Thi s isan indicati on that the conversi on e£ ciency into solitons decreases once the soliton thresho ld is crossed and the excessfunda mental electromagnetic ...
V ol . 1 0 3 ( 2003)
A CT A P HY SIC A P O LO N IC A A
No . 2{ 3
P ro ceed in gs of t h e I n t ern a ti o nal W o rk sho p N O A '02 , Êu k ≤ci n 200 2
Secon d H ar m on ic G en era tion , B eam D y n am i cs an d Sp atia l Soliton G en eration in P erio d ic ally P ole d K Ti OPO 4 H . K im, L. J ank ovi c an d G. St ege man Scho ol of O pt i cs/ C REO L , Uni versi t y of Centr al Fl ori da, USA
S. Ca rra sc o and L. To rne r Insti t ute of Pho to ni c Sciences, Uni versi ty Po l i tecni ca de Cata l uny a Ba rcelona , Spai n
M. Ka t z and D. Eg er El ectro -Opti c D i v. , Soreq NR C, Isra el Spat ial solito ns w ere in vesti gate d in perio di call y poled K T iOPO 4 under condition s of second har monic generation and many new features associated w ith soliton generation in quasi- ph ase-matched samples w ere observed . T he e˜ects on the second harmonic generation tuning curves of the beam narrow ing mechanisms resp onsibl e for soliton generation w ere found to b e dramatic. It w as demonstrated exp erimentall y that the mutual collapse of the fundamental and harmonic beams is a complex phenomenon. PACS numb ers: 42.65.J x , 42. 65. K y, 42.65. T g
1. I n t r o d u ct io n The famil y of spati al sol i tons, i .e. beams of Ùnite cross-section whi ch pro pagate wi tho ut di ˜ra cti on ha s gro wn substanti al l y i n the l ast decade [1]. Qua dra ti c spati al sol i tons are uni que m embers of the sol i to n fam il y because they requi re the co-exi stence o f two (o r even three) frequency com p onents m utua ll y coupl ed (2) thro ugh the second order nonl ineari ty d i j k [ 1 À 3]. They ha ve b een observed i n (1 07)
108
H . K i m et al .
b oth wa vegui des and bul k m edia and m any of thei r pro p erti es are no w well understo od [4{ 10]. The thresho l d i ntensi ti es for spati al sol i to n genera ti on dep end ( 2) i nv ersely on the e˜ecti ve nonl i neari ty used, d e˜ . It i s well kno wn i n the no nl i near opti cs communit y tha t the most e£ cient frequency conversi on i s obta i ned wi th ( 2) p eri odi cal l y p ol ed materi al s, wi th Li NbO 3 (PPLN) for whi ch d e ˜ > 1 5 pm / V and i ndeed the l owest qua dra ti c sol i to n thresho l ds ha ve been rep orted i n PPLN. However, Li NbO 3 su˜ers fro m pho toref ra cti ve dam age and usual ly requi res opera ti on at tem pera tures > 1 0 0 C, especial l y when hi gh i ntensi ti es are used f or sol i to n studi es where tem p eratures i n excess o f 1 5 0 C are needed [5]. Ano ther ferro electri c, na mely KTi OPO 4 , do es no t su˜er fro m pho to refra cti ve da mage, even at ro om tem p erature. It ha s b een p eri odi cal l y p ol ed (to pro duce p eri odi cal l y p oled KTi OPO 4 (PPKTP)) and second harm oni c generati on (SH G ) ha s b een rep orted [11{ 13]. The opti cal qua li ty of such crysta l s i s excell ent and SHG exp eri ments ha ve yi elded l arg e conversi on e£ ciencies. Thi s shoul d l ead to rea sona bl y l ow sol i ton thresho lds of a f ew G W / cm 2 . Here we revi ew our experi m ents on the pro perti es of qua dra ti c soli to ns i n PPKTP and thei r i mpa ct o n SHG e£ ciency and tuni ng curves. £
£
