Patent Publication Number: US-2022219259-A1

Title: Spectrally broadening ultrashort-pulse compressor

Description:
TECHNICAL FIELD OF THE INVENTION 
     The present invention relates in general to compression of the duration of ultrashort laser pulses, in particular to techniques for achieving such compression based on spectral broadening in a solid-state medium and with minimal degradation of the laser beam parameters. 
     DISCUSSION OF BACKGROUND ART 
     Laser pulses are considered ultrashort when their duration is less than a few picoseconds. Ultrashort laser pulses may attain very high peak powers. Ultrashort laser pulses are commonly used in scientific, industrial, and medical applications. Scientific applications include time-resolved studies of molecular dynamics and chemical reactions. In industrial applications, the ultrashort laser pulses are typically used for cutting, drilling, marking, or other machining that benefits from the unique material interaction properties of ultrashort laser pulses. Since the pulse duration is orders of magnitude shorter than the characteristic timescales for thermal diffusion in the material, the heat-affected zone does not extend far beyond the laser beam, and ultrashort laser pulses are therefore capable of processing material in a very precise fashion. One of the best-known medical applications of ultrashort laser pulses is in LASIK eye surgery, where femtosecond laser pulses may be used as the cutting-tool to make incisions in the cornea. 
     Many of these applications benefit from the pulse duration being as short as possible. Ultrashort laser pulses are typically generated in a resonator having a gain medium and operating in a mode-locked condition. Outside the resonator, and possibly after further amplification, the pulse duration may be reduced in a pulse compressor. However, the minimum obtainable pulse duration is limited by the time-bandwidth product and therefore by the spectral bandwidth of the laser pulses. The pulse duration is fundamentally limited by the amplification bandwidth of the gain medium used to generate and/or amplify the ultrashort laser pulses. However, gain medium properties also affect other important parameters, such as the average power of the generated and amplified beam of ultrashort laser pulses. It is often not possible to find a gain medium that has the amplification bandwidth sufficient to achieve a certain pulse duration while also satisfying other important requirements. 
     In some situations, the spectral bandwidth of the ultrashort laser pulses is insufficient to achieve a desired pulse duration, even if the pulse is compressed to the limit defined by the minimum time-bandwidth product. In these situations, a spectral broadening system may be invoked to deliberately broaden the spectral bandwidth of the pulse, whereafter the pulse can be compressed to a shorter duration. The most widely used method for such compression of an ultrashort laser pulse entails first broadening the spectrum of the pulse, which generates a chirped pulse characterized by a spectral gradient as a function of time, and then compressing the chirped pulse by dechirping the spectral gradient with a dispersive optic to temporally overlap all spectral components of the pulse. The spectral-broadening step takes place in a nonlinear medium and is based on self-phase modulation of the ultrashort laser pulse induced by the Kerr effect. 
     Optical fibers are a common choice for the nonlinear medium. However, in the case of high-intensity ultrashort laser pulses, the peak power may exceed the critical power in solid-state media for run-away self-focusing, caused by the Kerr effect, with detrimental effect on the laser beam profile and potential damage of the solid-state medium. Therefore, the more common choice for compression of high-intensity ultrashort laser pulses is a long gas-filled hollow-core optical fiber, where the laser beam propagates primarily in the gas and the gas serves as the nonlinear medium. 
     Solid-state solutions have been proposed. In one such scheme, disclosed by Kung et al. in U.S. Pat. No. 9,971,229, the laser beam is directed through a series of thin plates. Each plate imparts some degree of spectral broadening through Kerr-effect-induced self-phase modulation. Each plate is sufficiently thin that the beam waist produced by the related Kerr lens is after that the laser beam has exited the plate. The plates are distanced from each other to allow the laser beam to come to a focus and expand again before reaching the next plate. 
     SUMMARY OF THE INVENTION 
     Disclosed herein are systems and methods for compression of ultrashort laser pulses based on deliberate spectral broadening in spectrally broadening, bulk-optics. The spectrally broadening bulk-optics used herein are solid-state bulk media. The use of solid-state bulk media, as opposed to a gas-filled optical fiber or other gas-based solutions, eliminates the need for a gas supply and related hardware and offers a more compact solution. The ultrashort pulse is alternatingly (a) spectrally broadened in a broadening bulk-optic (or a set of broadening bulk-optics) while developing a chirp and (b) temporally compressed by a dechirping dispersive optical element. As the ultrashort pulse progresses through this series of alternating spectral broadening and temporally compressing dechirping, additional focusing optics control the laser beam size such that the spot size is greater at each successive repetition of spectral broadening. This scheme may be realized in a multipass configuration, wherein the laser beam makes multiple passes through the same broadening bulk-optic or the same set of broadening bulk-optics. Alternatively, the scheme may be realized in an unfolded configuration, wherein the laser beam passes through a series of broadening bulk-optics (or sets of broadening bulk-optics), with a dispersive optical element after each broadening bulk-optic (or set of broadening bulk-optics) and with focusing optics interspersed in the series. 
     Generally speaking, perfect temporal compression of a spectrally broadened ultrashort pulse requires a dispersive optic with a group delay dispersion that perfectly matches the chirp introduced by spectral broadening. The chirp introduced in each pass through the broadening bulk-optic(s) may be mostly linear, however with smaller nonlinear components as well. We have realized that, if the ultrashort pulse is allowed to make multiple passes through the broadening bulk-optic(s) before being dechirped, these nonlinear components tend to increase in amplitude and become increasingly nonlinear, ultimately resulting in a chirp that is too complex for even approximate dechirping. Significant dechirping imperfection amounts to unsatisfactory temporal compression. We have found that the present scheme of dechirping the ultrashort pulse after every pass through the broadening bulk-optic(s) helps minimize any compounded nonlinear effect of repeatedly applied spectral broadening. In this scheme, good pulse compression may be achieved with dispersive optics characterized by a linear group delay dispersion. 
     As the duration of the ultrashort pulse is shortened by each repetition of the spectral broadening and temporal compression steps, the peak power of the ultrashort pulse increases. The increase in spot size for each successive repetition of the spectral broadening step serves to achieve substantial spectral broadening in each pass while keeping the peak intensity below the damage threshold of the broadening bulk-optic(s) and keeping the B-integral (the on-axis nonlinear phase shift incurred through self-phase modulation in the broadening bulk-optic(s)) below the limit for run-away self-focusing and related undesirable outcomes. These undesirable outcomes may include a substantial change in the output beam parameters, filamentation formation causing degradation of the beam parameters, and failure to satisfactorily compress the pulse. In some implementations, the spot size increase is designed to keep the peak intensity below the damage threshold, and keep the B-integral below a limit of about 2.0 radians (rad) to avoid self-focusing and other related undesirable outcomes. The spot size increase may also be tuned to maintain a similar B-integral for each pass through the broadening bulk-optic(s), so as to maximize the spectral broadening achieved in each pass while avoiding self-focusing and other related undesirable outcomes. 
     In operation, the present scheme is likely to experience variation in the input beam properties, for example variation in pulse energy and other parameters that affect the power of the Kerr lens in the broadening bulk-optic(s). To reduce the sensitivity of the beam size progression through the system to such variation, the present scheme may be implemented in a near-imaging-condition configuration, where the propagation distance between subsequent passes through the broadening bulk-optic(s) is only slightly detuned from the propagation distance required to reimage the beam within the broadening bulk-optic(s) from pass to pass. The present scheme may further utilize adjustment of the thickness of the broadening bulk-optic(s) so as to stabilize the B-integral in the presence of changes to the input pulse energy or power. 
