We have analyzed the multi-mode interference effect depending on the wavelength and the polarization states of input beam in a multi-mode Ti:LiNbO3 waveguide at about 1300 nm region. The transmitted optical signal of a Ti:LiNbO3 waveguide shows the periodic oscillation as a function of input wavelength. The measured average periodicity of the oscillation in TM and TE polarization beams were about 18 nm and 48 nm, respectively. Actually, the periodicity is determined by the refractive index difference between the two modes (fundamental and first modes). Therefore, we have explained the experimental results with the theoretical calculations which are derived from a quasi-analytical technique based on the effective-refractive-index method and the equation of coupling length determined by the mode phase factor in the multi-mode waveguide.
© 2009 Optical Society of America
A Ti:LiNbO3 waveguide can provide various good optical properties such as a low propagation loss, electro-optic, thermo-optic, acousto-optic, and nonlinear optic effects. Therefore, a Ti:LiNbO3 waveguide has been well utilized in various optical devices such as electro-optic modulator , optical switch [2,3], polarization mode dispersion compensator , wavelength converter , optical logic gate , Solc-type wavelength filter , and compact laser sources . In basic, an integrated optic device based on a waveguide can give various functionalities by using not only integration elements (metal electrodes, acoustical absorber, photorefractive grating, periodically reversed microdomains, etc.) but also waveguide mode conditions. However, most of the Ti:LiNbO3 waveguide devices have been exploited based on a single-mode waveguide. Indeed, graded index waveguide such as a multi-mode Ti:LiNbO3 waveguide can make a multi-mode interference (MMI) effect  which reproduce an input optical field profile along the waveguide in single or multiple images with certain periodic intervals. A MMI effect is mainly adapted in a functional photonic integrated circuits based on semiconductor [10,11] and fiber devices [12,13] for optical switches [10,14], sensors [11-13], wavelength division multiplexer [15,16] and so on. However, up to now, no research has been reported on a MMI effect in a Ti:LiNbO3 waveguide except the application device for temperature insensitive comb filter . In this paper, we have observed and analyzed the MMI effect depend on the wavelength and the polarization states of input beam in a Ti:LiNbO3 waveguide which has multi-mode guiding condition at about 1300 nm region. We have also compared the experimental results and the theoretical calculations which are derived from the refractive indices (fundamental and first modes) of a Ti:LiNbO3 waveguide  and the beat length determined by the mode phase factor in the multi-mode waveguide .
2. Ti:LiNbO3 waveguide fabrication and experimental setup
A 33-mm long Ti:LiNbO3 channel waveguide of with a 7-µm width was fabricated by diffusing 980-nm thick Ti stripes upon the −Z face of 0.5-mm thick Z-cut LiNbO3 substrate along the X axis of crystal line. The diffusion time of Ti stripes was 9 h which is 1.5 h more than that of a usual single-mode Ti:LiNbO3 waveguide (@ 1550 nm) fabrication process . Such a long diffusion time allows a multi-mode guiding at 1300 nm wavelength region. The multi-mode guiding condition (TEM00, TEM10) at about 1300 nm was confirmed by using a broadband light source and an infrared camera. The full width at half-maximum (FWHM) of the horizontal and vertical direction were measured to be 5.5 and 2.8 µm, respectively. The propagation losses for TM and TE modes of waveguide were determined by the Fabry-Perot method to be 0.11 dB/cm and 0.05 dB/cm, respectively (@ 1550 nm) .
The experimental setup to analyze the MMI effect in a Ti:LiNbO3 waveguide is shown in Fig. 1. An optical signal from a home-made wavelength swept fiber laser (WSFL) which has 10 mW average power and 15 kHz repetition ratio was collimated and end-fire coupled into the Ti:LiNbO3 waveguide by a ×10 objective lens. The polarization direction of input beam was adjusted 45° against optical axis of the Ti:LiNbO3 waveguide and the output signal from the waveguide was separated by polarization beam splitter into a TM- and a TE- polarized beams. These two orthogonal polarized beams were monitored by an optical spectrum analyzer. A 1% of the transmitted TM-polarized beam was reflected by a beam splitter and monitored by an infrared camera. During the experiment, the temperature of the Ti:LiNbO3 waveguide was kept at 25°C by Peltier device.
3. MMI effect in a Ti:LiNbO3 waveguide
Figure 2 shows the mode profiles of output beam (TM polarization) from the Ti:LiNbO3 waveguide which were measured at the IR camera in a Fig. 1. As shown in Fig. 2, the input beam profile was reproduced according to the wavelength of WSFL in single or multiple images (see the movie file Media 1). Such kind of a self-imaging effect can be explained by the MMI in a multi-mode graded index waveguide . In other words, the interference between TEM00-and TEM10-modes leads to the periodic oscillation in waveguide images. To measure the optical spectrum of a 99% of the TM-polarized beam, the transmitted beam was coupled into a single-mode fiber which is connected to the optical spectrum analyzer (OSA) by ×10 objective lens.
