Patent Publication Number: US-7903701-B2

Title: Intracavity harmonic generation using a recycled intermediate harmonic

Description:
TECHNICAL FIELD 
     This disclosure relates to harmonic lasers, and more particularly, to efficient intracavity harmonic generation. 
     BACKGROUND INFORMATION 
     Laser systems are employed in a variety of applications including communications, medicine, and micromachining. These applications utilize a variety of laser wavelengths and output powers. In particular, high power laser beams having an ultraviolet (UV) wavelengths are widely used. Currently, there is no commercially available gain medium that directly generates UV laser beams. Thus, UV laser beams are typically generated through nonlinear processes, such as harmonic generation. 
     Two such harmonic generation configurations include an extracavity harmonic generation configuration and an intracavity harmonic generation configuration. The extracavity configuration generates harmonics outside of a resonant laser cavity. In other words, a laser beam is generated in a resonant laser cavity and directed to a crystal positioned external to the cavity. The intracavity configuration generates harmonics inside of a resonant laser cavity, which is generally more efficient than an extracavity configuration. 
     Generating a laser beam that is a third or higher harmonic of a fundamental frequency entails generating first a laser beam that is a second harmonic of the fundamental frequency. The extracavity and intracavity configurations generally do not convert all of the second harmonic beam to a third or higher order harmonic beam. Thus, the unused portion of the second harmonic beam reduces the overall efficiency of the laser system and reduces the power of the resultant third or higher order harmonic beam. 
       FIG. 1  is a schematic diagram of a known intracavity configuration for generating a third harmonic laser beam. A laser  100  employs a laser medium  102  positioned along an optical path  104  of a laser cavity  106  formed by end mirrors  108  and  110 , optical pumping input couplers  112  and  114 , and an output coupler  116 . Laser  100  is pumped with two laser diode pumps  118  and  120 . An optical fiber  122  directs laser radiation generated by laser diode pump  118  into laser cavity  106  through optical pumping input coupler  114 . Likewise, an optical fiber  124  directs laser radiation generated by laser diode pump  120  into laser cavity  106  through optical pumping input coupler  112 . A Q-switch  126 , such as an acousto-optic Q-switch (AO-QS), is positioned along optical path  104  and is driven at an appropriate pulse repetition rate (PRR) to obtain short energetic pulses from laser  100 . As a laser beam  130  having a fundamental wavelength resonates within cavity  106  between end mirrors  108  and  110 , laser medium  102  amplifies laser beam  130 . 
     A second harmonic generation (SHG) crystal  140  is positioned along optical path  104 . As laser beam  130  passes through SHG crystal  140 , SHG crystal  140  generates a second harmonic laser beam  142  having half the wavelength of laser beam  130 . As laser beam  130  and second harmonic laser beam  142  pass through a third harmonic generation (THG) crystal  150 , which is also positioned along optical path  104 , THG crystal  150  generates a third harmonic laser beam  152  having one-third the wavelength of laser beam  130 . Although second harmonic laser beam  142  reflects off end mirror  110 , a portion of second harmonic laser beam  142  that is not used in generating third harmonic laser beam  152  exits cavity  106  as an unused, wasted second harmonic laser beam  144  via output coupler  116 . Wasting unused second harmonic laser beam  144  lowers the conversion efficiency from the fundamental harmonic to the third harmonic and lowers the total power of third harmonic laser beam  152  that might otherwise be obtained. Third harmonic laser beam  152 , which has the desired wavelength (e.g., a UV wavelength of 355 nm), exits cavity  106  as an output laser beam  154  via output coupler  116 . Thus, output coupler  116  is highly reflective for laser beam  130  and antireflective for second harmonic laser beam  142  and third harmonic laser beam  152 . Laser beams  130 ,  142  and  152  are shown axially offset from one another for illustration purposes. 
     U.S. Pat. No. 5,943,351 of Zhou et al. describes one attempt to improve the efficiency of generating a third or higher harmonic beam from a second harmonic beam. As shown in  FIG. 2 , a laser  200  includes a cavity having a first mirror  210 , a lasing rod  220 , an acousto-optic Q-switch  222 , a second mirror  250 , a SHG crystal  230 , a third mirror  252 , a THG crystal  232 , and a fourth mirror  254 . A main cavity is formed by mirrors  210  and  254 , which cause oscillation of a fundamental beam  212  at 1064 nm using a Nd:YAG rod  220 . Mirrors  250  and  254  form a first sub-cavity for the intracavity second harmonic generation (i.e., 532 nm) to create a second harmonic beam  214  therein, and mirrors  252  and  254  form a second sub-cavity for the third harmonic generation (i.e., 355 nm) to create a third harmonic beam  216 . 
     Use of mirror  250  results in a significantly deteriorated third or higher order harmonic beam quality because a beam mode of second harmonic beam  214  is different each time second harmonic beam  214  passes through THG crystal  232 . For example, after second harmonic beam  214  is generated by SHG crystal  230 , second harmonic beam  214  converges as it propagates from SHG crystal  230  toward mirror  254  (assuming fourth mirror  254  is coincident with a beam waist). As second harmonic beam  214  passes through THG crystal  232 , a portion of second harmonic beam  214  will be used to generate third harmonic beam  216  and a portion of second harmonic beam  214  will remain unused. After the unused portion of second harmonic beam  214  reflects off mirror  254 , the unused portion of second harmonic beam  214  diverges as it propagates from mirror  254  toward mirror  250 . After the unused portion of second harmonic beam  214  reflects off mirror  250 , the unused portion of second harmonic beam  214  keeps diverging in a direction toward mirror  254  and continues to diverge as it subsequently propagates between mirrors  250  and  254  (assuming mirror  250  is a flat mirror). Thus, each time the unused portion of second harmonic beam  214  passes through THG crystal  232 , the beam mode (e.g., beam radius and beam divergence) of the unused portion of second harmonic beam  214  will be different, which results in a significantly deteriorated third or higher order harmonic beam quality and conversion efficiency. Accordingly, as noted in column 9, lines 6-9 of Zhou, the beam quality factor (i.e., M 2  factor) of the third or higher order harmonic beam is 1.6, which is likely larger than what could be achieved without recycling the unused portion of second harmonic beam  214 . As a point of reference, a diffraction-limited Gaussian beam has an M 2  factor of 1.0. 
     Thus, the present inventors have recognized a need for a system and method for recycling an unused portion of an intermediate harmonic beam (e.g., a second harmonic beam) to improve higher order harmonic beam generation efficiency (e.g., third or higher order harmonic beam generation efficiency) without sacrificing higher order harmonic beam quality. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram of a prior art laser that generates a third harmonic laser beam without recycling an unused portion of a second harmonic laser beam. 
         FIG. 2  is a schematic diagram of a prior art laser that recycles an unused portion of a second harmonic laser beam when generating a third harmonic laser beam, but which results in a significantly deteriorated third harmonic beam quality. 
         FIG. 3  is a schematic diagram of a laser that efficiently generates a third harmonic laser beam by recycling an unused portion of a second harmonic laser beam while maintaining third harmonic beam quality, according to one embodiment. 
         FIG. 4A  is a graphical representation of a prior art lowest-order Gaussian beam diverging away from its waist. 
         FIG. 4B  is a schematic diagram illustrating various prior art optical elements and associated transfer matrices. 
         FIG. 4C  is a graph illustrating a change in third harmonic beam power over a range of pulse repetition rates. 
         FIGS. 5 ,  6 ,  7 , and  8  are schematic diagrams of lasers that efficiently generate a fourth harmonic laser beam by recycling an unused portion of a second harmonic laser beam while maintaining fourth harmonic beam quality, according to various embodiments. 
         FIG. 9  is a schematic diagram of a laser that efficiently generates a fourth harmonic laser beam by recycling an unused portion of a second harmonic laser beam and an unused portion of a third harmonic laser beam while maintaining fourth harmonic beam quality, according to one embodiment. 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     With reference to the above-listed drawings, this section describes particular embodiments and their detailed construction and operation. The embodiments described herein are set forth by way of illustration only. In light of the teachings herein, skilled persons will recognize that there may be equivalents to what is expressly or inherently taught herein. For example, variations can be made to the embodiments described herein and other embodiments are possible. 
     For the sake of clarity and conciseness, certain aspects of components or steps of certain embodiments are presented without undue detail where such detail would be apparent to skilled persons in light of the teachings herein and/or where such detail would obfuscate an understanding of more pertinent aspects of the embodiments. 
       FIG. 3  is a schematic diagram of a laser  300  that efficiently generates a third harmonic laser beam  372  by recycling an unused portion of a second harmonic laser beam while maintaining third harmonic beam quality, according to one embodiment. Laser  300  employs a laser medium  302  positioned along an optical path  304  of a standing-wave resonant laser cavity  306  formed by end mirrors  308  and  310 , an optical input coupler  312 , and an optical energy coupler  314 . Preferably, laser medium  302  is interposed between optical input coupler  312  and optical energy coupler  314 , but laser medium  302  may be positioned elsewhere along optical path  304 . Laser medium  302  preferably comprises a conventional solid-state lasant or gain medium, such as YAG, YLF, YVO 4 , YALO, sapphire, alexandrite, or CrLiSAF compositions, that is doped with Nd, Yb, Er, Cr, or Tm. Laser medium  302  preferably produces laser radiation or laser beam energy having an infrared (IR) fundamental wavelength, such as 750-800 nm, 1064 nm, 1047 nm, or 1320 nm, but a variety of other wavelengths may be produced, such as visible wavelengths. In addition, other laser media or types of lasers could be employed including, a gas, CO 2 , excimer, or copper vapor laser. Other elements known in the art, such as the structural frame for holding components are included with laser  300 , but not shown in  FIG. 3  for clarity of illustration. 
     Laser  300  is optically pumped by a laser diode pump  318 . An optical fiber  320  directs laser radiation generated by laser diode pump  318  through a lens assembly  322  comprising a pair of plano-convex lenses  323  and  324 , which focus the laser radiation onto laser medium  302  through optical pumping input coupler  312 . Another laser diode pump may be provided along with an associated optical fiber and lens assembly to pump laser  300  through optical energy coupler  314 . Thus, laser  300  may be pumped through input coupler  312 , optical energy coupler  314 , or both. According to a preferred embodiment, laser radiation generated by laser diode pump  318  has a wavelength of about 880 to 900 nm. Preferably, optical pumping input coupler  312  is interposed between end mirror  308  and optical energy coupler  314 , but optical pumping input coupler  312  may be positioned elsewhere along optical path  304 . As a laser beam  330  (e.g., laser beam energy) having a fundamental wavelength resonates or oscillates within resonant laser cavity  306  between end mirrors  308  and  310 , laser medium  302  amplifies laser beam  330  to a selected optical power. Providing a single resonant laser cavity  306  to build up laser beam  330  helps ensure that the fundamental laser beam energy inside resonant laser cavity  306  is self-phase locked. If the fundamental laser beam energy is generated by another resonant laser cavity and injected into cavity  306 , a feedback loop may be necessary to phase lock the incoming fundamental laser beam energy with the fundamental laser beam energy inside cavity  306 . While laser medium  302  is preferably continuous wave (CW) pumped by a diode laser or diode laser array, any conventional laser pumping device or laser pumping scheme can be employed. For example, laser medium  302  may be end-pumped or side-pumped. In addition, laser  300  may be a pulsed pumped laser. 
     A Q-switch  332  is positioned within resonant laser cavity  306  along optical path  304 . Preferably, Q-switch  332  is interposed between end mirror  308  and optical pumping input coupler  312 , but Q-switch  332  may be positioned elsewhere along optical path  304 . Q-switch  332  is driven at an appropriate pulse repetition rate (PRR) to obtain short energetic pulses from laser  300 . Thus, Q-switch  332  helps generate high intensity pulses, which help improve higher order harmonic generation efficiency. According to some embodiments, Q-switch  332  may be omitted. Q-switch  332  preferably comprises an acousto-optic Q-switch (AO-QS), but may comprise another device that can be quickly switched between low-loss and high-loss states, such as an electro-optic Q-switch, a mechanical Q-switch, or a passive Q-switch. 
