- Email: [email protected]
Theory of high efficiency third harmonic generation of high power Nd-glass laser radiation
Volume 34, numl~r 3
OPTICS COMMUNICATIONS
September 1980
THEORY OF HIGH EFFICIENCY THIRD HARMONIC GENERATION OF HIGH POWER Nd-GLASS LASER RADIATION R.S. CRAXTON Laboratory .for Laser l:)wrgetics, University of Rochester, Rochester. New York 14623. USA Received 12 May 1980
Overall energy efficiencies on the order of 80~; are predicted for the conversion of 1.054 um laser radiation to the third harmonic. A new tripling configuration is used in which the fundamental beam is incident upon the first of two KDP Type 11cr3.'stalslinearly polarized at an angle of tan -l 1/x/~ to the ordinary direction. This configuration is compared with alternative tripling schemes.
i. Introduction The frequency up-conversion of high power, high ener~,. ND-glass laser beams for fusion applications has recently attracted considerable attention, because shorter wavelength radiation appears to offer substantial advantages in terms of increased absorption and improved energy coupling between the laser radiati~m and the target [1 ]. Such beams have been frequencx doubled [2.3] and quadrupled [2] using KDP cryst~s. with overall efficiencies of 5 0 - 7 0 % and 20% respectively, for use in laser fusion experiments, but to date frequency tripling has been neglected. Even for lasers of lower powers tripling has received very little attention, althou~! overall efficiencies of the order of 10% for short milli-Joule pulses of a modelocked Nd : YAG laser [4] and 30% for 150 ps pulses at I GW/cm 2 [5] have been obtained. The tripling process involves doubling a fraction of of the fundamental light of frequency co in a first ,ut,~,u,,~, ) cr),sta~, and then mixing the 26o light so produced with the unconverted co li~lt in a second crystal (the tripler). Efficient tripling depends on the fundamental and second harmonic photons emerging from the first crystal in a ratio of 1 : 1 over a broad intensity range. This paper demonstrates that this requirement can be achieved by means of an appropriate choice of polarization angle input to the doubler, it is then possible to design tripling systems for high power 474
laser beams which have overall conversion e fficiencies of the order of 80%, use a minimum of optical components, and are relatively insensitive to laser beam divergence. In a companion paper [6] it is confirmed experimentally that these predicted high efficiencies can be achieved; this result is expected to have a significant impact on the future development of laser fusion research.
2. Requirements for efficient tripling The equations governing frequency mixing in nonlinear crystals such as KDP have been known for some time [7] and are well approximated by: dE 1/dZ = --iK1E3E ~ e x p f - i A k • Z) - ½"t'l E! ,
(I)
dE2/dZ =-ilf21,.'3E ~ exp(--iAk • Z)--~-72b.'2 ,
(2)
dE3/dZ = iK3I:'IE 2 exp(iAk • Z ) ~ - 3 , 3 E 3 .
(3)
ltere the E~ are the complex electric vectors of waves propagating in the Z direction with frequencies 60/where 603 = 601 + 602' The electric field of wave/ is the real part of F/exp(i60/t - ikjZ), and the phase naismatch Ak = k 3 - (k 1 + k2) is proportional to tile deviation/x0 of the beam path from the phasematching direction. The 7j are absorption coefticients, which we take to be 0.04 cm--I for !.054/am radiation
Volume 34, number 3
OPTICS COMMUNICATIONS
and zero at the shorter wavelengths [8]. For tripling, 602 = 2601,603 = 3601, K2 ~ 2K1, and K 3 ~ 3K 1. (For 1.054/am radiation, a value o f K 1 = 1.4 X 10 -6 V -1 has been found to give dose agreement with experiment [6].) These equations are readily integrated for single rays. Solutions for rays of different intensities are combined using a computer code "MIXER", which is thereby able to propagate a complete beam through the two crystals taking full account of the beam's temporal and spatial dependence. In order to provide maximum insight into the overall tripling process, most solutions presented in this paper are for single rays. This theory is believed to be reliable because full MIXER calculations have, to date, agreed closely with experimental measurements for doubling and tripling, with different input polarizations, tilt angles, and crystal lengths [6]. Aside from the small absorption terms in eqs. ( 1 ) (3), the solutions all scale as K2L,Vq,where I is the incident intensity and L is the crystal thickness. Our discussion will be confined to an example of practical interest (9 mm thick crystals, and I ~ 4 GW/cm2), but our results can easily be scaled for other cases *. Additionally, our discussion assumes that all optical surfaces are coated to eliminate reflection losses. Single ray solutions for the tripler alone are shown in fig. 1 for a typical crystal, for the phase-matched case (Ak = 0). The efficiency is referred to the total energy input to the triplet in to and 260. The curves correspond to various percentages (or "mixes")M of second harmonic energy in the triplet input. The solutions are sensitive to the parameter M. For M = 67% the input photons at 60 and 260 are matched I : 1 and, in principle, arbitrarily high tripling efficiencies may be attained. For other values of M, after a ray has propagated a finite (optimum) distance Zop t into the crystal, the number of photons in one of the components depletes to zero, and as Z increases beyond Zo~,t the mixing process reverses and the 360 light reconvertes. This effect was described by Armstrong et al. [7]. Successful tripling is made difficult because the distance Zop t is a sensitive function of both M and intensity; for example, fig. 1 shows that input at 4 For beams of higher intensities where small crystal thicknesses may be impractical, the nonlinear coefficientKI may be adjusted instead by cutting the crystal at a non-optimum azimuthal angle.
