1. Introduction

Nonlinear materials processing for photonic device fabrication using near-IR femtosecond pulses has emerged as an active area of research because it is possible to fabricate localized, clean, three-dimensional structures in a wide range of materials without the need for linear absorption [1–12]. A variety of devices in glasses such as waveguides [1,2,11], couplers [3,10,12], gratings [4], 3D structures [5,12], active waveguides [6], and void structures [7–9] have been successfully fabricated. Previous studies have been done with either high-intensity femtosecond pulses from laser amplifiers [1–9] or with unamplified pulses directly from oscillators [10–12]. Fundamental investigations on the properties of the nonlinear process, such as threshold energy and refractive index change inside the structures, and their dependence on the exposure parameters and materials have been also reported. The fabrication of single mode waveguides has been demonstrated using systematic analysis of the guided mode. However, in previous demonstrations of functional photonic devices, such as waveguide couplers, only the appearance of multiple outputs has been used to assess device fabrication and detailed investigations of device function have not yet been reported. In order to demonstrate the feasibility of nonlinear fabrication techniques using femtosecond laser for practical photonic devices, a comprehensive investigation the device function is crucial. Our group has previously demonstrated the fabrication of single mode devices such as waveguides, X-couplers, and 3D waveguides using a high-energy femtosecond oscillator, and investigated the refractive index properties by optical coherence tomography (OCT) and guided mode analysis [12]. Laser oscillators have many advantages over amplifier systems, and moreover, the higher pulse repetition rate enables faster and more efficient waveguide fabrication. In this paper, we demonstrate the fabrication of coupled mode devices. The behavior of device function is investigated by varying structural parameters such as interaction length and separation. Results are compared to coupled mode theory. Mach-Zehnder interferometers were also fabricated and spectral filtering was demonstrated using an unbalanced path length interferometer. These devices constitute the basic building blocks of photonic devices.

Studies were performed using a novel, extended cavity mode-locked Ti:Al₂O₃ laser [13]. Since the total output power of a laser is limited, pulse energies can be increased by increasing the laser cavity length and reducing the pulse repetition rate. Using 4 MHz repetition rate Ti:Al₂O₃ laser, pulses of up to 100 nJ can be generated with 80 fs pulse duration. Waveguides were fabricated inside glass by focusing the femtosecond laser beam and translating the glass perpendicular to the incident beam to write the waveguides. The fabrication parameters were similar to those reported previously [12]. The high repetition rate laser pulses were tightly focused inside a 1 mm glass plate (Corning 0215 glass) using a high numerical aperture microscope objective. The absorption edge of the glass was approximately 300 nm. Single mode waveguides were fabricated by controlling the waveguide size and mode structure as a function of the incident pulse energy, focal spot size, and scan speed. Typical exposure parameters necessary to obtain single mode devices were pulse energies of ~20 nJ and scan speeds of ~10 mm/s. The index differential for single mode structures was measured by OCT to be 10^-3 [12]. Higher exposure energies or lower scan speeds resulted in the fabrication of larger, multimode waveguide structures and index differences of up to 10^-2, with void like structures from laser induced breakdown observable at the highest pulses and slowest scan speeds. Properties of the devices were characterized by coupling laser light into the waveguides, and investigating guided beam properties. The edges of the glass substrate were polished to optical quality in order to enable efficient coupling.

 

Fig. 1. (a) Phase contrast microscopic image of one of the directional couplers. The aspect ratio of the image is compressed by a factor of 5 in the horizontal direction in order to better visualize the directional coupler. The arrow in the image indicates the interaction region. (b) Schematic of the coupler. Separation, d, and interaction length, L, are varied with the fixed total length, 25 mm. He-Ne laser is guided into one of the input ports, and the output power at two ports is measured. The coupling ratio between the two waveguide ports, R, is obtained.

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Directional couplers were fabricated in which two single mode waveguides have an interaction region without intersecting. Figs. 1(a) and (b) show the phase contrast microscopic top view and the schematic of a directional coupler. The coupler has an interaction region of length L where the two waveguides are parallel and closely spaced with a separation d. Each waveguide has two bends of 1^°. In order to test the dependence of the directional couplers on the fabrications parameters and to confirm that it is consistent with coupled mode theory, we fabricated a series of couplers with different combinations of the waveguide separations, d, and interaction lengths, L. The total length of the waveguide devices was kept constant at 25 mm. When a He-Ne laser beam (633 nm) was coupled into one input port, power was transferred to the two output ports, suggesting coupling between the two waveguides.

In order to determine the waveguide coupling ratio, R, it is necessary to account for the waveguide transmission efficiencies, g, and the transmission of the waveguide bends, b. To measure the coupling ratio, R, a series of four measurements must be performed on each waveguide coupler. Light was coupled into each of the two input ports, ports- i=1,2, and the power transferred to each of the two output ports, ports-j=1,2, was measured. The transfer functions, P_ij, where power is coupled into port-i and emitted at port-j, are described by Eqs. (1a)–(1d).