     A nonlinear medium  340  is positioned along optical path  304  within resonant laser cavity  306 . Preferably, nonlinear medium  340  is interposed between optical energy coupler  314  and end mirror  310 , but nonlinear medium  340  may be positioned elsewhere along optical path  304 . Nonlinear medium  340  is preferably oriented such that laser beam  330  strikes an optical surface of nonlinear medium  340  at normal incidence. As laser beam  330  passes through nonlinear medium  340 , nonlinear medium  340  generates a laser beam  342  having a wavelength that is a fraction of the fundamental wavelength. In other words, nonlinear medium  340  interacts with and converts at least a portion of the laser beam energy at the fundamental wavelength propagating along optical path  304  into laser beam energy having a wavelength that is a harmonic fraction of the fundamental wavelength. Preferably, nonlinear medium  340  comprises a nonlinear crystal adapted to generate a second harmonic wavelength from the fundamental wavelength, such as AgGaS 2  (silver gallium selenite), AgGaSe 2 , BBO, BIBO (bismuth triborate), KTA (potassium titanyle arsenate, KTiOAsO 4 ), KTP, KDP (potassium dihydrogen phosphate, KH 2 PO 4 ), KD*P/KDP, LiNbO 3  (lithium niobate), LiIO 3  (lithium iodate), LBO, and derivatives thereof. Preferably, the nonlinear crystal is configured for type II phase matching, but the nonlinear crystal may also be configured for type I phase matching if additional components are used, such as one or more waveplates. In addition, the nonlinear crystal may comprise a Brewster cut crystal. Thus, nonlinear medium  340  preferably generates laser beam  342  having one-half the wavelength of laser beam  330  (i.e., a frequency that is twice the fundamental frequency of laser beam  330 ). 
     An anti-reflection (AR) coating may optionally be applied to nonlinear medium  340 . For example, a single or multilayer dielectric coating having AR characteristics at the fundamental and second harmonic wavelengths may be applied to an optical surface of nonlinear medium  340 . 
     A higher order harmonic nonlinear medium  350  is also positioned along optical path  304  within resonant laser cavity  306 . Thus, higher order harmonic nonlinear medium  350  is in optical association with resonant laser cavity  306 . Preferably, higher order harmonic nonlinear medium  350  is interposed between nonlinear medium  340  and end mirror  310 , but higher order harmonic nonlinear medium  350  may be positioned elsewhere along optical path  304 . Higher order harmonic nonlinear medium  350  is preferably oriented such that laser beam  330 , laser beam  342 , or both, strike an optical surface of higher order harmonic nonlinear medium  350  with normal incidence. Higher order harmonic nonlinear medium  350  converts laser radiation or energy having a harmonic wavelength, such as a first harmonic, second harmonic, third harmonic, or a combination of one or more of the first, second, or third harmonics, into laser radiation having one or more selected harmonic wavelengths, such as a second harmonic, third harmonic, fourth harmonic, or fifth harmonic. In a preferred embodiment, higher order harmonic nonlinear medium  350  converts laser radiation having a fundamental wavelength and laser radiation having a second harmonic wavelength into laser radiation having a third harmonic wavelength. Thus, as laser beam  330  and laser beam  342  pass through higher order harmonic nonlinear medium  350 , higher order harmonic nonlinear medium  350  generates a laser beam  352  having one-third of the wavelength of laser beam  330  (i.e., a frequency that is three times the fundamental frequency of laser beam  330 ). Higher order harmonic nonlinear medium  350  may comprise any of the nonlinear crystals described with reference to nonlinear medium  340  and may comprise the same or different nonlinear crystal as nonlinear medium  340 . The nonlinear crystal may be configured for either type I or type II phase matching. 
     Although laser beams  330 ,  342  and  352  propagate superimposed along at least a portion optical path  304  (e.g., laser beams  330  and  342  are superimposed between optical energy coupler  314  and end mirror  310  and laser beams  330 ,  342  and  352  are superimposed between end mirror  310  and an intracavity output coupler  370 ), laser beams  330 ,  342  and  352  are shown axially offset from one another for illustration purposes. 
     An AR coating may optionally be applied to higher order harmonic nonlinear medium  350 . For example, a single or multilayer dielectric coating having AR characteristics at the fundamental wavelength, second harmonic wavelength, and third harmonic wavelength may be applied to an optical surface of higher order harmonic nonlinear medium  350 . 
     Optical energy coupler  314  is positioned within resonant laser cavity  306  along optical path  304 . Thus, optical energy coupler  314  is positioned in optical association with laser medium  302  and nonlinear medium  340 . Preferably, optical energy coupler  314  is interposed between laser medium  302  and nonlinear medium  340 , but optical energy coupler  314  may be positioned elsewhere along optical path  304 . According to a preferred embodiment, optical energy coupler  314  is a flat mirror adapted to be reflective of laser beam  330  and anti-reflective of laser beam  342 . Thus, optical energy coupler  314  essentially separates laser beam  330  and laser beam  342 . For example, a single or multilayer dielectric coating having highly reflective (HR) characteristics, such as over about 99.5% reflection, at the fundamental wavelength and having AR characteristics, such as less than about 5% reflection, at the second harmonic wavelength may be applied to optical energy coupler  314 . According to a preferred embodiment, a coating is applied to both opposing optical surfaces of optical energy coupler  314 . A first coating having HR characteristics, such as over about 99.5% reflection, at the fundamental wavelength (e.g., 1064 nm) at about a 45° angle of incidence and having AR characteristics, such as less than about 5% reflection, at the second harmonic wavelength (e.g., 532 nm) at about a 45° angle of incidence is applied to an optical surface of optical energy coupler  314  facing laser medium  302  and nonlinear medium  340 . A second coating having AR characteristics, such as less than about 1% reflection, at the second harmonic wavelength (e.g., 532 nm) at about a 45° angle of incidence is applied to an optical surface of optical energy coupler  314  facing a curved mirror  360 . Thus, optical energy coupler  314  directs almost all of the incident fundamental laser beam energy between laser medium  302  and nonlinear medium  340 . In addition, optical energy coupler  314  directs almost all of the incident second harmonic laser beam energy out of resonant laser cavity  306 . Thus, the second harmonic laser beam energy not reflected by optical energy coupler  314  exits resonant laser cavity  306  as residual second harmonic laser beam energy (e.g., the portion of laser beam  342  between optical energy coupler  314  and curved mirror  360 ). Separation of the second harmonic laser beam energy from the fundamental laser beam energy at optical energy coupler  314  may help prevent damage to laser medium  302  and other optical components. 
     Residual second harmonic laser beam energy is reflected back into resonant laser cavity  306  by mode-matching optics  360 , which are set in optical association with optical energy coupler  314 . According to a preferred embodiment, mode-matching optics  360  comprise a curved mirror. The reflected residual second harmonic laser beam energy propagates through higher order harmonic nonlinear medium  350  so that higher order harmonic nonlinear medium  350  can convert at least a portion of the reflected residual second harmonic laser beam energy into laser beam  352 . Thus, the residual second harmonic laser beam energy is recycled (instead of being wasted) and is used to generate additional higher order laser beam energy (e.g., third harmonic laser beam energy). 
     As will be described in more detail below, a radius of curvature and position of curved mirror  360  are selected so that a beam mode (e.g., beam radius and beam divergence) of the reflected residual second harmonic laser beam energy substantially matches a beam mode of residual second harmonic laser beam energy. In other words, a radius of curvature and position of curved mirror  360  (e.g., relative to optical energy coupler  314 ) are selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam (e.g., between end mirror  310  and curved mirror  360 ). Maintaining a substantially uniform beam radius and beam divergence of laser beam  342  helps improve higher order laser beam energy (e.g., third harmonic laser beam energy) generation efficiency without affecting a beam mode of the higher order laser beam energy and without significantly deteriorating a beam quality of the higher order laser beam energy. 
     For example, laser  300  may be designed such that end mirror  310  is positioned coincident with a beam waist of the fundamental laser beam energy. Thus, as the fundamental laser beam energy propagates from optical energy coupler  314  to end mirror  310 , the fundamental laser beam energy converges. In addition, as the second harmonic laser beam energy, which is generated by nonlinear medium  340  from at least a portion of the fundamental laser beam energy, propagates from nonlinear medium  340  to end mirror  310 , the second harmonic laser beam energy converges. A portion of the second harmonic laser beam energy is not used by higher order harmonic nonlinear medium  350  to generate the third harmonic laser beam energy. Thus, the unused portion of the second harmonic laser beam energy is reflected by end mirror  310  and propagates back-and-forth between mirrors  310  and  360 . 
     After being reflected by end mirror  310 , the fundamental laser beam energy begins diverging and the unused portion of the second harmonic laser beam energy begins diverging. Thus, the fundamental laser beam energy diverges as it propagates from end mirror  310  toward optical energy coupler  314 . Nonlinear medium  340  generates additional second harmonic laser beam energy as the diverging fundamental laser beam energy passes through the nonlinear medium  340  in a direction toward optical energy coupler  314 . Thus, the residual second harmonic laser beam energy may comprise the additional second harmonic laser beam energy (e.g., the second harmonic laser beam energy generated as the fundamental laser beam energy passes through nonlinear medium  340  in a direction toward optical energy coupler  314 ) and the unused portion of the second harmonic laser beam energy. The unused portion of the second harmonic laser beam energy also diverges as it propagates from end mirror  310  toward optical energy coupler  314 . If mirror  360  were flat, the second harmonic laser beam energy would continue to diverge after being reflected by mirror  360 . Thus, each time the second harmonic laser beam energy passed through higher order harmonic nonlinear medium  350 , a beam mode (e.g., beam radius and beam divergence) of the second harmonic laser beam energy would be different and a significantly deteriorated third harmonic laser beam energy beam quality would result. Accordingly, a radius of curvature and position of curved mirror  360  are selected so that a beam mode of the second harmonic laser beam energy reflected by curved mirror  360  substantially matches a beam mode of the incoming second harmonic laser beam energy (e.g., a beam mode of the second harmonic laser beam energy that is generated by nonlinear medium  340  from the fundamental laser beam energy). In other words, the position and concave shape of curved mirror  360  are selected such that the second harmonic laser beam energy begins converging (e.g., in a similar or identical manner as the second harmonic laser beam energy generated by nonlinear medium  340 ) after being reflected by curved mirror  360 . 
     A radius of curvature and position of curved mirror  360  can vary based on the design of laser  300 . For example, if nonlinear medium  340  generates a Gaussian second harmonic beam, a radius of curvature of curved mirror  360  is selected to substantially match a radius of curvature of a Gaussian beam wavefront of the second harmonic beam at the position of curved mirror  360 . In other words, the radius of curvature of the Gaussian beam wavefront is first calculated for a given position of curved mirror  360 . Then, the radius of curvature of curved mirror  360  is selected to substantially match the calculated radius of curvature of the Gaussian beam wavefront (at the position of curved mirror  360 ) so that the second harmonic mode of the reflected beam is preserved. The radius of curvature of a Gaussian beam wavefront at a particular position can be determined from the wavelength of the beam, beam radius at the beam waist, beam quality factor (M 2 ), and distance from the beam waist. 
     For example,  FIG. 4A  is a graphical representation of a lowest-order Gaussian beam (i.e., where M 2 =1) diverging away from its waist. A normalized field pattern of the Gaussian beam at another plane z is expressed by Equation 1 and Equation 2, where w 0  is the waist radius and the complex radius of curvature q(z) is related to the spot size w(z) and the radius of curvature R(z) of the wavefront at any plane z by the definition in Equation 3. 
     
