Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics

Transcription

1 Appl. Phys. B 71, (2) / Digitl Oject Identifier (DOI) 1.17/s34426 Applied Physics B Lsers nd Optics Bck-side-coted chirped mirrors with ultr-smooth rodnd dispersion chrcteristics N. Mtuschek, L. Gllmnn, D.H. Sutter, G. Steinmeyer, U. Keller Ultrfst Lser Physics Lortory, Institute of Quntum Electronics, Swiss Federl Institute of Technology, ETH Hönggererg HPT, 893 Zürich, Switzerlnd (Fx: +41-1/ , E-mil: sguenter@iqe.phys.ethz.ch) Received: 19 April 2/Pulished online: 6 Septemer 2 Springer-Verlg 2 Astrct. We demonstrte new technique for the design of chirped mirrors with extremely smooth dispersion chrcteristics over n extended ultr-rodnd wvelength rnge. Our pproch suppresses spectrl dispersion oscilltions, which cn led to unwnted strong spectrl modultions nd limit the ndwidth of mode-locked lser pulses. Dispersion oscilltions re significntly reduced y coting the chirped mirror structure on the ck side of sustrte, providing idel impednce mtching etween coting nd mient medium. An nti-reflection coting my e dded on the front side of the sustrte, geometriclly seprted from the chirped mirror. The chirped mirror structure nd the nti-reflection coting re non-interfering nd cn e independently designed nd optimized. The seprtion of oth coting sections provides much etter solution for the impednce-mtching prolems thn previous pproches to chirped mirror design. We show y theoreticl nlysis nd numericl simultions tht minimum dispersion oscilltions re chieved if the index of the sustrte is identicl to the index of one of the coting mterils nd if doule-chirping is used for the chirped mirror structure. Bsed on this nlysis, we design mirror tht supports ndwidth of 22 THz with group dely dispersion oscilltions of out 2 fs 2 (rms), n order-of mgnitude improvement compred to previous designs of similr ndwidth. In first experimentl demonstrtion of ck-side-coted (BASIC) mirrors, we chieve nerly trnsform-limited nd virtully unchirped pulses of 5.8fs durtion from Kerr-lens mode-locked Ti:spphire lser. BA- SIC mirrors re prticulrly suited for higher-order dispersion compenstion schemes. They support the extremely rod spectr of few-cycle pulses nd promise to provide clen pulse shpes in this regime. PACS: Re; Bh; Wc Over the lst few yers, ultr short-pulse genertion in the su-1-fs regime hs dvnced t rpid pce, leding to su- 6-fs pulses directly from lser oscilltor [1, 2]. Retrcing the history of ultr fst oscilltors, it ecomes cler tht severl mjor steps towrds shorter pulse durtion hve een enled y improved dispersion-compenstion schemes. The longstnding world record of 6-fs pulse durtion ws chieved y optimized third-order dispersion in the pulse compressor [3]. The first su-1-fs lser pulses directly from Ti:spphire lser were otined y using silver mirrors s high reflectors nd prism pir for dispersion compenstion [4]. This lser ws operted t wvelength where the prism pir provides simultneous second- nd third-order dispersion compenstion. However, considerle residul higher-order contriutions prevented this lser from producing pulses shorter thn 8.5 fs. The invention of chirped mirrors y Szipöcs et l. set new milestone in ultr short-pulse genertion [5]. Tody, ultr short-pulse genertion in the 5-fs regime generlly relies on chirped mirrors. Chirped mirrors re used in Ti:spphire lsers, externl compression schemes [6, 7], nd opticl prmetric mplifiction [8]. Ultimtely, chirped mirrors do not support n unrestricted ndwidth nd limit further pulse shortening [9]. Thus, further improvements in chirped mirror design nd friction re mndtory for the genertion of shorter pulses. The sic ide ehind chirped mirror is reltively simple: lyers with incresing thickness (e.g., qurter-wve lyers with grdully incresing Brgg wvelength) re stcked such tht longer wvelengths penetrte deeper into the mirror structure, producing negtive group-dely dispersion (GDD). More precisely, the GDD of chirped mirror is designed such tht the dispersion of other elements in the lser cvity is compensted for. Additionlly, these mirrors provide n enhnced high-reflectnce ndwidth compred to stndrd dielectric qurter-wve mirrors. In comintion, this llows for more efficient compenstion of higher-order dispersion over roder spectrl rnge thn prisms do. In [1] 7.5-fs pulses were reported for prism-less ring oscilltor using only chirped mirrors, which provided dispersion compenstion over ndwidth ν 85 THz (71 89 nm). However, chirped mirrors hve one importnt drwck. The designed mirror dispersion shows unwnted spectrl oscilltions round the trget function. These oscilltions re cused y n impednce mismtch of the chirped mirror