     In one aspect, an ultrashort-pulse compressor includes one or more broadening bulk-optics arranged to intersect a propagation path of an ultrashort-pulsed laser beam multiple times so as to spectrally broaden a pulse of the laser beam during each of multiple passes through the one or more broadening bulk-optics. The compressor also includes one or more dispersive optics for compressing a duration of the pulse after each of the multiple passes. In addition, the compressor includes a plurality of focusing elements for focusing the laser beam between the multiple passes. The focusing elements are arranged around the one or more broadening bulk-optics. Distances between the one or more broadening bulk-optics and the focusing elements along the propagation path are detuned from imaging such that a spot size of the laser beam, at the one or more broadening bulk-optics, is greater at each successive one of the multiple passes. 
     In another aspect, a method for compressing an ultrashort pulse of a laser beam includes repeating a group of steps of (a) spectrally broadening and chirping the pulse in one or more broadening bulk-optics, (b) dechirping the pulse, after the spectrally broadening step, to compress duration of the pulse, and (c) focusing the laser beam to set a spot size of the laser beam on the one or more broadening bulk-optics in the spectrally broadening step. Propagation distances of the laser beam between the spectrally broadening step of each successive repetition of the group of steps are detuned from imaging such that the spot size is greater for each successive repetition of the group of steps. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of the specification, schematically illustrate preferred embodiments of the present invention, and together with the general description given above and the detailed description of the preferred embodiments given below, serve to explain principles of the present invention. 
         FIG. 1  illustrates a method for compressing an ultrashort pulse of a laser beam, according to an embodiment. The method alternates between steps of spectrally broadening the pulse in one or more broadening bulk-optics and steps of dechirping and temporally compressing the pulse. The method manipulates the size of the laser beam such that the spot size of the ultrashort pulse at the broadening bulk-optic(s) increases for every repetition of the spectrally broadening step. 
         FIG. 2  illustrates a multipass ultrashort-pulse compressor configured to perform the method of  FIG. 1 , with each repetition of the spectrally broadening step utilizing the same broadening bulk-optic or the same set of broadening bulk-optics, according to an embodiment. 
         FIGS. 3A and 3B  are cross-sectional side-views that illustrate the transverse profile of a laser beam (shaded) when propagating through one example of a near-imaging-condition embodiment of the compressor of  FIG. 2  configured with a positive detuning parameter. 
         FIG. 4  shows the equivalent cross sections of the laser beam that are imaged onto successive passes through the broadening bulk-optic(s) of an example of a near-imaging-condition embodiment of the compressor of  FIG. 2  configured with a negative detuning parameter. 
         FIGS. 5A-F  illustrate simulated properties of an alternative compressor configured at the imaging condition, corresponding to the detuning parameter being zero. 
         FIGS. 6A-E  illustrate simulated properties of a near-imaging-condition embodiment of the compressor of  FIG. 2  characterized by a non-zero detuning parameter. 
         FIGS. 7A-D  illustrate simulated properties of another near-imaging-condition embodiment of the compressor of  FIG. 2  characterized by a non-zero detuning parameter. 
         FIGS. 8A and 8B  illustrate a broadening bulk-optic pair including two wedge-shaped broadening bulk-optics having a combined thickness that is adjustable by changing positions of the wedge-shaped broadening bulk-optics relative to each other, according to an embodiment. 
         FIGS. 9A and 9B  provide an example that demonstrate adjustment of broadening bulk-optic thickness to compensate for a pulse energy change. 
         FIG. 10  illustrates a multipass ultrashort-pulse compressor configured to perform the method of  FIG. 1 , with the last dechirping and temporal compression step performed by a separate dispersive optic, according to an embodiment. 
         FIG. 11  illustrates a multipass ultrashort-pulse compressor configured to perform the method of  FIG. 1  in a ring-scheme, according to an embodiment. 
         FIG. 12  illustrates an ultrashort-pulse compressor configured to perform the method of  FIG. 1 , with each repetition of the spectral broadening step being performed by a separate respective broadening bulk-optic, according to an embodiment. 
         FIG. 13  is a flowchart of a method for compressing an ultrashort pulse of a laser beam, according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring now to the drawings, wherein like components are designated by like numerals,  FIG. 1  illustrates one method  100  for compressing an ultrashort pulse of a laser beam. Method  100  alternates between steps  110  of spectrally broadening the pulse and steps  120  of temporally compressing the pulse. Each instance of step  110  is followed by a corresponding instance of step  120 . Step  110  takes place in a broadening bulk-optic, or a set of broadening bulk-optics, by virtue of self-phase modulation. Step  110  introduces a spectral chirp where the instantaneous frequency of the pulse changes over time. For example, the instantaneous frequency of the pulse may increase from its leading end to its trailing end. Step  120  utilizes one or more dispersive optical elements that dechirp the pulse to compress its duration. In one example, step  120  uses a chirped mirror. In another example, step  120  uses a set of prisms. Each pair of steps  110  and  120  cooperate to shorten the pulse duration. 
     Method  100  receives an ultrashort laser pulse  180  having a pulse duration τ 0  (e.g., a FWHM duration). A first instance of step  110  spectrally broadens laser pulse  180  and thereby generates a chirped laser pulse  182 ( 1 ), whereafter a first instance of step  120  dechirps chirped laser ( 1 ) to produce a temporally compressed laser pulse  184 ( 1 ) having a shorter pulse duration τ 1 . The dechirping in step  120  may fully or only partly compensate for the chirp introduced in step  110 . Next, a second repetition of steps  110  and  120  cooperate to form a temporally compressed laser pulse  184 ( 2 ) having an even shorter pulse duration τ 2 . In the example depicted in  FIG. 1 , method  100  performs at least three repetitions of steps  110  and  120 . More generally, method  100  may perform two or more repetitions of steps  110  and  120 . 
     The chirp introduced in step  110  is likely to have some degree of nonlinearity. As compared to an alternative method performing dechirping only after the last one of several passes through the broadening bulk-optic(s), the alternation between steps  110  and  120  in method  100  helps minimize any compounded nonlinearity induced by repeatedly applied spectral broadening. In one embodiment, the dispersive optical element(s), utilized by step  120 , are characterized by a linear group delay dispersion. At least in part by virtue of the alternation between steps  110  and  120 , this embodiment of method  100  may achieve good temporal compression even when the chirp introduced in step  110  exhibits nonlinearity and/or when the dechirping in step  120  is not perfect. 
     Method  100  further manipulates the size of the laser beam such that the spot size  180 S of the ultrashort pulse at the broadening bulk-optic(s) increases for every repetition of step  110 . As illustrated in  FIG. 1 , the spot size  180 S at each non-first repetition of step  110  is greater than the spot size  180 S at the preceding iteration of step  110 . As discussed above, this increasing spot size  180 S enables method  100  to achieve substantial spectral broadening in each iteration of step  110  while keeping the peak intensity below the damage threshold of the broadening bulk-optic(s) and keeping the B-integral below the limit for run-away self-focusing and other, related undesirable outcomes. In one embodiment, the beam-size manipulation of method  100  keeps the B-integral below approximately 2.0 radians (rad) to avoid these undesirable outcomes. We have found that, when the B-integral is kept below 2.0 rad, the final compressed laser pulses, generated by compressor  200 , may have a beam quality M 2  of less than 2.0. Method  100  may maintain a similar B-integral for each repetition of step  110 , so as to maximize the spectral broadening achieved in each repetition of step  110  while staying below the damage threshold and avoiding self-focusing and other related undesirable outcomes. For example, method  100  may keep the multiple B-integrals, respectively associated with the multiple repetitions of step  110 , within ±30% of the average B-integral thereof. 