September 1980
'°1 iz
6o
|
INPUT INTENSITY ( ~ + 2~0 ) (GW/cm 2)
Fig. 1. Tripling efficiency of a 9 mm thick ptlase-matched KDP Type II crystal as a function of total input intensity, for variouspercentages ("mixes")M of second harmonic in the input. A small absorption of 0.04 cm-l is included for the fundamental GW/cm 2 will convert with an efficiency of 25% if M = 50% but 80% if M " 60%. A typical laser pulse, gaussian in time with some modulation in space, can be expected to contain a combination of input intensities and s ~cond harfnonic mixes. Moderately high tripling efficiencies can be attained by working at intensities sufficiently low that reconversion never occurs for any value of M, but in order to attain high efficiencies at high input intensities it is necessary to select a doubler which will provide an input mix M to the tripler that is approximately 67% for all rays in the beam in space and time. This may be achieved by using any of three distinctly different doubling configurations illustrated schematically in fig. 2 and described in the following section.
3. Three configurations for efficient tripling
a) The first "Angle-Detuned" scheme (fig. 2a) involves conventional Type II doubling, except that the crystal is tilted at some small angle A0 d off the phase-matched direction. Equal numbers of photons at the fundamental frequency 60 are incident in each of the ordinary (o)and extraordinary (e) directions. For a suitable choice of A0 d and doubler thickness, 475
\.,hmw 3-=,mmdwr 3
OPTICS COMMUNICATIONS
~xxo oul of every three incident photons combine to fore1 a 2,.,3 photon in the e-direction. The remaining ~adialiou al ~.o emerges as two linearly polarized ,:,,mip,mcuts of equal magnitude in the o- and e-directrans, but with a relative phase difference due to the hh-eftingencc o f the doubler. Jacobs has recently .,Jmwn [~l that this may be removed by temperature 1uniug lhe doubler, or by passing the emerging radialion through a special waveplate (one that is simultaneously a quarter wave for w radiation with its fast axis at 45 ° m the o- and e-directions of the crystal. and a whole wave lbr 2 ~ radiation). A second special waveplate is then required to render the co and 2 ~ electric vectors orthogonal for input to the Tx pe !1 tripler 191. These waveplates would not be required ifa Ty'pe I doubler were used, but the sensi;ixi|.~ ~ tO angular mismatch would then be greatly increased (by a factor of about 2.7). h l In the second "'Polarization-Mismatch'" scheme tfi,g. 2b L which is ideal for existing high power Nd-glass laser systems and which has been demonstrated in ref. 161. the incident electric field at ¢.o is linearly polarized al an angle 0p = tan- 1 !/x/~-to the o-direction. This m>ures that two o-photons are input for every e-photon" i;n a suitable choice of doubler thickness the e-photon ~,,d[ comlm~e with the o-photon to give one e photon .a 2~c. leaving one o-photon at co unconverted. Tiffs ,i,,~;!~lel ,~utput is suitably polarized for direct input ", ,~ 1\ :~e II 'ripler. t,bviatin-the need {br irttermediate ~t~icai components. AW residual fundamental e~hotons pass through the tripler without further inter-
September 1980
i
o TYPEII SPECIAL OOUBLER(DETUNEO) WAVEPLATES
TYPEII TRIPLER
(a) ANGLE-DETUNING SCHEME
i
e
2~
G
TYPEII OOUBLER
TYPEII TRIPLER
(b) POLARIZATION-MISMATCH SCHEME
w
e
2w
~
TYPE I DOUBLER
o
TYPE I TRIPLER
(C) POLARIZATION-BYPASS SCHEME Fig. 2. Idealized doubler performance for three tripling schemes. In each case 2 out ot 3 ordinary photons at w are converted in the doubler to one extraordinary photon at 2~. The unconverted photons at to emerge, respectively, (a) elliptical polarized at 45 ° to the o- and c-axes, (b} plane polarized parallel to the o-axis, and (c) plane polarized parallel to the c-axis.
4. Etficiency predictions
acUO|l.