2. S am p l es an d exp er i m ental ap p ar at u s KTP i s a bi axi al ferro electri c crysta l . Its ferro electri c do mai ns, and hence the second order no nl i neari ty were peri o di call y reversed al ong the z -axi s at Soreq usi ng the l ow tem p erature pol i ng techni que [11]. W hen z -p ol ari zed l i ght i s pro pagated al ong the crysta l ' s x -axi s, pha se-matchi ng can b e a chieved when the p eri odi city Ê sati sÙes the wa ve vecto r matchi ng condi ti on  k = 0 wi th  k = 2 k ( ! ) À k (2 ! ) + 2 m ¤ = Ê , where m i s an i nteg er, and k (2 ! ) and k ( ! ) are the wa ve vecto rs at the harm oni c and funda menta l frequenci es respecti vel y. The ef(2 ) (2 ) fecti ve no nl i near coe£ cient ha s been m easured to b e 9.5 pm / V ( d e ˜ = 2 d 3 3 = ¤ ) (2 )
whi ch corresp onds to a d 3 3 = 1 4 :9 pm = V [ 12; 13] . The sampl e was 1 cm long and the wi dth and thi ckness of the PPKTP were 4.5 m m and 0.5 mm (a l ong z ) respect i vely. The measured SHG band wi dth was 0.2 nm , i n good agreement wi th theo ry a nd an i ndi cati on tha t pha se-m atchi ng i s real i zed over the f ul l l ength of the crysta l [12, 13]. Because b oth the pro pagati on and p ol ari zati ons l i e al ong the crysta l axes of a bi axi al crysta l , the pha se-m atchi ng i s non- cri ti cal l y-pha se-m atc hed (NCP M). Thi s i s i ndi cated in Fi g . 1 where the e˜ect of the QPM i s to m o ve the appro pri ate cut of the f unda m enta l refra cti ve i ndex surf ace to ta ng ency wi th the second ha rm oni c one, case (b). For a p ol i ng peri o d of 8 : 9 9 ñ m and m = 1 , pha se-m atchi ng for a 1064 nm funda m ental b eam o ccurs aro und 4 2 : 7 C [14]. At tem pera tures l ower tha n the pha se-m atchi ng tem p erature ( T < T P M , Fi g. 1a), there a re two di recti ons sym m etri cal ly di splaced i n angl e fro m the x - axi s at whi ch pha se-m atchi ng occurs, wherea s for T > T PM the i ntera cti on canno t be pha se-m atched at any ang l e. £
Secon d Har moni c Generat i on
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Let us no te tha t for on-a xi s pro pa gati on, there i s no spati al wal k- o˜ b etween the funda m ental and ha rm oni c beams. Howev er, for pro pa gati on at an ang l e ˚ f rom the x - axi s, the wal k- o˜ is dom i nated by the QPM condi ti on gi vi ng an ang ul ar wa lk- o˜ ê = sin 1 [ sin ˚ 2 ¤ = k (2 ! )Ê ] [15]. À
Fig. 1.
Cuts of the index ellip soids in the
x
À
y
dashed line is meant to simulate approximately
plane for the z - polari zed F W and SH . T he the e˜ect of the nonlin ear QPM grating
on phase-matching. Part (a ) is for temp eratures below temp erature, (c)
Fig. 2.
T >
Tpm
T pm
, (b) is at the phase- matching
.
Measurement apparatus
and the parameters for the laser system and beams.
The experi m ental system used i s shown schemati cal ly i n Fi g. 2. The l aser source was an EKSMA Nd :Y AG 10 Hz system . Pul se energ ies of onl y a few ñ J were requi red for the exp eri ments. The output b eam fro m the l aser was spati al l y Ùltered gi vi ng a m easured Ga ussian b eam qua l i ty facto r M 2 of 1 : 0 1 : 1 . The l aser's ful l - wi dth hal f maxi mum spectra l band wi dth wa s measured to vary f rom 0.11 (5 0 ps pul ses) to 0.14 nm (2 5 ps pul ses), ab out a facto r of two sm all er tha n the SHG band wi dth. The funda m ental b eam was f ocused onto the i nput crysta l facet by lenses to mi ni mum beam wai sts w 0 = 4 0 (f or appro xi mati ng the pl ane wa ve l im it) and 1 6 1 8 ñ m (f or the na rro w b eam exp eri ments), as measured by the kni fe-edge
110
H . Ki m et al .
m etho d. A mi ni mum wai st of 1 6 : 4 ñ m gives 6.9 di ˜ra cti on l engths of pro pa gati o n at M 2 = 1 . The i ndi vi dua l b eams emergi ng fro m the back face of the crysta l were di rected onto a camera and a detecto r to m easure thei r pul se energy and spati al di stri buti on.