       FIG. 2  illustrates one multipass ultrashort-pulse compressor  200  configured to perform method  100 , with each repetition of step  110  utilizing the same broadening bulk-optic or the same set of broadening bulk-optics. Compressor  200  includes a broadening bulk-optic  210  (or, alternatively, a set of broadening bulk-optics  210 ), two chirped mirrors  220 , and two concave mirrors  230  with positive optical power. Chirped mirrors  220  are end mirrors of a multipass arrangement that is folded by concave mirrors  230  and contains broadening bulk-optic(s)  210 . Compressor  200  receives a laser beam  290  of ultrashort pulses. Laser beam  290  then passes through broadening bulk-optic(s)  210  several times while traveling between chirped mirrors  220 . Concave mirror  230 ( 1 ) folds the propagation path in a first leg  280 ( 1 ) of the multipass arrangement between broadening bulk-optic(s)  210  and chirped mirror  220 ( 1 ), and concave mirror  230 ( 2 ) folds the propagation path in a second leg  280 ( 2 ) of the multipass arrangement between broadening bulk-optic(s)  210  and chirped mirror  220 ( 2 ). Chirped mirrors  220  have no or only insignificant optical power. In one embodiment, chirped mirrors  220  are planar mirrors. Chirped mirrors  220 ( 1 ) and  220 ( 2 ) may be a matched pair with mutually cancelling group-delay-dispersion ripples, at least to a good approximation, so as to optimize the dechirping and resulting temporal compression. Each chirped mirror  220  may be characterized by a linear group delay dispersion. 
     Broadening bulk-optic(s)  210  is (are), for example, made of sapphire, fused silica, yttrium-aluminum garnet (YAG), or calcium fluoride (CaF 2 ). The wavelength of laser beam  290  may be between 700 and 1100 nanometers (nm). In one scenario, laser beam  290 , as first received by compressor  200 , is generated using a ytterbium doped gain medium (e.g., using a Yb:YAG laser) and has a center wavelength of about 1030 nm. In this scenario, laser beam  290  may be characterized by a pulse energy in the range between 10 microJoules (μJ) and a few milliJoules (mJ) and a pulse duration in the range between 500 and 1000 femtoseconds (fs). In another scenario, laser beam  290 , as first received by compressor  200 , is generated using a titanium-sapphire gain medium and has a center wavelength of about 800 nm. In this scenario, laser beam  290  may be characterized by a pulse energy in the range between 0.1 and 1 mJ and a pulse duration of less than 100 fs, for example between 5 and 50 fs. Laser beam  290  may be linearly or circularly polarized. 
     In the embodiment depicted in  FIG. 2 , compressor  200  includes mirrors  250  and  252 . Mirror  250  couples laser beam  290  into the multipass arrangement by directing laser beam  290  onto a propagation path that makes the several passes through broadening bulk-optic(s)  210  before being intercepted and ejected from the multipass arrangement by mirror  252 . The placement of mirrors  250  and  252  may differ from that shown in  FIG. 2 , for example to accommodate an even number of passes through broadening bulk-optic(s)  210 . As will be understood by a person of ordinary skill in the art, other schemes for coupling laser beam  290  into the multipass arrangement and ejecting laser beam  290  from the multipass arrangement are possible. For example, laser beam  290  may enter or leave the multipass arrangement via an opening in one of concave mirrors  230 , or laser beam  290  may enter or leave the multipass arrangement along a propagation path that passes by one of concave mirrors  230  with no need for deflection by an additional beam steering mirror. 
     Each pass of laser beam  290  through broadening bulk-optic(s)  210  corresponds to a respective repetition of spectral-broadening step  110  of method  100 . After each pass through broadening bulk-optic(s)  210 , one of chirped mirrors  220  reflects laser beam  290 . Each such reflection corresponds to a respective repetition of dechirping and temporal-compression step  120  of method  100 . Whereas the same broadening bulk-optic  210  (or the same set of broadening bulk-optic(s)  210 ) performs each repetition of step  110 , chirped mirrors  220 ( 1 ) and  220 ( 2 ) alternately perform step  120 . 
     Between each pair of subsequent passes through broadening bulk-optic(s)  210 , one of concave mirrors  230  focuses laser beam  290  to increase its spot size at the next pass through broadening bulk-optic(s)  210 , until laser beam  290  is ejected from the multipass arrangement. In each leg  280 , the optical power of concave mirror  230  cooperates with distance  270 , from concave mirror  230  to broadening bulk-optic(s)  210 , and distance  272 , from concave mirror  230  to an associated chirped mirror  220 , to cause the spot size of laser beam  290  at broadening bulk-optic(s)  210  to be greater for each successive pass through broadening bulk-optic(s)  210 . Herein, each of distances  270  and  272  is measured along the propagation path of laser beam  290 . The segments of the propagation path, associated with a distance  270  or a distance  272 , may be straight (as depicted) or have folds not shown in  FIG. 2 . 
     While  FIG. 2  depicts an embodiment configured for three passes  212 ( 1 - 3 ) of laser beam  290  through broadening bulk-optic(s)  210 , compressor  200  may be configured for a different number of passes (two, four, or more) through broadening bulk-optic(s)  210 , without departing from the scope hereof. The optimum number of passes depends on several factors. For example, a certain compression factor may, at least in principle, be achieved with relatively low B-integrals when the number of passes is relatively high. On the other hand, each pass is likely associated with some loss (e.g., 1-5%), and each additional pass adds alignment complexity. It may therefore, at least in some scenarios, be favorable to keep the number of passes to no more than 15, 10, or even 5. 
     Compressor  200  may be configured to receive laser beam  290  as a collimated laser beam. For this purpose, compressor  200  may include a lens  240  (or a collection of lenses, for example including a telescope) that focuses laser beam  290  onto broadening bulk-optic(s)  210 . Similarly, compressor  200  may include a lens  242  (or a collection of lenses) that collimates laser beam  290  after ejection from the multipass arrangement. In addition, laser beam  290  does not need to follow the exact propagation path drawn in  FIG. 2 . For example, passes  212  may be offset from each other in the dimension orthogonal to the plane of  FIG. 2  rather than within the plane of  FIG. 2 . 
     Compressor  200  may be integrated in a laser apparatus that further includes an ultrashort-pulse laser source  260  generating laser beam  290 . 
     In one embodiment of compressor  200 , referred to as the near-imaging-condition embodiment, the propagation path length between successive passes through broadening bulk-optic(s)  210  is 2(2f+x) in each leg  280 . Here, f is the focal length of the instance of concave mirror  230  located in the respective leg  280 , and x is a non-zero detuning parameter that may be negative or positive. The absolute value of x is less than 0.1f. A propagation path length of 4f (corresponding to x=0) would correspond to concave mirror  230  imaging the spot size of laser beam  290  at one pass through broadening bulk-optic(s)  210  onto the next pass through broadening bulk-optic(s)  210 , such that the spot size remains the same for sequential passes. In the near-imaging-condition embodiment, the propagation path length between successive passes through broadening bulk-optic(s)  210  is slightly detuned from 4f such that the spot size at broadening bulk-optic(s)  210  increases as a function of pass number, as indicated in  FIG. 1 . 
