Abstract:
The invention features methods and systems for optical correlation of ultrashort optical waveforms, e.g., pulses. The optical waveform passes through a diffractive optic, e.g., a mask or grating, to generate multiple sub-beams corresponding to different diffractive orders. At least two of the sub-beams are then imaged onto the sample to produce a desired crossing pattern. To perform the correlation, the diffracted sub-beams are variably delayed relative to one another prior to overlapping on the sample. The sample generates a signal beam in response to the overlapping sub-beams, the signal beam providing a correlation between the sub-beams for each of the variable delays.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 60/092,688, filed Jul. 14, 1998, the contents of which are incorporated herein by reference. 
    
    
     STATEMENT AS TO FEDERALLY SPONSORED RESEARCH 
     This invention was made with government support under Grant Number CHE-9713388 awarded by the National Science Foundation. The government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     The invention relates to optical correlation techniques for characterizing materials and optical waveforms. 
     Modern laser technology permits the routine generation of ultrashort optical pulses, i.e., pulses having a duration of less than about 1 psec. Some lasers can even generate pulses as short as about 10 fsec. More generally, modern laser systems can produce ultrafast optical waveforms that have features as short as ultrafast pulses, e.g., a terahertz train of ultrashort pulses. See, e.g., U.S. Pat. Nos. 5,682,262 and 5,719,650. Such ultrashort waveforms (including single pulse waveforms) can be used to probe chemical and physical phenomena in atoms, molecules, and materials. Unfortunately, the time scales for such measurements and for the optical waveforms themselves exceed the bandwidth of most, if not all, electronic detectors. As a result, many measurements involve optical correlation techniques in which two or more waveforms overlap on a sample or non-linear optical crystal. 
     SUMMARY OF THE INVENTION 
     The invention features methods and systems for optical correlation of ultrashort optical waveforms, e.g., pulses. The optical waveform passes through a diffractive optic, e.g., a mask or grating, to generate multiple sub-beams corresponding to different diffractive orders. At least two of the sub-beams are then imaged onto the sample to produce a desired crossing pattern. To perform the correlation, the diffracted sub-beams are variably delayed relative to one another prior to overlapping on the sample. The sample generates a signal beam in response to the overlapping sub-beams, the signal beam providing a correlation between the sub-beams for each of the variable delays. 
     In general, in one aspect, the invention features a method for autocorrelating an optical waveform. The method includes: passing an input beam containing the optical waveform through a diffractive mask to form at least two sub-beams; delaying one of the sub-beams relative to the other sub-beam; and imaging the two sub-beams onto a non-linear optical crystal to allow the two sub-beams to spatially overlap with one another. The diffractive mask defines the object plane and the non-linear optical crystal defines the image plane. The overlapping sub-beams are delayed relative to one another, and the non-linear optical crystal generates a signal beam in response to the overlapping sub-beams. 
     The method can include any of the following features. The method can further include measuring the intensity of the signal beam and repeating the measuring step for each of multiple delays between the sub-beams. The method can further include spectrally resolving the signal beam and measuring the intensity of the spectrally resolved signal beam, and repeating the resolving and measuring steps for each of multiple delays between the sub-beams. The non-linear optical crystal can generate the signal beam by second harmonic generation or by any other non-linear optical mechanism. The delaying step can include introducing material into a path of one of the sub-beams. The imaging step can include passing the sub-beams through a pair of lenses. The optical waveform can have temporal features shorter than about 1 psec, shorter than about 300 fsec, or even shorter than about 50 fsec. The optical waveform can be an optical pulse. The two sub-beams can correspond to different orders of diffraction for the diffractive mask. 
     In general, in another aspect, the invention features an optical autocorrelator for characterizing an an optical waveform. The autocorrelator includes: a diffractive mask which during operation diffracts an input beam carrying the optical waveform into at least two sub-beams; an optical delay assembly positioned in the path of a first of the two sub-beams, wherein during operation the optical assembly introduces a variable delay between the two sub-beams; a non-linear optical crystal; an optical imaging system which during operation images the two sub-beams onto the non-linear optical crystal to allow the two sub-beams to spatially overlap one another, the diffractive mask defining a object plane and the non-linear optical crystal defining the image plane; and an analyzer which during operation measures an intensity of a signal beam produced by the non-linear optical crystal in response to the two overlapping sub-beams. 
     The autocorrelator can include any of the following features. The autocorrelator can further include a controller connected to the optical delay assembly and the analyzer, wherein during operation the controller causes the optical delay assembly to introduce multiple delays between the two sub-beams and records the intensity of the signal beam for each of the multiple delays. The optical delay assembly can include an optical window positioned in the path of the first sub-beam and a rotation stage supporting and adjustably orienting the optical window, the adjustable orientation of the optical window defining the variable delay between the two sub-beams. The autocorrelator can further include a stationary optical window positioned in the path of the second of the two sub-beams to impart a fixed delay to the second sub-beam. The analyzer can include a grating and a multi-element photodetector, wherein during operation the grating spectrally resolves the signal beam on the photodetector and the photodetector records the spectrally resolved intensity of the signal beam. Alternatively, the analyzer can be a photodetector. The non-linear optical crystal can generate the signal beam by second harmonic generation or by any other non-linear optical technique. The optical imaging system can include a pair of lenses and the optical delay assembly can be positioned between the pair of lenses. Each of the pair of lenses can be a spherical lens. The two sub-beams can correspond to different orders of diffraction for the diffractive mask. 
     Embodiments of the invention include many advantages. For example, the correlation technique optimizes the overlap of the two sub-beams on the sample (e.g., the non-linear crystal) and thereby greatly simplifies alignment and robustness of the optical correlation system. 
     Other features, aspects, and advantages follow. 
    
