Two-dimensional coherent superposition of blue ... - OSA Publishing

Two-dimensional coherent superposition of blueshifted signals from an array of highly nonlinear waveguiding wires in a photonic-crystal fiber Ming-Lie Hu,1 Yan-Feng Li,1 Lu Chai,1 Qirong Xing,1 Lyubov V. Doronina,2 Anatoly A. Ivanov,3 Ching-Yue Wang,1 and Aleksei M. Zheltikov2,3 1

Ultrafast Laser Lab, School of Precision Instruments and Optoelectronics Engineering, Key Laboratory of Optoelectronic Information Technical Science, Tianjin University, 300072 Tianjin, P.R. China 2 Physics Department, M.V. Lomonosov Moscow State University, 119992 Moscow, Russia 3 International Laser Center, M.V. Lomonosov Moscow State University, 119899 Moscow, Russia e-mail: [email protected]

Abstract: Frequency-shifted dispersive optical waves generated as a result of soliton dynamics of 30-fs Ti: sapphire-laser pulses in an array of waveguiding wires, implemented on a platform of a photonic-crystal fiber (PCF), are shown to produce regular stable interference patterns with high visibility, indicating a high coherence of frequency-shifted fields. For a hexagonal array of waveguides built into a silica PCF, the field intensity at the main peak of a six-beam interference pattern was found to be a factor of 22 higher than the intensity of a frequency-shifted signal from an individual waveguide in the array and 3.7 times higher than the field intensity attainable through an incoherent superposition of the same fields. ©2008 Optical Society of America OCIS codes: (190.4370) Nonlinear optics, fibers; (190.7110) Ultrafast nonlinear optics.

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Received 5 May 2008; revised 28 Jun 2008; accepted 28 Jun 2008; published 10 Jul 2008

21 July 2008 / Vol. 16, No. 15 / OPTICS EXPRESS 11176

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1. Introduction Highly nonlinear photonic-crystal fibers (PCFs) [1, 2] offer a new platform for the creation of fiber-format sources and components for ultrafast optics, nonlinear microspectroscopy, and optical information technologies. Through the past few years, these fibers have been advantageously used for a precise calibration of frequency combs in optical metrology [3], carrier--envelope phase control in ultrafast science [4] and attosecond technology [5], supercontinuum generation [6 – 8], tunable frequency shifting [9], generation of correlated photon pairs [10, 11], as well as pump--seed synchronization in the amplification of ultrashort light pulses, including few-cycle field waveforms [12]. Coherence properties of new frequency components generated through nonlinear-optical interactions in PCFs are of key significance for applications of PCF-based supercontinuum sources and frequency shifters in high-precision spectroscopy and frequency-comb metrology, ultrafast optics, coherence control, as well as imaging using coherent Raman processes. Young’s type interference experiments performed by Bellini and Hänsch [13] demonstrated that two supercontinuum fields generated by two phase-locked ultrashort laser pulses are also phase-locked within a broad spectral range. Following this seminal work, coherence properties of supercontinua generated in PCFs have been systematically studied for several important regimes of nonlinear-optical transformations of light pulses in these fibers. Gu et al. [14] have presented an accurate quantitative measurement of the wavelength dependence of coherence across the supercontinuum spectrum and verified their experimental data by detailed numerical simulations. Lu and Knox [15] have applied a method of delayed pulses to demonstrate the mutual coherence of supercontinua generated by individual pulses in trains of femtosecond pulses. Fundamental noise limitations inherent in supercontinuum generation in PCFs have been revealed by Corwin et al. [16]. Experimental results presented by Gross et al. [17] suggest a strong dependence of coherent properties of supercontinuum radiation on the regime of nonlinear-optical transformation of the light field in a PCF, with the coherence degree lowering for higher laser peak powers. Here, we demonstrate that the coherence of frequency-shifted dispersive waves generated as a result of soliton dynamics of ultrashort light pulses in an array of waveguiding wires built into a PCF can be used for a coherent addition of these signals. The physics behind this coherent addition of signals from small-diameter waveguides involves well-known effects of spectral transformation of ultrashort pulses in highly nonlinear waveguides and interference of coherent optical fields. However, the study presented in this work addresses an issue of whether noise amplification, inherent in nonlinear-optical processes, and phase shifts in individual waveguides, which are difficult to control without a precise waveguide dispersion engineering, leave any chance at all for a coherent addition of signals from individual #95787 - $15.00 USD

