## Electro-optically reconfigurable waveguide superimposed gratings

Optics Express, Vol. 9, Issue 10, pp. 483-489 (2001)

http://dx.doi.org/10.1364/OE.9.000483

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### Abstract

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.

© Optical Society of America

## 1. Introduction

1. M. Kulishov, P. Cheben, X. Daxhelet, and S. Delprat, “Electro-optically induced tilted phase gratings in waveguide,” J. Opt. Soc. Am. B **18**, 457–464 (2001). [CrossRef]

2. M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. , **38**, 7356–7363 (1998). [CrossRef]

## 2. Electrostatics

*V*

_{0}and ±(

*V*

_{0}+Δ

*V*), 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**:

**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 [1

1. M. Kulishov, P. Cheben, X. Daxhelet, and S. Delprat, “Electro-optically induced tilted phase gratings in waveguide,” J. Opt. Soc. Am. B **18**, 457–464 (2001). [CrossRef]

**c**is an electric field with periodicity

*l*, and nonzero constant field component.

## 3. Electro-optics

*x*-direction, the refractive index is proportional to the

*x*-component of the electric strength vector

*E*

_{x}=-

*∂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 Δ

*V*voltage, one can switch between the two gratings (Δ

*V*=0,

**b**is OFF; Δ

*V*=-

*2V*

_{0},

**c**is OFF), or both of them can be activated with arbitrary weighting factor, when -

*2V*

_{0}<Δ

*V*<0.

## 4. Guided wave interaction

*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:

*a*

_{1},

*a*

_{2}and

*a*

_{3}are the complex modal amplitudes; Δ

*β*

_{12}=

*β*

_{1}-

*σ*

_{11}-

*β*

_{2}+

*σ*

_{22}-

*π*/

*l*; Δ

*β*

_{13}=

*β*

_{1}-

*σ*

_{11}-

*β*

_{3}+

*σ*

_{33}-2

*π*/

*l*; Δ

*β*

_{23}=

*β*

_{2}-

*σ*

_{22}-

*β*

_{3}+

*σ*

_{33}-

*π*/

*l*;

*σ*

_{ii}are “dc” coupling coefficients and

*κ*

_{ij}are the “ac” cross-coupling coefficients

*i*,

*j*=(1,3) [3

3. T. Erdogan, “Fiber grating spectra,” Journ. of Lightwave Techn. , **15**, 1277–1294 (1997). [CrossRef]

*β*

_{23}=Δ

*β*

_{13}-Δ

*β*

_{12}and the resulting equation can be reduced to:

*R, S*and

*P*are

*R*(

*z*)=

*a*

_{1}

*exp*[-

*j*(Δ

*β*

_{12}+Δ

*β*

_{13})

*z*/

*2*];

*S*(

*z*)=

*a*

_{2}

*exp*[-

*j*(Δ

*β*

_{13}-Δ

*β*

_{12})

*z*/

*2*];

*P*(

*z*)=

*a*

_{3}

*exp*[-

*j*(Δ

*β*

_{12}-Δ

*β*

_{13})

*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 Δ

*V*voltage, because

*κ*

_{12}is proportional to Δ

*V*/

*2V*

_{0}and

*κ*

_{13}and

*κ*

_{23}are proportional to (1+Δ

*V*/

*2V*

_{0}) and for the EO active core and non EO-active cladding they can be expressed in the following way:

*µ*

_{0}is the vacuum permeability;

*c*is the speed of light in vacuum;

*k*

_{0}is the free-space wave number;

*r*is the electro-optic coefficient;

*n*

_{o}is the intrinsic refractive index of the core (ordinary for TE modes and extraordinary for TM modes);

*e*

_{jt}is the transverse electric field of the

*j*-th mode (j=1,2,3), and finally

**b**and

**c**gratings. The “dc” coupling coefficients,

*σ*

_{11},

*σ*

_{22}and

*σ*

_{33}are the linear functions of (1+Δ

*V*/

*2V*

_{0}) :

*E*

_{x0}is the constant component of the electric field induced by the grating

**c**in the core.

## 5. Discussion

*λ*. Choosing appropriate values of the Δ

*V*(0 or -

*2V*

_{0}) the rejection bands can be selectively switched ON and OFF, or they can be activated simultaneously with precise control of their reflectivity (dynamic range) (-

*2V*

_{0}<Δ

*V*<0).