    
     BRIEF DESCRIPTION OF DRAWINGS 
     FIGS. 1 a  and  1   b  are schematic diagrams of overlapping femtosecond beams (a) crossed at an angle to one another and (b) crossed using a diffraction grating G and confocal imaging system L 1  and L 2 ; 
     FIGS. 2 a  and  2   b  are CCD images of the interference patterns produced by two overlapping 30 fs pulses corresponding to the arrangements of FIGS. 1 a  and  1   b , respectively; 
     FIG. 3 is a schematic of an optical autocorrelator based on the beam crossing technique of FIG. 1 b;    
     FIG. 4 is an autocorrelation curve obtained by rotating a 150 micron glass slide in one arm of the autocorrelator of FIG.  3  and monitoring the noncollinear SHG output versus the rotation angle; and 
     FIG. 5 is a schematic diagram of an analyzer for spectrally resolving a correlation signal beam. 
    
    
     DETAILED DESCRIPTION 
     Crossing Ultrashort Pulses 
     In many applications, ultrashort optical pulses, e.g., pulses on the order of 100 fs, crossed at nonzero angle overlap only over a small region in space. This limitation can be overcome by using diffraction orders of a grating. We consider the arrangement in which, upon diffraction of a femtosecond pulse by a grating, two beams corresponding to the first-order diffraction maxima are recombined at the image plane by a system of two confocal lenses. In this arrangement, the beams overlap over the their full aperture with the short duration of the pulses being preserved. 
     Various ultrafast optical techniques involve crossing of two or more femtosecond pulses in a medium. Referring to FIG. 1 a , beams  200  and  202  including pulses  204  and  206 , respectively, can be crossed with one another using beamsplitters and mirrors. However, the shorter the pulses, the smaller the area  210  over which the pulses overlap. For two beams crossed at the angle θ, the size of the overlap area is given by cπ/sin(θ/2), where c is the speed of light in the medium, and π is the pulse duration. For example, for a 30 fs pulse duration and a moderate angle such as 5° the beams overlap only within a strip approximately 200 microns wide. The number of interference fringes produced by two beams is independent of the angle and, for transform-limited pulses, is roughly 2cπ/λ, where λ is the optical wavelength. With 30 fs pulses at λ=800 nm, only about 20 interference fringes can be produced. 
     These limitations can be overcome if we cross diffraction orders of a grating using system  250  shown in FIG. 1 b . Pulses corresponding to different diffraction orders propagate at different angles and have parallel pulse fronts. In system  250 , a beam  260  containing femtosecond pulse  262  is transmitted through a diffraction grating  264  to form at least two sub-beams  290  and  292 , corresponding to the first diffraction orders. The sub-beams are imaged by two confocal lenses L 1  and L 2  or other suitable optics, e.g., reflective optics, with grating  264  being placed in the front focal plane of the first lens. The lenses have focal lengths f 1  and f 2 , respectively, and are denoted by reference numerals  266  and  268 , respectively. A spatial filter  270  transmits the two sub-beams and blocks other sub-beams corresponding to different orders of diffraction. The sub-beams are then recombined at image plane I (denoted by reference numeral  272 ). System  250  not only provides pulse overlap in the image plane I, but also preserves short pulse duration. 
     Let us assume that the incident pulse in FIG. 1 b  is transform-limited with a Gaussian temporal profile, and that the polarization is perpendicular to the plane of the drawing. The electric field of the incident beam is given by: 
      E=E 0 exp[−(t−z/c) 2 /τ 0   2 )exp[iω 0 (t−z/c)]= 
     