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waveguides. We will show that, for a hexagonal array of six waveguides built into a silica PCF, such a coherent addition can be observed, allowing the field intensity at the main maximum of the resulting six-beam interference pattern to be enhanced by a factor of 22 relative to the intensity of a frequency-shifted signal from an individual waveguide in the array and by a factor of 3.7 with respect to the intensity attainable through an incoherent superposition of the same fields. 2. Experimental Experiments were performed with the use of a Ti: sapphire oscillator pumped with a 5-W second-harmonic output of a diode-pumped Nd: YVO4 laser. Chirped mirrors and a prism pair were used for dispersion compensation inside the laser cavity, providing a flat profile of group-delay dispersion over a broad spectral band. Such a laser oscillator can deliver pulses with a typical temporal width of about 30 fs, an energy up to 5 nJ at a pulse repetition rate of 100 MHz and a central wavelength of 800 nm. A silica PCF designed and fabricated for the purpose of this study is shown in Fig. 1(a). Laser radiation in our experiments was coupled into microchannel waveguides located at the nodes of a hexagon (encircled with a dashed line in Fig. 1(a)) defining unit cells of the cladding lattice. At the wavelength of 500 nm, the effective mode area for each of these waveguides was estimated as 4 μm2. As shown in earlier work [18], such small-core waveguiding wires in PCF cladding provide a high nonlinearity, due to the strong confinement of the light field in a small-diameter core, leading to efficient spectral transformation of ultrashort pulses. The group-velocity dispersion (GVD) passed through zero around 700 nm for each of the waveguiding wires forming a waveguide array in our experiments, providing anomalous dispersion for 800-nm laser pulses. No special technology was employed in this work to ensure a nanoscale precision of PCF geometry engineering, as only the basic PCF fabrication processes were employed. The most appropriate waveguide arrays have been then selected for our experiments out of many available PCF samples. This difficulty can be resolved by using recently demonstrated PCF fabrication technologies enabling a reliable reproduction of nanoscale features in PCFs [19]. Spectral intensity, arb. units

(a)

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0.30 0.25 0.20 0.15 0.10 0.05 0.00 400

600

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Fig. 1. (a) Scanning electron-microscope image of the photonic-crystal fiber with a hexagonal unit cell of the cladding encircled by a dashed line. (b) Spectra of Ti: sapphire-laser pulses transmitted through two adjacent microchannel waveguides in the hexagonal unit cell of the PCF cladding are shown by filled and open circles. The dashed line shows the spectrum of the input pulse.

3. Results and discussion In experiments, laser radiation was simultaneously coupled into the waveguiding wires in the PCF cladding by focusing a laser beam into a spot with a diameter of about 15 μm on the front end of the fiber in such a way as to provide a uniform illumination of all the nodes of hexagonal unit cell of the PCF cladding. An imaging system and a camera were used to control the illumination of the front end of the fiber. The energy of laser radiation coupled into each waveguide of a hexagonal array under these conditions was estimated as 30 pJ. #95787 - $15.00 USD