3 . P h ys ics o f SH G b eam d yn am i cs In order to understa nd the SHG and sol i to n generati on at hi gh i ntensi ti es, i t i s necessary to Ùrst understa nd how the com bina ti on of b eam no nl i neari ty and pro pagati on a˜ects the b eam shap e. As the i nput i ntensi ty i s i ncreased, two di fferent no nl i near b eam na rro wi ng (o r bro adeni ng ) \ mechani sms" come into pl ay. The Ùrst i s the uni versal na rro wi ng m echani sms associated wi th all nonl i near wa ve m i xi ng i ntera cti ons [16]. In bri ef sum m ary, i n no nl i near opti cs the no nl i near pol ari zati on i nduced i s al ways due to the pro duct of Ùelds. W hen the Ùelds ha ve Ùnite cross-secti on, the no nl i near p ol ari zati on deri ved f rom pro ducts of the Ùelds i s by deÙniti on na rro wer tha n the mi xi ng Ùelds and hence the electro m agneti c wa ves generated are al so na rro wer. For SHG by G aussian b eams for exampl e, p the generated harm oni c (up- conversi on) i s 2 narro wer tha n the funda m enta l. For do wn- conv ersion, whi ch also al ways occurs, the regenerated funda m ental i s al so na rro wer tha n either the funda m ental or ha rm oni c. Theref ore thi s mechani sm l eads to b oth b eams narro wed in space. If thi s mutua l f ocusing i s m ore e˜ecti ve tha n the usual di ˜ra cti on, b oth b eam cross-secti ons are reduced on pro pagati on. The second \ m echani sm" i s cal l ed \ cascadi ng " [16, 17]. It i s based on the di ˜erence between the funda m ental and harm oni c pha se vel ociti es whi ch, i n the case of i deal pl ane wa ves, o ccurs away fro m pha se-match. On thi s regard, l et us no te tha t Ùnite b eam s conta i n a conti nuo us spectrum of pl ane wa ves, thus wi th Ùnite pum p beams cascadi ng o ccurs al so at no m ina l pha se-ma tchi ng . As energy Ûows back and forth b etween the two wa ves, the pha se fro nt of the slower wa ve reta rds tha t o f the faster wa ve and e˜ecti vely slows down the faster wa ve, i .e. i t i nduces an a ddi ti ona l pha se shift. Since the amount of energy exchang e depends on the l ocal i nput i ntensi ty , the pha se shif t vari es al ong the b eam ' s tra nsverse di stri buti on, i .e. i ntensi ty pro Ùle. The resulti ng pha se fro nt b ecomes either concave or convex, l eadi ng to either a self- defocusi ng or self- focusi ng e˜ect, respecti vely. Al tho ug h i n actua l f act there i s no refra cti ve index chang e associated wi th thi s no nl o cal e˜ect, i t can b e i nterpreted as an e˜ecti ve no nl i near refra cti ve i ndex n 2 , i .e. n 2 ; e ˜ wi th an equi va lent i ndex chang e  n e ˜ = n 2 ; e ˜ I (r ) , where I ( r ) i s the ra di al dependence of the b eam intensi ty . Fo r  k > 0 , self-fo cusing occurs and wi th  k < 0 defo cusing ta kes pl ace. Thus for  k > 0 b eam na rro wi ng i s enha nced, wherea s for  k < 0 the two \ mechanism s" tend to i nterf ere wi th one ano ther. Let us no ti ce tha t the sim pl e pi cture sketched ab ove o b vi ously hol ds onl y i n the l i m it where a l arge funda m enta l frequency signa l i ntera cts very weakl y wi th i ts sma ll second- harm oni c, under condi ti o ns where the energy conversi on between
Second Har moni c Generat i on
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them i s negl ig ibl e. Otherwi se, the ab ove self- focusi ng self-defocusi ng pi cture does no t ho l d. For exam pl e, under condi ti ons of a second- ha rm oni c compa ra bl e to the funda m ental, self- focusing ta kes pl ace at b oth sides of pha se-matchi ng, as was observed exp eri menta l l y i n the Ùrst observ ati on of quadra ti c sol i to ns [4]. It i s now well understo od tha t spati al sol i to ns consisti ng of coupl ed funda m enta l and ha rm o ni c beams exist [1{ 3, 16, 17]. They are the nonl i near eigenm odes i n the l i m it i n whi ch the di ˜ra cti on l ength L d = ¤ w 20 n= Ñ v a c