       
          =(E 0 τ 0 /2π ½ )exp[−(ω−ω 0 ) 2 τ 0   2 /4]exp[iω(t−z/c)]dω  (1) 
       
     
     where ω 0  is the central frequency of light, τ 0  is related to the FWHM pulse duration τ by τ 0 =τ (2*ln2) ½  and the distance z is measured from the front focal plane of the lens L 1 . Consider one of the plane waves, comprising the integral in Eq. (1), in which the electric field is given by exp[iω(t−z/c)]. Upon diffraction of this plane wave by the grating, a wave diffracted into the nth order is given by 
     
       
         E n (ω)=A n exp[iωt−i(ω 2 /c 2 −q n   2 ) ½ z−iq n x],  (2) 
       
     
     where x is the vertical coordinate measured, e.g., from the optical axis, q n  is the diffraction wave vector expressed through the grating period Λ by q n =2πn/Λ, and A n  is the complex amplitude which depends on whether the grating is a phase or amplitude one and on the grating profile (we assume a symmetric grating so that A n =A −n ). For a phase grating, A n  is ω-dependent, but for a small frequency spread, e.g., δω/ω 0 &lt;&lt;1, this dependence is weak and can be ignored. For embodiments in which this condition is not met, amplitude gratings may be preferable. 
     Disregarding diffraction by the lens system, the plane wave in Eq. (2) will be transformed by the imaging system into a plane wave 
      E n (ω)=A n exp[iωτ−i(ω 2 /c 2 −q n   2 ) ½ z′+i(q n /M)x−iLω/c],  (3) 
     where the distance z′ is measured now from the image plane, M=f 2 /f 1  is the magnification factor of the imaging system, and an additional phase term Lω/c is due to the optical path L from the point (x=0, z=0) to its image at (x=0, z′=0). The electric field yielded by the nth-order diffraction at the output of the imaging system is given by the superposition of plane waves, 
     
       
         E n =(E 0 τ 0 /2π ½ ) A n exp(iq n x/M)ƒdωexp[−(ω−ω 0 ) 2 τ 0   2 /4)]×exp[iωt′−i(ω 2 /c 2 −q n   2 /M 2 ) ½ z′]  (4) 
       
     
     where t′=t−Lω/c. 
     The integral in Eq. (4) is independent of x. Therefore, the planes of equal amplitude in a pulse are parallel to the plane z′=0. Assuming a small frequency spread, δω/ω 0 &lt;&lt;1, and small angles, q n /M&lt;&lt;ω 2 /c 2 , one gets the following result for the duration of the pulse:                τ   =       (       τ   o   2     +       z   ′2            4        c   2          q   n   4           M   4          ω   o   6          τ   0   2             )       1   /   2         ,           (   5   )                                
     i.e., the pulses are compressed to the original duration τ 0  as they approach the image plane. 
     Exactly in the image plane z′=0, the electric field given by the two beams corresponding to ±1 orders of diffraction is given by 
     
       
         E=2A 1 E 0 cos(q 1 x/M) exp(−t′ 2 /τ 0   2 )exp(iω 0 t′).  (6) 
       
