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We demonstrate broadband light generation in diamond pumped by two-color femtosecond laser pulses. We find that phase matching plays a critical role in the output angle and frequency of the generated sidebands. When a third femtosecond probe pulse is applied to the crystal in the boxed Coherent anti-Stokes Raman Scattering geometry, a two-dimensional array of multi-color beams is generated through the Raman, four-wave mixing, and six-wave-mixing processes. We test the mutual coherence between the generated sidebands. Such coherence, maintained over the broad spectrum, opens possibilities for synthesis of subfemtosecond light waveforms.
©2008 Optical Society of America
1. Introduction
In the past, mutually coherent equidistant frequency sidebands have been obtained by broadband collinear Raman generation in molecular gases[1]. The essence of this technique is the preparation of a large molecular coherence by two laser fields tuned close to a Raman resonance. It has been argued that the resultant sidebands can be used to synthesize optical pulses as short as a fraction of a femtosecond (fs) [2]. While the sideband generation in gasses occurred collinearly with the driving beams, when two-color laser fields are applied to the crystal, it is necessary to use non-collinear geometry, due to the large dispersion present in the crystal. We have reported broadband sideband generation in Raman-active crystal PbWO4 [3, 4]. Coherent high-order anti-Stokes scattering has also been observed in crystals such as YFeO 3, KTaO3 and KNbO3 when two color fs pulses were used [5, 6, 7]. Recently Matsubara et al. has demonstrated promising Fourier synthesis using multiple coherent anti-Stokes Raman scattering (CARS) signals obtained in a LiNbO3 crystal at room temperature, and generated isolated pulses with 25 fs duration at 1 kHz repetition rate [8].
The motivation to use diamond for broadband generation is two-fold. Firstly, diamond has a single strong narrow Raman line at a very large frequency shift (1332 cm -1), compared to the other crystals [9]. Secondly, diamond is isotropic and the refractive index is well known, which makes it easier (compared to the other Raman crystal PbWO4 we studied) to do some theoretical calculation and thus help us understand the complicated experimental results.
Beside the above-mentioned two reasons, diamond also has several remarkable properties which are desirable for broadband generation [10]. First of all, diamond is capable of transmitting over an unusually broad spectral range (from x-ray region to the microwave and mm wavelengths) and has the widest electromagnetic bandpass of any material. This broadband transmission is essential for the sideband generation. Second, it has an extremely high thermal conductivity (five times that of copper) and is extremely chemically inert. Therefore, it is not easy to get damaged by a laser and it does not require much protection from moisture. At last, diamond has the highest atom density of any material. A high density means a high Raman gain.
Back in 1963, diamond was chosen as a representative covalent crystal for the stimulated Raman scattering (SRS) experiment, one anti-Stokes (AS) and two Stokes (S) were observed when a high intensity (20 MW/cm2) Ruby laser was used for excitation [11]. Natural diamonds are costly and scarce. Recently, synthesis of large area diamonds at lower pressure using chemical vapor deposition (CVD) technique was developed. Kaminskii et al. used CVD diamond for Raman laser converter based on SRS and reported observation of SRS (up to 1 S and 3 AS sidebands) in diamond (using both ns and ps pulses) [12, 13]. In this article we present the study of the Raman generation in diamond using fs pulses.
2. Broadband coherent light generation in diamond driven by two-color femtosecond laser field
Our amplified fs laser (Mira + Legend, Coherent) has an output of 1 mJ per 35 fs pulse (at 1 kHz repetition rate) at the fundamental 805 nm wavelength. It is used to pump two computer-controlled optical parametric amplifiers (OPerA, Coherent). Signal and idler pulses obtained from the two OPAs are then frequency doubled or mixed with the fundamental to produce up to 30 μJ per 50 fs pulse at tunable visible wavelengths. The pulses are attenuated to 1 to 3μJ per pulse (typically 3μJ for the pump and 1μJ for the Stokes) and focused to about 100μm spot size at the sample. The corresponding laser intensity is right below the onset of strong self phase modulation (SPM). The thickness of the sample is about 500μm.
First we apply two-color excitation. The experimental setup is about the same as the one we used for PbWO4 crystal [3].We start with an angle 3.6° (the phase matching angle between the peak wavelengthes) between the pair of beams.