       
         
           
             
               
                 
                   
                     
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     The initial value (e.g., at the beam waist) of the complex radius of curvature q(z) is shown in Equation 4, where λ is the wavelength of the radiation in which the beam is propagating. 
     
       
         
           
             
               
                 
                   
                     
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     Thus, if the complex radius of curvature q(z) at a position z is known, the radius of curvature of the wavefront R can be calculated from q(z). 
     The complex radius of curvature q(z) of a beam changes as the beam propagates through various optical elements, such as free space or a lens. Thus, the complex radius of curvature q(z) after propagating through one or more optical elements can be calculated from a complex radius of curvature q(z) at a known position. A beam waist size, beam waist location, and beam quality factor (M 2 ) can be obtained through measurement. Thus, the complex radius of curvature q(z) can be calculated at the beam waist using Equation 4. 
     An optical element can be expressed mathematically by a transfer matrix, such as the transfer matrix shown in Equation 5.  FIG. 4B  illustrates various optical elements and associated transfer matrices. Transfer matrices for other optical elements are well known. Optical element  410  represents free space, optical element  420  represents a plane dielectric interface, optical element  430  represents a plane dielectric slab having a thickness L, and optical element  440  represents a spherical mirror. Matrix multiplication rules can be applied to derive a single transfer matrix for a multi-element system. 
     
       
         
           
             
               
                 
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     The complex radius of curvature q2 of a beam after propagating through an optical element can be calculated from the known complex radius of curvature q1 (e.g., at the beam waist) and the optical element transfer matrix as shown in Equation 6. 
     
       
         
           
             
               
                 
                   
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     If the beam quality factor (M 2 ) is greater than one, the wavefront radius of curvature is the same as the lowest-order Gaussian beam having a waist radius shown in Equation 7, where w 0  is the beam waist radius. 
     
       
         
           
             
               
                 