2 51 structure to the mient medium, which is typiclly ir [11]. Generlly, the mplitude of dispersion oscilltions drmticlly increses with mirror ndwidth. A powerful pproch to ddress this prolem of mirror design is the concept of doule-chirped mirrors (DCMs, [12]). In DCM the different mtching prolems re solved y different multilyer susections [13] (see Fig. 1). The technique of doule-chirping mtches the impednce of the chirped mirror section to the low-index or high-index mteril of the coting, which is then mtched to mient ir y n dditionl nti-reflection (AR) coting on top of the chirped mirror. With one of our first sets of DCMs we redily generted pulses of 6.5-fs durtion [14]. Dispersion compenstion ws otined over ndwidth of ν 115 THz (68 92 nm). Further improvements in oth the theoreticl understnding of DCM [13] nd in the lyer-deposition ccurcy finlly resulted in pulses in the two-opticl-cycle regime. The dispersion-compenstion ndwidth supported y the DCMs used in these lsers is ν 18 THz [1]. Inevitle imperfections of the AR coting, which provides the impednce-mtching step to ir, re the limiting fctor in such ultr-rodnd designs. Consequently, the enormous ndwidth cn only e supported with n dditionl cncelltion of dispersion oscilltions. This cncelltion is chieved y using comintions of DCMs under different ngles of incidence [1, 15]. Alterntively, suitly designed comintions of mirrors from different coting runs my e used [16]. Becuse of the limiting trde-off oundry condition etween dispersion-oscilltion mplitude nd ndwidth, further improvement of pulse ndwidth with conventionl chirped mirrors ppers to e difficult. In this pper, we present novel chirped mirror concept, which suppresses detrimentl dispersion oscilltions y design. This is chieved y geometriclly seprting the chirped mirror stck from the AR section, coting them on two non-interfering surfces of Fig. 1,. Schemtic comprison of stndrd doule-chirped mirror (DCM) nd ck-side-coted (BASIC) mirror. Both pproches employ the sme functionl components: sustrte, chirped mirror structure nd n nti-reflection coting (AR). In the conventionl pproch, AR nd chirped mirror re coted on top of ech other, wheres in cse of the BA- SIC mirror the AR coting is optionl nd coted on the opposite side of the sustrte. In, the sudivision of the chirped coting into doulechirp section, simple-chirp section nd n unchirped qurter-wve section is shown s descried in [13]. The sme pproch cn e employed for the chirped mirror structure of the BASIC mirror. In the following, exmples of BASIC mirrors with simple-chirped cotings re lso considered sustrte. As illustrted in Fig. 1, the AR coting is coted on the front side nd the chirped mirror on the ck side of the sustrte. Therefore, we cll these mirrors ck-side-coted (BASIC) mirrors. This pproch provides etter solution to impednce-mtching prolems in chirped mirror design. The refrctive index of the sustrte cn e chosen close to the refrctive index of the low-index coting mteril. This voids the discontinuity of the refrctive index t the interfce to ir. In such sitution, dispersion oscilltions cn e lmost completely removed [17]. Other thn stndrd DCMs, BASIC mirrors do not require the front AR coting for reduction of dispersion oscilltions. A nerly totl elimintion of dispersion oscilltions is chieved y non-interfering front nd ck surfces. For the BASIC mirror, the AR is merely required to void dditionl losses, i.e., to increse the net reflectnce of the mirror. Consequently, the BASIC technique llows compenstion for the dispersion over much roder ndwidth with strongly reduced dispersion oscilltions. As consequence of the pssge through ulk mteril, the dditionl positive mteril dispersion reduces the chievle negtive net GDD of such mirror. Hence, the most powerful pplictions of BASIC mirrors re comintions with other dispersion-compensting elements like grting or prism sequences, which llow for pre-compenstion of secondorder dispersion. BASIC mirrors cn then e used to correct