i s compa ra bl e to or l arg er tha n the pa ra m etri c gai n l ength L p g = cn= [ ! d 2e˜ j E (! ; x = 0 ) j ] . Sta bl e sol i to ns f or typ e I pha se-ma tch consist of i n phase funda m ental and harm oni c com p onents, wi th a speciÙc har moni c t o fundament al Ùeld rat i o whi ch dep ends on the i nput i ntensi ty and the i ni ti al wa ve vecto r mi sm atch [1À 3 ; 1 6 À 19]. Because they are the nonl inear eigenm odes, abo ve a thresho l d i ntensi ty any i nput wi ll evol ve wi th di sta nce i nto a quadra ti c sol ito n [20, 21]. Typi cal ly onl y a f unda m enta l input i s used and the second- ha rm oni c i s al l owed to evol ve wi th di sta nce under the i nÛuence of the usual coupl ed m ode equa ti ons whi ch descri b e SHG [4]. These equati ons l ead near the i nput facet of the crysta l to an SH b eam whi ch is ¤ = 2 out of pha se wi th the funda menta l. Thus b oth the rel ati ve pha se a s well as the rel ati ve am pl i tudes of the FW and SH need to evol ve i nto val ues appro pri ate to a quadra ti c sol i to n. Thi s m eans tha t the well - known pl ane sol uti ons for the FW and SH, for exam pl e SHG / ta nh 2 ( z = L pg ) on pha se-m atch, are no l onger val i d. Instea d the Ùeld ampl itudes underg o oscil l ato ry b ehavi or whi ch dam ps out wi th di stance as the sol i to n pro p erti es are appro ached asym pto ti cal l y. A typi cal exam pl e i s shown i n Fi g. 3. Let us no te tha t the exchang e of energy occurs wi th di sta nce between the FW and SH on phase-mat ch. Thi s pro cess i s not i nherentl y a di aba ti c and the excess FW and SH energy is ra di ated away m aking the sol i to n generati on e£ ciency l ess tha n 100% .
Fig. 3.
T ypical evolution of the intensity
of a FW and a SH beam with propagatio n
distance w hen only the FW is input.
The b eam wai st al so underg oes m ore com pl ex dyna m ics tha n m onoto ni c na rro wi ng of both b eam s [2 2]. Sho wn i n Fi g. 4 are the measured f unda m enta l and ha rm oni c b eam wai sts at the output facet of the sampl e for vari ous pha se-m i smatches [23]. The overa l l b eam narro wi ng is clear. Ho wever, dependi ng on the sign of the wa ve vector mi smatch, the b eam wai st of one of the two beams ini ti al ly b ecomes l arg er before i t narro ws do wn [22 , 23]. Let us no te tha t the freel y di ˜ra cti ng fun-
112
H . Ki m et al .
da m ental i s exp ected to ha ve a wi dth of ¤ 9 0 ñ m at the end of the crysta l . The i nci dent i ntensi ty at whi ch b oth b eams, FW and SH, reach a pl ateau wi th wi dths compa ra bl e to the i nput b eam wi dth i s deÙned as the spati al sol i to n thresho l d [24]. For PPKTP on pha se-m atch i t occurs at a p eak i ntensi ty of ¤ 3 G W = cm 2 for an i nput 1/ e2 b eam size of 1 6 : 4 ñ m . Howev er, l et us no te tha t the thresho ld for the nonl inear beam dyna mi cs to set i n i s an order of ma gni tude sm al l er, o f order 0.4 G W / cm 2 , or l ess.
Fig. 4.
FW and SH output
(a) Fundamental, (Â
kL
C ondition s:
beam sizes (FWH M) as a function
(b) second harmonic.
), circles:
of input
In both (a) and (b), squares:
C ( m, pulse w idth 50 ps,
), triangles:
C (
T
intensity . =
49 : 0
£
C ).
.
Awa y fro m pha se-match, the deta i l s dep end on whether the narro wi ng m echani sms i nterf ere constructi vel y or destructi vel y, i .e.  k > 0 or  k < 0 , respecti vely. Let us no te tha t f or  k > 0 sol i to ns are form ed at about the sam e thresho l d as for on pha se-m atch, whereas for the equi v alent j  k L j wi th  k < 0 soli to ns are no t form ed o ver the i ntensi ty ra ng e i nvesti gated. Thi s i s veri Ùed by b oth the i ntensi ty dep endence of the harm oni c output, and the o utput b eam pro Ùles. Thi s b ehavi or for negati ve pha se-m i smatch has b een attri buted to thi rd order e˜ects, Kerr nonl inear refra cti on and tw o pho to n absorpti on i n KTP [23 ].