<)The ttlird "'Polarization-Bypass'" scheme (fig. 2c) i., similar to tile "'Polarization-Mismatch" scheme except that Type I crystals are used. In this case the c-photon at frequency co passes straight through the doubler. The o-photons are converted as completely as possible to the secc.nd harmoni.: by use of the maximum doubler length compatible with the intrinsic ,,,,c, gen,.c ol the input beam. Tills scheme is parti,.ularlv appropriate Ibr use with crystals permitting temperature tuning at a phvsenmtching angle of 90 ° (e.g. CDA): such crystals are less sensitive to angular mismatch than angle-tuned Type I KDP crystals. The output is suitable lbr direct input to a Type 1 triplet, but could go into a Type II tripler if first passed through a special waveplate to render the co- and 2co-electric ~ectors ortl~ogonal. 476
Each scheme operates in a regime where the doubler efficiency is nearly constant when plotted against intensity. Characteristic curves appropriate for ~dl three schemes are shown in fig. 3 and are described separately below.
4.1. The angle-detuned scheme 'File dotted curves in fig. 3 apply to conventional Type !1 doubling (0p = 45 °) with detuning angles A0 d - 0 and 800/arad. The latter curve is appropriate for tile "Angle-Detuned" tripling scheme as it exhibits conversion values close to 67% over a wide intensity range, and overall tripling efficiencies in excess of 80% are attainable using this curve. Similar curves would apply to a Type I doubler, except thai to obtain the
Volume 34, number 3
1°°i
OPTICS COMMUNICATIONS
T
"
I
I
I
,..,,....,.,...,.......-
,--"
~
~
100 ~
] ~
~
~
~''" ~0d 0
I/// // /'//"
--~--~---....
AOd : 800 prad
1 i -
>. (..) Z
.
V-'
~
~ A
/I~
September 1980 V--
i
~
|
gle Rays \
80
i
b_
m
60
z .J
m i,-
40
.J
0Q
---- : .
Q:
20
0
_
0
t
2
........
i
4
.
.
.
.
I
. . . .
6
20
/
8
10
INTENSITY ON DOUBLER (FW/cm ~)
Fig. 3. Doubling efficiency of a 9 mm thick phase-matched KDP Type !! crystal as a function of input intensity. Dotted curves - conventional configuration (Op = 45°) with AOd = 0 and 800 prad. Solid curves - "Polarization-Mismatch" scheme lop ~ 35°), wifll A0d = 0 and 300 prad. A small absorption ol" 0.04 cm-t is included for the fundamental. same efficiency at a given intensity the crystal would be i .35 times as thick and the 800 prad curve would be labelled 300 prad. (For a given thickness a KDP Type ! crystal is twice as sensitive to mismatch; for equivalent conversion efficiency it is therefore 2.7 times as sensitive.) The "'Angle-Detuned" scheme is sensiti~e to angular mismalch in any case, and is theretbre better suited to Type ii doubling.
4.2. 771epolerization-mismatch scheme The solid curves of fig. 3 correspond to the new "Polarization-Mismatch'" scheme, for detuning angles A0 d = 0 and 300 prad. The o- and e-haput photons are not mixed one for one and, similarly to the tripler, reconversion sets in when the fundamental extraordinary wave has been depleted. The choice of 0p = tat,,-1 ! / , / ) - ( ~ 3 5 o) ensures that the upper solid curve A0 d = 0 peaks near 67%. In fig. 3, all rays in the range 2 - 5 GW/cm 2 are doubled with an efficiency in the 60--70% rat)go and subsequently tripled with overall efl'iciencies in the 8 0 - 9 0 % range according to fig. I. Note tha; rays which are doublet with less than the optimum doubling efficiency rid = 67% enter the tripler with a more favorable mix M than rid; for example, a ray at 5 GW/cm 2 has ~d = 60% but enters the tripler with M = 63% and emerges with an overall
2
4
6
8
10
INTENSITY INPUT TO DOUBLER (GW/cm:)
Fig. 4. Overall tripling efficiency of a two-stage "Polarization-Mismatch" system cemprising two 9 mm thick KDP Type II crystals. Solid line: single ray at indicated intensity. Dashed line: average over gaussian temporal profile whose peak intensity is given by the abscissa. efficiency of 90%. This is due in part to the loss of reconverted extraordinary photons (which do not interact in the tripler and therefore are ignored in the definition of M), but also to the small absorption of the fundamental 1.054 pin radiation. Tile lower solid line in fig. 3 illustrates the low sensitivity of this configuration to angle mismatch. Characteristics for the overall "Polarization-Mismatch" tripling system are shown in fig. 4. Here the ~fficiency is defined as the third harmonic energy output from the tripler divided by the fundamental energy input to the doubler. The solid curve applies to single rays, as in figs. 1 and 3; the dashed curve is derived from the solid curve by averaging over a beam which is flat in space and gaussian in time. All rays inci. dent upon the doubler in the range 2.8 to 5 GW/cm 2 are tripled with overall efficiencies of over 90%; gaussian beams whose peak intensities lie between 2.2 and 5 GW/cm 2 are tripled with overall efficiencies from 70 to 86%. (Note the sharp drop in the solid curve above 5 GW/cm2: beams propagated through the system with higher peak intensities suffer progressively more severe distortions. For thicker triplers this drop becomes steeper.) MIXER calculations indicate that the overall tripling efficiency is not significantly degraded if the beam has an additional +25% spatial modulation, and 80% tripling is still achievable provided that the most intense ray in the beam has an 477