higher-order dispersion. This pper widely uses the nottion nd results derived in [11, 13, 18]. We will use this formlism to specificlly ddress the prolem of impednce mtching etween the chirped mirror coting nd mient medium. In Sect. 1, we give the exct description of multilyer interference cotings y exct coupled-mode equtions [18]. Additionlly, we extend this description y inclusion of the mient medium. This is chieved with similrity trnsformtion of the coefficient mtrix of the coupled-mode equtions. The trnsformtion mtrix is the Fresnel mtrix, which descries the refrctive-index step etween mient medium nd the firstlyer mteril. We introduce the method of equivlent lyers s n lterntive exct description of chirped mirror structure [19 21]. The design prmeters of coupled-mode theory nd the equivlent-lyer design prmeters re strongly relted to ech other [22]. As one result, the normlized equivlent (refrctive) index (Herpin index) is the reciprocl of the chrcteristic impednce for the equivlent trnsmission-line model derived in [11]. With respect to the synthesis prolem of chirped mirror, this correspondence illustrtes the equivlence of mtching the impednce with mtching the equivlent index. In Sect. 2 we discuss three exmples, which differ in the degree of impednce mtching chieved. In prticulr, we investigte mtching to n mient medium with refrctiveindexwhichisthegeometricvergeofthehighnd low-index mteril. Although this is very intuitive nd common choice (see, e.g., [23]), it does not provide perfect impednce mtching s will ecome cler in the comprison with our finl exmple. In Sect. 3 we present prcticl exmple of BASIC design. This design covers ndwidth of 22 THz (61 nm) for dispersion compenstion in Ti:spphire lser. The design exhiits residul GDD oscilltions of 5 fs 2 (pek-to-pek) or 2 fs 2 (rms), which is n order-of-mgnitude improvement compred to previous design techniques. We compre dispersion nd reflectnce of mnufctured cotings nd designs. Experimentl results re

3 511 descried in Sect. 4. Using these mirrors in our lser, we generte pulses of 5.8-fs durtion, which re then crefully chrcterized y spectrl phse interferometry for direct electricfield reconstruction (SPIDER, [24, 25]). The deposition ccurcy of ion-em sputtering presently sets limit to the dispersion oscilltions of the mnufctured cotings. In the ner future, it is to e expected tht improved in-situ monitoring schemes llow for n improved control of lyer thickness to fully exploit the potentil of the BASIC pproch. n(m) n 2 n 1 n(m) A B 1_ φ 2 1 φ=φ 1 + φ 2 φ 2 1_ φ 2 1 A B m 1 Exct coupled-mode theory nd equivlent lyers for the description of multilyer interference cotings n Exclusion of the mient medium A inry multilyer coting consists of sequence of lternting lyers with low nd high refrctive indices n 1 nd n 2, respectively. For the theoreticl considertions nd exmples in Sects. 1 nd 2, we ssume tht the refrctive indices re rel quntities without ny wvelength dependence. We compose the chirped Brgg structure y symmetriclly defined unit cells ccording to Fig. 2 [11]. The vlues for the refrctive indices my e chosen ritrrily nd n 2 > n 1 is not required. Figure 2 shows the resulting refrctive-index profile for sequence of three unit cells. Appliction of stndrd coupled-mode theory on such multilyer stck neglects emedding of this structure into n mient medium, e.g., ir on the top nd the sustrte t the ottom side of the coting. Usully the influence of index discontinuities t oth ends of the coting is tken into ccount fter the coupled-mode differentil eqution system hs een solved. An exct description for chirped Brgg grtings is given y exct coupled-mode equtions of the form [11] ( ) ( )( ) d A(m) δ(m) κ(m) A(m) = i. (1) dm B(m) κ(m) δ(m) B(m) In (1) A nd B represent slowly vrying normlized mplitudes of the right- nd leftwrd propgting wves, respectively. The qusi-continuous vrile m determines the position inside the mirror nd counts the symmetric unit cells. By convention, m is negtive numer. κ nd δ denote the coupling nd detuning coefficients, respectively. For non-uniform grting structures these coefficients re loclly defined for ech unit cell. As we hve recently proven, chirped mirror is exctly descried y the coupled-mode equtions (1), even for ritrrily lrge refrctive-index differences of the lyer mterils, provided the normlized