The sim pl est experi m ent, usual l y the Ùrst one do ne wi th any no nl i near crysta l , i s to m easure the SHG tuni ng curve at l ow i nput p owers. Because the wavel ength of the l aser is Ùxed i n the present case, the two opti ons are to either vary the inci dence a ngl e at the pha se-m atchi ng tem pera ture, or to va ry the tem perature for i nci dence al ong the x -axi s. These two tuni ng curves wi l l gi ve di ˜erent resul ts b ecause the Ùrst exhi bi ts sym metry ab out the x -axi s, whereas the second ha s no sym m etry ab out the pha se-m atchi ng tem pera ture, see Fi g. 1. The uni que
Second Har moni c Generat i on
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.. .
aspect of the present exp eri ments i s tha t the i nput i ntensi ti es used here are su£ ci entl y l arg e, and the i nput beam di am eters sm al l enough, to get signi Ùcant b eam na rro wi ng o ver the length of the crysta l and the com pl ex beam dyna mi cs tha t accom pani es i t. 4.1. SHG temperat ur e tuni ng cur ves W hen the i nput b eam di am eter i s l arg e enough so tha t the crysta l l ength is at l east one di ˜ra cti on l ength l ong , the l ow p ower tuni ng curves resembl e the usual sinc 2 (Â k L= 2 ) response for b oth cases. The tem p erature tuni ng resul t i s shown i n Fi g. 5a. Howev er, when the i nput b eam di ameter is reduced, the di ˜erence i n SHG response due to the di ˜erent pha se-m i smatc h condi ti ons shown i n Fi gs. 1a and c i s evident. For T < T p m , there are two , sym m etri cal ly di splaced wi th angl e, pha se-matchi ng condi ti ons. The ang ul ar spectrum associated wi th a na rro w i nput b eam has com p onents whi ch can theref ore b e pha se-m atched so tha t the SHG tuni ng curve has a bro ader response to wards lower tem pera tures, agai n as exhi bi ted i n Fi g. 5a. Thi s asym metry e˜ect has b een kno wn since the earl y da ys of nonl i near opti cs [ 2 5 , 26]. L
Fig. 5.
T uning curves for SH G obtained by temp erature tuning around the low intensity
phase- match temp erature ( 4 2 : 7 C ). (a) Input intensity £
input beam w idths (solid line: 1 |
0.02 GW/cm
2
, 2|
16 : 4
2.0 GW/cm
m, dotted line: 41 2
,3 |
6. 1 GW/cm
of 11 kW /cm
2
m). (b) Di˜erent 2
, 4 |
for tw o di˜erent input intensity:
12. 0 GW/cm
2
.
The SHG tuni ng curve on pha se-m atch as a functi on of increasing input i ntensi ty i s shown i n Fi g. 5b. The m ost sal i ent f eature i s the bro adeni ng of the response curves wi th i ncreasing i ntensi ty . The reason i s caused by the self-pha se m odul ati on i nduced by cascadi ng bel ow the thresho l d for soli to n f orm atio n and by the generati on of spati al sol ito ns ab ove the thresho ld. Once a sol i to n i s form ed, i t pro pagates uncha ng ed to the end of the sam pl e. Hence the SHG occurs e˜ecti vel y no t o ver the who l e sampl e, but over the di sta nce requi red to establ i sh a sol i to n. Thi s requi res a few para m etri c gai n l engths L pg . Thus the e˜ecti ve l ength, at l east
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H . Ki m et al .
as far as SHG is concerned, decreases wi th i ncreasing i ntensi ty and becom es l ess tha n L . Hence the ba nd wi dth becom es l arg er tha n i n the smal l conversi on li m i t. Thi s ha s b een veri Ùed by deta i l ed mo deli ng [14]. W hen  k < 0 , the situa ti on i s considera bl y more com pl ex. As di scussed b efore, L pg conti nues to decrease wi th i ncreasing i ntensi ty . Thi s l eads to a pro gressivel y m ore ra pi d excha nge of energy b etween the f unda m enta l and ha rm oni c wi th di stance. Bel ow the thresho l d for sol i to n form atio n, the two m echani sm s, no nl i near opti cal b eam narro wi ng vi a wave m i xing and b eam spreadi ng due to cascadi ng , com p ete wi th one another. Furtherm ore, the i nterf erence i s di ˜erent o ver di ˜erent pa rts o f the b eam pro Ùle whi ch l eads to the l arg e am oun t of \ no i se" on thi s side of the tuni ng curve rel ati ve to  k > 0 regi on. Thi s resul ts in very compl ex b eam pro Ùles. Let us note tha t at the hi ghest input i ntensi ty recorded, thi s chaoti c regi on actua l l y leads to a maxi mum i n the SHG whi ch equa ls the on- pha se ma tch val ue for whi ch soli to n generati on occurred.