The transmission efficiency for guiding is g _1,2 and the transmission of the waveguide bends is b _1,2 for waveguides 1 and 2, respectively. The input light power is P ₀ and P ₀ g _1,2 is the power guided in the waveguides 1,2 including the input coupling efficiency and the propagation loss in the waveguide. The propagation loss is assumed to be uniform and identical for the two waveguides. Each waveguide has two bends and each bend is assumed to have the same transmission efficiency. Four measurements, P_ij (i,j=1,2), were performed for each coupler. The coupling ratio, R, was obtained using Eqs. (1a) to (1d). It is important to note that the value of the coupling ratio, R, can be determined from these equations without the need to assume specific values for g _1,2 or b _1,2. A value for the transmission of the waveguide bends, b= 0.73, and the ratio of the guiding efficiencies, g ₁/g ₂ =1, are measured assuming b=b ₁=b ₂. As expected, these values were almost constant and consistent for all couplers irrespective of separations and interaction lengths.

Directional couplers operate using coupled mode effects. When two waveguides are brought into close proximity with a small separation between them, the mode that is guided in one of the waveguides can couple into the other waveguide by evanescent field interaction. The operation of these devices can be described by coupled mode theory [14]. The two waveguides are aligned in the z direction and have the same radius, a, with a separation, d, and an interaction length L. In this model, the waveguides are assumed to be lossless with uniform properties. The coupling ratio, R, between the input power P ₁(0) at the input (z=0) of waveguide-1, and the power P ₂(L) coupled into waveguide-2 at a position z=L can be described by Eq. (2a) using coupled mode theory.

where

Here C ₁₂ is a coupling coefficient between two waveguides which is a function of an overlap integral of the eigenmodes; β ₁ and β ₂. are the propagation constants of the modes, which is a function of the wavelength and the refractive index [14]. Eq. (2a) shows that the coupling ratio, R, oscillates with a characteristic length, π/γ as a function of interaction length L. This oscillation of power between the two waveguides is characteristic of coupled mode behavior. When both waveguides have similar propagation constants, β ₁=β ₂, the power oscillation is governed only by the coupling coefficient, C ₁₂. The coupling coefficient, C ₁₂ is proportional to an overlap integral of the eigenmodes. Since the eigenmodes are evanescent and exponentially decaying outside the waveguide, the coupling coefficient depends strongly on the separation, d, between the two waveguides. When the separation d between the waveguides becomes larger, the overlap between the eigenmodes is reduced and the coupling coefficient is smaller, so the oscillation of power between the two waveguides becomes slower.

 

Fig. 2. (a) Interaction length dependence of the coupling ratio for waveguide separations d = 8 μm, (b) d = 10 μm, and (c) d = 12 μm. Experimental results (dots) and their best-fit results to sinusoidal curves (lines) are shown. The period of the oscillation increases from (a) 5, to (b) 9, to (c) 14 mm, which is consistent with the coupled mode theory.

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In order to experimentally confirm that the directional couplers function as coupled mode devices, a series of over thirty couplers were fabricated with different waveguide separations, d, and interaction lengths, L. Four measurements were performed on each of the thirty couplers in order to determine the coupling ratio, R. Figs. 2(a) to (c) summarize the results of these measurements and show the measured values of R for each of ten different interaction lengths, L, ranging between 0 and 10 mm for waveguide separations of (a) d=8, (b) d=10, and (c) d=12 μm. The oscillatory behavior of the coupling ratio as a function of interaction length, L, that is predicted by coupled mode theory is clearly evident. The lines in the figures show best-fit sinusoidal functions. To the best of our knowledge, this is the first demonstration of the oscillatory behavior of the coupling coefficient and confirms coupled mode operation in devices fabricated by femtosecond nonlinear material processing.

When the separation, d, between the waveguides increases from (a) 8 μm, to (b) 10 μm to (c) 12 μm, the oscillation period of the coupling ratio, R, increases from (a) 5 mm, to (b) 9 mm, to (c) 14 mm. This behavior is consistent with coupled mode theory, which predicts that decreasing in coupling coefficient, C ₁₂, yields an increase in the oscillation period of the coupling ratio with interaction length. The scatter in the data points is due to the fact that each measurement is performed on a separate directional coupler device. Small variations in the coupling of the incident beam and measurements of the coupling ratio are possible when measurements are performed on multiple devices. The error in the measurement of the coupling ratio is estimated to ±0.02 from the noise in the light intensity and the reproducibility of the alignment. The fluctuation of data in Fig. 2(c) looks more significant than the others because the measurement of weak signal is required. In addition, it is important to note that the coupling coefficient, C ₁₂, is exponentially dependent on the separation, d, between the two waveguides. Thus micron scale variations in the waveguide separations can produce fluctuations in the coupling ratio. Given these factors, the scatter in the measured coupling ratios is relatively small.

