A new concept of electro-optically reconfigurable waveguide superimposed gratings with their widely controllable transmission characteristics are present. It is based on two types of superimposed refractive index gratings, electronically induced in an electro-optically active core. These gratings can be independently switched ON and OFF or both simultaneously activated with the controllable weighting factor. In this case its transmission characteristics represent double-dip rejection band spectrum with independent control of the dip positions and bandwidth. This simple concept opens opportunities for developing a number of tunable devices for integrated optics by use of the proposed design as a building block.
©2001 Optical Society of America
Optical gratings are very useful components in optical telecommunication. They are employed in many applications like gain flatteners for optical amplifiers, dispersion compensators, optical sensors and so on. Technologies that provide real time optimization and control will be essential for enabling the next generation of optical network. Recently Kulishov et el demonstrated the designs of electro-optically (EO) induced gratings with tilted  and untilted  distributions of refractive index with adjustable parameters. However, the coupling properties of these multiple waveguide gratings have not yet been studied.
The general view of the EO induced grating is shown in Fig.1(a) in cross-section. The top and bottom electrode structures are under periodical electric potentials ±V0 and ±(V0 +ΔV), and they are shifted by δ(0≤δ≤2l) in respect with one another. From electrostatic point of view this problem is equivalent to superposition of two independent problems shown in Fig.1(b) and Fig.1(c). Due to the linear nature of Laplace’s equation, the solution of the electrostatic problem a can be written as a linear combination of the solutions for the electrostatic problems b and c:
Solution for the structure b describes an electric field distribution inside the waveguide’s core and cladding, that is repeated with the period 2l and does not have constant component of the electric field, where l is the electrode pitch. For δ≠0, l or 2l this field is asymmetrical in respect to the x-direction . In turn, the solution for the structure c is an electric field with periodicity l, and nonzero constant field component.
In the case of EO active cladding and/or core, periodical distribution of the refractive index will be formed as a response to the periodical distribution of the electric field. For Pockels type EO materials, such as EO crystals or EO polymers poled in x-direction, the refractive index is proportional to the x-component of the electric strength vector Ex =-∂φ/∂x with different proportionality factors for TE or TM polarized guided waves. Therefore two types of phase gratings will be induced in the waveguide: the first one (b-grating) with the period 2l that could be made tilted when δ≠0, l; and the second one (c-grating) with the period l. Because of the nonzero constant electric field, the second grating contains also EO induced constant component of the refractive index. Using different magnitude of the ΔV voltage, one can switch between the two gratings (ΔV=0, b is OFF; ΔV=-2V0 , c is OFF), or both of them can be activated with arbitrary weighting factor, when -2V0 <ΔV<0.
4. Guided wave interaction
The parameters of the waveguide structure should be chosen in such way that it supports only one core guided mode with a propagation constant β 1 and a number of cladding guided modes. Beating length between the core mode and the i-th cladding mode (propagation constant β 2) should be two times the beating length between the core mode and the j-th cladding mode (propagation constant β 3): 2(β 1-β 2)≈β 1-β 3. In this case the core guided mode will interact with the gratings out-coupling into co-propagating cladding modes (β 1>β 2>β 3). The interaction process is described by the following set of coupled wave equations:
where a 1, a 2 and a 3 are the complex modal amplitudes; Δβ12 =β1 -σ11 -β2 +σ22 -π/l; Δβ13 =β1 -σ11 -β3 +σ33 -2π/l; Δβ23 =β2 -σ22 -β3 +σ33 -π/l; σii are “dc” coupling coefficients and κij are the “ac” cross-coupling coefficients i,j=(1,3) . For the problem under consideration Δβ23 =Δβ13 -Δβ12 and the resulting equation can be reduced to:
where the new amplitudes R, S and P are R(z)=a 1 exp[-j(Δβ12 +Δβ13 )z/2]; S(z)=a2 exp[-j(Δβ13 -Δβ12 )z/2]; P(z)=a3 exp[-j(Δβ12 -Δβ13 )z/2]. We can analytically solve this set of first-order differential equations with boundary conditions R(0)=1 and S(0)=P(0)=0. Note that the cross-coupling coefficients can be controlled through ΔV voltage, because κ12 is proportional to ΔV/2V0 and κ13 and κ23 are proportional to (1+ΔV/2V0 ) and for the EO active core and non EO-active cladding they can be expressed in the following way:
where µ0 is the vacuum permeability; c is the speed of light in vacuum; k0 is the free-space wave number; r is the electro-optic coefficient; no is the intrinsic refractive index of the core (ordinary for TE modes and extraordinary for TM modes); ejt is the transverse electric field of the j-th mode (j=1,2,3), and finally and are the core cross-section distributions of the first harmonics of the electrostatic fields for b and c gratings. The “dc” coupling coefficients, σ11 , σ22 and σ33 are the linear functions of (1+ΔV/2V0 ) :
where Ex0 is the constant component of the electric field induced by the grating c in the core.