Volume 34, number 3
OPTICS COMMUNICATIONS
intensity less than 5 GW/cm 2. A major advantage of this scheme is that it operates on the angle-tuning peaks of two Type II crystals and is therefore relatively insensitive to angular mismatch. If both crystals are detuned by 300/arad, the gaussian beam peaking at 4 GW/cm 2 would be converted with an overall efficiency of 65% rather than 84%. The intrinsic divergence and pointing errors of the experimental system described in ref. [6] were substantially less than 300 mad. Rizzo [10] has pointed out a further advantage of the "'Polarization-Mismatch" scheme: the whole tripling process may take place in a single crystal I~.'d in a double pass mode, whereby the output of the doubling stage is reflected almost normally back into the crystal (after passing in each direction through a quarter-waveplate to rotate each component through r,/2. so that the 2co beam returns to the crystal in the o-direction as required for tripling). Tiffs is possible because the optimum tripling system uses crystals of equal thickness and because the phase-matclffng an#es for Type I1 doubling and tripling are approximate. 13' equal (~59 ° in KDP for 1.054 tam radiation). (The Type I "'Polarization-Bypass" scheme can similarly be double-passed.)
4.3. The polarizathm-byl~ass scheme The "'Polarization-Bypass'" scheme uses the Type I version of the upper dotted curve of fig. 3, with the abscissa referring just to the o-polarized input; the curve saturates at 100% (in the absence of absorption), providing in pri~ciple a means of attaining an input n~x of 67% to the tripler over an arbitrarily wide range of intensities. This scheme would be suitable lbr tripling a shaped pulse [ I I ]. The asymptotic curve of a Type ii doubler could also be used if one third of the input were split off before the doubler and recombined with the second harmonic before being sent to the triplet.
5. Conclusion We have shown that by operating a KDP Type II doubling crystal of an appropriate thickness in a new "'Polarization-Mismatch" con2guration, where the fundamental beam is input polarized at tan - / 1 [x/~(~35 °) to the ordinary direction, the emerging fun478
September 1980
damental and second harmonic photons are matched one for one over a broad intensity range thereby providing optimal input to a tripling crystal. This new scheme is relatively insensitive t o angle " ~!smatch and makes possible the conversion of existing high power Nd-glass laser systems to the third harmonic with overall energy efficiencies of the order of 80%. The theory has been confirmed experimentally [6].
Acknowledgements The author has benefitted from many valuable discussions with his colleagues, particularly S.D. Jacobs, J.E. Rizzo, W. Seka and R. Boni, whose skilled experimentation turned the 80% pipe-dream into a practical eality. Encouragement from J. Soures and R.L. McCrory, and useful conversations with A. Budgor, W. Lee Smith and L.D. Siebert are also acknowledged. This work was partially supported by the following sponsors: Exxon Research and Engineering Company, General Electric Company, Northeast Utilities Service Company, New York State Energy Research and Development Authority, The Standard Oil Company (Ohio), The University of Rochester, and Empire State Electric Energy Research Corporation. Such support does not imply endorsement of the content by any of the above parties.
References [1] F. Amiranoff. R. Benattar, R. Fabbro, E. I:abre, C. Garban, C. Poperies, J. Virmont and M. Weinfcld, Bull. Amer. Phys. Soc. 24 (1979) 1069. [2] C. Loth, D. Bruneau and E. Fabre, Appl. Optics 19 (1980) 1022. [3[ KMS Fusion Annual Reports (1977, 1978). [4] A.H. Kung, J.F. Young, G.C. Bjorklund and S.E. ltarris, Phys Rev. Lett. 29 (1972) 985. [5] D.T. Attwood, E.L. Pierce and L.W. Coleman, Optics Comm. 15 (1975) 10. [6] W. Seka, S.D. Jacobs, J.E. Rizzo, R. Boni and R.S. Cra×ton, Optics Comm. 34 (1980) 469. [7] J.A. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, Phys. Rev. 127 (1962) 1918. [8] S.D. Jacobs, private communication. The absorption in KDP at 1.054 ~m is between 4 and 5% per cm. [9] S.D. Jacobs, J.E. Rizzo, W. Seka, R.S. Craxton, R.E. Hopkins, R. Boni and T. Nowicki, Topical Meeting on Inertial Confinement Fusion, San Diego (1980). [lOl J.E. Rizzo, private communication. [11] R.J. Mason, Nucl. Fusion 15 (1975) 1031.