     The propagation path length between successive passes through broadening bulk-optic(s)  210  alternates between (a) approximately twice the sum of distances  270 ( 1 ) and  272 ( 1 ) in leg  280 ( 1 ) and (b) approximately twice the sum of distances  270 ( 2 ) and  272 ( 2 ) in leg  280 ( 2 ). In a symmetric implementation of the near-imaging-condition embodiment, concave mirrors  230 ( 1 ) and  230 ( 2 ) have the same focal length f, distances  270 ( 1 ) and  270 ( 2 ) are identical, and distances  272 ( 1 ) and  272 ( 2 ) are identical. In one asymmetric implementation, concave mirrors  230 ( 1 ) and  230 ( 2 ) have the same focal length f, but the detuning parameter x in leg  280 ( 1 ) is different from the detuning parameter x in leg  280 ( 2 ) due to a difference between distances  270 ( 1 ) and  270 ( 2 ) and/or a difference between distances  272 ( 1 ) and  272 ( 2 ). In another asymmetric implementation, concave mirrors  230 ( 1 ) and  230 ( 2 ) have different focal lengths f 1  and f 2 . In this asymmetric implementation, the ratio of the detuning parameter to the focal length may be the same in legs  280 ( 1 ) and  280 ( 2 ) or differ between legs  280 ( 1 ) and  280 ( 2 ). The performance of these symmetric and asymmetric implementations is similar, in terms of pulse compression. However, other considerations, such as spatial constraints, may deem one of these implementations advantageous over the others. 
     In one example of the near-imaging-condition embodiment, the detuning is implemented between broadening bulk-optic(s)  210  and concave mirrors  230 . In this example, distance  270  is f+x, and distance  272  is f, in each leg  280 . This configuration results in concave mirror  230  and chirped mirror  220 , in each leg  280 , forming an imaging system with unity magnification, which may simplify the process of designing compressor  200  as compared to when all or part of the detuning is implemented in distance  272 . As discussed above, the two legs  280  may be configured with different values of f and/or x. 
       FIGS. 3A and 3B  illustrate in cross section the transverse profile  300  of laser beam  290  (shaded) when propagating through one example of the near-imaging-condition embodiment of compressor  200  with a positive detuning parameter x. In this example, legs  280 ( 1 ) and  280 ( 2 ) are identical, and the detuning is implemented in the segments between broadening bulk-optic(s)  210  and concave mirrors  230 , such that distances  270 ( 1 ) and  270 ( 2 ) are f+x and distances  272 ( 1 ) and  272 ( 2 ) are f.  FIG. 3A  shows transverse profile  300  of laser beam  290  from entry into compressor  200  through three passes through broadening bulk-optic(s)  210 .  FIG. 3A  schematically indicates the locations, along the propagation path of laser beam  290 , of broadening bulk-optic(s)  210 , chirped mirrors  220 , concave mirrors  230 , and a telescope  350 . Telescope  350  is an embodiment of lens  240 .  FIG. 3B  shows the equivalent cross sections of laser beam  290  that concave mirrors  230  image onto successive passes through broadening bulk-optic(s)  210 . For simplicity,  FIGS. 3A and 3B  do not take into consideration potential focusing imparted by Kerr-effect lensing in broadening bulk-optic(s)  210 . Each of  FIGS. 3A and 3B  indicates a longitudinal axis  394  of laser beam  290 . In each of  FIGS. 3A and 3B , transverse profile  300  may correspond to the 1/e 2  diameter of laser beam  290 . Detuning parameter x and the resulting increase in spot size  380  for successive passes through broadening bulk-optic(s)  210 , indicated in  FIG. 3A , are exaggerated for clarity. 
     Telescope  350  demagnifies laser beam  290  in broadening bulk-optic(s)  210  for a first pass with a spot size  380 ( 1 ). Laser beam  290  forms a waist at this first pass through broadening bulk-optic(s)  210 . After the first pass through broadening bulk-optic(s)  210 , laser beam  290  propagates a distance f+x to concave mirror  230 ( 1 ). Since the distance f+x exceeds the focal length f, the two reflections by concave mirror  230 ( 1 ) causes laser beam  290  to form a waist before the second pass through broadening bulk-optic(s)  210 . This waist is an image of the original waist of laser beam  290  in the first pass through broadening bulk-optic(s)  210 , and therefore has spot size  380 ( 1 ). (Chirped mirror  220  does not affect the focusing properties of laser beam  290 .) Laser beam  290  then diverges before reaching broadening bulk-optic(s)  210  again for a second pass. Consequently, the spot size  380 ( 2 ) of laser beam  290  at the second pass through broadening bulk-optic(s)  210  is greater than spot size  380 ( 1 ). Concave mirror  230 ( 2 ) repeats this behavior for the third pass through broadening bulk-optic(s)  210 , such that the spot size  380 ( 3 ) at the third pass exceeds spot size  380 ( 2 ). In embodiments of compressor  200  configured for more than three passes through broadening bulk-optic(s)  210 , the increase in spot size  380  at broadening bulk-optic(s)  210  continues for each additional pass. 
       FIG. 3B  shows caustic  392  of laser beam  290  at the first pass through broadening bulk-optic(s)  210 , as defined by telescope  350 . The shaded part of caustic  392  indicates the portion of caustic  392  that matches transverse profile  300 , and dashed lines indicate the extrapolation of caustic  392  into regions where other optical elements of the compressor cause transverse profile  300  to deviate from caustic  392 . Caustic  392  has a waist with spot size  380 ( 1 ). The subsequent two reflections by concave mirror  230 ( 1 ) image a cross section of caustic  392 , a distance 2x downstream from the waist, onto the second pass through broadening bulk-optic(s)  210 . This cross section has spot size  380 ( 2 ). Next, the two reflections by concave mirror  230 ( 2 ) image a cross section of caustic  392 , a distance 4x downstream from the waist, onto the third pass through broadening bulk-optic(s)  210 . This cross section has spot size  380 ( 3 ). 
       FIG. 4  shows the equivalent cross sections of laser beam  290  that concave mirrors  230  image onto successive passes through broadening bulk-optic(s)  210 , when detuning parameter x is negative. In this case, distance f+x is less than the focal length f of concave mirrors  230 . Therefore, the cross sections of laser beam  290 , imaged onto broadening bulk-optic(s)  210  by concave mirrors  230 , are upstream of the waist of caustic  392 . Specifically, the second pass through broadening bulk-optic(s)  210  corresponds to a cross section of caustic  392  that is a distance 2x upstream from the waist and has spot size  380 ( 2 ), and the third pass through broadening bulk-optic(s)  210  corresponds to a cross section that is a distance 4x upstream from the waist and has spot size  380 ( 3 ). 
     In a modification of compressor  200 , the pair of concave mirrors  230  is replaced by a pair of lenses with the same focusing power as concave mirrors  230  and located in the same places as concave mirrors  230 . In this modification, the multipass arrangement is linear rather than folded. The folded arrangement is, however, more compact, which may be beneficial in many practical implementations subject to spatial constraints. The replacement of concave mirrors  230  with lenses does not alter the transverse profile of laser beam  290  as it propagates through the compressor. 
       FIGS. 5A-F  illustrate simulated properties of an alternative compressor configured at the imaging condition, corresponding to detuning parameter x being zero. In contrast to compressor  200 , the compressor addressed by  FIGS. 5A-F  is, by virtue of zero detuning, configured to maintain the same spot size at each pass through broadening bulk-optic(s)  210 . Otherwise, this compressor is similar to compressor  200 . 
       FIG. 5A  shows the transverse profile  500  of laser beam  290  from entry into the compressor through six passes through broadening bulk-optic(s)  210 , disregarding any Kerr-effect lensing in broadening bulk-optic(s)  210 .  FIG. 5B  shows the transverse profile  502  of laser beam  290  from entry into the compressor through six passes through broadening bulk-optic(s)  210 , with Kerr lensing in broadening bulk-optic(s)  210  taken into account. Each of  FIGS. 5A and 5B  schematically indicates the positions of broadening bulk-optic(s)  210 , chirped mirrors  220 , and concave mirrors  230 . 
