## Abstract

We investigate electro- and thermo-optic effects on multi-wavelength *S̆*olc filters based on *χ*
^{(2)} nonlinear quasi-periodic photonic crystals. The multi-wavelength *S̆*olc filters are composed of two building blocks A and B, in which each containing a pair of antiparallel poled domains, arranged as a Fibonacci sequence. The transmittances at filtering wavelengths can be modulated from 0 to 100 % by applying an external voltage but the filtering wavelengths are unchanged. The filtering wavelengths can be tuned by varying temperature. As temperature decreases, the filtering wavelengths increase (~-0.45 nm/°C).

© 2008 Optical Society of America

## Corrections

Chul-Sik Kee, Yeung Lak Lee, and Jongmin Lee, "Electro- and thermo-optic effects on multi-wavelength Solc filters based on χ^{(2)}nonlinear quasi-periodic photonic crystals: Erratum," Opt. Express

**17**, 10773-10773 (2009)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-17-13-10773

## 1. Introduction

*χ*
^{(2)} nonlinear photonic crystals are nonlinear materials whose the direction of second-order nonlinear coefficient *χ*
^{(2)} is periodically modulated in space with keeping its magnitude constant. [1] *χ*
^{(2)} nonlinear photonic crystals can be fabricated by poling periodically ferroelectric materials such as lithium niobate (LiNbO_{3}) which has attracted much attention because of its large electro-optic and nonlinear optical coefficients. Periodically poled lithium niobates (PPLNs) have been expected to be very useful in the enhancement of conversion efficiency in optical second harmonic generation and other nonlinear optical processes. [2, 3, 4, 5]

It has been in recent demonstrated that a PPLN acts as a birefringent spectral filter, a folded *Š*olc filter composed of alternating birefringent plates. [6, 7] Tunable *Š*olc filters based on PPLNs have also been proposed. The tuning ability is due to the electric-optic, thermo-optic, and photovoltaic effects of LiNbO_{3}. Thus, the filtering wavelengths can be tuned by applying external voltage, varying temperature, and illuminating ultra-violet light. [8, 9, 10, 11, 12]

Quasi-periodic structures have given rise to some interesting phenomena which are not observed in periodic structures. [13, 14] For example, high efficient quasi-phase-matched third harmonic generation in a *χ*
^{(2)} nonlinear quasi-periodic photonic crystal that is a quasi-periodically poled lithium niobate (QPLN) with a Fibonacci sequence was reported. [15] Recently, the multi-wavelength filtering of a QPLN with a Fibonacci sequence was proposed theoretically. [16] Thus, the filtering characteristics of QPLN with a Fibonacci sequence are expected to be tuned by applying external voltage, varying temperature, and illuminating ultra-violet light.

In this paper, we investigate electro- and thermo-optic effects on the filtering characteristics of a QPLN with a Fibonacci sequence. An external dc- field rotates the poled axes to the field direction and modulates transmittances at the filtering wavelengths. The variation of filtering wavelengths can be achieved by varying the sample temperature. As the temperature decreases, the filtering wavelengths increase (~-0.45 nm/°C). Thus, the filtering transmittance and wavelength can be practically controlled by varying the dc-field and temperature.

## 2. Results and discussion

Figure 1 shows the schematic of experimental set-up for a QPLN *S̆*olc filter that is placed between two crossed polarizers. There is a rocking angle *ϕ* (-*ϕ*) between the optical axis *Z* and a positive (negative) domain. The input light polarized along the *Z* direction rotates 4*ϕ* after passing through the first positive and negative domains and finally, 4*Nϕ* after passing through all the domains of *N* sets. If this final rotation angle is 90°, the rotated light is polarized along the *Y* direction and passes through the rear polarizer completely. The presence of an external dc-field along *Y* axis deforms the refractive index ellipsoid, and consequently the *Y* and the *Z* axes of the *Z*-cut lithium niobate rotate by a small angle *θ* about the *X* axis. The rotation angle *θ* is given by

where *E* is the field intensity and *γ*
_{51} is the electro-optic coefficient. Note that the coefficient *γ*
_{51} changes its sign in the negative domains because of the 180° rotation of the crystal structure.

The transmission spectrum of a QPLN *S̆*olc filter under the dc-field can be simulated by the Jones matrix. [17] The components of the Jones matrix for a domain **D** are given by

where Ω=(*ϕ*+*θ*) is an angle between axis of a domain and Z-axis, and Γ is the phase retardation of a domain. For *λ*, Γ=*π*(*n _{e}*-

*n*)

_{o}*l*/

*λ*, where

*l*is the thickness of a domain, and

*n*and

_{e}*n*are the extraordinary and the ordinary index of LiNbO

_{o}_{3}, respectively. [17] The matrix multiplication for

*N*domains results in the Jones matrix for a QPLN

*S̆*olc filter,

**M**. The transmissivity of the filter is

**T**=|

**M**

_{21}|

^{2}because the input and output lights are linearly polarized in the

*Z*and

*Y*directions, respectively. Thus, the transmission intensity at a given wavelength can be electrically modulated by an external voltage because the dc-field varies Ω.

A QPLN with a Fibonacci sequence has two building blocks A and B of length *l _{A}* and

*l*, respectively. They are ordered in a Fibonacci sequence according to the production rule

_{B}*F*=

_{i}*F*

_{i-1}|

*F*

_{i-2}for

*i*≥3, with

*F*

_{1}={A} and

*F*

_{2}={AB}. Each block has a domain of length

*l*

_{A1}(

*l*

_{B1}) with positive ferroelectric domain and a domain of length

*l*

_{A2}(

*l*

_{B2}) with negative ferroelectric domain. This structure is of benefit to fabricate a QPLN

*S̆*olc filter because poling the same size domains is easy in the electric poling technique.