                   
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     EXAMPLE 
     By way of example, end mirror  310  of  FIG. 3  is positioned coincident with the beam waist and curved mirror  360  is positioned 282 mm from end mirror  310 . A radius of curvature of the Gaussian beam wavefront (of the second harmonic beam) is calculated to be 400 mm at a distance of 282 mm from end mirror  310 . Thus, a concave shape is chosen for curved mirror  360 , and the radius of curvature of curved mirror  360  is selected to be 400 mm. Accordingly, a beam radius and beam divergence for the second harmonic beam are maintained everywhere along the beam path as the second harmonic beam propagates back-and-forth between end mirror  310  and curved mirror  360  inside of a cavity defined by end mirror  310  and curved mirror  360 . 
     In operation, a laser having the configuration shown in  FIG. 3  produced a third harmonic beam having a beam quality factor (M 2 ) of 1.2 and an average power of 2.73 watts. Curved mirror  360  had a radius of curvature of 400 mm and was positioned 282 mm from end mirror  310 . The fundamental beam wavelength was 1064 nm, the second harmonic wavelength was 532 nm, and the third harmonic wavelength was 355 nm. Laser medium  302  was pumped with a 30 watt laser diode  318 , and Q-switch  332  was driven at a pulse repetition rate (PRR) of 100 kHz. By recycling the second harmonic laser beam, the third harmonic beam power was increased by about 35%. As shown in  FIG. 4C , varying the pulse repetition rate resulted in a slight variation in the third harmonic beam power (average power). The third harmonic beam power ranged from about 3.14 watts at a 70 kHz PRR to about 2.73 watts at a 100 kHz PRR. 
     If a position of curved mirror  360  is adjusted (e.g., moved closer to end mirror  310 ), a radius of curvature of the Gaussian beam wavefront at the adjusted position can be calculated so that a corresponding radius of curvature can be selected for curved mirror  360 . In a similar vein, a position of curved mirror  360  can be adjusted to compensate for changes in the radius of curvature of curved mirror  360 . In addition, if end mirror  310  is changed from a flat mirror to a curved mirror, a position of curved mirror  360 , a radius of curvature of curved mirror  360 , or both, may be adjusted accordingly so that the second harmonic mode of the beam reflected between mirrors  310  and  360  is preserved. While examples have been provided in which curved mirror  360  is selected to have a concave shape, it is possible that curved mirror  360  is selected to have a convex shape (e.g., to make the reflected beam diverge upon reflection) or a flat shape (e.g., a radius of curvature may be selected to be infinity). 
     In addition, mode-matching optics comprising one or more lenses and curved mirror  360  may be used to match a beam mode of the reflected residual second harmonic laser beam energy with a beam mode of the residual second harmonic laser beam energy. For example, a lens (or more than one lens) may be interposed between optical energy coupler  314  and curved mirror  360 . Thus, curved mirror  360  may have a radius of curvature set to infinity and the lens may have a radius of curvature (e.g., to make the beam converge or diverge), a position (e.g., relative to curved mirror  360 ), or both, selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy (e.g., the second harmonic laser beam energy reflected by curved mirror  360 ) are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam (e.g., between end mirror  310  and curved mirror  360 ). The lens may also cooperate with the radius of curvature of curved mirror  360  to help preserve the second harmonic beam mode (e.g., if a radius of curvature of curved mirror  360  is not set to infinity). For example, a combination of a radius of curvature of curved mirror  360 , a position of curved mirror  360 , a radius of curvature of the lens, and a position of the lens may be selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam. While the lens may be interposed between curved mirror  360  and optical energy coupler  314 , the lens (or more than one lens) may be positioned anywhere along the path of the second harmonic laser beam energy to preserve the second harmonic beam mode. In addition, according to one embodiment, the mode-matching optics comprise one or more lenses that are used to match a beam mode of the reflected residual second harmonic laser beam energy with a beam mode of the residual second harmonic laser beam energy. 
     Curved mirror  360  is preferably positioned outside of resonant laser cavity  306 , but may be positioned within resonant laser cavity  306 . Positioning curved mirror  360  outside of resonant laser cavity  306  can simplify the procedure of aligning curved mirror  360  with end mirror  310 . For example, the position of curved mirror  360  can be adjusted without impacting laser beam  330 . In addition, positioning curved mirror  360  outside of resonant laser cavity  306  can simplify the design of curved mirror  360 . For example, the radius of curvature of curved mirror  360  can be selected based on the second harmonic laser beam without regard to the fundamental laser beam. Further, curved mirror  360  can be designed to have HR characteristics for the second harmonic laser beam energy without having to design curved mirror  360  to have HR characteristics or AR characteristics for other wavelengths, such as the fundamental laser beam energy or a third or higher harmonic laser beam energy. For example, a coating having HR characteristics, such as over about 99.8% reflection, at the second harmonic wavelength (e.g., 532 nm) may be applied to an optical surface of curved mirror  360  facing optical energy coupler  314 . Thus, a more reflective, less complex, and possibly less expensive single or multilayer dielectric coating may be applied to curved mirror  360 . In addition, positioning additional elements inside resonant laser cavity  306  increases cavity loss, which results in reduced output power. Further, positioning additional elements inside resonant laser cavity  306  may cause reliability issues. Thus, positioning curved mirror  360  outside of resonant laser cavity  306  helps create a more efficient and more reliable laser that has a higher output power. 
     According to one embodiment, an intracavity output coupler  370  is positioned within resonant laser cavity  306  along optical path  304 . Thus, output coupler  370  is positioned in optical association with nonlinear medium  340  and higher order harmonic nonlinear medium  350 . Preferably, output coupler  370  is interposed between nonlinear medium  340  and higher order harmonic nonlinear medium  350 , but output coupler  370  may be positioned elsewhere along optical path  304 . According to a preferred embodiment, output coupler  370  is a flat mirror adapted to be reflective of laser beam  352  and anti-reflective of laser beams  330  and  342 . Thus, output coupler  370  essentially separates laser beam  352  from laser beams  330  and  342 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the third harmonic wavelength and having AR characteristics, such as less than about 1% reflection, at the fundamental wavelength and second harmonic wavelength may be applied to output coupler  370 . According to a preferred embodiment, a coating is applied to both opposing optical surfaces of output coupler  370 . A first coating having HR characteristics, such as over about 95% reflection, at the third harmonic wavelength (e.g., 355 nm) at about a 10° angle of incidence, and having AR characteristics, such as less than about 1% reflection at the fundamental wavelength (e.g., 1064 nm) and less than about 3% reflection at the second harmonic wavelength (e.g., 532 nm) at about a 10° angle of incidence is applied to an optical surface of output coupler  370  facing nonlinear medium  350 . A second coating having AR characteristics, such as less than about 0.5% reflection, at the fundamental wavelength (e.g., 1064 nm) and second harmonic wavelength (e.g., 532 nm) at about a 10° angle of incidence is applied to an optical surface of output coupler  370  facing nonlinear medium  340 . Thus, output coupler  370  directs almost all of the incident third harmonic laser beam energy out of resonant laser cavity  306 . In addition, output coupler  370  permits almost all of the incident fundamental laser beam energy and the second harmonic laser beam energy to propagate through output coupler  370 . Thus, the reflected third harmonic laser beam energy exits resonant laser cavity  306  through output coupler  370  as an output third harmonic laser beam  372 . Separation of the third harmonic laser beam energy from the fundamental laser beam energy and the second harmonic laser beam energy at output coupler  370  may help prevent damage to nonlinear medium  340 , laser medium  302 , and other optical components from the third harmonic laser beam energy. The third harmonic laser beam energy may exit resonant laser cavity  306  in other ways. For example, end mirror  310  may include a coating having AR characteristics at the third harmonic wavelength (e.g., 355 nm) to allow the third harmonic laser beam energy to exit resonant laser cavity  306  through end mirror  310 . In addition, a reflective, dispersive, or transmissive output coupler may be positioned in optical association with higher order harmonic nonlinear medium  350  to allow the third harmonic laser beam energy to exit resonant laser cavity  306 . 
     According to a preferred embodiment, end mirror  308  is a flat mirror adapted to be reflective of laser beam  330 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 99.8% reflection at the fundamental wavelength (e.g., 1064 nm) at normal incidence may be applied to end mirror  308 . End mirror  310  is preferably a flat mirror adapted to be reflective of laser beams  330 ,  342 , and  352 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 99.5% reflection at the fundamental wavelength (e.g., 1064 nm) at normal incidence and over about 99% reflection at the second harmonic wavelength (e.g., 532 nm) and the third harmonic wavelength (e.g., 355 nm) at normal incidence may be applied to end mirror  310 . Optical input coupler  312  is preferably a flat mirror adapted to be reflective of laser beam  330  and transmissive of the laser radiation generated by laser diode pump  318 . For example, a single or multilayer dielectric coating having highly reflective (HR) characteristics, such as over about 99.5% reflection, at the fundamental wavelength and having AR characteristics, such as less than about 2% reflection, at the wavelength generated by laser diode pump  318  may be applied to optical input coupler  312 . According to a preferred embodiment, a coating is applied to both opposing optical surfaces of optical input coupler  312 . A first coating having HR characteristics, such as over about 99.5% reflection, at the fundamental wavelength (e.g., 1064 nm) at about a 45° angle of incidence and having AR characteristics, such as less than about 2% reflection, at the wavelength generated by laser diode pump  318  (e.g., about 880 to 900 nm) at about a 45° angle of incidence is applied to an optical surface of optical input coupler  312  facing laser medium  302 . A second coating having AR characteristics, such as less than about 0.8% reflection, at the wavelength generated by laser diode pump  318  (e.g., about 880 to 900 nm) at about a 45° angle of incidence is applied to an optical surface of optical input coupler  312  facing lens assembly  322 . 
       FIG. 5  is a schematic diagram of a laser  500  that efficiently generates a fourth harmonic laser beam  572  by recycling an unused portion of a second harmonic laser beam while maintaining fourth harmonic beam quality, according to one embodiment. Laser  500  is similar to laser  300  described with reference to  FIG. 3  but is adapted to generate fourth harmonic laser beam  572 . 
     Laser  500  incorporates a higher order harmonic nonlinear medium  550  in place of higher order harmonic nonlinear medium  350 . Higher order harmonic nonlinear medium  550  is adapted to convert laser radiation or energy having the second harmonic wavelength into laser radiation having a fourth harmonic wavelength. Higher order harmonic nonlinear medium  550  is positioned along optical path  304  within resonant laser cavity  306 , which is formed by end mirror  308  and an end mirror  510 . Thus, higher order harmonic nonlinear medium  550  is in optical association with resonant laser cavity  306 . Preferably, higher order harmonic nonlinear medium  550  is interposed between nonlinear medium  340  and end mirror  510 , but higher order harmonic nonlinear medium  550  may be positioned elsewhere along optical path  304 . Higher order harmonic nonlinear medium  550  is preferably oriented such that laser beam  330 , laser beam  342 , or both, strike an optical surface of higher order harmonic nonlinear medium  550  at normal incidence. As laser beam  342  passes through higher order harmonic nonlinear medium  550 , higher order harmonic nonlinear medium  550  generates a laser beam  552  having one-quarter of the wavelength of laser beam  330  (i.e., a frequency that is four times the fundamental frequency of laser beam  330 ). Higher order harmonic nonlinear medium  550  may comprise any of the nonlinear crystals described with reference to nonlinear mediums  340  and  350  and may comprise the same or different nonlinear crystal as nonlinear mediums  340  and  350 . According to a preferred embodiment, higher order harmonic nonlinear medium  550  comprises a doubling crystal that is configured for type I phase matching, but the crystal may also be configured for type II phase matching. 