       FIGS. 5C-F  pertain to the case where Kerr lensing is taken into account, and plot respective parameters for five successive passes through broadening bulk-optic(s)  210 .  FIG. 5C  plots the 1/e 2  beam radius at broadening bulk-optic(s)  210  in terms of its deviation δw NLC  from the beam radius at the first pass through broadening bulk-optic(s)  210 .  FIG. 5D  plots the B-integral of the beam in broadening bulk-optic(s)  210 .  FIG. 5E  plots the focal length f Kerr  of the Kerr lens induced in broadening bulk-optic(s)  210 .  FIG. 5F  plots the pulse duration Δt after dechirping and compression by chirped mirrors  220 . 
       FIGS. 5A-F  are all based on an example where the focal length f of concave mirrors  230  is 150 millimeters (mm), each distance  272  is 150 mm, the pulse energy of laser beam  290  is 2 mJ, the input pulse duration of laser beam  290  is 300 femtoseconds, the wavelength of laser beam  290  is 1030 nm, broadening bulk-optic(s)  210  have a nonlinear refractive index of 3×10 −16  centimeters squared per watt (cm 2 /W) and a total thickness at each pass of 0.9 mm, and the pulse shortening factor per pass (after dechirping) is assumed to be 0.8. The pulse shortening factor is defined as the ratio of (a) the pulse duration after the pass (and after subsequent dechirping) to (b) the pulse duration before the pass. 
     Referring first to  FIG. 5A , a pair of telescope lenses  550  and  552  form a waist in laser beam  290  where laser beam  290  is incident on broadening bulk-optic(s)  210  for the first pass therethrough. Laser beam  290  makes its first pass through broadening bulk-optic(s)  210 , propagates a distance f to concave mirror  230 ( 1 ), a distance f to chirped mirror  220 ( 1 ), a distance f back to concave mirror  230 ( 1 ), and then a distance f to broadening bulk-optic(s)  210 . These propagation distances, a total of 4f, match the imaging condition. Laser beam  290  therefore forms a waist again where laser beam  290  is incident on broadening bulk-optic(s)  210  for the second pass therethrough, with the waist size at the second pass being the same as at the first pass. This evolution of transverse profile  500  repeats between each pair of successive passes through broadening bulk-optic(s)  210 , such that the size of laser beam  290  is the same each time laser beam  290  encounters broadening bulk-optic(s)  210 . 
     Referring now to  FIG. 5B , Kerr lensing in broadening bulk-optic(s)  210  affects the evolution of transverse profile  502 . As compared to transverse profile  500 , the Kerr lens focuses laser beam  290 , such that laser beam  290  converges rather than diverges when leaving broadening bulk-optic(s)  210 . Between each pair of successive passes through broadening bulk-optic(s)  210 , laser beam  290  forms a waist after the preceding pass through broadening bulk-optic(s)  210  and attains a largest size at the second encounter of concave mirror  230  before the next pass through broadening bulk-optic(s)  210 . As laser beam  290  propagates through the compressor, the waist size decreases and the largest size increases. However, because concave mirrors  230  are arranged at the imaging condition, concave mirrors  230  still image the same beam size onto broadening bulk-optic(s)  210  at each pass, despite the effect of Kerr lensing. This is evidenced by  FIG. 5C , which shows that the spot size at broadening bulk-optic(s)  210  is the same for each pass. Because the spot size at broadening bulk-optic(s)  210  remains the same for each pass, the B-integral increases (see  FIG. 5D ) and the Kerr focal length decreases (see  FIG. 5E ) for successive passes. The B-integral grows from approximately 1.1 rad at the first pass to more than 2.0 rad already at the fourth pass, and this growth accelerates for successive passes. As the B-integral increases beyond about 2.0 rad, undesirable effects, such as run-away self-focusing, filamentation, and damage to the optical elements, tend to occur. Additionally, as the pulse duration decreases (see  FIG. 5F ), the peak intensity may approach or exceed the damage threshold. 
       FIGS. 6A-E  illustrate simulated properties of one near-imaging-condition embodiment of compressor  200 , characterized by a non-zero detuning parameter x.  FIGS. 6A-E  demonstrate the advantage of the spot size increase that takes place when operating slightly detuned from the imaging condition.  FIGS. 6A-E  pertain to an example of compressor  200  that is identical to the compressor of  FIGS. 5A-F  except for being configured with a detuning parameter x of −40 mm, corresponding to a relative detuning |x|/f of 0.067. 
       FIG. 6A  is a plot similar to that of  FIG. 5B  and shows the transverse profile  600  of laser beam  290  from entry into compressor  200  through six passes through broadening bulk-optic(s)  210 , with Kerr-effect lensing in broadening bulk-optic(s)  210  taken into account.  FIGS. 6B-E  plot respective parameters for five successive passes through broadening bulk-optic(s)  210 .  FIG. 6B  plots the 1/e 2  beam radius w NLC  at broadening bulk-optic(s)  210 .  FIG. 6C  plots the B-integral.  FIG. 6D  plots the focal length f Kerr  of the Kerr lens induced in broadening bulk-optic(s)  210 .  FIG. 6E  plots the pulse duration Δt after dechirping and compression by chirped mirrors  220 . 
     Transverse profile  600  may at a first glance appear quite similar to transverse profile  502  of  FIG. 5A . However, by virtue of the detuning from imaging condition, the spot size at broadening bulk-optic(s)  210  increases for each pass (see  FIG. 6B ). As a consequence, the B-integral has a maximum value of less than 1.3 and actually decreases after the third pass. Related hereto, the Kerr focal length increases after the second pass. A B-integral of less than 1.3 is far less likely to lead to the undesirable effects mentioned above, such as run-away self-focusing, filamentation, and damage to the optical elements. The average B-integral, over the five passes, is approximately 1.0, and each of the five B-integrals is within approximately 20% of this average value. In addition, the increase in the spot size (see  FIG. 6B ) may prevent the peak intensity from approaching or exceeding the damage threshold when the pulse duration decreases (see  FIG. 6E ). 
       FIGS. 6A-E  pertain to just one example of a near-imaging-condition embodiment of compressor  200 . Similarly advantageous properties, or even more advantageous properties, may be achieved with other parameters. For example, the thickness of broadening bulk-optic(s)  210  may be decreased to operate with lower B-integrals. Additionally, the number of passes may be decreased or increased as needed to achieve a certain overall pulse compression. 
       FIGS. 7A-D  illustrate simulated properties of another near-imaging-condition embodiment of compressor  200 , characterized by a non-zero detuning parameter x. The  FIGS. 7A-D  embodiment is identical to the  FIGS. 6A-E  embodiment except that the thickness of broadening bulk-optic(s)  210  is reduced from 0.9 mm to 0.5 mm.  FIGS. 7A-D  plot respective parameters for ten successive passes through broadening bulk-optic(s)  210 .  FIG. 7A  plots the 1/e 2  beam radius w NLC  at broadening bulk-optic(s)  210 .  FIG. 7B  plots the B-integral.  FIG. 7C  plots the focal length f Kerr  of the Kerr lens induced in broadening bulk-optic(s)  210 .  FIG. 7D  plots the pulse duration Δt after dechirping and compression by chirped mirrors  220 . 
