is illustrated in Fig. 2.

Fig. 2: Transfer function of a Mach–Zehnder modulator, push–pull driven bya harmonic waveform.

Furthermore, Fig. 2 also shows that the non–linearitydepends on the choice of the bias voltage, therefore, as it willbe discussed later, such type of MZI modulator yields a rather limited dynamic range, mostly due to the third ordernonlinearity.

Several approaches to linearization have been proposed,including cascaded MZI schemes, dual MZ schemes anddirectional coupler based modulators, among many others [2].A rather interesting linearization scheme, based on a ring-assisted MZI, the so called Ring-Assisted Mach-Zehnder(RAMZ), as shown in Fig. 3, was proposed by Xie et al. [3].The principle of operation of a RAMZ relies on the fact thatthe phase response of a ring resonator can be tuned bychanging the coupling coefficient between the ring resonatorand the optical waveguide, or Mach–Zehnder arm, to each it is connected. At a certain coupling strength, the transferfunction a RAMZ becomes rather linear, namely its third-order derivative vanish at a given bias point of the transferfunction.

Such schematics do not require complicated electronics oroptical control, once the appropriated coupling is determinedby fabrication. As an example, using the parameters specifiedin [1], one concludes that a RAMZ provides a Spur–FreeDynamic Range (SFDR) which is at least 18 dB higher thanthe one attained with a standard MZ modulator, as the oneshown in Fig.1

Fig. 3: Ring assisted Mach–Zehnder modulator (RAMZ).

However, from the point of view of microwave bandwidth,the MZI configuration displays better performance than theRAMZ. Regarding to the bandwidth issue, one may be able to enhance the RAMZ performance by using a multi-ringassisted MZI schematics, in which both, the ring resonantfrequencies and the coupling coefficients are properly chosen. As an example, a Double- Ring Assisted Mach–Zehnder (D-RAMZ), is shown in Fig. 4 [2].

Fig. 4: Double-Ring assisted Mach–Zehnder (D-RAMZ).

Like a conventional MZI, the D-RAMZ operates at aquadrature point, i.e., a quarter-period delay exists betweenthe two arms. In addition, the effective optical length of thering resonator is required to be an odd-multiple of the halfoperating wavelength in order to make it work in an off–resonance state.

It can be shown that if two identical rings are coupled oneach arm of a MZ with a push–pull schemes operation, theoutput intensity of the D-RAMZ can be rewritten as a function of the refractive index change inside the ring resonators. Asfar as SFDR is concerned, the D-RAMZ’s performanceoutplays both, the standard MZI and the single ring assistedMZI (RAMZ) [2].

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Recently, a few researchers reported results showing thatthe Birefringent Gires–Tournois Interferometer (BGTI), basedon electrooptic materials, has the potential to enables rathersimpler optical modulation schemes capable of yielding SFDR as good as the D-RAMZ. Motivated by such results theauthors of this paper undertook the challenge of developing aphysics–mathematical model for a BGTI modulator aiming tobe able to identify its similarity with respect to the D-RAMZbased modulator, as far as optical transfer function isconcerned.

III – BIREFRINGENT GIRES–TOURNOIS MODULATOR

The scheme of the proposed modulator, shown in Fig. 5,comprises standard components of electrooptic modulators(polarizer, circulator and waveplate) and a GTI made of aLiNbO3 Pockel’s cell, as shown in Fig. 5. In such modulator,the optical circular feeds the GTI with a linearly polarizedcontinuous–wave laser beam which propagates along the z–axis of the LiNbO3; the optical polarization is at an angle of45° to the x–crystal’s axis. Furthermore, the voltage sourceacts on the Pockel’s cell giving rise to y – axis orientedelectric field, which induces anti - symmetrical changes to the refractive indexes seen by each optical polarization inside thePockel’s .

Fig. 5: Schematic representation of an intensity modulator based onBirefringent Gires–Tournois interferometer.

Under the above stated conditions the refractive indexesseen by the x-polarized and y-polarized waves, respectively,inside the Pockel’s cell are:

322

322

1212

x o o o

y o o o

n n n r E n n

n n n r E n n

⎧ = + = + Δ⎪⎪⎨⎪ = − = − Δ⎪⎩

(2)

where on and 22r are, respectively, the ordinary refractiveindex and the relevant linear electrooptic coefficient ofLiNbO3 and E is the y–axis oriented modulation electric field.

Taking into account the Pockel’s cell front surface mirrorreflectance, ? , one can readily shows that the x and ycomponents of the total optical electric field steaming out of

the BGTI are given by:

( ),

,0,

,,

2 1

j x yj tx yout

x y j x yx y

eE eE

e

φω

φ

−

−

⎛ ⎞Γ +⎜ ⎟= ⎜ ⎟

⎜ ⎟+ Γ⎝ ⎠

(3)

where:

( )

( )

0

0

4 4

4 4

x o

y o

nn n L L

nn n L L

π πφ φλ λπ πφ φλ λ

Δ⎧ = + Δ = +⎪⎪⎨ Δ⎪ = + Δ = −⎪⎩

(4)

where 0E , λ and ω are, respectively the amplitude, the freespace wavelength and the frequency of the CW laser.

Some physical insight on the BGTI’operation as well asuseful mathematical simplifications on its modeling areobtained when one rewrites (3) as folow:

( ) 0, ,exp

2out

x y x yE

E j tω⎡ ⎤= − Φ⎣ ⎦ (5)

where,

0,

1 22arctg tan1 2x y

n Lφ πλ

⎡ ⎤− Γ Δ⎛ ⎞Φ = − +⎢ ⎥⎜ ⎟+ Γ ⎝ ⎠⎣ ⎦(6)

The above given result, (6), reveals one of the mostattractive features of the BGTI, namely, the fact that the phase of each optical polarization inside the LiNbO3 has a nonlineardependence on both the front surface mirror reflectance, ? ,and the electrooptic induced refractive index change.