OPTICS COMMUNICATIONS
September 1980
THEORY OF HIGH EFFICIENCY THIRD HARMONIC GENERATION OF HIGH POWER Nd-GLASS LASER RADIATION R.S. CRAXTON Laboratory .for Laser l:)wrgetics, University of Rochester, Rochester. New York 14623. USA Received 12 May 1980
Overall energy efficiencies on the order of 80~; are predicted for the conversion of 1.054 um laser radiation to the third harmonic. A new tripling configuration is used in which the fundamental beam is incident upon the first of two KDP Type 11cr3.'stalslinearly polarized at an angle of tan -l 1/x/~ to the ordinary direction. This configuration is compared with alternative tripling schemes.
i. Introduction The frequency up-conversion of high power, high ener~,. ND-glass laser beams for fusion applications has recently attracted considerable attention, because shorter wavelength radiation appears to offer substantial advantages in terms of increased absorption and improved energy coupling between the laser radiati~m and the target [1 ]. Such beams have been frequencx doubled [2.3] and quadrupled [2] using KDP cryst~s. with overall efficiencies of 5 0 - 7 0 % and 20% respectively, for use in laser fusion experiments, but to date frequency tripling has been neglected. Even for lasers of lower powers tripling has received very little attention, althou~! overall efficiencies of the order of 10% for short milli-Joule pulses of a modelocked Nd : YAG laser [4] and 30% for 150 ps pulses at I GW/cm 2 [5] have been obtained. The tripling process involves doubling a fraction of of the fundamental light of frequency co in a first ,ut,~,u,,~, ) cr),sta~, and then mixing the 26o light so produced with the unconverted co li~lt in a second crystal (the tripler). Efficient tripling depends on the fundamental and second harmonic photons emerging from the first crystal in a ratio of 1 : 1 over a broad intensity range. This paper demonstrates that this requirement can be achieved by means of an appropriate choice of polarization angle input to the doubler, it is then possible to design tripling systems for high power 474
laser beams which have overall conversion e fficiencies of the order of 80%, use a minimum of optical components, and are relatively insensitive to laser beam divergence. In a companion paper [6] it is confirmed experimentally that these predicted high efficiencies can be achieved; this result is expected to have a significant impact on the future development of laser fusion research.
2. Requirements for efficient tripling The equations governing frequency mixing in nonlinear crystals such as KDP have been known for some time [7] and are well approximated by: dE 1/dZ = --iK1E3E ~ e x p f - i A k • Z) - ½"t'l E! ,
(I)
dE2/dZ =-ilf21,.'3E ~ exp(--iAk • Z)--~-72b.'2 ,
(2)
dE3/dZ = iK3I:'IE 2 exp(iAk • Z ) ~ - 3 , 3 E 3 .
(3)
ltere the E~ are the complex electric vectors of waves propagating in the Z direction with frequencies 60/where 603 = 601 + 602' The electric field of wave/ is the real part of F/exp(i60/t - ikjZ), and the phase naismatch Ak = k 3 - (k 1 + k2) is proportional to tile deviation/x0 of the beam path from the phasematching direction. The 7j are absorption coefticients, which we take to be 0.04 cm--I for !.054/am radiation
Volume 34, number 3
OPTICS COMMUNICATIONS
and zero at the shorter wavelengths [8]. For tripling, 602 = 2601,603 = 3601, K2 ~ 2K1, and K 3 ~ 3K 1. (For 1.054/am radiation, a value o f K 1 = 1.4 X 10 -6 V -1 has been found to give dose agreement with experiment [6].) These equations are readily integrated for single rays. Solutions for rays of different intensities are combined using a computer code "MIXER", which is thereby able to propagate a complete beam through the two crystals taking full account of the beam's temporal and spatial dependence. In order to provide maximum insight into the overall tripling process, most solutions presented in this paper are for single rays. This theory is believed to be reliable because full MIXER calculations have, to date, agreed closely with experimental measurements for doubling and tripling, with different input polarizations, tilt angles, and crystal lengths [6]. Aside from the small absorption terms in eqs. ( 1 ) (3), the solutions all scale as K2L,Vq,where I is the incident intensity and L is the crystal thickness. Our discussion will be confined to an example of practical interest (9 mm thick crystals, and I ~ 4 GW/cm2), but our results can easily be scaled for other cases *. Additionally, our discussion assumes that all optical surfaces are coated to eliminate reflection losses. Single ray solutions for the tripler alone are shown in fig. 1 for a typical crystal, for the phase-matched case (Ak = 0). The efficiency is referred to the total energy input to the triplet in to and 260. The curves correspond to various percentages (or "mixes")M of second harmonic energy in the triplet input. The solutions are sensitive to the parameter M. For M = 67% the input photons at 60 and 260 are matched I : 1 and, in principle, arbitrarily high tripling efficiencies may be attained. For other values of M, after a ray has propagated a finite (optimum) distance Zop t into the crystal, the number of photons in one of the components depletes to zero, and as Z increases beyond Zo~,t the mixing process reverses and the 360 light reconvertes. This effect was described by Armstrong et al. [7]. Successful tripling is made difficult because the distance Zop t is a sensitive function of both M and intensity; for example, fig. 1 shows that input at 4 For beams of higher intensities where small crystal thicknesses may be impractical, the nonlinear coefficientKI may be adjusted instead by cutting the crystal at a non-optimum azimuthal angle.