     
     The interference pattern with the period MΛ/2 extends over the entire image plane. Thus we have two pulses overlapping in the image plane over the area limited only by the aperture of the optical system. 
     In terms of the space-time picture, the full overlap  280  results from the tilted pulse fronts as shown in FIG. 1 b . In terms of spectral components, a diffracted beam consists of components with different wave vector directions. However,the x-component of the wave vector is the same for all the spectral components. Therefore, when the two beams are crossed, the difference in the x-component of the wavevector Δk x =2q 1 /M is well defined, resulting in a well-defined periodic interference pattern. 
     In an experiment, we used 30 fs pulses of an amplified Ti:sapphire system at λ=800 nm and compared the beams crossing with a beamsplitter and mirrors as in FIG. 1 a , and that of FIG. 1 b . In the latter case, we used a phase grating with the period λ=10 microns, and two spherical lenses with focal lengths 15 cm. FIG. 1 c  shows the interference pattern produced by crossing the beams as in FIG. 1 a , which contains, as expected, only about 20 high-contrast interference fringes. In contrast, the grating set-up of FIG. 1 b  resulted in a fringe pattern spreading all over the laser spot. A portion of this pattern is shown in FIG. 1 d.    
     The arrangement shown in FIG. 1 b  makes it possible for the femtosecond pulses to overlap in time and space over the full aperture of the beams. Although in the arrangement considered here, the two beams were obtained from a single one, a similar arrangement can be used to optimize the overlap of two beams of different wavelength or polarizations. The techniques has many advantages. One obvious advantage is that the signal in wave mixing measurements can be collected from a larger area, which should be helpful if the signal is weak and the excitation intensity is limited by the damage threshold of the medium. A more fundamental issue is accurate definition of Δk x  for propagating material excitations. To be specific, let us consider impulsive stimulated Raman scattering on phonon-polaritons, where two crossed beams are used to excite phonon polariton modes at the wavevectors equal to +/−Δk x , and the resulting standing wave is detected via diffraction of a probe pulse. By crossing pulses as in FIG. 1 a , one can only produce a limited number of polariton periods, equal to the number of the interference fringes. Consequently, the signal due to the standing wave dies out as the counter-propagating waves leave the excitation region, making it difficult to accurately measure the polariton frequency, attenuation, and nonlinear effects. Using the grating arrangement of FIG. 1 b  to produce an unlimited number of interference fringes would be advantageous for this and other experimental techniques using ultrashort pulses to excite propagating material excitations. 
     The system can also be easily adapted to correlate the two sub-beams with one another by introducing a variable delay between the two sub-beams. For example, substantially transparent optical material positioned between lenses  266  and  268  along the path of one of the sub-beams would introducing extra optical path length to one of the sub-beams. 
     Optical Autocorrelator 
     The techniques described above can be used in an optical autocorrelator for characterizing ultrashort optical waveforms, e.g., measuring the pulse duration of ultrashort optical pulse. FIG. 3 illustrates a schematic for such an autocorrelator  100 . 
     An ultrashort optical beam  12  containing optical waveform  10  is incident on a diffractive optic  15  that diffracts beam  12  into at least two orders, e.g., diffracted order +1 and −1, to form diffracted beams  14  and  16 . Diffractive optic  15  can be a mask or grating that imparts amplitude modulation, phase modulation, or both, and which may be reflective or transmissive. Suitable diffractive optics are described, e.g., in U.S. Pat. No. 5,734,470, the contents of which is incorporated by reference. 
     A pair of lenses  18  and  20  image the profile of diffracted beams  14  and  16  immediately after diffractive optic  15  onto a non-linear optical crystal  22 , e.g., a crystal of LiTaO 3 , LiNbO 3 , KTP, or KDP. Thus, as described above, the diffracted beams  14  and  16  spatially overlap completely in the plane of non-linear optical crystal  22 , i.e., the image plane defined by lenses  18  and  20 , without any loss of temporal resolution. Thus, the autocorrelator is substantially alignment-free. 
     The non-linear optical crystal generates the second harmonic of diffracted beams  14  and  16 , which exit the crystal as beams  24  and  26 , respectively. In addition, the non-linear crystal generates a second harmonic signal beam  28  having an intensity proportional to the temporal and spatial overlap of beams  14  and  16  in crystal  22 . An analyzer  30 , e.g., a photodiode, measures the intensity of signal beam  28 . A spatial filter  32  and a spectral filter  34  prevent beams  24  and  26  and scattered fundamental light, respectively, from reaching analyzer  30 . In other embodiments, non-linear mechanisms different from second harmonic generation can be used. For example, non-linear optical crystal  22  may generate a signal beam for spatially and temporally overlapping beams  14  and  16  based on, e.g., self-diffraction, polarization rotation, or difference-frequency mixing. As described above, use of diffractive element  15  increases the overlap of beams  14  and  16  relative to conventional beam crossing, so the signal beam  28  is stronger, thereby increasing the sensitivity of the autocorrelator. 