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Propagating in the regime of anomalous dispersion inside the PCF, these pulses tended to evolve toward solitons with a pulse width of 50 fs and an energy of 20 pJ. The corresponding soliton peak power of 400 W agrees very well with the estimate for the peak power of the fundamental soliton supported by waveguiding wires with the cross-section geometry as defined by the PCF image in Fig. 1(a). The second-order dispersion β2 and nonlinearity γ are estimated as β2 ≈ 900 fs2/cm and γ ≈ 90 W−1km−1 for these waveguides, enabling formation of fundamental solitons with a peak power [20] Ps = β 2 γ −1τ −2 ≈ 0.4 kW. High-order dispersion tends to perturb solitons, making them unstable with respect to emission of nonsolitonic, dispersive waves [21 – 23]. In our experiments, these instabilities were manifested as intense radiation giving rise to a prominent feature in the spectrum of PCF output centered at approximately 460 nm (Fig. 1(b)). As the dispersion profiles of waveguiding wires chosen for our experiments were very similar to each other (due to the similarity of the geometry and sizes of the waveguides), radiation spectra measured for laser pulses transmitted through these waveguides were also very similar, exhibiting almost indistinguishable dispersive-wave features in their high-frequency parts (cf. the spectra shown by open and filled circles in Fig. 1(b)). The mutual coherence of blue-shifted dispersive waves emitted by solitons in two different channel waveguides was verified by a two-beam, Young’s type interference experiment. Letting radiation from two adjacent waveguides pass through a pinhole diaphragm and blocking radiation from the remaining part of the fiber, we observed a well-resolved interference pattern (see Fig. 2(a)). A one-dimensional cut of this interference pattern, measured with the use of a CCD array (shown by filled circles in Fig. 2(c)), features characteristic fringes with a visibility ξ = (I max − I min ) (I max + I min ) ≈ 0.8, where Imax and Imin are the maximum and minimum intensities in the interference pattern. The field intensity at the main maximum of the interference pattern was a factor of η2 ≈ 3.6 higher than the intensity of the blue-shifted signal from each of the waveguides. Deviations from an ideal interference pattern with ξ = 1 and η = 4 expected for two identical spatially separated sources of fully coherent radiation with a given wavelength (Fig. 2(b) and the dashed line in Fig. 2(c)) can be mainly attributed under our experimental conditions to a partial longitudinal (i.e., temporal) decoherence of blue-shifted signals from channel waveguides. To quantify the coherence degree of blue-shifted signals generated by two waveguides, we introduce the correlation coefficient σ = 2 E1 E 2 E12 + E 22 , where E1 and E2 are the amplitudes of the blue-shifted fields generated by the first and second waveguides. The time-averaged field intensity at the main peak of the interference pattern produced by blue-shifted signals from these two waveguides is then given by I 2 = I 2 (1 + σ ) , where I 2 = c (E12 + E22 )(8π )−1 and c is the speed of light. For blue-shifted signals with equal amplitudes, E1 = E 2 = E0 , we have

−1 I 2 = 2I 0 , where I 0 = cE02 (4π ) , and the field intensity at the maximum of the interference

pattern ranges from 2I 0 for incoherent signals (σ = 0) to 4I 0 for fully coherent fields. With the correlation coefficient σ estimated from the observed interference pattern as σ ≈ ξ ≈ 0.8, we find I 2 ≈ 3.6 I 0 , which agrees well with the results of our measurements. Figure 3(a) presents an interference pattern produced by blue-shifted, 460-nm signals generated in all the six channel waveguides of a hexagonal unit cell of the PCF cladding (encircled by a dashed line in Fig. 1(a)). The power of a blue-shifted signal at the output of each waveguide was estimated as 0.3 mW. The most striking and most remarkable feature of this interference pattern is its triangular symmetry, inherited from the symmetry of the waveguide-array source. The time-averaged field intensity at the main peak of the interference pattern produced by six blue-shifted signals can be represented as I 6 = I 6 1 + σ ij cos Δ ij ,

where I 6 = c(8π )−1 #95787 - $15.00 USD

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∑

6

i =1

Ei2 , σ ij = 2 Ei E j

∑

6 i =1

( ∑

i≠ j

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Ei2 , and the phases Δ ij = ϕ ij + δ ij include



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Intensity, arb. units

both the spatial phase shifts ϕ ij = k (ri − r j ) , where k is the wave vector of the blue-shifted signal and ri and rj are the radius vectors of the output ends of waveguide sources viewed from the observation point corresponding to the main peak of the interference pattern, and the differences δij between the phase shifts acquired by blue-shifted dispersive-wave signals in the ith and jth waveguides. As the waveguides in the considered hexagonal array have nearly identical dispersion profiles, as indicated by nearly identical spectra of blue-shifted signals (Fig. 1(b)), and have equal lengths, the phase shifts δij are small. 1.0

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Fig. 2. Experimental (a) and computer-simulated (b) two-beam interference patterns produced by 460-nm frequency-shifted signals from two adjacent waveguides in the hexagonal waveguide array. (c) One-dimensional cut of the two-beam interference pattern produced by 460-nm frequency-shifted signals: (filled circles) experimental results, (dashed line) simulations for two fully coherent sources of 460-nm radiation.