September 1980
'°1 iz
6o
|
INPUT INTENSITY ( ~ + 2~0 ) (GW/cm 2)
Fig. 1. Tripling efficiency of a 9 mm thick ptlase-matched KDP Type II crystal as a function of total input intensity, for variouspercentages ("mixes")M of second harmonic in the input. A small absorption of 0.04 cm-l is included for the fundamental GW/cm 2 will convert with an efficiency of 25% if M = 50% but 80% if M " 60%. A typical laser pulse, gaussian in time with some modulation in space, can be expected to contain a combination of input intensities and s ~cond harfnonic mixes. Moderately high tripling efficiencies can be attained by working at intensities sufficiently low that reconversion never occurs for any value of M, but in order to attain high efficiencies at high input intensities it is necessary to select a doubler which will provide an input mix M to the tripler that is approximately 67% for all rays in the beam in space and time. This may be achieved by using any of three distinctly different doubling configurations illustrated schematically in fig. 2 and described in the following section.
3. Three configurations for efficient tripling
a) The first "Angle-Detuned" scheme (fig. 2a) involves conventional Type II doubling, except that the crystal is tilted at some small angle A0 d off the phase-matched direction. Equal numbers of photons at the fundamental frequency 60 are incident in each of the ordinary (o)and extraordinary (e) directions. For a suitable choice of A0 d and doubler thickness, 475
\.,hmw 3-=,mmdwr 3
OPTICS COMMUNICATIONS
~xxo oul of every three incident photons combine to fore1 a 2,.,3 photon in the e-direction. The remaining ~adialiou al ~.o emerges as two linearly polarized ,:,,mip,mcuts of equal magnitude in the o- and e-directrans, but with a relative phase difference due to the hh-eftingencc o f the doubler. Jacobs has recently .,Jmwn [~l that this may be removed by temperature 1uniug lhe doubler, or by passing the emerging radialion through a special waveplate (one that is simultaneously a quarter wave for w radiation with its fast axis at 45 ° m the o- and e-directions of the crystal. and a whole wave lbr 2 ~ radiation). A second special waveplate is then required to render the co and 2 ~ electric vectors orthogonal for input to the Tx pe !1 tripler 191. These waveplates would not be required ifa Ty'pe I doubler were used, but the sensi;ixi|.~ ~ tO angular mismatch would then be greatly increased (by a factor of about 2.7). h l In the second "'Polarization-Mismatch'" scheme tfi,g. 2b L which is ideal for existing high power Nd-glass laser systems and which has been demonstrated in ref. 161. the incident electric field at ¢.o is linearly polarized al an angle 0p = tan- 1 !/x/~-to the o-direction. This m>ures that two o-photons are input for every e-photon" i;n a suitable choice of doubler thickness the e-photon ~,,d[ comlm~e with the o-photon to give one e photon .a 2~c. leaving one o-photon at co unconverted. Tiffs ,i,,~;!~lel ,~utput is suitably polarized for direct input ", ,~ 1\ :~e II 'ripler. t,bviatin-the need {br irttermediate ~t~icai components. AW residual fundamental e~hotons pass through the tripler without further inter-
September 1980
i
o TYPEII SPECIAL OOUBLER(DETUNEO) WAVEPLATES
TYPEII TRIPLER
(a) ANGLE-DETUNING SCHEME
i
e
2~
G
TYPEII OOUBLER
TYPEII TRIPLER
(b) POLARIZATION-MISMATCH SCHEME
w
e
2w
~
TYPE I DOUBLER
o
TYPE I TRIPLER
(C) POLARIZATION-BYPASS SCHEME Fig. 2. Idealized doubler performance for three tripling schemes. In each case 2 out ot 3 ordinary photons at w are converted in the doubler to one extraordinary photon at 2~. The unconverted photons at to emerge, respectively, (a) elliptical polarized at 45 ° to the o- and c-axes, (b} plane polarized parallel to the o-axis, and (c) plane polarized parallel to the c-axis.
4. Etficiency predictions
acUO|l.