The optical intensity at the output of the BGTI modulatorshown in Fig. 1, when one takes into account the static phaseshift due to the optical wave plate, is given by:

( )

( )

20

2 30

1 sin2

11 ...2 6

out x y

x y x y

EI

E

⎡ ⎤= + Φ − Φ =⎣ ⎦

⎡ ⎤= + Φ − Φ − Φ − Φ +⎢ ⎥⎣ ⎦

(7)

Based on (7) one may estimates the third-order spurious-free dynamic range (SFDR) of a BGTI by carrying out thewell known two-tone third order intermodulation analysis.However, in order to find out the similarities between theBGTI and D – RAMZ, as results predicted in this paper, oneshould make sure that ( ) ( )0 2 1 0,1, 2...k kφ π= + = [3]. Under such condition, one expands xΦ and yΦ in powers of nΔ toobtain:

( )( )

( )

( )( )

( )

33

3

33

3

14 1 1 41 3 1

14 1 1 41 3 1

x

y

n L n L

n L n L

π πλ λ

π πλ λ

⎧ − Γ Γ− Γ ⎛ ⎞Φ = Δ + Δ⎪ ⎜ ⎟+ Γ ⎝ ⎠⎪ + Γ⎪⎨

− Γ Γ⎪ − Γ ⎛ ⎞Φ = − Δ − Δ⎜ ⎟⎪ + Γ ⎝ ⎠ + Γ⎪⎩

(8)

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The results given by (7) and (8) shows that the BGTI’soptical transfer function mathematical dependence withrespect to mirror reflectance , ? , is exactly equal to thatexhibited by the D-RAMZ with respect to the couplingcoefficient between the ring resonator and the opticalwaveguide [2]. Hence, from the point of view of transferfunction, one concludes that the BGTI and D – RAMZ,behave on a rather similar fashion. Therefore, one shouldexpect that a BGTI based modulator offers high linearity withno requirements for complicated and precise fabricationprocess, when compared to D – RAMZ.

IV – NUMERICAL SIMULATIONS

By combining equations (7) and (8), one readily obtains anexpression for the optical intensity at the BGTI output, whichgives its dependence on both the front surface mirrorreflectance, ? , and the modulation voltage, ( )v t :

20 01 21 sin 4arctg cot

2 1 2outE nI L

φ πλ

⎧ ⎫⎧ ⎫⎡ ⎤− Γ Δ⎛ ⎞⎪ ⎪ ⎪⎪= − +⎨ ⎨ ⎬⎬⎢ ⎥⎜ ⎟+ Γ ⎝ ⎠⎪ ⎪⎣ ⎦⎪ ⎪⎩ ⎭⎩ ⎭(9)

The graphic representation of this equation as a function of the static voltage level, the so called static characteristic curve, allows one to readily investigates the improvement inmodulation linearity. For instance, as shown in Fig. 6, onesees a higher linear behavior when 0.5Γ = compared to theone displayed with 0Γ = .

Fig. 6: Graphic representation of modulator static characteristic curve, asgiven by (9).

To this aim, one has to expand its optical transfer function,as given by (9), in a Taylor series about a zero DC biasvoltage. Such procedure was adopted in this paper, and aseven–order polynomial was considered a satisfactoryapproach. By substituting the two–tone voltage

( ) ( )1 2sen senv t A t tω ω= + into the polynomial, the amplitude of the fundamental signal at angular frequency 1ω and theintermodulation (IM) distortion at angular frequency 1 22ω ω− ,which is usually named third–order intermodulation distortion, were calculated with multinomial theorem:

( )

( )

1

1 2

3 5 7

1 3 5 7

3 5 72

3 5 7

9 25 12254 4 64

3 25 7354 8 64

m m m m

m m m

v v v vI a a a a

V V V V

v v vI a a aV V V

ω

π π π π

ω ω

π π π

−

⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ = + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎨

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩

(10)

where the coefficients 1 3 5 7, , ea a a a only depend on the B-GTI front surface mirror reflectance, ? .

Numerical simulation of SFDR was carried out usingLiNbO3 material and data and setting the noise floor to be –160 dBm. Fig. 4 shows the signal power and IM power of for BGTI’s either having 0.5Γ = or 0.0Γ = , as a function ofthe input signal power.

Fig. 4: Output signal power and IM power of BGTI”s, with 0=Γ , dashedline, and 5,0=Γ , continuous line.

V – CONCLUSIONS

This paper dealt with the similarities between BGTI and D - RAMZ. It was shown that a BGTI based on a special tailored LiNbO3 substrate may yields a SFDR as high as that issued by a much more cumbersome D - RAMZ structure. It isworthwhile to point out that among the many potentialapplications of B-GTI, steams out those related to opticalsensors of electromagnetic field and optical signal waveformgenerators.

REFERENCES

[1] H. Tazawa, W. H. Steier, “Bandwidth of Linearized Ring ResonatorAssisted Mach-Zehnder Modulator” IEEE Photonics Letters ,pp. 1851-1853, Sep. 2005.

[2] X. Xie, J. Khurgin, J. Kang, F. Chow, “Linearized Mach-ZehnderIntensity Modulator”, IEEE Photonics Tech. Letters, vol. 15, no. 04, pp. 531-533, April 2003.

[3] N. Reingand, I. Shpantzer, T. Achian, A. Kaplan, A. Greenblatt, G.Harston, P.S. Cho, “Novel Design for the Broadband Linearized OpticalIntensity Modulator”, MilCom 2003 IEEE, vol. 2, pp. 1208 - 1212,October 2003.

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