Fig. 6.
SH G pulse energy e£ciency
temp erature
¤
42 :7
£
versus input fundamental
intensity
at the Q PM
C.
The va ri atio n i n the SHG e£ ciency on pha se-m atch i s shown i n Fi g. 6. (The tuni ng curves di scussed previ ously were no rm al ized to the same m axi mum value.) The m aximum SHG e£ ciency peaks at ab out 50%, at ab out the soli to n thresho l d. Let us no te ho wever tha t the SHG e£ ciency dro ps af ter i t achi eves i ts p eak val ue. Thi s i s an i ndi cati on tha t the conversi on e£ ci ency i nto sol i to ns decreases once the sol i to n thresho l d i s cro ssed and the excess funda menta l electro m agneti c energy i s ra di ated away and no t converted i nto spati al l y l ocal ized second ha rm oni c. 4. 2. A ngle t uni ng cur ves As m entio ned previ o usly, the x - axi s b eing a crysta l axi s i s al so a sym m etry axi s. Thus for ang l e detuni ng away fro m the x -axi s i n one of the pri nci pal pl anes x À y or x À z , the SHG pro p erti es, i ncl udi ng qua dra ti c sol i to ns shoul d b e sym m etri cal ab out the x -axi s. Results are shown for  k = 0 (Fi g. 7),  k < 0 (Fi g. 8) and
Second Har moni c Generat i on
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(Fi g. 9). The response curves al l exhi bi t sym m etry a bout the x -axi s, as exp ected. They al so all exhi bi t i ncreases i n ang ul ar band wi dth for SHG wi th i ncreasing i nput i ntensi ty whi ch occurs for the same reasons as di scussed previ ousl y for the tem perature tuni ng case. Fi gure 7 i ndi cates tha t at l arg e i nput i ntensi ti es on pha se-match the band wi dth op ens up to ab out Ï 8 (externa l ang le of i ncidence). Let us no te, howev er, tha t as i ndi cated in Fi g. 6, the SHG e£ ciency dro ps wi th i ncreasing i ntensi ty so tha t there i s an e£ ciency- ba nd- wi dth tra de-o˜. Si mi la r bro adeni ng etc. o ccurs fo r  k < 0 , see Fi g. 8. The m ost i nteresti ng aspect of these curves i s the mi ni mum i n the SHG tha t o ccurs al ong the x -axi s  k > 0
£
Fig.
7.
(T 2 |
SH G tuning 0
, 3 |
2.0 GW/cm
, 4 |
6.1 GW/cm
SH G tuning curve versus incidence angle in the
angle corresp onds to the 3 |
x
angle corresp onds to phase-match. Input intensity:
0.2 GW/cm
Fig. 8.
curve versus incidenc e angle in the
6.1 GW/cm
, 4 |
-axis. Input intensity:
19. 7 GW/cm
.
1|
,5 |
À
plane at
y
1|
0.02 GW/cm
19.7 GW/cm
plane at 0.2 GW/cm
42 : 7
C ;
.
C ( ;2|
£
2. 0 GW/cm
). ,
116
H . Ki m et al .
at a rel ati vely l ow i nput i ntensi ty of 2.0 G W / cm 2 . At thi s intensi ty stro ng b eam di storti on ha s al ready set i n and there i s ag ain an i nterpl ay b etween the cascadi ng and wa ve mi xi ng m echani sm s so tha t energy osci ll ates b etween the funda m ental and ha rm oni c o ver di stances com pa ra bl e to the sam pl e l ength. Thi s i nterpl ay l eads to the m i ni mum al ong the p oli ng axi s at tha t pa rti cul ar i nput i ntensi ty . Let us no te tha t there i s no spati al sol i to n generati on o ccurri ng at thi s i ntensi ty .
Fig. 9.
SH G detuning curves at ro om temp erature for di˜erent
tensities. 1 |
Fig. 10.