When the two waveguides have identical structures and refractive indices, the amplitude of the oscillation in the coupling ratio is unity, corresponding to a complete transfer of power between the two waveguides. However, the experimental measurement shows that the amplitude of the oscillation in coupling ratio decreases as the waveguide separation increases. If there are differences in waveguides, then there will be a mismatch of the propagation constants and β ₁–β ₂ will be nonzero. If the coupling coefficient, C ₁₂, becomes small compared to the mismatch in propagation constants, β ₁–β ₂, the magnitude of the oscillation in the coupling ratio will decrease. This can be seen from Eqs. (2b) and (2a). Differences in the waveguide properties may be caused by fluctuations of the laser exposure during the fabrication process. Also fluctuations in the nonlinear fabrication process might produce nonuniform waveguide properties or propagation loss as function of length. These parasitic effects are not well described by the simple coupled mode theory model presented here, which assumes uniform coupling and no loss. Further investigations including these effects are required to fully understand the detailed properties of the waveguide couplers. However, the data presented conclusively demonstrate that the operation of the directional couplers is the result of coupled mode interaction.

 

Fig. 3. Wavelength dependence of the coupling ratio for the coupler with L=5 mm and d=2 μm. Experimental results (dots) and their best-fit results to sinusoidal curves (lines) are shown.

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The previous measurements required the fabrication and measurement of multiple directional couplers with varying interaction lengths, L, in order to permit comparison with coupled mode theory. In order to avoid the effects of variations in the fabrication and measurement process between multiple devices, it is helpful to perform measurements on single devices as a function of wavelength. Measurements of the coupling ratio were performed using broadband light from a modelocked Ti:Al₂O₃ laser [13]. The broadband light was coupled into the waveguide directional coupler, and the wavelength dependence of the output was measured by a spectrometer. Fig. 3 shows the wavelength dependence of the coupling ratio, R, for the directional coupler with a separation of d=2 μm between waveguides and an interaction length of L=5 mm. The measurements show that the coupling ratio oscillates as a function of wavelength, as expected. The large wavelength independent background is probably the result of scattered light which is detected by the spectrometer. The oscillation period of the coupling ratio with wavelength was observed to change with changes in the waveguide separation, exhibiting a similar behavior to Fig. 2. These results are the first systematic measurements of mode coupling behavior in devices fabricated by femtosecond nonlinear material processing and demonstrate the feasibility of this technique for photonic device fabrication.

 

Fig. 4. (a) Schematic representation of Mach-Zehnder Interferometer with phase-contrast microscope images of waveguides; (b) Input (red line) and output (black line) spectra from broadband Ti:Al₂O₃ and, (c) Normalized output spectrum (black line) demonstrating the filtering effect of the interferometer compared to theoretical model (red line).

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The fabrication and characterization of a Mach-Zehnder interferometer filter is demonstrated as another example of a more complex photonic device. The interferometer consists of two X-couplers placed back-to-back, with crossing angles of 2° as shown in Fig. 4(a). The path length difference between the two arms is approximately 10 μm. Light coupled into the interferometer is split into the two arms of the interferometer at the first X-coupler, travels different path lengths, and will either constructively or destructively interfere at the second X-coupler. The unbalanced path length Mach-Zehnder interferometer functions as a wavelength dependent filter. The frequency or wavelength dependence of the interferometer can be measured using a broadband light source [15]. Fig. 4(b) shows the input spectrum with a FWHM of 130 nm, together with the output spectrum. Fig. 4(c) shows the wavelength transfer function in the crossed interferometer arm. The transfer function is constructed by normalizing the output spectrum by the input spectrum. The interferometric fringes are clearly observed in the normalized output and demonstrate the spectral filtering behavior expected from an unbalanced interferometer. The reduction in oscillations at shorter wavelengths may be due to the larger fluctuations of the incident light spectrum in this wavelength range or the possible onset of higher order modes in these waveguides. The red line in Fig. 4(c) shows the theoretically predicted wavelength dependence of the transfer function for a path length difference of 9.3 μm. The experimental measurements are in close agreement with the theory. The difference between design arm path length difference and actual path length from the wavelength measurement is only ~0.7 μm out of 16.5 mm or a fraction of 4x10^-5. This variance is probably due to small errors in the translation of the motorized stages used for fabrication, or slight differences in the refractive index between the two waveguides.

In conclusion, we have demonstrated the fabrication of coupled mode devices and interferometers using nonlinear femtosecond materials processing in glass. Oscillation of the power between the two waveguides has been observed as a function of the waveguide interaction length and coupling coefficient as well as a function of wavelength. These results are consistent with coupled mode theory and demonstrate that the directional couplers operate by mode coupling. The fabrication of an unbalanced Mach-Zehnder interferometer is also demonstrated as an example of a more complex device. The operation of the unbalanced interferometer as a wavelength filter is demonstrated and is in agreement with theory. These results demonstrate that practical photonic device fabrication is possible using femtosecond nonlinear materials processing.

We gratefully acknowledge helpful discussions with Drs. Chris Schaffer, Hermann Haus, Eric Mazur, Wayne Knox, and Wolfgang Drexler as well as S.H. Cho for developing the laser used in this study. K. Minoshima was visiting from the National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan. A.M. Kowalevicz is also with the Harvard University Division of Engineering and Applied Sciences. This research was supported in part by the Air Force Office of Scientific Research contract F49620-01-1-0084 and the Medical Free Electron Laser Program, contract AFOSR F49620-01-1-0186.