     Identical glass slides  36  and  38  are positioned between lenses  18  and  20  to receive and transmit diffracted beams  14  and  16 , respectively. Slide  36  is fixed normal to beam  14  and slide  38  is mounted on a motorized rotation stage  40 , which allows beam  16  to intersect slide  38  over a range of incident angles θ. When beam  16  is normal to slide  38 , i.e., θ=90°, beams  14  and  16  temporally overlap completely and maximize the intensity of signal beam  28  generated by crystal  22 . As the angle θ differs from θ=90°, beam  16  travels through a path length in slide  38  that is larger than that of beam  14  in slide  36 . Thus, beam  16  is delayed relative to beam  14  and their temporal overlap in crystal  22  decreases, thereby reducing the intensity of signal beam  28 . For example, for glass slide  38  having a thickness of about 150 microns and being oriented at angle θ of about 27°, the delay is about 20 fs. Larger delays can be achieved by increase the difference angle θ from 90° or using thicker slides. The precise delay between the two beams can be determined from their difference in optical path length. Furthermore, since glass slide  38  has substantially parallel faces, the direction of beam  16  is unaffected by slide  38 . Thus, beams  14  and  16  spatially overlap completely in the plane of crystal  22  over the range of angles for θ. 
     To scan through a range of delays, a controller  44  rotates rotation stage  40  using a drive signal  45 . At the same time, controller  44  receives a signal  46  from analyzer  30  indicative of the intensity of signal beam  28 . Controller  44  records an autocorrelation of input beam  12  by monitoring signal  46  as a function of the drive signal  45 , which can be converted to a delay time between beams  14  and  16 . For example, assuming waveform  10  has an intensity profile I(t) then the correlation signal S(τ) is proportional to the integral of I(t)I(t+τ), where τ is the delay time between the two beams and the integral is taken over all times t. 
     FIG. 4 illustrates an autocorrelation of a 30 fs, 800 nm pulse from a Ti:sapphire laser system recorded using the autocorrelator described herein, except that glass slide  38 , which was 150 microns thick, was rotated manually. 
     Referring again to FIG. 3, autocorrelator  100  can also include a mask  70  positioned before lens  18  for transmitting sub-beams corresponding to selected orders of diffraction, e.g., −1 and +1, and blocking other orders of diffraction. In addition, where optical waveform  10  includes multiple, well-separated frequencies, e.g., ω 1 , and ω 2 , mask  70  can be used to select among the different wavelengths of the sub-beams. For example, mask  70  could select the +1 order for ω 1  and the −1 order for ω 2 . In this case, the correlation signal S(τ) would no longer be an autocorrelation of I(t), but a correlation between the ω 1  component of I(t) and the ω 2  component of I(t). Mask  70  can also be positioned between lenses  18  and  20 , or between lens  20  and non-linear crystal  22 . 
     As shown in FIG. 5, analyzer  30  can include a grating  90  that diffracts signal beam  28  into its spectral components  94  and directs them to a multielement detector  92 , which records the intensities of the spectral components  94 . If necessary, imaging optics can be positioned between grating  90  and multielement detector  92 . Measuring a correlation signal beam as a function of both delay and spectral frequency can provide additional information about waveform  10 , see, e.g., R. Trebino and D. J. Kane in  J. Opt. Soc. Am ., A10:1101 (1993). Alternatively, analyzer  30  can be a single-element detector, which measures the intensity of all spectral components of the signal beam. 
     Also, in other embodiments, slide  36 , like slide  38 , can be supported by a motorized rotation stage and oriented under the control of controller  44  so that both positive and negative delays can be introduced between beams  14  and  16 . Alternatively, slide  36  can retain a fixed orientation at a non-normal offset angle or can be thicker than slide  38  so that beam  16  precedes beam  14  for θ=90θ and follows beam  14  for other angles, e.g., angles less than 60°. Furthermore, in other embodiments, other means for introducing a delay between beams  14  and  16  can be used. For example, one or both of the glass slides may be replaced with a series of reflective optics or an etalon, which may be under the control of a motorized translation or rotation stage. 
     Furthermore, in other embodiments, optics different from lenses  18  and  20  may be used to image diffracted beams  14  and  16  onto non-linear optical crystal  22 . For example, curved reflective optics can be used, which may be advantageous for cases in which the glasses in lenses  18  and  20  introduce significant dispersion into beams  14  and  16 , thereby stretching their pulse durations. In addition, reflective optics can be used to form a more compact, folded geometry. Also, in other embodiments, one or more lenses or reflective optics can be used to image the diffracted beams onto the crystal. 
     Other aspects, advantages, and modifications are within the scope of the following claims.