     The reduction in the thickness of broadening bulk-optic(s)  210  at least initially results in a reduction of the B-integral and an increase of the Kerr focal length, because both the B-integral and the power of the Kerr lens are proportional to the thickness of broadening bulk-optic(s)  210 . However, the Kerr focal length associated with one pass through broadening bulk-optic(s)  210  affects the spot size at the next pass, and the overall impact of the thickness reduction is therefore not a simple scaling. As seen by comparing  FIG. 7A  to  FIG. 6B , the growth of the spot size at broadening bulk-optic(s)  210 , as a function of pass number, is slower when the thickness is reduced. In the  FIGS. 7A-D  embodiment, the Kerr focal length takes on a minimum value of approximately 1300 mm at the third pass and then increases for each subsequent one of the ten passes. The B-integral is approximately 0.6 rad at the first pass through broadening bulk-optic(s)  210 , increases to almost 1.0 rad at the fifth pass, whereafter the increase slows to keep the B-integral to less than 1.1 rad for the remaining five passes. The B-integral averages approximately 0.92 rad, and each of the ten B-integrals are within 25% of the average value. 
     The results of  FIGS. 7A-D  also serve as an illustration of the impact of a decrease in pulse energy. The B-integral is, at least to a good approximation, proportional to both the thickness of broadening bulk-optic(s)  210  and the peak intensity of laser beam  290 . The power of the Kerr lens is proportional to both the thickness of broadening bulk-optic(s)  210  and the peak power of laser beam  290 . Therefore, a decrease in pulse energy by a certain factor has at least approximately the same impact as a decrease in thickness by the same factor. 
     Addressing now specifically the spot size increase as a function of pass number, the results shown in  FIGS. 6A-E  and  FIGS. 7A-D  demonstrate, by example, that concave mirrors  230  and the relatively small detuning parameter x reduce the sensitivity of near-imaging-condition embodiments of compressor  200  to changes in pulse energy.  FIG. 6B  shows an increase in spot size by about 5% from the first to the second pass, and an increase by about 50% from the first to the fifth pass.  FIG. 7A  shows an increase in spot size by about 2.5% from the first to the second pass, and an increase by about 20% from the first to the fifth pass. In both cases, the spot size increase is steady but not dramatic. Additionally, the difference in the spot size growth rate, as a function of pass, between  FIG. 6B  and  FIG. 7A  is noticeable but not drastic, despite the thickness reduction by almost 50% being equivalent to a decrease in pulse energy by almost 50%. In contrast, consider a multipass compressor that does not include concave mirrors  230  but instead relies solely on the Kerr lens induced in broadening bulk-optic(s)  210  to maintain a relatively similar spot size for each pass (with the laser beam forming a waist between each pass). In this non-imaging compressor, a decrease in Kerr focal length by about 50%, due to a pulse energy change, would result in the spot size approximately tripling between the first pass and the second pass rather than staying relatively similar. Thus, in the near-imaging-condition embodiment of compressor  200 , concave mirrors  230  have a strong stabilizing effect, rendering the near-imaging-condition embodiment of compressor  200  relatively insensitive to changes in pulse energy. Furthermore, as will be discussed in the following, compressor  200  may be implemented with a pair of broadening bulk-optic(s) having adjustable combined thickness, which may be used to compensate for variation in pulse energy. 
       FIGS. 8A and 8B  illustrate one broadening bulk-optic pair  800  including two wedge-shaped broadening bulk-optics  810  having a combined thickness that is adjustable by changing positions of the wedge-shaped broadening bulk-optics relative to each other. Broadening bulk-optic pair  800  is an embodiment of broadening bulk-optic(s)  210 . Each broadening bulk-optic  810  has a thicker end with thickness  820  and a thinner end with thickness  822 , and a length  824  between the thinner and thicker ends. Each of broadening bulk-optics  810  intersect laser beam  290 , such that laser beam  290  experiences the combined thickness of broadening bulk-optics  810  for each pass. One or both of broadening bulk-optics  810  are moveable along directions  860 , such that the combined thickness is adjustable. 
       FIGS. 8A and 8B  depict two example configurations with relatively smaller and larger combined thicknesses, respectively. Laser beam  290  experiences the combination of thicknesses  826 ( 1 ) and  826 ( 2 ) in the  FIG. 8A  configuration. In  FIG. 8B , broadening bulk-optics  810  have been moved more toward each other, such that laser beam  290  experiences the combination of larger thicknesses  826 ′( 1 ) and  826 ′( 2 ). 
     Although not shown in  FIGS. 8A and 8B , broadening bulk-optic pair  800  may be incorporated in an assembly that includes translation stages to translate broadening bulk-optics  810  along directions  860 . The translation stages may be motorized. Embodiments of compressor  200  implementing broadening bulk-optic pair  800  may be configured to actively adjust the relative positions of broadening bulk-optics  810  in response to changes in, e.g., pulse energy, so as to stabilize the performance of compressor  200  in the presence of such changes. 
     Each broadening bulk-optic  810  has a wedge angle  828 . Wedge angle  828  may be in the range between 2 and 10 degrees, and/or may be set to provide a desired clear aperture size while maintaining a manufacturable thickness  822  (e.g., 0.3 mm or more) at the thinner end. In one example, the combined thickness of broadening bulk-optic pair  800  is adjustable between a smaller combined thickness of about 0.5-1.0 mm and a larger combined thickness of 8-10 mm. The extent of broadening bulk-optics  810  orthogonal to the plane of  FIGS. 8A and 8B  may be set to accommodate several non-overlapping passes of laser beam  290  offset from each other in the dimension orthogonal to the plane of  FIGS. 8A and 8B . In one example, the extent of broadening bulk-optics  810  orthogonal to the plane of  FIGS. 8A and 8B  is at least 30 mm, for example between 30 and 60 mm. Broadening bulk-optics  810  may have an anti-reflective coating. In addition, for a small wedge angle  828 , broadening bulk-optic pair  800  may be implemented in compressor  200  at approximately Brewster&#39;s angle relative to laser beam  290 . 
       FIGS. 9A and 9B  demonstrate adjustment of broadening bulk-optic thickness to compensate for a pulse energy change.  FIGS. 9A and 9B  plot transverse profiles  900  and  902  of laser beam  290  when propagating through one near-imaging-condition embodiment of compressor  200  that implements broadening bulk-optic pair  800 . In this embodiment of compressor  200 , the focal length of each concave mirror  230  is 600 mm, detuning parameter x is −30 mm, laser beam  290  completes five passes through broadening bulk-optic pair  800 , and a telescope lens pair  950  and  952  demagnifies laser beam  290  to an initial waist before the first pass through broadening bulk-optic pair  800 . Transverse profile  900  is calculated for a pulse energy of 2.0 mJ and a combined thickness of broadening bulk-optic pair  800  of 0.5 mm. Transverse profile  902  is calculated for a pulse energy of 0.2 mJ and a combined thickness of broadening bulk-optic pair  800  of 5.0 mm. The 10× thickness increase perfectly compensates for the 10× reduction in pulse energy, such that transverse profiles  900  and  902  are identical. Furthermore, the B-integrals and Kerr focal lengths are the same for the two scenarios associated with transverse profiles  900  and  902 . 
     Referring again to  FIG. 2 , compressor  200  may implement broadening bulk-optic pair  800  (of  FIG. 8 ) together with a photodetector  264  and a controller  266 . Photodetector  264  monitors the pulse energy of laser beam  290 , and controller  266  adjusts the combined thickness of broadening bulk-optic pair  800  according to the pulse energy measured by photodetector  264 . Compressor  200  may include a beamsplitter  262  that directs a small fraction of each pulse of laser beam  290  toward photodetector  264 . Photodetector  264  may also include functionality to monitor the size of laser beam  290 . Controller  266  may be configured to adjust the location(s) of lens(es)  240 , according to measurements performed by photodetector  264 , to stabilize the transverse profile of laser beam  290  directed into the multipass arrangement, so as to make compressor  200  robust to variation in the transverse profile of laser beam  290  provided to compressor  200  from, e.g., laser source  260 . 