For the case of a PPLN *S̆*olc filter, the filtering wavelength is given by *λ _{m}*=(

*n*-

_{e}*n*)Λ/(2

_{o}*m*-1)(

*m*=1,2,3…), where Γ is the period of PPLN. The formula of the filtering wavelength can be extended from a periodic structure to a quasi-periodic structure. The filtering wavelengths of the QPLN

*S̆*olc filter with a Fibonacci sequence are given by

where Λ_{p,q}=*L*/(*p*+*qτ*), *p* and *q* are integer indices of the quasi-periodicity, *τ*=(1+√5)/2 which is so-called the golden ratio, and *L*=*τl*
_{A}+*l*
_{B} is the average lattice parameter. [18]

Figure 2 shows the transmission spectrum of a QPLN *S̆*olc filter consists of 1000 building blocks of A and B arranged as a Fibonacci sequence in the range of wavelength from 900 to 1700 nm when an external voltage *V*
_{ex} is zero. *l*
_{A}=24*µ*m, *l*
_{B}=17:5*µ*m, and *l*
_{A2}=*l*
_{B2}=*l*=11*µ*m. The rocking angle *ϕ* was assumed to be *ϕ*=*π*/8*N*, where *N*=1000. To consider the wavelength dependence of *n _{e}* and

*n*, the Sellmeier equation was employed in simulations. The temperature of the sample was assumed to be 24 °C. The insets denote the quasiperiodic indices corresponding to the filtering wavelengths,

_{o}*λ*

_{1,1}=1572.1 nm,

*λ*

_{0,2}=1297.7 nm,

*λ*

_{2,1}=1173.2 nm,

*λ*

_{1,2}=1018.5 nm, and

*λ*

_{3,0}=1398.9 nm. The main filtering wavelength corresponds to

*λ*

_{1,1}. The simulated filtering wavelengths were matched well to the theoretical ones from Eq. (6). [16] The transmittance at

*λ*

_{1,2}is about 25 % and the transmittance at

*λ*

_{2,1}and

*λ*

_{0,2}are about 10 %. However, the transmittance at

*λ*

_{3,0}are very low.

The transmittances at the filtering wavelengths can be varied by applying an external voltage. Figure 3 shows the transmission spectra around *λ*
_{1,2} when *V*
_{ex}=0.0 (black line), 1.5 (red line), and 2.5 kV (blue line). The experimental value of *θ*, 0.017°/kV, was employed in simulations. [8] The sample width is assumed to be 1 mm. One can see that the transmittance at *λ*
_{1,2} can be modulated from 20 to 100 % when *V*
_{ex} varies from 0 to 2.77 kV.

We have systematically investigated the electro-optics effect on the filtering efficiency of the QPLN *S̆*olc filter by varying *V*
_{ex} from -1.32 to 3.3 kV. Figure 4 shows the dependence of the transmittances at filtering wavelengths as a function of *V*
_{ex}. The insets denote the filtering wavelengths and the quasi-periodic indices corresponding to the wavelengths. One can see that the transmittances at the filtering wavelengths are almost zero when *V*
_{ex}=-1.32 kV. This is because the rotated angle *θ*(=0.017°/kV×-1.32 kV=-0.022°) cancels out the rocking angle *ϕ*(=180°/8*N*=0.022°). The transmittance at *λ*
_{1,1} changes with oscillatory behavior when *V*
_{ex} varies. The oscillatory dependence of filtering intensity has been also found in PPLN *S̆*olc filters under an external dc-field because the intensity is proportional to sin^{2}(4*Nϕ*). [8] The transmittance at *λ*
_{1,1} (*λ*
_{1,2}) becomes the maximum when *V*
_{ex}=0.36 kV (2.77 kV). The transmittance at *λ*
_{1,2} is larger than that at *λ*
_{1,1} when *V*
_{ex} is from 0.85 to 3.1 kV. For the case of *V*
_{ex}=1.7 kV, the transmittance at *λ*
_{1,2} is over 85%, while the transmittance at *λ*
*1,1* is zero. The transmittances of *λ*
_{1,1} and *λ*
_{1,2} is nearly the same when *V*
_{ex} is around 0.85 and 3.1 kV. The transmittances at other filtering wavelengths of *λ*
_{2,1}, *λ*
_{0,2}, and *λ*
_{3,0} increase as *V*
_{ex} increases. The voltage-dependent filtering intensity would be useful in implementing the tunable multi-wavelength QPLN *S̆*olc filter. However, *V*
_{ex} can not change the filtering wavelengths because it changes the directions of poles only.

The tuning of filtering wavelength can be achieved by varying the sample temperature *T*. Figure 5 shows the transmission spectra around *λ*
_{2,1} (a) and *λ*
_{1,1} (b) when *T*=24 (black line), 23 (red line), 22 (green line), and 21 °C (blue line). To consider the temperature-dependent *n _{e}* and

*n*of lithium niobate, the Sellmeier equation was employed in simulations. The transmission spectrum shifts to long wavelength as

_{o}*T*decreases. The tuning ratio is about -0.45 nm/°C which is good agreement with the experimental value of the PPLN case. [8]

## 3. Conclusion

We studied the electro- and thermo-optic effects on the filtering intensity and wavelengths of a QPLN multi-wavelength *S̆*olc filter. The filtering intensity is practically modulated by applying an external voltage but the wavelengths is not changed. The tuning of filtering wavelengths can be achieved by varying the sample temperature.

## Acknowledgments

This work is supported by Ministry of Knowledge Economy of Korea through Technology Infrastructure Building Program and Ministry of Education, Science and Technology of Korea through Photonic 2020 and APRI-Research Program of GIST.

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