The typical transmission spectrum of the waveguide with the both activated refractive index distributions is presented in Fig.2. The waveguide parameters could be designed to position the peak wavelengths at a given separation Δλ. Choosing appropriate values of the ΔV (0 or -2V0 ) the rejection bands can be selectively switched ON and OFF, or they can be activated simultaneously with precise control of their reflectivity (dynamic range) (-2V0 <ΔV<0).
If the both b and c distributions are activated and momentum conditions are satisfied Δβ12 =Δβ13 =0, the core guided wave β1 is coupled into the cladding modes β2 and β3 . However, the momentum condition for the coupling between the cladding modes Δβ23 will be automatically satisfied that generally enables coupling between these modes. This situation is depicted in Fig.3.
Coupling between the two cladding modes is proportional to κ23 coupling coefficient. Normally it is much smaller than κ12 and κ13 , however it still prevent us from precise control over energy ratio out-coupled into the two cladding modes. This problem can be resolved within our design concept by using the potential application scheme presented in Fig.4a. The symmetry of TE2 and TE3 cladding modes (Fig.4d) and index distribution of the gratings (proportional to the x-component of the electrostatic field) results in vanishing the overlap integral (6) for κ23 , and the coupling between the coupling modes becomes impossible (Fig.4e).
For transmission gratings, power is exchanged between cladding and core modes periodically, so the grating length governs the bandwidth of the spectra. In our EO reconfigurable periodical structure, the length of the gratings can be easily controlled, because coupling does not occur without the presence of the voltage at the electrodes. Therefore, beating length of the both structures b and c might be adjusted independently through proper electric potential distribution, that gives an additional means to control the device.
For example, because of 2(β 1-β 2)≈β1 -β3 in our design, when the both gratings, b and c, are activated along the whole electrode structure length, the sort-wavelength dip (left one) is always about twice narrower than the long-wavelength (right) one, as it can be seen fromFig.2. In Fig.5 the blue curve demonstrates the case with the transmission spectrum where both gratings, b and c, activated along the whole length L with κ12 ≈κ13 , whereas the red curve represents the spectrum where the electrode fingers are run off by potentials that the first half of the length (L/2) are both grating are activated with κ12 ≈2κ13 (ΔV=-1.4 V0 ), and the second half (L/2) the grating c is only activated (ΔV=-2 V0 ). Side-lobes in the spectrum can also be substantially suppressed through spatial modulation (appodization) of the bias voltage V0 =V0 (z).
The bias voltage V0 also lends us one more parameter to control it is the peak separation through “dc” coupling coefficient adjustments. The transmission spectrum can be shifted towards longer or shorter wavelength just changing σ11 , σ22 and σ33 through V0 adjustment. For the case of the EO-active core and non-EO cladding with the induced index change only in the core, we find that σ11 ≫σ22 , σ33 (since normally confinement factor for a cladding mode is small). Even traditional EO materials, such as LiNbO3, can provide a substantial tuning range , not to mention newly advanced EO materials, polymer dispersed liquid crystals or organically modified glasses .
Another peculiarity of the proposed design is that full energy exchange between core and cladding modes can be achieved even with nonzero momentum mismatch Δβ12 ≠0 and Δβ13 ≠0 which is impossible for the single grating. The similar process takes place for superimposed Bragg gratings, where one of the mode acts as an information-transfer mediator .
A novel EO tunable filter has been demonstrated. The device is based on EO induction of two types of the periodical refractive index distributions with the opportunity to independently activate each of them or both simultaneously with the controllable weighting factor. In this case its transmission characteristics represent two peak rejection band spectrum with independent control of the peak positions and bandwidth. This simple concept opens opportunities for developing a number of tunable devices for integrated optics by use of the proposed design as a building block.
References and links
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2. M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. , 38, 7356–7363 (1998). [CrossRef]
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5. J. Zhao, X. Shen, and Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume index gratings”, Optics & Laser Technology , 33, 23–28 (2001) [CrossRef]