       FIG. 10  illustrates one multipass ultrashort-pulse compressor  1000  configured to perform method  100 , with the last dechirping and temporal compression step performed by a separate dispersive optic  1022 . Compressor  1000  is similar to compressor  200 , but (a) further includes dispersive optic  1022  and (b) is configured to direct laser beam  290  to dispersive optic  1022  instead of one or chirped mirrors  220  after the last pass through broadening bulk-optic(s)  210 . Dispersive optic  1022  may be a chirped mirror or a transmissive dispersive optic, e.g., a prism pair. In compressor  1000 , mirror  252  is arranged to direct laser beam  290  to dispersive optic  1022 , for example via collimating lens  242 . As compared to compressor  200 , compressor  1000  may offer improved dechirping and resulting temporal compression by implementing a dispersive optic  1022  that is superior to chirped mirrors  220  but possibly also more complex. For example, dispersive optic  1022  may be a prism pair, whereas replacement of chirped mirrors  220  by prism pairs may not be practical inside the multipass arrangement. 
       FIG. 11  illustrates one multipass ultrashort-pulse compressor  1000  configured to perform method  100  in a ring-arrangement. Compressor  1000  is similar to compressor  200  except for chirped mirrors  220  being replaced by a single dechirp module  1120  that accepts multiple passes of laser beam  290  from concave mirror  230 ( 1 ), dechirps and temporally compresses each of these passes, and directs the laser beam toward concave mirror  230 ( 2 ). 
     In the example depicted in  FIG. 11 , the successive passes of laser beam  290  through broadening bulk-optic(s)  210  are offset from each other in the dimension orthogonal to the plane of  FIG. 11 . As laser beam  290  propagates through the multipass arrangement of compressor  1100 , the propagation path of laser beam  290  shifts in the dimension orthogonal to the plane of  FIG. 11 . As indicated in  FIG. 11 , compressor  1100  may include mirror  252  arranged to intercept laser beam  290  after a last pass through broadening bulk-optic(s)  210 . Without departing from the scope hereof, the successive passes may instead (or also) be offset from each other in the plane of  FIG. 11 . 
     In compressor  1000 , each pass of laser beam  290  through broadening bulk-optic(s)  210  propagates in the same direction. For comparison, in compressor  200 , passes of laser beam  290  through broadening bulk-optic(s)  210  alternate between two opposite propagation directions. 
     One embodiment of dechirp module  1120  includes a prism pair that transmits laser beam  290  on its path from concave mirror  230 ( 1 ) to concave mirror  230 ( 2 ). Another embodiment of dechirp module  1120  includes a single chirped mirror that receives laser beam  290  from concave mirror  230 ( 1 ) and reflects laser beam  290  toward concave mirror  230 ( 2 ). This embodiment of dechirp module  1120  may include additional mirrors arranged to ensure that laser beam  290  is incident on the single chirped mirror at near-normal incidence. 
     In an alternative embodiment, not shown in  FIG. 11 , the last dechirping and temporal compression step in compressor  1100  is performed by a separate dispersive optic outside the multipass arrangement, such as dispersive optic  1022 . In this alternative embodiment, mirror  252  intercepts the laser beam  290  after its last pass through broadening bulk-optic(s)  210  and before reaching dechirp module  1120 . 
     As compared to compressors  200  and  1000 , compressor  1100  allows for (a) the use of prism pairs in each repetition of dechirping and temporal-compression step  120 , which may be deemed advantageous in some situations, or (b) elimination of one chirped mirror to operate with only a single chirped mirror. Designs with only a single chirped mirror may be cheaper, but do not offer the performance benefits provided by matched chirped mirror pairs (with mutually cancelling ripples in the group delay dispersion profile). 
       FIG. 12  illustrates one ultrashort-pulse compressor  1200  configured to perform method  100 , with each repetition of spectral-broadening step  110  being performed by a separate respective broadening bulk-optic  1210 ( 1 ). Compressor  1200  includes a plurality of broadening bulk-optics  1210  or sets of broadening bulk-optics  1210 , a respective plurality of chirped mirrors  1220 , and a respective plurality of concave mirror pairs  1230 A and  1230 B. Each instance of broadening bulk-optic  1210  depicted in  FIG. 12  may be a set of broadening bulk-optics, such as broadening bulk-optic pair  800 , without departing from the scope hereof. Compressor  1200  may further include one or both of lenses  240  and  242 , functioning as discussed above in reference to  FIG. 2 . The transverse profile of laser beam  290 , as it propagates through compressor  1200 , is the same as when laser beam  290  propagates through compressor  200 . However, laser beam  290  does not make multiple passes through or via the same optical elements in compressor  1200 . 
     In operation, laser beam  290  passes through broadening bulk-optic(s)  1210 ( 1 ) to undergo a first repetition of spectral-broadening step  110 , then propagates to concave mirror  1230 A( 1 ) for reflection thereby toward chirped mirror  1220 ( 1 ). Laser beam  290  is then reflected by chirped mirror  1220 ( 1 ) to undergo a first repetition of dechirping and temporal-compression step  120 . The reflection by chirped mirror  1220 ( 1 ) directs laser beam  290  to concave mirror  1230 B( 1 ) for reflection thereby. The reflection by concave mirror  1230 B( 1 ) directs laser beam  290  toward broadening bulk-optic(s)  1210 ( 2 ) and, from there, onwards to a second instance of concave mirror  1230 A, chirped mirror  1220 , and concave mirror  1230 B. This sequence may continue with additional instances of broadening bulk-optic(s)  1210 , concave mirror  1230 A, chirped mirror  1220 , and concave mirror  1230 B. In the example shown in  FIG. 12 , compressor  1200  has three instances of this set of optical elements such that laser beam  290  undergoes three repetitions of steps  110  and  120 . Without departing from the scope hereof, compressor  1200  may instead be configured to perform two, four, or more repetitions of steps  110  and  120 . Each concave mirror pair  1230 A and  1230 B, except for the last such concave mirror pair, ensures that the spot size of laser beam  290 , when incident on each non-first instance of broadening bulk-optic(s)  1210 , is greater than at the previous instance of broadening bulk-optic(s)  1210 . 
     In each instance, concave mirror  1230 A is distance  270  downstream from broadening bulk-optic(s)  1210 , chirped mirror  1220  is distance  272  downstream from concave mirror  1230 A, and concave mirror  1230 B is distance  272  downstream from chirped mirror  1220 . Each non-first instance of broadening bulk-optic(s)  1210  is distance  270  downstream from a preceding instance of concave mirror  1230 B. Herein, each of distances  270  and  272  are measured along the propagation path of laser beam  290 . The segments of the propagation path, associated with a distance  270  or a distance  272 , may be linear or have folds not shown in  FIG. 12 . Distances  270  and  272  may be configured as is compressor  200 , for example to realize a near-imaging-condition embodiment of compressor  1200 . Broadening bulk-optic  1210  may have a common thickness or different thicknesses. 
     In a first alternative embodiment, concave mirrors  1230 A and  1230 B are replaced by respective lenses with the same focal length, and each chirped mirror  1220  is repositioned to receive laser beam  290  from a lens replacing a concave mirror  1230 A and reflect laser beam  290  toward a lens replacing a concave mirror  1230 B. In a second alternative embodiment, each chirped mirror  1220  is replaced by a transmissive dispersive optic, such as a prism pair, also configured to perform dechirping and temporal-compression step  120 . In this second alternative embodiment, each instance of concave mirror  1230 B (or an equivalent lens) is repositioned to receive laser beam  290  as transmitted by the transmissive dispersive optic replacing a preceding instance of chirped mirror  1220 . 