In Fig. 3(b), we plot a computer-simulated interference pattern produced by a hexagonal array of N = 6 fully coherent identical radiation sources with σij = 0, δij = 0, Ei = E0 (with i running from 1 to 6), and I 6 = 6I 0 . In this case, the time-averaged field intensity at the main peak of the interference pattern is N2 = 36 times higher than the intensity of a blue-shifted signal generated by each individual waveguide source, I6 = 36I0. Deviations of the experimental interference pattern from the computer-generated carpet in Fig. 3(b) originate from a partial decoherence of the blue-shifted signals produced in different waveguides, small displacements of waveguide sources from the nodes of an ideal hexagon, nonmonochromaticity of blue-shifted signals, variations in the intensity of the blue-shifted signal from one waveguide to another, and nonzero phase shifts δij. The field intensity at the maximum of the experimental six-beam interference pattern is η6 ≈ 22 times higher than the intensity of the blue-shifted, 460-nm signal from individual waveguides in the array and 3.7 higher than the intensity (6I0) attainable through an incoherent addition of signals with the same individual intensities I0. The maximum field intensity achieved at the center of the sixbeam interference pattern was ≈4 107 W/cm2 (intensity in Fig. 3(b) is given in arbitrary units). The experimental intensity enhancement factor is, however, 10% lower than the value of η 6 = 6 + 30σ ≈ 30, expected for an interference of six equal-amplitude fields Ei = E0 generated by an array of six identical waveguides with δij = 0 and mutual correlation coefficients assumed to be equal to each other and estimated as 3σij ≈ σ ≈ 0.8 from the two-beam interference experiment. The deviation of the experimental intensity enhancement factor η6 from η 6 is mainly explained under conditions of our experiments by nonzero phase shifts δij and inevitable variations in the intensity of the blue-shifted signal from one waveguide to another. In agreement with theoretical expectations, variations in the geometry of channel waveguides were found to change the central wavelength of dispersive-wave emission and the phase shifts δij in individual waveguides. The mutual correlation coefficients σij, however, are mainly controlled by the properties of the input field, including its spectral bandwidth and the #95787 - $15.00 USD

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level of noise, rather than the geometry of channel waveguides. Accordingly, arrays of channel waveguides with slightly different geometries were observed to generate partially coherent dispersive-wave emission signals at slightly different central wavelengths. A typical interference pattern generated by such an array of waveguides is presented in Fig. 3(c). With a careful control over the phase shifts δij in individual waveguides by means of advanced PCF technologies, such waveguide arrays in PCF structures should allow the synthesis of broadband radiation fields with precisely controlled spectral phase profiles. This strategy of broadband lightwave engineering offers much promise for the development of fiber-format sources of high-power few- and single-cycle light pulses. 1.0

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Fig. 3. Experimental (a) and computer-simulated (b) six-beam interference patterns produced by 460-nm frequency-shifted signals from the hexagonal waveguide array. Simulations were performed for a hexagonal array of fully coherent identical radiation sources with σij = 0, δij = 0, E = E , and I 6 = 6I 0 ; x and y are the transverse coordinates in the plane of the interference i

0

pattern and L is the distance from this plane to the end of the fiber. (c) Experimental multibeam interference pattern generated by an array of channel waveguides with slightly different geometries.

4. Conclusion

We have demonstrated that a coherent addition of blue-shifted dispersive waves generated by solitons in an array of highly nonlinear waveguiding wires in a silica PCF allows a radical local enhancement of field intensity at the main maximum of the resulting interference pattern. Results of our experiments suggest a promising method for the generation of highintensity frequency-shifted light fields through a coherent addition of nonlinear signals generated in specifically designed arrays of structure-integrated waveguides implemented on the platform of photonic-crystal fibers. Photonic-crystal fibers capable of synthesizing highquality two-dimensional interference patterns as a result of interference of controllably frequency-shifted light fields in the blue visible and UV ranges also offer much promise for a fabrication of two- and three-dimensional micro- and nanostructures, including photoniccrystal structures, by means of photolithography and laser micromachining techniques. Acknowledgments

We are grateful to K.V. Dukel’skii and V.S. Shevandin for fabricating fiber samples. This study was supported in part by the Russian Foundation for Basic Research (projects 06-0239011, 06-02-16880, 07-02-91215, 07-02-12175, 08-02-90061, and 05-02-90566), Award no. RUP2-2695 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union, the Federal Research Program of the Ministry of Science and Education of Russian Federation, National Basic Research Program of China (Grant Nos. 2003CB314904 and 2006CB806002), National High Technology Research and Development Program of China (Grant No. 2007AA03Z447), National Natural Science Foundation of China (Grant No. 60678012), NSFC-RFBR program (No. 60711120198) and the Program for New Century Excellent Talents in University (Grant No. NCET-07-0597).

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