2
0.02 GW/cm
V arious output
phase- matching the two peaks
peak (9
(8 : 8
,2|
0.9 GW/cm
2
,3|
6. 45 GW/cm
fundamental 2
,4|
2
.
2
.
beam pro Ùles, upp er FW and low er SH , on the original £
), the secondary peak
(9 :4
£
) and in the minimum
) for angle tuning at ro om temp erature. T he fundamental
tensity is 9.7 GW/cm
input in-
12. 9 GW/cm
betw een input in-
Second Har moni c Generat i on
Fig. 11.
A samplin g of output
117
.. .
beam proÙles, upp er FW and low er SH , for di˜erent
input intensities on the phase-matching
peak
(8 : 84
£
) at ro om temp erature.
Ang ul ar detuni ng correspondi ng to the scenari o in Fi g. 1a i s shown i n Fi g. 9, i .e. Â k > 0 al ong the x -axi s. As expected , there are tw o m axi m a at Ï 8 : 8 correspondi ng to the two pha se-m atchi ng condi ti ons i n Fi g. 1a. Increa sing i nput i ntensi ty agai n leads to bro adeni ng of the SHG respo nse curves, a uni versa l feature for al l these results. At the hi ghest intensi ti es, the angul ar (externa l) band wi dth for SHG i s in excess of Ï 1 0 degrees. One of the mo re interesti ng features i n these curves i s the spli tti ng of the p eaks i nto doubl ets whi ch occurs aro und 9.7 G W / cm 2 i nput i ntensi ty . The o utput b eam s pro Ùles obta i ned near the two p eaks, and i n the mi nimum b etween them , shown i n Fi g. 10, g ive some clues as to what i s ha pp eni ng. As i ndi cated previ ousl y, the QPM- i nduced wal k- o˜ occurs away fro m the x -axi s, and the i nterna l wal k- o˜ ang le i s ab out 0.15 degrees at these p eaks. In f act i t i s clear fro m Fi g. 10 tha t in the mi nimum regi on the funda menta l whi ch carri es m ost of power i s spl i t spati al l y by the wal k-o ˜ i nto two hal ves, and at the p eaks i t i s a sing l e b eam. The e˜ects of the wal k- o˜, the energy exchang e b etween the funda m ental and harm oni c and Ùnal l y sol i ton generati on are clear i n Fi g. 11. At l ow i ntensi ti es there i s energy excha nge b etween the beams and the wal k- o˜ l eads to tw o FW and SH b eams. The beam na rro wi ng m echanism s tend to pul l the di ˜erent b eam s i nto a com m on spati al pro pa gati on di recti on [27]. Thi s ul ti m atel y results i n the generati on of well -lo cal ized spati al sol i to ns wi th a thresho l d of ab out 4 G W / cm 2 wi th the o ccasiona l excepti on where the p eaks are spli t as di scussed previ ousl y. £
5 . Su m m ar y The com bi na ti on of hi gh intensi ti es of m ulti - G W/ cm 2 and beams wi th wa i sts i n the 10' s of mi crons l eads to l arge chang es i n the output b eam intensi ty di stri buti ons for ha rm oni c generati on in PPKTP . The thresho ld for soli to n form ati on whi ch occurs due to a ba lance b etween b eam na rro wi ng and no rm al di ˜ra cti on i s ab out 4 G W / cm 2 at NCPM for 16.4 mi cron b eam wai sts, G aussian b eam qua l i ty facto r M x2 ¤ M y2 ¤ 1 : 1 , at the i nput. Pri or to and duri ng sol ito n form atio n,
118
H . Ki m et al .
there a re a numb er of contri buti ng mechani sms i ncl udi ng di ˜ra cti on, b eam na rro wi ng due to no nl i near wave m i xing and cascadi ng, wa ve vector m ism atch and QPM- i nduced wal k- o˜. The result i s com pl ex beam dyna mi cs for b oth the funda m enta l and second harm oni c. The SHG e£ ciency p eaks at about 5 0% on pha se-m atch due to soli to n generati on and then fal ls wi th i ncreasing i ntensi ty as the e£ ciency of sol i to n generati on decreases slowl y. The band wi dths of the SHG tuni ng curves al ways i ncrease wi th i ncreasing intensi ty since the para m etri c gai n l ength whi ch vari es inversel y wi th the input Ùeld li m i ts the l ength over whi ch the SH signa ls gro w. Ac kn owl ed gm en t s The exp eri menta l research i n the US was supported by an Arm y Research O£ ce MUR I, and the col l ab orati on between the Am eri can and Spa ni sh i nv esti gato rs wa s m ade p ossibl e by the Com mi ssion for Scienti Ùc Excha ng e b etween the USA and Spai n. S.C. and L. T. were supported by the G eneral i ta t de Cata l uny a and by the Spani sh G overnm ent under gra nt TIC2 000-1010.