The measured optical spectrum of transmitted TM-polarized beam is shown in Fig. 3. The black scatter-line indicates the optical spectrum which was measured at position A (after Lens3 in Fig. 1) through the iris diaphragm. As the same way, the other spectrum (blue scatter-line) in Fig.3 was measured at position B. The measured optical spectra of the direct and the mirror images  show a periodic oscillation as a function of wavelength of input beam. The measured periodicity of both spectra were about 18.1 nm and phase difference between two fringe spectra was about 180°, which means that this waveguide can be used for not only a wavelength filter  but also functional devices such as MMI coupler , multimode splitter , optical switch , wavelength division multiplexer  and so on.
The period and the number of oscillation in a certain sample (Ti:LiNbO3) length L, can be calculated by the following equation;
where n0 and n1 are the effective refractive indices of the fundamental and first-order modes, i indicates polarization direction (ordinary or extraordinary) of input beam and the coupling length Lc is defined by
where β0 and β1 are the propagation constants of the fundamental and first-order modes and λ0 is the wavelength of input beam. The values of the effective refractive indices in a Ti:LiNbO3 waveguide are given following equation;
where nsub and δn(x,y) are the refractive index of bulk LiNbO3 and refractive index change by Ti diffusion, respectively. x and y indicate ordinary and extraordinary directions. The value of δn(x,y) used in theoretical calculation was induced by a quasi-analytical technique based on the effective-refractive-index method .
Figure 4(a) and (b) show the calculated effective refractive indices of extraordinary wave and the number of oscillation in a 33-mm Ti:LiNbO3 waveguide as a function of input wavelength at room temperature. Actually, when light propagate through a Ti:LiNbO3 waveguide, the input beam profile is reproduced according to not only the wavelength of input beam (see. Fig. 2) but also the position of waveguide. Therefore, longer length of waveguide can provides larger number of periodic oscillation (Eq. (1)). In our case, the number of oscillation was calculated more than 50 at about 1300 nm region as you can see in Fig. 4(b). The values of both refractive indices (fundamental and first-order modes) decreased when wavelength increased. However, the refractive index difference between two modes was almost constant to the wavelength and it was about 2.1×10-3. Therefore, the periodicity of the oscillation showed a slight decrease as a function of the wavelength and the calculated average value was about 18.1 nm which is almost same value of the measured periodicity in Fig. 3.
In the case of TE polarization, the calculated refractive index difference between two modes (fundamental and first modes) was about 1.38×10-3 which is smaller than that of TM polarization case (see Fig. 6(a)). Therefore, the periodicity of the oscillation was longer than that of TM polarization case. Fig. 5 shows the optical spectra as a function of wavelength at position A and B (after Lens4 in Fig. 1). The periodicity of both spectra were rapidly reduced as increased of input beam wavelength and the periodicity varied from 55 nm to 42 nm at 1300 nm wavelength region. Such kind of long period and rapid decrease of the periodicity are due to the small index difference between two modes (fundamental and first-order modes).
The theoretical calculation results on effective refractive indices of ordinary wave are shown in Fig. 6(a). As shown in Fig. 6(b), the number of oscillation was less than that of TM polarization case (Fig. 4(b)). The calculated periodicity was about 41.1 nm which is a little bit shorter than that of the measured value in Fig. 5.
The MMI effect in a multi-mode Ti:LiNbO3 waveguide have been analyzed depending on the wavelength and the polarization states of input beam at 1300 nm region. The periodicity of the oscillation which is induced by a MMI effect in a Ti:LiNbO3 waveguide is decided by the refractive index difference between two modes (fundamental and first modes) and the length of a waveguide. The measured average periodicity of the oscillation in TM- and TE- polarized beams were about 18 nm and 48 nm, respectively. The difference of the periodicity depending on the polarization of input beam was explained by a quasi-analytical technique based on the effective-refractive-index method and the equation of coupling length determined by the mode phase factor in the multi-mode waveguide. The theoretical calculation shows good agreement with the experimental results. We believe that such kind of a MMI Ti:LiNbO3 waveguide device which is integrated with electrodes, acoustical absorber, photorefractive grating, periodically reversed microdomains and so on, can be used for a future multi-functional integrated devices.
This work was supported by the Korea Science and Engineering Foundation (KOSEF) NCRC grant funded by the Korea government (MEST) (No. R15-2008-006-02001-0) and Asian Laser Center Program through a grant provided by the Gwangju Institute of Science & Technology in 2009..
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