**b**and

**c**distributions are activated and momentum conditions are satisfied Δ

*β*

_{12}=Δ

*β*

_{13}=0, the core guided wave

*β*

_{1}is coupled into the cladding modes

*β*

_{2}and

*β*

_{3}. However, the momentum condition for the coupling between the cladding modes Δ

*β*

_{23}will be automatically satisfied that generally enables coupling between these modes. This situation is depicted in Fig.3.

*κ*

_{23}coupling coefficient. Normally it is much smaller than

*κ*

_{12}and

*κ*

_{13}, 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 TE

_{2}and TE

_{3}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

*κ*

_{23}, and the coupling between the coupling modes becomes impossible (Fig.4e).

**b**and

**c**might be adjusted independently through proper electric potential distribution, that gives an additional means to control the device.

*β*

_{1}-

*β*

_{2})≈

*β*

_{1}-

*β*

_{3}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

*κ*

_{12}≈

*κ*

_{13}, 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

*κ*

_{12}≈

*2κ*

_{13}(Δ

*V*=-

*1.4 V*

_{0}), and the second half (

*L*/2) the grating

**c**is only activated (Δ

*V*=-

*2*

*V*

_{0}). Side-lobes in the spectrum can also be substantially suppressed through spatial modulation (appodization) of the bias voltage

*V*

_{0}=

*V*

_{0}(

*z*).

*V*

_{0}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

*σ*

_{11},

*σ*

_{22}and

*σ*

_{33}through

*V*

_{0}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

*σ*

_{11}≫

*σ*

_{22},

*σ*

_{33}(since normally confinement factor for a cladding mode is small). Even traditional EO materials, such as LiNbO

_{3}, can provide a substantial tuning range [2

2. M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. , **38**, 7356–7363 (1998). [CrossRef]

4. P. Cheben, F. del Monte, D. J. Worsfold, D. Carrisson, C.P. Grover, and J.D. Mackenzie, “A photorefractive organically modified silica glasses with high optical gain,” Nature , 2000, **408**, 64–66 (2000). [CrossRef] [PubMed]

*β*

_{12}≠0 and Δ

*β*

_{13}≠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 [5

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]

## 6. Conclusion

## References and links

1. | M. Kulishov, P. Cheben, X. Daxhelet, and S. Delprat, “Electro-optically induced tilted phase gratings in waveguide,” J. Opt. Soc. Am. B |

2. | M. Kulishov, “Interdigitated electrode-induced phase grating with an electrically switchable and tunable period,” Appl. Opt. , |

3. | T. Erdogan, “Fiber grating spectra,” Journ. of Lightwave Techn. , |

4. | P. Cheben, F. del Monte, D. J. Worsfold, D. Carrisson, C.P. Grover, and J.D. Mackenzie, “A photorefractive organically modified silica glasses with high optical gain,” Nature , 2000, |

5. | J. Zhao, X. Shen, and Y. Xia, “Beam splitting, combining and cross coupling through multiple superimposed volume index gratings”, Optics & Laser Technology , |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(160.2100) Materials : Electro-optical materials

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 17, 2001

Published: November 5, 2001

**Citation**

Mykola Kulishov and Xavier Daxhelet, "Electro-optically reconfigurable waveguide superimposed gratings," Opt. Express **9**, 483-489 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-483

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### References

- M. Kulishov, P. Cheben, X. Daxhelet, S. Delprat, "Electro-optically induced tilted phase gratings in waveguide," J. Opt. Soc. Am. B 18, 457-464 (2001). [CrossRef]
- M. Kulishov, "Interdigitated electrode-induced phase grating with an electrically switchable and tunable period," Appl. Opt. 38, 7356-7363 (1998). [CrossRef]
- T. Erdogan, "Fiber grating spectra," J. Lightwave Techn. 15, 1277-1294 (1997). [CrossRef]
- P. Cheben, F. del Monte, D. J. Worsfold, D. Carrisson, C. P. Grover, J. D. Mackenzie, "A photorefractive organically modified silica glasses with high optical gain," Nature 2000, 408, 64-66 (2000). [CrossRef] [PubMed]
- J. Zhao, X. Shen, Y. Xia, "Beam splitting, combining and cross coupling through multiple superimposed volume index gratings," Opt. & Laser Technol. 33, 23-28 (2001) [CrossRef]

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