     Although laser beams  330 ,  342  and  552  propagate superimposed along at least a portion optical path  304  (e.g., laser beams  330  and  342  are superimposed between optical energy coupler  314  and end mirror  510  and laser beams  330 ,  342  and  552  are superimposed between end mirror  510  and an intracavity output coupler  570 ), laser beams  330 ,  342 , and  552  are shown axially offset from one another for illustration purposes. 
     An AR coating may optionally be applied to higher order harmonic nonlinear medium  550 . For example, a single or multilayer dielectric coating having AR characteristics at the fundamental wavelength, second harmonic wavelength, and fourth harmonic wavelength may be applied to an optical surface of higher order harmonic nonlinear medium  550 . 
     End mirror  510  is similar to end mirror  310 , but is adapted to be reflective of laser beams  330 ,  342 , and  552 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 99% reflection, at the fundamental wavelength, the second harmonic wavelength, and the fourth harmonic wavelength may be applied to end mirror  510 . 
     According to one embodiment, intracavity output coupler  570  is positioned within resonant laser cavity  306  along optical path  304 . Intracavity output coupler  570  is similar to intracavity output coupler  370 , but is adapted to be reflective of laser beam  552  and anti-reflective of laser beams  330  and  342 . Thus, output coupler  570  essentially separates laser beam  552  from laser beams  330  and  342 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the fourth harmonic wavelength and having AR characteristics, such as less than about 1% reflection at the fundamental wavelength and second harmonic wavelength may be applied to output coupler  570 . Thus, output coupler  570  directs almost all of the incident fourth harmonic laser beam energy out of resonant laser cavity  306 . In addition, output coupler  570  permits almost all of the incident fundamental laser beam energy and the second harmonic laser beam energy to propagate through output coupler  570 . Thus, the reflected fourth harmonic laser beam energy exits resonant laser cavity  306  through output coupler  570  as an output fourth harmonic laser beam  572 . Separation of the fourth harmonic laser beam energy from the fundamental laser beam energy and the second harmonic laser beam energy at output coupler  570  may help prevent damage to nonlinear medium  340 , laser medium  302 , and other optical components from the fourth harmonic laser beam energy. The fourth harmonic laser beam energy may exit resonant laser cavity  306  in other ways. For example, end mirror  510  may include a coating having AR characteristics at the fourth harmonic wavelength to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306  through end mirror  510 . In addition, a reflective or transmissive output coupler may be positioned in optical association with higher order harmonic nonlinear medium  550  to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306 . 
     The radius of curvature and position of curved mirror  360  (e.g., relative to optical energy coupler  314 ) is selected as described with reference to  FIG. 3 . Thus, the radius of curvature and position of curved mirror  360  are selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam (e.g., between end mirror  510  and curved mirror  360 ). In addition, mode-matching optics similar or identical to that described with reference to  FIG. 3  may be used to match a beam mode of the reflected second harmonic laser beam energy with a beam mode of the incoming second harmonic laser beam energy. Maintaining a substantially uniform beam radius and beam divergence of laser beam  342  helps improve fourth harmonic laser beam energy generation efficiency without affecting a beam mode of the fourth harmonic laser beam energy and without significantly deteriorating a beam quality of the fourth harmonic laser beam energy. 
       FIG. 6  is a schematic diagram of a laser  600  that efficiently generates a fourth harmonic laser beam  672  by recycling an unused portion of a second harmonic laser beam while maintaining fourth harmonic beam quality, according to another embodiment. Laser  600  is similar to laser  300  described with reference to  FIG. 3  but is adapted to generate fourth harmonic laser beam  672  with a higher order harmonic nonlinear medium  650  positioned outside of resonant laser cavity  306 . 
     Laser  600  incorporates higher order harmonic nonlinear medium  650  instead of higher order harmonic nonlinear medium  350 . Higher order harmonic nonlinear medium  650  is adapted to convert laser radiation or energy having the second harmonic wavelength into laser radiation having a fourth harmonic wavelength. Higher order harmonic nonlinear medium  650  is positioned outside of resonant laser cavity  306 , which is formed by end mirror  308  and an end mirror  610 , so that higher order harmonic nonlinear medium  650  is in optical association with resonant laser cavity  306 . Preferably, higher order harmonic nonlinear medium  650  is positioned on a side of end mirror  610  opposite nonlinear medium  340 . Higher order harmonic nonlinear medium  650  is preferably oriented such that laser beam  342  strikes an optical surface of higher order harmonic nonlinear medium  650  with normal incidence. As laser beam  342  passes through higher order harmonic nonlinear medium  650 , higher order harmonic nonlinear medium  650  generates a laser beam  652  having one-quarter of the wavelength of laser beam  330  (i.e., a frequency that is four times the fundamental frequency of laser beam  330 ). Higher order harmonic nonlinear medium  650  may comprise any of the nonlinear crystals described with reference to nonlinear mediums  340  and  350  and may comprise the same or different nonlinear crystal as nonlinear mediums  340  and  350 . According to a preferred embodiment, higher order harmonic nonlinear medium  650  comprises a doubling crystal that is configured for type I phase matching, but the crystal may also be configured for type II phase matching. 
     Although laser beams  330  and  342  propagate superimposed along at least a portion optical path  304  (e.g., laser beams  330  and  342  are superimposed between optical energy coupler  314  and end mirror  610 ), laser beams  330  and  342  are shown axially offset from one another for illustration purposes. In addition, although laser beams  342  and  652  propagate superimposed between end mirror  610  and a curved extracavity output coupler  670 , laser beams  342  and  652  are shown axially offset from each other for illustration purposes. 
     An AR coating may optionally be applied to higher order harmonic nonlinear medium  650 . For example, a single or multilayer dielectric coating having AR characteristics at the fundamental wavelength, second harmonic wavelength, and fourth harmonic wavelength may be applied to an optical surface of higher order harmonic nonlinear medium  650 . 
     End mirror  610  is similar to end mirror  310 , but is adapted to be reflective of laser beams  330  and  652  and anti-reflective of laser beam  342 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the fundamental wavelength and fourth harmonic wavelength and having AR characteristics, such as less than about 1% reflection, at the second harmonic wavelength may be applied to end mirror  610 . In addition, a coating may be applied to both opposing optical surfaces of end mirror  610 . Thus, end mirror  610  reflects almost all of the incident fundamental laser beam energy back toward nonlinear medium  340  and reflects almost all of the incident fourth harmonic laser beam energy toward an curved output coupler  670 . In addition, end mirror  610  allows almost all of the incident second harmonic laser beam energy to pass through end mirror  610 . 
     A curved extracavity output coupler  670  is positioned outside of resonant laser cavity  306  so that output coupler  670  is in optical association with higher order harmonic nonlinear medium  650 . According to a preferred embodiment, output coupler  670  is positioned such that higher order harmonic nonlinear medium  650  is interposed between end mirror  610  and output coupler  670 . Output coupler  670  is adapted to be reflective of laser beam  342  and anti-reflective of laser beam  652 . Thus, output coupler  670  essentially separates laser beam  652  from laser beam  342 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the second harmonic wavelength and having AR characteristics, such as less than about 1% reflection, at the fourth harmonic wavelength may be applied to output coupler  670 . Thus, output coupler  670  reflects almost all of the incident second harmonic laser beam energy back toward higher order harmonic nonlinear medium  650  and curved mirror  360 . In addition, output coupler  670  permits almost all of the incident fourth harmonic laser beam energy to propagate through output coupler  670  and thus out of a cavity  680  formed by end mirror  610  and output coupler  670 . Thus, the transmitted fourth harmonic laser beam energy exits cavity  680  through output coupler  670  as an output fourth harmonic laser beam  672 . The fourth harmonic laser beam energy may exit cavity  680  in other ways. For example, a reflective or transmissive output coupler may be positioned in optical association with higher order harmonic nonlinear medium  650  to allow the fourth harmonic laser beam energy to exit cavity  680 . 
     A radius of curvature and position of output coupler  670  (e.g., relative to end mirror  610 ) is selected in a similar manner as described with reference to curved mirror  360  of  FIG. 3 . Thus, the radius of curvature and position of output coupler  670  are selected so that a beam mode (e.g., beam radius and beam divergence) of the reflected second harmonic laser beam energy substantially matches a beam mode of the incoming second harmonic laser beam energy (e.g., the portion of the second harmonic laser beam energy that was not used to generate the fourth harmonic laser beam energy). In other words, a radius of curvature and position of output coupler  670  are selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam (e.g., between output coupler  670  and curved mirror  360 ). Maintaining a substantially uniform beam radius and beam divergence of laser beam  342  helps improve higher order laser beam energy (e.g., fourth harmonic laser beam energy) generation efficiency without affecting a beam mode of the higher order laser beam energy and without significantly deteriorating a beam quality of the higher order laser beam energy. 
     For example, laser  600  may be designed such that end mirror  610  is positioned coincident with a beam waist of the fundamental laser beam energy. Thus, as the fundamental laser beam energy propagates from optical energy coupler  314  to end mirror  610 , the fundamental laser beam energy converges. In addition, as the second harmonic laser beam energy, which is generated by nonlinear medium  340  from at least a portion of the fundamental laser beam energy, propagates from nonlinear medium  340  to end mirror  610 , the second harmonic laser beam energy converges. 
     After being reflected by end mirror  610 , the fundamental laser beam energy begins diverging as it propagates from end mirror  610  toward optical energy coupler  314 . After passing through end mirror  610 , the second harmonic laser beam energy begins diverging and continues to diverge as it propagates toward higher order harmonic nonlinear medium  650 . A portion of the second harmonic laser beam energy is not used by higher order harmonic nonlinear medium  650  to generate the fourth harmonic laser beam energy. Thus, the unused portion of the second harmonic laser beam energy is reflected by output coupler  670  and propagates back-and-forth between output coupler  670  and curved mirror  360 . 