Fig. 1. Broadband generation in diamond with two input pulses (λ1= 630 nm and λ2= 584 nm, δv= 1250 cm-1) crossed at angles of 3° and 5.8°. Top: Generated beams projected onto a white screen. The two pump beams, S 1 and the first few AS beams are attenuated (after the sample) by a neutral-density filter. The AS 2 spot clearly shows two different colors, with blue corresponding to the Raman generation and green to the FWM signal. Bottom: Normalized spectra of the generated sidebands.
By applying two 50 fs pulses (at λ1= 630 nm, λ2= 584 nm, and δv= 1250 cm-1), we obtain generation of up to 16 AS and 2 S sidebands. The highest frequency generated is in the UV region at a wavelength of 301 nm. Similar to the sideband generation in PbWO4, the sidebands are not equally spaced in frequency [3]. To see the angle dependence, we vary the crossing angle between the two applied laser beams (we use angles of 3, 3.8, 4, 4.5 and 5.8 °), and record changes in the generated spectrum (Fig. 1, only 3 and 5.8 ° are shown for clarity). We find that the generation is more efficient at a 3° beam crossing angle. The energy conversion from pump to AS 1 is 3%, 2% to the S1 and 0.5% to AS 2. The total energy conversion from pump to sidebands is 14 %. The sidebands generated using 3° crossing angle overlap substantially in spectrum with each other.
From Fig. 1, we can clearly see that the instantaneous four-wave-mixing (FWM) signal coexists with Raman generation in the lower orders of the sidebands. The AS 2 beam spot shows two colors: blue and green. By measuring the frequency shift from the preceding sideband, we deduce that the blue is due to Raman generation, while the green beam corresponds to the FWM signal.
We find that sidebands generated at 5.8° have a larger (about twice) frequency spacing compared to the 3° case when we plot the sideband frequency versus the sideband order as shown in Fig. 2 (a). Also the sidebands come out at a much larger angle spacing as shown in the Fig. 2 (b) (we define the Stokes output angle as 0 degree). This is more obvious for low-order sidebands which are more affected by the strong FWM signal. This shows that the initial phase matching condition between the two pump beams decides the generated sideband output angles and frequencies.
To prove that the sideband generation is from the Raman rather than the FWM process, we keep the pump frequency the same while tuning the Stokes frequency from 820 cm -1 to 2608 cm-1 and record the sideband frequency. The result is shown in Fig. 3. When Δv varies from 820 cm-1 to 2608 cm-1, the output angle doesn’t vary much. Also, the frequency vary much less compared to the change of Δv. These facts are clear indications that the signals are generated mainly through the Raman process. The competing FWM process only affects the sidebands slightly. We find that for optimal sideband generation, we should choose a pair of pulses which are on resonance with the Raman transition and cross at an angle that satisfies the phase matching condition. In this case, the low-order sidebands are very strong because of the boost from the FWM process.
Fig. 2. The sideband frequency as a function of the sideband order (a) and as a function of the sideband output angle (b) with the two input beams (λ1=630 nm and λ2=584 nm, δv= 1250 cm-1) crossing at two different angles, 3° (square) and 5.8° (round). The sidebands generated at 5.8° have a larger (about twice) frequency spacing compared to the 3° case.
We note that the sideband generation is very sensitive to the polarization of the pump beams. The conversion efficiency is maximized when the pump and Stokes beams have parallel polarization. No sidebands are generated when the polarizations of the pump and Stokes beams are perpendicular to each other.
3. Phase matching calculation
Apparently, phase matching plays a critical role in the generation of multiple spectral sidebands in Raman-active crystals (as compared to the collinear Raman generation in gasses [1, 2]). Due to the material dispersion, phase matching is optimized when different frequency components propagate at different angles.
We perform calculations that support our qualitative understanding of the role of phase matching using a pair of Gaussian pulses, which are very close to those we measure in experiment using a spectrometer. We take the spectral intensity of an n’s AS sideband as given by
Here k⃗ is the wavevector (k = nv/c), vR is the Raman shift and c is the speed of light. The refractive index of diamond n is calculated by using the generalized Cauchy dispersion formula[14]:
Fig. 3. The Sideband frequency and output angle at the different detuning (Δv vary from 820 cm-1 to 2608 cm-1) between the pump (fixed at 594 nm) and Stokes pulses. The output angles do not vary much, nor does the frequencies of the sidebands, although Δv varies a lot.
with λ in units of μm. This formula fits well with the available experimental data[10, 15]. The k-vector of the Raman excitation k⃗R is determined by the directions of the two input beams. We assume that the angle between the low-order sidebands is approximately the same.