<)The ttlird "'Polarization-Bypass'" scheme (fig. 2c) i., similar to tile "'Polarization-Mismatch" scheme except that Type I crystals are used. In this case the c-photon at frequency co passes straight through the doubler. The o-photons are converted as completely as possible to the secc.nd harmoni.: by use of the maximum doubler length compatible with the intrinsic ,,,,c, gen,.c ol the input beam. Tills scheme is parti,.ularlv appropriate Ibr use with crystals permitting temperature tuning at a phvsenmtching angle of 90 ° (e.g. CDA): such crystals are less sensitive to angular mismatch than angle-tuned Type I KDP crystals. The output is suitable lbr direct input to a Type 1 triplet, but could go into a Type II tripler if first passed through a special waveplate to render the co- and 2co-electric ~ectors ortl~ogonal. 476
Each scheme operates in a regime where the doubler efficiency is nearly constant when plotted against intensity. Characteristic curves appropriate for ~dl three schemes are shown in fig. 3 and are described separately below.
4.1. The angle-detuned scheme 'File dotted curves in fig. 3 apply to conventional Type !1 doubling (0p = 45 °) with detuning angles A0 d - 0 and 800/arad. The latter curve is appropriate for tile "Angle-Detuned" tripling scheme as it exhibits conversion values close to 67% over a wide intensity range, and overall tripling efficiencies in excess of 80% are attainable using this curve. Similar curves would apply to a Type I doubler, except thai to obtain the
Volume 34, number 3
1°°i
OPTICS COMMUNICATIONS
T
"
I
I
I
,..,,....,.,...,.......-
,--"
~
~
100 ~
] ~
~
~
~''" ~0d 0
I/// // /'//"
--~--~---....
AOd : 800 prad
1 i -
>. (..) Z
.
V-'
~
~ A
/I~
September 1980 V--
i
~
|
gle Rays \
80
i
b_
m
60
z .J
m i,-
40
.J
0Q
---- : .
Q:
20
0
_
0
t
2
........
i
4
.
.
.
.
I
. . . .
6
20
/
8
10
INTENSITY ON DOUBLER (FW/cm ~)
Fig. 3. Doubling efficiency of a 9 mm thick phase-matched KDP Type !! crystal as a function of input intensity. Dotted curves - conventional configuration (Op = 45°) with AOd = 0 and 800 prad. Solid curves - "Polarization-Mismatch" scheme lop ~ 35°), wifll A0d = 0 and 300 prad. A small absorption ol" 0.04 cm-t is included for the fundamental. same efficiency at a given intensity the crystal would be i .35 times as thick and the 800 prad curve would be labelled 300 prad. (For a given thickness a KDP Type ! crystal is twice as sensitive to mismatch; for equivalent conversion efficiency it is therefore 2.7 times as sensitive.) The "'Angle-Detuned" scheme is sensiti~e to angular mismalch in any case, and is theretbre better suited to Type ii doubling.
4.2. 771epolerization-mismatch scheme The solid curves of fig. 3 correspond to the new "Polarization-Mismatch'" scheme, for detuning angles A0 d = 0 and 300 prad. The o- and e-haput photons are not mixed one for one and, similarly to the tripler, reconversion sets in when the fundamental extraordinary wave has been depleted. The choice of 0p = tat,,-1 ! / , / ) - ( ~ 3 5 o) ensures that the upper solid curve A0 d = 0 peaks near 67%. In fig. 3, all rays in the range 2 - 5 GW/cm 2 are doubled with an efficiency in the 60--70% rat)go and subsequently tripled with overall efl'iciencies in the 8 0 - 9 0 % range according to fig. I. Note tha; rays which are doublet with less than the optimum doubling efficiency rid = 67% enter the tripler with a more favorable mix M than rid; for example, a ray at 5 GW/cm 2 has ~d = 60% but enters the tripler with M = 63% and emerges with an overall
2
4
6
8
10
INTENSITY INPUT TO DOUBLER (GW/cm:)
Fig. 4. Overall tripling efficiency of a two-stage "Polarization-Mismatch" system cemprising two 9 mm thick KDP Type II crystals. Solid line: single ray at indicated intensity. Dashed line: average over gaussian temporal profile whose peak intensity is given by the abscissa. efficiency of 90%. This is due in part to the loss of reconverted extraordinary photons (which do not interact in the tripler and therefore are ignored in the definition of M), but also to the small absorption of the fundamental 1.054 pin radiation. Tile lower solid line in fig. 3 illustrates the low sensitivity of this configuration to angle mismatch. Characteristics for the overall "Polarization-Mismatch" tripling system are shown in fig. 4. Here the ~fficiency is defined as the third harmonic energy output from the tripler divided by the fundamental energy input to the doubler. The solid curve applies to single rays, as in figs. 1 and 3; the dashed curve is derived from the solid curve by averaging over a beam which is flat in space and gaussian in time. All rays inci. dent upon the doubler in the range 2.8 to 5 GW/cm 2 are tripled with overall efficiencies of over 90%; gaussian beams whose peak intensities lie between 2.2 and 5 GW/cm 2 are tripled with overall efficiencies from 70 to 86%. (Note the sharp drop in the solid curve above 5 GW/cm2: beams propagated through the system with higher peak intensities suffer progressively more severe distortions. For thicker triplers this drop becomes steeper.) MIXER calculations indicate that the overall tripling efficiency is not significantly degraded if the beam has an additional +25% spatial modulation, and 80% tripling is still achievable provided that the most intense ray in the beam has an 477