     If output coupler  670  were flat, the unused portion of the second harmonic laser beam energy would continue to diverge after being reflected by output coupler  670 . Thus, each time the unused portion of the second harmonic laser beam energy passed through higher order harmonic nonlinear medium  650 , a beam mode (e.g., beam radius and beam divergence) of the unused portion of the second harmonic laser beam energy would be different and a significantly deteriorated fourth harmonic laser beam energy beam quality would result. Accordingly, a radius of curvature and position of output coupler  670  are selected so that a beam mode of the unused portion of the second harmonic laser beam energy reflected by output coupler  670  substantially matches a beam mode of the incoming second harmonic laser beam energy (e.g., a beam mode of the second harmonic laser beam energy that is generated by nonlinear medium  650  from the second harmonic laser beam energy). In other words, the position and concave shape of output coupler  670  are selected such that the unused portion of the second harmonic laser beam energy begins converging (e.g., in a similar or identical manner as the second harmonic laser beam energy generated by nonlinear medium  340  as the fundamental laser beam energy propagates from end mirror  610  toward optical energy coupler  314 ) after being reflected by output coupler  670 . 
     A radius of curvature and position of output coupler  670  can vary based on the design of laser  600 . For example, if nonlinear medium  340  generates a Gaussian second harmonic beam, a radius of curvature of output coupler  670  is selected to substantially match a radius of curvature of a Gaussian beam wavefront of the second harmonic beam at the position of output coupler  670 . In other words, the radius of curvature of the Gaussian beam wavefront is first calculated for a given position of output coupler  670 . Then, the radius of curvature of output coupler  670  is selected to substantially match the calculated radius of curvature of the Gaussian beam wavefront (at the position of output coupler  670 ) so that the second harmonic mode of the reflected beam is preserved. The radius of curvature of a Gaussian beam wavefront at a particular position can be determined from the wavelength of the beam, beam radius at the beam waist, beam quality factor (M 2 ), and distance from the beam waist as previously described with reference to  FIG. 3 . 
     If a position of output coupler  670  is adjusted (e.g., moved closer to end mirror  610 ), a radius of curvature of the Gaussian beam wavefront at the adjusted position can be calculated so that a corresponding radius of curvature can be selected for output coupler  670 . In a similar vein, a position of output coupler  670  can be adjusted to compensate for changes in the radius of curvature of output coupler  670 . While examples have been provided in which output coupler  670  is selected to have a concave shape, it is possible that output coupler  670  is selected to have a convex shape (e.g., to make the reflected beam diverge upon reflection) or flat shape (e.g., a radius of curvature may be selected to be infinity). In addition, mode-matching optics similar or identical to that described with reference to  FIG. 3  may be used to match a beam mode of the reflected second harmonic laser beam energy (e.g., reflected by output coupler  670 ) with a beam mode of the incoming second harmonic laser beam energy. 
     The radius of curvature and position of curved mirror  360  (e.g., relative to optical energy coupler  314  and output coupler  670 ) is selected as described with reference to  FIG. 3 . For example, after the unused portion of the second harmonic laser beam energy is reflected by output coupler  670 , the unused portion of the second harmonic laser beam energy begins converging as it propagates toward nonlinear medium  650 . As the unused portion of the second harmonic laser beam energy passes through nonlinear medium  650  again, nonlinear medium  650  generates additional fourth harmonic laser beam energy (which is reflected by end mirror  610  and exits cavity  680  via output coupler  670 ). A portion of the unused portion of the second harmonic laser beam energy is unused and propagates toward end mirror  610 . The twice unused portion of the second harmonic laser beam energy will converge as it propagates from nonlinear medium  650  to end mirror  610 . After the twice unused portion of the second harmonic laser beam energy propagates through the end mirror  610 , the twice unused portion of the second harmonic laser beam energy begins to diverge. The twice unused portion of the second harmonic laser beam energy continues diverging as it propagates from end mirror  610  to curved mirror  360 . In addition, the fundamental laser beam energy diverges as it propagates from end mirror  610  toward optical energy coupler  314 . Nonlinear medium  340  generates additional second harmonic laser beam energy as the diverging fundamental laser beam energy passes through nonlinear medium  340  in a direction toward optical energy coupler  314 . The second harmonic laser beam energy not reflected by optical energy coupler  314  exits resonant laser cavity  306  as residual second harmonic laser beam energy. Thus, residual second harmonic laser beam energy may comprise the additional second harmonic laser beam energy (e.g., the second harmonic laser beam energy generated as the fundamental laser beam energy passes through nonlinear medium  340  in a direction toward optical energy coupler  314 ) and the twice unused portion of the second harmonic laser beam energy. 
     If mirror  360  were flat, the residual second harmonic laser beam energy would continue to diverge after reflecting off mirror  360 . Thus, as the residual second harmonic laser beam energy passed through nonlinear medium  650 , a beam mode (e.g., beam radius and beam divergence) of the residual second harmonic laser beam energy would be different than a beam mode of the second harmonic laser beam energy generated by nonlinear medium  340  and a significantly deteriorated fourth harmonic laser beam energy beam quality would result. Accordingly, a radius of curvature and position of curved mirror  360  are selected so that a beam mode of the residual second harmonic laser beam energy reflected by curved mirror  360  substantially matches a beam mode of the incoming second harmonic laser beam energy (which may include the second harmonic laser beam energy generated by nonlinear medium  340  from the fundamental laser beam energy in a direction toward optical energy coupler  314 ). In other words, the position and concave shape of curved mirror  360  are selected such that the residual second harmonic laser beam energy begins converging (e.g., in a similar or identical manner as the second harmonic laser beam energy generated by nonlinear medium  340  in a direction toward  610 ) after being reflected by curved mirror  360 . 
     A radius of curvature and position of curved mirror  360  can vary based on the design of laser  600 . For example, if nonlinear medium  340  generates a Gaussian second harmonic beam, a radius of curvature of curved mirror  360  is selected to substantially match a radius of curvature of a Gaussian beam wavefront of the second harmonic beam at the position of curved mirror  360 . In other words, the radius of curvature of the Gaussian beam wavefront is first calculated for a given position of curved mirror  360 . Then, the radius of curvature of curved mirror  360  is selected to substantially match the calculated radius of curvature of the Gaussian beam wavefront (at the position of curved mirror  360 ) so that the second harmonic mode of the reflected beam is preserved. The radius of curvature of a Gaussian beam wavefront at a particular position can be determined from the wavelength of the beam, beam radius at the beam waist, beam quality factor (M 2 ), and distance from the beam waist as previously described with reference to  FIG. 3 . 
     If a position of curved mirror  360  is adjusted (e.g., moved closer to optical energy coupler  314 ), a radius of curvature of the Gaussian beam wavefront at the adjusted position can be calculated so that a corresponding radius of curvature can be selected for curved mirror  360 . In a similar vein, a position of curved mirror  360  can be adjusted to compensate for changes in the radius of curvature of curved mirror  360 . While examples have been provided in which curved mirror  360  is selected to have a concave shape, it is possible that curved mirror  360  is selected to have a convex shape (e.g., to make the reflected beam diverge upon reflection) or flat shape (e.g., a radius of curvature may be selected to be infinity). In addition, mode-matching optics similar or identical to that described with reference to  FIG. 3  may be used to match a beam mode of the reflected second harmonic laser beam energy (e.g., reflected by curved mirror  360 ) with a beam mode of the incoming second harmonic laser beam energy. 
       FIG. 7  is a schematic diagram of a laser  700  that efficiently generates a fourth harmonic laser beam  772  by recycling an unused portion of a second harmonic laser beam while maintaining fourth harmonic beam quality, according to yet another embodiment. Laser  700  is similar to laser  300  described with reference to  FIG. 3  but is adapted to generate fourth harmonic laser beam  772 . 
     Laser  700  incorporates a mixing nonlinear medium  780  in addition to higher order harmonic nonlinear medium  350 . As described with reference to  FIG. 3 , higher order harmonic nonlinear medium  350  is positioned along optical path  304  within resonant laser cavity  306 . Preferably, higher order harmonic nonlinear medium  350  is interposed between mixing nonlinear medium  780  and an end mirror  710 . Higher order harmonic nonlinear medium  350  is preferably oriented such that laser beam  330 , laser beam  342 , or both, strike an optical surface of higher order harmonic nonlinear medium  350  with normal incidence. In a preferred embodiment, higher order harmonic nonlinear medium  350  converts laser radiation or energy having a fundamental wavelength and laser radiation or energy having a second harmonic wavelength into laser radiation or energy having a third harmonic wavelength. Thus, as laser beam  330  and laser beam  342  pass through higher order harmonic nonlinear medium  350 , higher order harmonic nonlinear medium  350  generates laser beam  352  having one-third of the wavelength of laser beam  330  (i.e., a frequency that is three times the fundamental frequency of laser beam  330 ). Higher order harmonic nonlinear medium  350  may comprise any of the nonlinear crystals described with reference to nonlinear medium  340  and may comprise the same or different nonlinear crystal as nonlinear medium  340 . The nonlinear crystal may be configured for either type I or type II phase matching. 
     According to one embodiment, mixing nonlinear medium  780  is adapted to convert or mix laser radiation having the fundamental wavelength and laser radiation having the third harmonic wavelength into laser radiation having a fourth harmonic wavelength. Alternatively, mixing nonlinear medium  780  may be adapted to convert or mix laser radiation having the second harmonic wavelength and laser radiation having the third harmonic wavelength into laser radiation having a fifth harmonic wavelength (i.e., to generate a fifth harmonic laser beam  772 ). Mixing nonlinear medium  780  is positioned along optical path  304  within resonant laser cavity  306 , which is formed by end mirror  308  and an end mirror  710 . Preferably, mixing nonlinear medium  780  is interposed between an intracavity output coupler  770  and nonlinear medium  350 . Mixing nonlinear medium  780  is preferably oriented such that laser beam  330 , laser beam  342 , laser beam  352 , or any combination thereof, strike an optical surface of mixing nonlinear medium  780  at normal incidence. As laser beam  330  and laser beam  352  passes through mixing nonlinear medium  780 , mixing nonlinear medium  780  generates a laser beam  782  having one-quarter of the wavelength of laser beam  330  (i.e., a frequency that is four times the fundamental frequency of laser beam  330 ). Mixing nonlinear medium  780  may comprise any of the nonlinear crystals described with reference to nonlinear mediums  340  and  350  and may comprise the same or different nonlinear crystal as nonlinear mediums  340  and  350 . According to a preferred embodiment, mixing nonlinear medium  780  comprises a mixing crystal that is configured for type I phase matching, but the crystal may also be configured for type II phase matching. As shown in  FIG. 7 , nonlinear medium  340 , mixing nonlinear medium  780 , and nonlinear medium  350  are in optical series with one another. 