We find that when the angle is smaller than the phase matching angle, the peak frequency shift of the generated AS 1 from the pump pulse is 1163 cm -1, which is smaller than the Raman shift. The opposite happens when an angle larger than the phase matching angle is applied, as shown in Fig. 4. The peak frequency shift of the generated AS 1 from the pump pulse is 1301 cm -1 for input angle of 3.6° and 1466 cm-1 at 4.6°. This agrees with our experimental observations in diamond. The peak frequencies of the two input pulses phase match at 3.6 °. Therefore, when the two input pulses cross at 5.8°, the generated spectrum has a larger frequency spacing than that of the 3°. This calculation may also explain the gradually decreasing frequency spacing between the sidebands, since due to the normal medium dispersion, the optimum beam-crossing angle increases with increasing sideband frequency.
4. The measurement of the coherence decay time for CARS/CSRS
The unequal spacing of the sidebands generated raised the question of the Raman modes of the crystal we use. Using a pair of visible pulses as pump (584 nm) and Stokes (635 nm)and using a UV pulse (also about 50 fs width) at 318 nm as a probe, we get the CARS signal at 305.8 nm (δv= 1250 cm-1) and coherent Stokes Raman scattering (CSRS) signal at 332 nm (δv= 1326 cm-1) as shown in Fig. 5. Undoubtedly, this is the strong Raman line in diamond. The width of the line in the SRS spectrum related to SRS-promoting vibration transition is ΔvR ≈2.5 cm-1. Therefore the phonon relaxation time T 2 = (πcΔvR)-1 ≈4.2 ps [12]. We measure the exponential decay of the CARS/CSRS signal as shown in Fig. 5 (bottom).We get the intensity of the CARS/CSRS signal by averaging over the center wavelengths. To remove the effect of the strong instantaneous FWM signal over the CARS/CSRS signal, we only fit the data about 2 ps after the 0 delay with an exponential decay and obtain decay time T 2 as 2.6 and 2.7 ps for CARS and CSRS, respectively.
Fig. 4. Theoretical calculation of the generated AS 1 when the two pumps cross at angle 2.6, 3.6 and 4.6 degree respectively, with the 3.6 degree corresponding to the phase matching angle between two input pulses. The input pulses is 630 nm and 581.23 nm which has a frequency difference of exactly Raman shift 1332 cm-1. The sample thickness used for calculation is 500μm.
Fig. 5. The CARS and CSRS signals observed in diamond using a UV probe pulse. Top, the spectra of the CARS (left, center wavelength is 305 nm) and CSRS (right, center wavelength is 332 nm) as a function of the probe delay. Bottom, the exponential decay of the CARS (left) and CSRS (right) signals intensity average over their center wavelengths. Log scale is used.
Fig. 6. The 2-D array generation in diamond with three input pulses (λpump= 720 nm, λStokes= 800 nm, and λprobe= 600 nm). The wavelengths of the sidebands are labeled in nm. The degenerated FWM signal (2Y-IR, a shorthand of 2ωY - ωIR) from the probe and Stokes pulses and the one (2Y-R) from the probe and pump pulses are much stronger than the Raman generation spots. They either superimpose or shift slightly. The six-wave-mixing signal (3Y-2R) from the pump and probe pulses is also visible.