     Without departing from the scope hereof, compressor  1200  may omit the last instance of concave mirror  1230 B, e.g.,  1230 B( 3 ) in the example shown in  FIG. 12 . 
     Compressor  1200  may be simpler to align than the multipass designs of compressors  200 ,  1000 , and  1100 . However, compressor  1200  requires more optical elements and may therefore be both expensive and bulky compared to compressors  200 ,  1000 , and  1100 , especially when a relatively large number of successive repetitions of steps  110  and  120  are required. 
       FIG. 13  is a flowchart of one method  1300  for compressing an ultrashort pulse of a laser beam. Method  1300  is an embodiment of method  100  and may be performed by compressor  200 ,  1000 ,  1100 , or  1200 . Method  1300  repeats a group of steps  1310 ,  1320 , and  1330 . Steps  1310  and  1320  are embodiments of steps  110  and  120 , respectively, of method  100 . 
     Step  1310  spectrally broadens and chirps an ultrashort pulse in one or more broadening bulk-optics. In one example of step  1310 , broadening bulk-optic(s)  210  or  1210  spectrally broadens and chirps an ultrashort pulse of laser beam  290 , as discussed above in reference to  FIGS. 2 and 12 . Each repetition of step  1310  is followed by a repetition of step  1320 . Step  1320  dechirps the chirped ultrashort pulse to compress its duration. In one example of step  1320 , chirped mirror  220  or  1220  dechirps and compresses the chirped ultrashort pulse, as discussed above in reference to  FIGS. 2 and 12 . Step  1330  focuses the laser beam to set the spot size of the laser beam on the broadening bulk-optic(s), such that the spot size is greater for each successive repetition of step  1310 , as discussed above in reference to  FIGS. 1 and 2 . 
     Step  1310  may implement one or both of steps  1312  and  1314 . When implementing step  1312 , step  1310  uses the same broadening bulk-optic (or the same set of broadening bulk-optics) to spectrally broaden the laser pulse. For example, step  1310  may use broadening bulk-optic(s)  210  as discussed above in reference to compressors  200 ,  1000 , and  1100 . When implementing step  1314 , step  1310  uses a pair of wedge-shaped broadening bulk-optics to spectrally broaden the laser pulse. For example, step  1310  may use broadening bulk-optic pair  800 . Step  1314  enables adjustment of the Kerr-effect focusing imparted by step  1310 . 
     Embodiments of method  1300  that implement step  1314  may further include a step  1302  of adjusting the combined thickness of the pair of wedge-shaped broadening bulk-optics, for example to compensate for a change in pulse energy, for example as discussed above in reference to  FIGS. 8, 9A, and 9B . 
     Step  1320  may implement a step  1322  of using a chirped mirror to dechirp and temporally compress the laser pulse. For example, step  1320  may utilize chirped mirrors  220  as discussed above in reference to  FIG. 2 , or chirped mirrors  1220  as discussed above in reference to  FIG. 12 . 
     Each non-first repetition of step  1330  may implement a step  1332  of using one or two concave mirrors to set the spot size at the broadening bulk-optic(s) in step  1310 . In one such example, performed by either one of compressors  200  and  1000 , concave mirror  230 ( 1 ) sets the spot size of laser beam  290  on broadening bulk-optic(s)  210  when laser beam  290  reaches broadening bulk-optic(s)  210  from leg  280 ( 1 ), while concave mirror  230 ( 2 ) sets the spot size of laser beam  290  on broadening bulk-optic(s)  210  when laser beam  290  reaches broadening bulk-optic(s)  210  from leg  280 ( 2 ). In another such example, performed by compressor  1200 , concave mirror pair  1230 A and  1230 B sets the spot size of laser beam  290  on each non-first instance of broadening bulk-optic(s)  1210 . 
     In one embodiment, method  1300  keeps the B-integral below 2.0 rad in each repetition of step  1310 , in order to avoid run-away self-focusing and other undesirable outcomes. In this embodiment, the spot sizes defined by step  1330  cooperate with the configuration of the broadening bulk-optic(s) used in step  1310  to keep the B-integral below 2.0 rad (step  1340 ). 
     In another embodiment, the spot sizes defined by step  1330  cooperate with the configuration of the broadening bulk-optic(s) used in step  1310  to keep the B-integral similar in each repetition of step  1310 . For example, method  1300  may keep the B-integral for each repetition of step  1310  to within ±30% of the average B-integral (as averaged over all repetitions of step  1310 ). This embodiment may serve to maximize the spectral broadening imparted by each repetition of step  1310  while staying below a desired maximum B-integral, such as 2.0 rad, to avoid self-focusing and other undesirable outcomes (step  1342 ). 
     Without departing from the scope hereof, the compressors and compression methods discussed above in reference to  FIGS. 1-13  may serve primarily to spectrally broaden the laser pulse. In such implementations, the final duration of laser pulses produced by the compressors and compression methods may or may not be shorter than the initial duration. In the context of method  100  (see  FIG. 1 ), the requirements to compression and dechirping in steps  120  of method  100  may be relaxed such that the duration τ of laser pulse  184  generated by a last repetition of step  120  exceeds the duration τ 0  of laser pulse  180 . 
     Also without departing from the scope hereof, the compressors and compression methods discussed above in reference to  FIGS. 1-13  may, in a modification, utilize dispersive broadening bulk optic(s) and omit separate dechirping. In such embodiments, the dispersion of the broadening bulk-optic(s) dechirpes the ultrashort laser pulse while it is being spectrally broadened. For example, the broadening bulk-optic(s) may be composed of, or include, a negative-dispersion material that dechirps the ultrashort laser pulse. The negative-dispersion material may be the same material that imparts spectral broadening in the broadening bulk-optic. With the use of such dispersive broadening optic(s), method  100  may be modified to perform steps  110 ( 1 ) and  120 ( 1 ) simultaneously, perform steps  110 ( 2 ) and  120 ( 2 ) simultaneously, etc. In another example, compressor  200  is modified to replace chirped mirrors  220  with standard non-chirping mirrors, and broadening bulk-optic(s)  210  has a dispersion that dechirps the ultrashort laser pulse as it undergoes spectral broadening therein. Other compressors and compression methods discussed above in reference to  FIGS. 1-13  may be similarly modified. Alternatively, the negative-dispersion material may be a reflective coating on a surface of the broadening bulk-optic. In one such example, based on a modification of compressor  200 , leg  280 ( 1 ) is omitted and the reflective, dispersive coating is on the side of broadening bulk-optic(s)  210  farthest from concave mirror  230 ( 2 ), such that broadening bulk-optic(s)  210  receives each pass of laser beam  290  from leg  280 ( 2 ) and the reflective, dispersive coating directs laser beam  290  back to leg  280 ( 2 ). 
     In another modification, the compressors and compression methods discussed above in reference to  FIGS. 1-13  may be configured as tools for spectrally narrowing a laser pulse. Such embodiments are based on bulk-optic(s) with self-phase modulation that spectrally narrows the ultrashort laser pulse instead of spectrally broadening the ultra-short laser pulse. In addition, in these embodiments, the detuning from imaging-condition may be reconfigured to cause a decreasing spot size at each successive pass through the spectrally narrowing bulk-optic(s). 
     The present invention is described above in terms of a preferred embodiment and other embodiments. The invention is not limited, however, to the embodiments described and depicted herein. Rather, the invention is limited only by the claims appended hereto.