     Although laser beams  330 ,  342 ,  352 , and  782  propagate superimposed along at least a portion optical path  304  (e.g., laser beams  330  and  342  are superimposed between optical energy coupler  314  and end mirror  710 , laser beams  330 ,  342  and  352  are superimposed between end mirror  710  and an intracavity output coupler  770 , and laser beams  330 ,  342 ,  352 , and  782  are superimposed between mixing nonlinear medium  780  and intracavity output coupler  770 ), laser beams  330 ,  342 ,  352 , and  782  are shown axially offset from one another for illustration purposes. 
     An AR coating may optionally be applied to higher order harmonic nonlinear medium  350 , mixing nonlinear medium  780 , or both. For example, a single or multilayer dielectric coating having AR characteristics at the fundamental wavelength, second harmonic wavelength, third harmonic wavelength, and fourth harmonic wavelength may be applied to an optical surface of higher order harmonic nonlinear medium  350 , mixing nonlinear medium  780 , or both. 
     End mirror  710  is similar to end mirror  310 , but may be adapted to be reflective of laser beam  782  in addition to laser beams  330 ,  342 , and  352 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 99% reflection, at the fundamental wavelength, the second harmonic wavelength, the third harmonic wavelength, and the fourth harmonic wavelength may be applied to end mirror  710 . 
     An intracavity output coupler  770  is positioned within resonant laser cavity  306  along optical path  304 , according to one embodiment. Output coupler  770  is similar to intracavity output coupler  370 , but is adapted to be reflective of laser beam  782  and anti-reflective of laser beams  330  and  342 . Output coupler  770  may be reflective or anti-reflective of laser beam  352 . Thus, output coupler  770  essentially separates laser beam  782  from laser beams  330  and  342 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the fourth harmonic wavelength and having AR characteristics, such as less than about 1% reflection, at the fundamental wavelength and second harmonic wavelength may be applied to output coupler  770 . Thus, output coupler  770  directs almost all of the incident fourth harmonic laser beam energy out of resonant laser cavity  306 . In addition, output coupler  770  permits almost all of the incident fundamental laser beam energy and the second harmonic laser beam energy to propagate through output coupler  770 . Thus, the reflected fourth harmonic laser beam energy exits resonant laser cavity  306  through output coupler  770  as an output fourth harmonic laser beam  772 . Separation of the fourth harmonic laser beam energy from the fundamental laser beam energy and the second harmonic laser beam energy at output coupler  770  may help prevent damage to nonlinear medium  340 , laser medium  302 , and other optical components. The fourth harmonic laser beam energy may exit resonant laser cavity  306  in other ways. For example, end mirror  710  may include a coating having AR characteristics at the fourth harmonic wavelength to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306  through end mirror  710  (e.g., if the positions of mixing nonlinear medium  780  and nonlinear medium  350  are interchanged). In addition, a reflective or transmissive output coupler may be positioned in optical association with higher order harmonic nonlinear medium  780  to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306 . 
     The radius of curvature and position of curved mirror  360  (e.g., relative to optical energy coupler  314 ) are selected as described with reference to  FIG. 3 . Thus, the radius of curvature and position of curved mirror  360  are selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam (e.g., between end mirror  710  and curved mirror  360 ). In addition, mode-matching optics similar or identical to that described with reference to  FIG. 3  may be used to match a beam mode of the reflected second harmonic laser beam energy (e.g., reflected by curved mirror  360 ) with a beam mode of the incoming second harmonic laser beam energy. Maintaining a substantially uniform beam radius and beam divergence of laser beam  342  helps improve fourth harmonic laser beam energy generation efficiency without affecting a beam mode of the fourth harmonic laser beam energy and without significantly deteriorating a beam quality of the fourth harmonic laser beam energy. 
       FIG. 8  is a schematic diagram of a laser  800  that efficiently generates a fourth harmonic laser beam  872  by recycling an unused portion of a second harmonic laser beam while maintaining fourth harmonic beam quality, according to still another embodiment. Laser  800  is similar to laser  700  described with reference to  FIG. 7  but nonlinear medium  350 , mixing nonlinear medium  780 , and the output coupler are positioned in a different optical arrangement. In  FIG. 7 , the components are arranged in an optical array, starting with end mirror  710 , followed by nonlinear medium  350 , followed by mixing nonlinear medium  780 , followed by output coupler  770 , and ending with nonlinear element  340 . In  FIG. 8 , the components are arranged in a different optical array, starting with an end mirror  810 , followed by mixing nonlinear medium  780 , followed by output coupler  870 , followed by nonlinear medium  350 , and ending with nonlinear element  340 . 
     As described with reference to  FIG. 7 , higher order harmonic nonlinear medium  350  converts laser radiation or energy having a fundamental wavelength and laser radiation or energy having a second harmonic wavelength into laser radiation or energy having a third harmonic wavelength. Thus, as laser beam  330  and laser beam  342  pass through higher order harmonic nonlinear medium  350 , higher order harmonic nonlinear medium  350  generates laser beam  352  having one-third of the wavelength of laser beam  330  (i.e., a frequency that is three times the fundamental frequency of laser beam  330 ). In addition, as described with reference to  FIG. 7  mixing nonlinear medium  780  is adapted to convert or mix laser radiation having the fundamental wavelength and laser radiation having the third harmonic wavelength into laser radiation having a fourth harmonic wavelength, according to one embodiment. Thus, as laser beam  330  and laser beam  352  pass through mixing nonlinear medium  780 , mixing nonlinear medium  780  generates a laser beam  782  having one-quarter of the wavelength of laser beam  330  (i.e., a frequency that is four times the fundamental frequency of laser beam  330 ). Alternatively, mixing nonlinear medium  780  may be adapted to convert or mix laser radiation having the second harmonic wavelength and laser radiation having the third harmonic wavelength into laser radiation having a fifth harmonic wavelength (i.e., to generate a fifth harmonic laser beam  872 ). 
     Although laser beams  330 ,  342 ,  352 , and  782  propagate superimposed along at least a portion optical path  304  (e.g., laser beams  330  and  342  are superimposed between optical energy coupler  314  and end mirror  810 , laser beams  330 ,  342  and  352  are superimposed between nonlinear medium  350  and end mirror  810 , and laser beams  330 ,  342 ,  352 , and  782  are superimposed between end mirror  810  and an intracavity output coupler  870 ), laser beams  330 ,  342 ,  352 , and  782  are shown axially offset from one another for illustration purposes. 
     End mirror  810  is similar to end mirror  310 , but is adapted to be reflective of laser beam  782  in addition to laser beams  330 ,  342 , and  352 . For example, a single or multilayer dielectric coating having HR characteristics, such as over about 99% reflection, at the fundamental wavelength, the second harmonic wavelength, the third harmonic wavelength, and the fourth harmonic wavelength may be applied to end mirror  810 . 
     An intracavity output coupler  870  is positioned within resonant laser cavity  306  along optical path  304 , according to one embodiment. Output coupler  870  is similar to intracavity output coupler  770 , but is adapted to be reflective of laser beam  782  and anti-reflective of laser beams  330 ,  342 , and  352 . Thus, output coupler  870  essentially separates laser beam  782  from laser beams  330 ,  342 , and possibly  352 . Output coupler  870  may be anti-reflective of laser beam  352  in one direction (e.g., as laser beam  352  propagates from nonlinear medium  350  toward end mirror  810 ) and be reflective of laser beam  352  in the other direction (e.g., as laser beam  352  propagates from end mirror  810  toward output coupler  870 ) so that the third harmonic laser beam energy that is not used by mixing nonlinear element  780  exits cavity  306 ). Alternatively, output coupler  870  may be anti-reflective of laser beam  352  in both directions (e.g., as laser beam  352  propagates from nonlinear medium  350  toward end mirror  810  and as laser beam  352  propagates from end mirror  810  toward nonlinear medium  350 ). A single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the fourth harmonic wavelength and having AR characteristics, such as less than about 1% reflection, at the fundamental wavelength, second harmonic wavelength, and third harmonic wavelength may be applied to output coupler  870 . Thus, output coupler  870  directs almost all of the incident fourth harmonic laser beam energy out of resonant laser cavity  306 . In addition, output coupler  870  permits almost all of the incident fundamental laser beam energy, the second harmonic laser beam energy, and possibly the third harmonic laser beam energy to propagate through output coupler  870 . Thus, the reflected fourth harmonic laser beam energy exits resonant laser cavity  306  through output coupler  870  as an output fourth harmonic laser beam  872 . Separation of the fourth harmonic laser beam energy from the fundamental laser beam energy, the second harmonic laser beam energy, and possibly the third harmonic laser beam energy at output coupler  870  may help prevent damage to nonlinear medium  340 , laser medium  302 , and other optical components from the fourth harmonic laser beam energy. The fourth harmonic laser beam energy may exit resonant laser cavity  306  in other ways. For example, end mirror  810  may include a coating having AR characteristics at the fourth harmonic wavelength to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306  through end mirror  810 . In addition, a reflective or transmissive output coupler may be positioned in optical association with higher order harmonic nonlinear medium  780  to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306 . 
     The radius of curvature and position of curved mirror  360  (e.g., relative to optical energy coupler  314 ) are selected as described with reference to  FIG. 3 . Thus, the radius of curvature and position of curved mirror  360  are selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam (e.g., between end mirror  810  and curved mirror  360 ). In addition, mode-matching optics similar or identical to that described with reference to  FIG. 3  may be used to match a beam mode of the reflected second harmonic laser beam energy (e.g., reflected by curved mirror  360 ) with a beam mode of the incoming second harmonic laser beam energy. Maintaining a substantially uniform beam radius and beam divergence of laser beam  342  helps improve fourth harmonic laser beam energy generation efficiency without affecting a beam mode of the fourth harmonic laser beam energy and without significantly deteriorating a beam quality of the fourth harmonic laser beam energy. 
       FIG. 9  is a schematic diagram of a laser  900  that efficiently generates a fourth harmonic laser beam  972  by recycling an unused portion of a second harmonic laser beam and a third harmonic laser beam while maintaining fourth harmonic beam quality, according to one embodiment. Laser  900  is similar to laser  800  described with reference to  FIG. 8 , but includes a curved mirror  910  for recycling the third harmonic laser beam energy that was not used by mixing nonlinear element  780  to generate the fourth harmonic laser beam energy. In an alternative embodiment, mixing nonlinear medium  780  may be adapted to convert or mix laser radiation having the second harmonic wavelength and laser radiation having the third harmonic wavelength into laser radiation having a fifth harmonic wavelength (i.e., to generate a fifth harmonic laser beam  972 ). 
     Although laser beams  330 ,  342 ,  352 , and  782  propagate superimposed along at least a portion optical path  304  (e.g., laser beams  330  and  342  are superimposed between optical energy coupler  314  and end mirror  810 , laser beams  330 ,  342  and  352  are superimposed between curved mirror  910  and end mirror  810 , and laser beams  330 ,  342 ,  352 , and  782  are superimposed between end mirror  810  and an intracavity output coupler  970 ), laser beams  330 ,  342 ,  352 , and  782  are shown axially offset from one another for illustration purposes. 