5. Broadband coherent light generation in diamond driven by three-color femtosecond laser field
When we apply a Stokes pulse (IR, 800 nm), a pump pulse (red, 728 nm) and a visible probe pulse (yellow, 600 nm) at the usual boxed-CARS geometry, in addition to the strong CARS and CSRS signals, we obtain a two dimensional (2-D) color array as shown in Fig. 6. Here the angle between the pump and Stokes beams is 5°. The probe beam is 11° from the pump beam and 16° from the Stokes beam. We propose the following explanation for the 2-D color array generation. The frequency spacing between the pump and Stokes pulses is around 1000 cm -1. As a result, the average sideband spacing is very regular, about 1040 cm -1. The sidebands in the first row are generated by the pump and Stokes pulses. The second row is generated by the CARS/CSRS and the high-order CARS processes. The FWM signal from the probe and Stokes/pump pulses leads to the generation of the third row. The degenerate FWM signal from the probe and Stokes pulses (2ωY - ωIR) and from the probe and pump (2ωY - ωR) pulses are much stronger than the generation spots due to Raman effect. They either superimpose or shift slightly. The six-wave-mixing signal (3ωY - 2ωR) from the pump and probe pulses results in the fourth row sideband generation. We measure the coherence decay time again so that we can compare it with that measured with a UV probe beam. The CARS signal (558.6 nm) has a frequency shift of 1235 cm-1 from the probe pulse and an exponential decay time of 2.5 ps. The CSRS signal (652.2 nm) has a frequency shift of 1329 cm-1 and an exponential decay time of 3.0 ps. These decay times are similar to the ones obtained with a UV probe pulse.
6. Coherence among the sidebands
At last, we test the mutual coherence among the generated sidebands through an interference measurement. We use three input beams (instead of two) and align them in one plane, such that the higher-order AS sidebands generated through different channels overlap in space. We first generate AS 2 (Green, λGreen= 545 nm) by focusing Red (λR= 690 nm) and IR (λIR= 802 nm) pulses into the diamond crystal. Then we send a third beam in the direction of the generated AS 1 beam, as shown in Fig. 7(top).We tune the frequency of this third applied field (Yellow, λYellow= 590 nm) such that the AS 1 generated from Red and Yellow fields coincides (both in space and frequency) with the AS 2 generated by Red and IR pulses. vGreen - vYellow= 1400 cm-1, which agrees well with the Raman shift of 1332 cm-1. When only one pair of pulses (either one) is applied to the crystal, the energy of the combined field (measured with a photodiode) is about 1 unit. When all three pulses are applied, the measured energy varies from 0 to 4 units from shot to shot as shown in Fig. 7 (we show 9 random shots), as a result of coherent addition of the two generated pulses. Since we don’t attempt to stabilize the phases of the three input fields, the interference of the generated “Green” pulses is partially constructive or destructive, at random. We conclude that the two pulses, generated through the two separated Raman processes, have good mutual coherence. This measurement confirms our expectation that the (highly-coherent) Raman process results in generation of mutually-coherent sidebands. Further proof is shown by a different experiment in our earlier paper [3]. Assuming that all the sidebands generated from 3° crossing in diamond are properly phased or locked to each other, we estimate that a single cycle pulse with width of 0.5 fs can be obtained.
Fig. 7. “Green” sideband energy under different conditions. Black solid line: all three input pulses are present. Red dashed line: IR and Red pulses present only. Blue dotted line: Yellow and Red pulses present only. Top: Input beam geometry.
7. Conclusions
We demonstrate efficient broadband nonlinear frequency conversion in diamond. We measure 7% energy conversion into the AS 1 and 1% into the AS 2 and S 1. The high-order CARS/CSRS and FWM/SWM signals are also observed. Our theoretical calculation shows that when the angle between the pump and Stokes beams is smaller than the optimal phase matching angle, the frequency spacing between the sidebands is smaller than the Raman shift and the opposite happens when the angle between the pump and Stokes beams is larger than the optimal phase matching angle. When a third femtosecond probe pulse is applied to the crystal in the boxed- CARS geometry, a 2-D array of multi-color beams is generated through the Raman, four-wave mixing, and six-wave-mixing processes. We also test the mutual coherence between the generated sidebands.
We believe that the generation can be optimized by choosing a pair of beams with proper wavelengths and using a lens with an optimal focal length so that the beams cross at the crystal with a phase matching angle. Also, a comparatively small dispersion means that the generated sidebands can be increased both in number and energy by using a thicker sample (say, 1 mm), since more Raman gain can be obtained.
Acknowledgments
We thank Phil Hemmer for providing us the diamond sample. We also thank Dmitry Pestov, Robert K. Murawski, Yuri Rostovtsev, Phil Hemmer for helpful discussions and technical assistance. This project is supported by the National Science Foundation (Grant no. PHY-0354897), the Texas Advanced Research Program (Grant no. 010366-0001-2007), and the Robert A. Welch Foundation (Grant no. A1547).
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