R ef er en ces [1] Review ed recently in Spati al Soli tons, Ed. S. T rillo, W. T orruellas, Springer- Verlag, Berlin 2001. [2] Yu. N . K aramzin, A .P. Sukhoruko v, Z h. Ek sp. T eor. Ph y s. 6 8, 834 (1975) ( Sov. Ph ys.-JETP 41, 414 (1976)). [3] G.I . Stegeman, in: Pr oc. N A T O A dvan ced Resear ch Wo rk shop on Soli ton Dri ven Ph ot on i cs, Ed. A .D. Boardman, A .P. Sukhoruko v, K luw er A cademic Publish ers, H olland 2001, p. 21. [4] W. E. T orruellas, Z. W ang, D. J . H agan, E. W. VanStryland, L. T orner, C.R. Menyuk, Ph y s. Rev . L ett. 74, 5036 (1995).
G. I. Stegeman,
[5 ] R. Schiek, Y . Baek, G. I . Stegeman, Ph ys. R ev. E 5 3, 1138 (1996). [6] P. Di T rapani, G. V aliulis, W. C hianglia , A . A dreoni, Ph ys. Rev . L ett. 80, 265 (1998). [7] M. T .G. C anv a, R. A . Fuerst, D. Bab oiu, G. I . Stegeman, G. A ssanto, 1683 (1997). [8] B. Bourliague t, V . C ouderc, D. C . H anna, C. De A ngelis,
A . Barthelemy , G.W. 1410 (1999).
[9 ] X . Liu, L. J . Qian, F. W. Wise, [10] R. Malendevi ch, L. Janko vic, Ch. Bosshard, P. Gunter,
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Smith,
4631 (1999). S.
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[11] G. Rosenman, A . Sklia r, D. Eger, M. Oron, M. K atz, (1998).
F uerst,
G. I.
Stegeman, 3650
[12] A . Englande r, R. La vi, M. K atz, M. Oron, D. Eger, E. Lebiush, G. Rosenman, A . Skliar, 1598 (1997).
Second Har moni c Generat i on
119
.. .
[13] A . A rie, G. Rosenman, A . K orenÙeld, A . Skliar, M. O ron, M. K atz, D. Eger, Op t. Let t. 23, 28 (1998). [14] H . K im, L. J anko vic, G. I. Stegeman, M. K atz, S. Carrasco, L. T orner, A ppl. Ph ys. L ett. 81, 2710 (2002). [ 15] M. M. F ej er, G. A . Magel, D. H . J undt, R. L. Byer, IEEE J. Quant. El ectron. 28, 2631 (1992). [16] Review ed in G. I. Stegeman, D. J . H agan, L. T orner, Op t. Qu antum El ectr on. 28, 1691 (1996). [17] G. A ssanto, G. I. Stegeman, Op t. E xpress 10, 388 (2002). [18] L. T orner, Opt. Commu n. 114, 136 (1995). [19] L. T orner, D. Mihalache, D. Mazilu, E. M. W right, W. E. T orruellas, G. I. Stegeman, 149 (1995). [20] L. T orner, C.R. Menyuk, G. I. Stegeman, [21] L. T orner, E.M. Wright, [ 22] P. Di T rapani, A . Bramati, G. Valiulis,
1615 (1994). 864 (1996).
S. Minardi, W. Chinaglia , C . C onti, S. T rillo, J . K ilius, 183902 (2001).
[23] H . K im, L. J anko vic, G. I. Stegeman, S. C arrasco, L. T orner, D. Eger, M. K atz, , in press. [24] R. A . F uerst, B. L. Law rence, W. E. T orruellas, G. I . Stegeman, (1997). [ 25] D. A . K leinman, A . A shkin, G. D. Boyd, [26] G.D. Boyd, D. A . K leinman, [27] L. T orner, D. Mazilu, D. Mihalache,
338 (1966). 3597 (1968). 2455 (1996).
19