     An intracavity output coupler  970  is positioned within resonant laser cavity  306  along optical path  304 , according to one embodiment. Output coupler  970  is similar to intracavity output coupler  870  described with reference to  FIG. 8 , but is adapted to be reflective of laser beam  782  and anti-reflective of laser beams  330 ,  342 , and  352 . In particular, output coupler  970  is anti-reflective of laser beam  352  in both directions (e.g., as laser beam  352  propagates from nonlinear medium  350  toward mixing nonlinear medium  780  and as laser beam  352  propagates from end mirror  810  toward curved mirror  910 ). Thus, output coupler  970  essentially separates laser beam  782  from laser beams  330 ,  342 , and  352 . A single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the fourth harmonic wavelength and having AR characteristics, such as less than about 1% reflection, at the fundamental wavelength, second harmonic wavelength, and third harmonic wavelength may be applied to output coupler  970 . Thus, output coupler  970  directs almost all of the incident fourth harmonic laser beam energy out of resonant laser cavity  306 . In addition, output coupler  970  permits almost all of the incident fundamental laser beam energy, the second harmonic laser beam energy, and the third harmonic laser beam energy to propagate through output coupler  970 . Thus, the reflected fourth harmonic laser beam energy exits resonant laser cavity  306  through output coupler  970  as an output fourth harmonic laser beam  972 . Separation of the fourth harmonic laser beam energy from the fundamental laser beam energy, the second harmonic laser beam energy, and the third harmonic laser beam energy at output coupler  970  may help prevent damage to nonlinear medium  340 , laser medium  302 , and other optical components from the fourth harmonic laser beam energy. The fourth harmonic laser beam energy may exit resonant laser cavity  306  in other ways. For example, end mirror  810  may include a coating having AR characteristics at the fourth harmonic wavelength to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306  through end mirror  810 . In addition, a reflective or transmissive output coupler may be positioned in optical association with higher order harmonic nonlinear medium  780  to allow the fourth harmonic laser beam energy to exit resonant laser cavity  306 . 
     Curved mirror  910  is positioned within resonant laser cavity  306  along optical path  304 . Curved mirror  910  is preferably adapted to be reflective of laser beam  352  (i.e., the third harmonic laser beam energy) and anti-reflective of laser beams  330  and  342 . Thus, curved mirror  910  essentially separates laser beam  352  from laser beams  330  and  342 . A single or multilayer dielectric coating having HR characteristics, such as over about 95% reflection, at the third harmonic wavelength and having AR characteristics, such as less than about 1% reflection, at the fundamental wavelength and second harmonic wavelength may be applied to curved mirror  910 . Thus, curved mirror  910  reflects almost all of the incident third harmonic laser beam energy back toward mixing nonlinear element  780  so that mixing nonlinear element  780  can generate additional fourth harmonic laser beam energy from the reflected third harmonic laser beam energy. In addition, curved mirror  910  permits almost all of the incident fundamental laser beam energy and the second harmonic laser beam energy to propagate through curved mirror  910 . Separation of the third harmonic laser beam energy from the fundamental laser beam energy and the second harmonic laser beam energy at curved mirror  910  allows the third harmonic laser beam energy to be recycled and may help prevent damage to nonlinear medium  340 , laser medium  302 , and other optical components from the third harmonic laser beam energy. 
     As will be described in more detail below, a radius of curvature and position of curved mirror  910  are selected so that a beam mode (e.g., beam radius and beam divergence) of the reflected third harmonic laser beam energy (i.e., the reflected unused third harmonic laser beam energy and the reflected third harmonic laser beam energy generated by nonlinear medium  350  in a direction toward curved mirror  910 ) substantially matches a beam mode of the incoming third harmonic laser beam energy. In other words, a radius of curvature and position of curved mirror  910  (e.g., relative to end mirror  810 ) is selected so that a beam radius and a beam divergence of the reflected third harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming third harmonic laser beam energy everywhere along a beam path of the incoming third harmonic laser beam (e.g., between end mirror  810  and curved mirror  910 ). Maintaining a substantially uniform beam radius and beam divergence of laser beam  352  helps improve higher order laser beam energy (e.g., fourth harmonic laser beam energy) generation efficiency without affecting a beam mode of the higher order laser beam energy and without significantly deteriorating a beam quality of the higher order laser beam energy. 
     For example, laser  900  may be designed such that end mirror  810  is positioned coincident with a beam waist of the fundamental laser beam energy. Thus, as the fundamental laser beam energy propagates from optical energy coupler  314  to end mirror  810 , the fundamental laser beam energy converges. In addition, as the second harmonic laser beam energy, which is generated by nonlinear medium  340  from at least a portion of the fundamental laser beam energy, propagates from nonlinear medium  340  to end mirror  810 , the second harmonic laser beam energy converges. Further, as the third harmonic laser beam energy, which is generated by nonlinear medium  350  from at least a portion of the fundamental laser beam energy and at least a portion of the second harmonic laser beam energy, propagates from nonlinear medium  350  to end mirror  810 , the third harmonic laser beam energy converges. A portion of the third harmonic laser beam energy will not be used by mixing nonlinear medium  780  to generate the fourth harmonic laser beam energy. Thus, the unused portion of the third harmonic laser beam energy will be reflected by end mirror  810  and propagate back-and-forth between mirrors  810  and  910 . 
     After being reflected by end mirror  810 , the fundamental laser beam energy begins diverging, the second harmonic laser beam energy begins diverging, and the unused portion of the third harmonic laser beam energy begins diverging. Thus, the fundamental laser beam energy and the second harmonic laser beam energy diverge as the fundamental and second harmonic energies propagate from end mirror  810  toward optical energy coupler  314 . Similarly, the unused portion of the third harmonic laser beam energy diverges as it propagates from end mirror  810  toward curved mirror  910 . The diverging fundamental laser beam energy and diverging second harmonic laser beam energy generate additional third harmonic laser beam energy as the diverging fundamental laser beam energy and second harmonic laser beam energy pass through nonlinear element  350 . If mirror  910  were flat, the unused portion of the third harmonic laser beam energy and the newly generated third harmonic laser beam energy would continue to diverge after being reflected by mirror  910 . Thus, each time the third harmonic laser beam energy passed through mixing nonlinear medium  780 , a beam mode (e.g., beam radius and beam divergence) of the third harmonic laser beam energy would be different and a significantly deteriorated fourth harmonic laser beam energy beam quality would result. Accordingly, a radius of curvature and position of curved mirror  910  are selected so that a beam mode of the third harmonic laser beam energy reflected by curved mirror  910  substantially matches a beam mode of the incoming third harmonic laser beam energy (e.g., a beam mode of the unused portion of the third harmonic laser beam energy and a beam mode of the third harmonic laser beam energy that is generated by nonlinear medium  350 ). In other words, the position and concave shape of curved mirror  910  are selected such that the third harmonic laser beam energy begins converging (e.g., in a similar or identical manner as the third harmonic laser beam energy generated by nonlinear medium  350  as the fundamental laser beam energy and second harmonic laser beam energy propagate from nonlinear medium  340  towards mixing nonlinear medium  780 ) after being reflected by curved mirror  910 . 
     A radius of curvature and position of curved mirror  910  can vary based on the design of laser  900 . For example, if nonlinear medium  350  generates a Gaussian third harmonic beam, a radius of curvature of curved mirror  910  is selected to substantially match a radius of curvature of a Gaussian beam wavefront of the third harmonic beam at the position of curved mirror  910 . In other words, the radius of curvature of the Gaussian beam wavefront is first calculated for a given position of curved mirror  910 . Then, the radius of curvature of curved mirror  910  is selected to substantially match the calculated radius of curvature of the Gaussian beam wavefront (at the position of curved mirror  910 ) so that the third harmonic mode of the reflected beam is preserved. The radius of curvature of a Gaussian beam wavefront at a particular position can be determined from the wavelength of the beam, beam radius at the beam waist, beam quality factor (M 2 ), and distance from the beam waist as described with reference to  FIG. 3 . 
     If a position of curved mirror  910  is adjusted (e.g., moved closer to end mirror  810 ), a radius of curvature of the Gaussian beam wavefront at the adjusted position can be calculated so that a corresponding radius of curvature can be selected for curved mirror  910 . In a similar vein, a position of curved mirror  910  can be adjusted to compensate for changes in the radius of curvature of curved mirror  910 . In addition, if end mirror  810  is changed from a flat mirror to a curved mirror, a position of curved mirror  910 , a radius of curvature of curved mirror  910 , or both, may be adjusted accordingly so that the third harmonic mode of the beam reflected between mirrors  810  and  910  is preserved. While examples have been provided in which curved mirror  910  is selected to have a concave shape, it is possible that curved mirror  910  is selected to have a convex shape (e.g., to make the reflected beam diverge upon reflection) or flat shape (e.g., a radius of curvature may be selected to be infinity). In addition, mode-matching optics similar or identical to that described with reference to  FIG. 3  may be used to match a beam mode of the reflected third harmonic laser beam energy (e.g., reflected by curved mirror  910 ) with a beam mode of the incoming third harmonic laser beam energy. 
     The radius of curvature and position of curved mirror  360  (e.g., relative to optical energy coupler  314 ) are selected as described with reference to  FIG. 3 . Thus, the radius of curvature and position of curved mirror  360  are selected so that a beam radius and a beam divergence of the reflected second harmonic laser beam energy are essentially the same as a beam radius and a beam divergence of the incoming second harmonic laser beam energy everywhere along a beam path of the incoming second harmonic laser beam (e.g., between end mirror  810  and curved mirror  360 ). In addition, mode-matching optics similar or identical to that described with reference to  FIG. 3  may be used to match a beam mode of the reflected second harmonic laser beam energy (e.g., reflected by curved mirror  360 ) with a beam mode of the incoming second harmonic laser beam energy. Maintaining a substantially uniform beam radius and beam divergence of laser beam  342  helps improve fourth harmonic laser beam energy generation efficiency without affecting a beam mode of the fourth harmonic laser beam energy and without significantly deteriorating a beam quality of the fourth harmonic laser beam energy. 
     Various embodiments of systems and methods have been described to efficiently generate a higher order harmonic laser beam by recycling a portion of an intermediate harmonic laser beam while maintaining higher order harmonic beam quality. It should be recognized that any higher order harmonic laser beam (e.g., second and higher harmonic laser beams) may be generated using any combination of the intermediate harmonic recycling techniques described herein. 
     It will be obvious to those having skill in the art that many changes may be made to the details of the above-described embodiments without departing from the underlying principles of the invention. The scope of the present invention should, therefore, be determined only by the following claims.