Volume 34, number 3
OPTICS COMMUNICATIONS
intensity less than 5 GW/cm 2. A major advantage of this scheme is that it operates on the angle-tuning peaks of two Type II crystals and is therefore relatively insensitive to angular mismatch. If both crystals are detuned by 300/arad, the gaussian beam peaking at 4 GW/cm 2 would be converted with an overall efficiency of 65% rather than 84%. The intrinsic divergence and pointing errors of the experimental system described in ref. [6] were substantially less than 300 mad. Rizzo [10] has pointed out a further advantage of the "'Polarization-Mismatch" scheme: the whole tripling process may take place in a single crystal I~.'d in a double pass mode, whereby the output of the doubling stage is reflected almost normally back into the crystal (after passing in each direction through a quarter-waveplate to rotate each component through r,/2. so that the 2co beam returns to the crystal in the o-direction as required for tripling). Tiffs is possible because the optimum tripling system uses crystals of equal thickness and because the phase-matclffng an#es for Type I1 doubling and tripling are approximate. 13' equal (~59 ° in KDP for 1.054 tam radiation). (The Type I "'Polarization-Bypass" scheme can similarly be double-passed.)
4.3. The polarizathm-byl~ass scheme The "'Polarization-Bypass'" scheme uses the Type I version of the upper dotted curve of fig. 3, with the abscissa referring just to the o-polarized input; the curve saturates at 100% (in the absence of absorption), providing in pri~ciple a means of attaining an input n~x of 67% to the tripler over an arbitrarily wide range of intensities. This scheme would be suitable lbr tripling a shaped pulse [ I I ]. The asymptotic curve of a Type ii doubler could also be used if one third of the input were split off before the doubler and recombined with the second harmonic before being sent to the triplet.
5. Conclusion We have shown that by operating a KDP Type II doubling crystal of an appropriate thickness in a new "'Polarization-Mismatch" con2guration, where the fundamental beam is input polarized at tan - / 1 [x/~(~35 °) to the ordinary direction, the emerging fun478
September 1980
damental and second harmonic photons are matched one for one over a broad intensity range thereby providing optimal input to a tripling crystal. This new scheme is relatively insensitive t o angle " ~!smatch and makes possible the conversion of existing high power Nd-glass laser systems to the third harmonic with overall energy efficiencies of the order of 80%. The theory has been confirmed experimentally [6].
Acknowledgements The author has benefitted from many valuable discussions with his colleagues, particularly S.D. Jacobs, J.E. Rizzo, W. Seka and R. Boni, whose skilled experimentation turned the 80% pipe-dream into a practical eality. Encouragement from J. Soures and R.L. McCrory, and useful conversations with A. Budgor, W. Lee Smith and L.D. Siebert are also acknowledged. This work was partially supported by the following sponsors: Exxon Research and Engineering Company, General Electric Company, Northeast Utilities Service Company, New York State Energy Research and Development Authority, The Standard Oil Company (Ohio), The University of Rochester, and Empire State Electric Energy Research Corporation. Such support does not imply endorsement of the content by any of the above parties.
References [1] F. Amiranoff. R. Benattar, R. Fabbro, E. I:abre, C. Garban, C. Poperies, J. Virmont and M. Weinfcld, Bull. Amer. Phys. Soc. 24 (1979) 1069. [2] C. Loth, D. Bruneau and E. Fabre, Appl. Optics 19 (1980) 1022. [3[ KMS Fusion Annual Reports (1977, 1978). [4] A.H. Kung, J.F. Young, G.C. Bjorklund and S.E. ltarris, Phys Rev. Lett. 29 (1972) 985. [5] D.T. Attwood, E.L. Pierce and L.W. Coleman, Optics Comm. 15 (1975) 10. [6] W. Seka, S.D. Jacobs, J.E. Rizzo, R. Boni and R.S. Cra×ton, Optics Comm. 34 (1980) 469. [7] J.A. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, Phys. Rev. 127 (1962) 1918. [8] S.D. Jacobs, private communication. The absorption in KDP at 1.054 ~m is between 4 and 5% per cm. [9] S.D. Jacobs, J.E. Rizzo, W. Seka, R.S. Craxton, R.E. Hopkins, R. Boni and T. Nowicki, Topical Meeting on Inertial Confinement Fusion, San Diego (1980). [lOl J.E. Rizzo, private communication. [11] R.J. Mason, Nucl. Fusion 15 (1975) 1031.