coupling coefficient nd detuning coefficient re properly defined. The exct coefficients cn e written s [18, 22] 2r κ(m) = α(ϕ(m), φ(m)) ( 1 r 2 ) 1 sin (φ(m) + φ(m)), (2) 2 1 δ(m) = α (φ(m), φ(m)) 1 r 2 { sin (φ(m)) +r 2 sin ( φ(m)) }, (3) n 1 Fig. 2. Refrctive-index profile of the symmetriclly defined unit cell. φ i,(i = 1, 2) denotes the phse shift in the different lyers, yielding totl opticl phse shift φ. The mplitude A(m) refers to rightwrd-propgting wves, B(m) to leftwrd-propgting wves. Resulting index profile for sequenceofthreeunitcells with r = n 2 n 1, n 2 + n 1 (4) φ(m) = φ 2 (m) + φ 1 (m), (5) φ(m) = φ 2 (m) φ 1 (m), (6) φ 1 (m) = 2π λ n 1d 1,m, (7) φ 2 (m) = 2π λ n 2d 2,m. (8) In (2) (8), r denotes the Fresnel reflectivity etween the djcent medi 1 nd 2, λ is the vcuum wvelength, nd d 1/2,m the physicl thickness of lyers 1 nd 2 of the m th unit cell. φ 1/2 (m) descries the opticl phse shift in medi 1 nd 2, respectively. Thus φ(m) gives the totl opticl phse shift of the m th unit cell, nd φ(m) is mesure of the duty cycle within this unit cell. The fctor α = γ/ sin(γ) is defined y the exct propgtion constnt ( ) F 2R 1 i ln γ = rctn δ 2 κ 2 = with rctn i ln F R + ) ( 1 F 2 R F R ( 1 F 2 R F R ( F R ; F R < 1 ; 1 F R ) + π ; < F R +1 ) F 2R 1 + π ; F R > +1 F R = 1 1 r 2 { cos (φ) r 2 cos ( φ) }, (1) m (9)

4 512 using the definitions s given in [22]. The piecewise definition of γ llows for the distinction of different stop-nd nd pssnd regions. Complex vlues of γ (for F R > 1) indicte the stop nds. In the first cse (F R < 1), the stop nds re centered t totl phse shifts φ tht re odd multiples of π. ThisincludesthecseofthefundmentlBrggwvelength t φ = π.inthefourthcse(f R > 1) the stop nds re centered t even multiples of π. In coupled-mode theory the coupling nd detuning coefficients re design prmeters, i.e., prticulr spectrl response chrcteristics of multilyer coting cn e chieved y suitle choice of these prmeters. For the design prolem of chirped mirror the concept of impednce mtching plys n importnt role. As derived in [11], the exct coefficients (2) nd (3) for the individul unit cells define chrcteristic impednce Z ccording to δ(m) κ(m) Z(m) = δ(m) + κ(m). (11) The oscilltions oserved in the dispersion properties of chirped mirror re cused y n impednce mismtch in the front prt of the mirror. The impednce cn e mtched y properly djusting the coupling coefficient s function of penetrtion depth [11]. Moreover, the GDD cn e independently designed y suitle choice of the detuning coefficient with the chirp lw [13]. It is importnt to note tht κ, δ, nd Z re not only functions of the considered unit cell m ; they lso depend on wvelength λ(see (7) nd (8)). Alterntively, multilyer cotings tht re composed of symmetriclly defined unit cells cn e descried y the method of equivlent lyers [26, 27]. According to Herpin s theorem, every symmetricl comintion of homogeneous lyers is equivlent, t one ritrry wvelength, to single homogeneous lyer [19]. This lyer, clled the equivlent lyer, is chrcterized y its equivlent (refrctive) index N e (Herpin index) nd its equivlent (phse) thickness Γ e.atthe design wvelength the sustitution of multiple lyers y n equivlent lyer conserves the opticl properties of the coting nd is therefore exct. In the picture of equivlent lyers, N e nd Γ e ct s design prmeters, similr to the sitution descried ove for the coupled-mode theory. Hence, we hve n lterntive formlism tht exctly descries multilyer coting y using set of two design prmeters. As recently shown in [22], there is forml equivlence etween coupled-mode theory nd the method of equivlent lyers. Reltions exist tht directly link the coupled-mode prmeters (κ, δ) with those descriing the equivlent lyer (N e, Γ e ). For the three-lyer comintion shown in Fig. 2, the following reltions re vlid: N e n 1 = 1 Z = = δ + κ δ κ sin(φ) +r 2 sin( φ) + 2r sin [(φ + φ) /2] sin(φ) +r 2 sin( φ) 2r sin [(φ + φ) /2], (12) cos (Γ e ) = cos (γ + π) = F R = 1 1 r 2 { cos (φ) r 2 cos ( φ) }, (13) where (2), (3), nd (9) (11) hve een used. Thus, the normlized equivlent index is the reciprocl of the chrcteristic impednce Z 1, nd the equivlent thickness is essentilly given y the exct propgtion constnt, γ. Inversion of(13) results in multi-vlued solution for Γ e.wetkethesolution derived in [22], which is comptile with the definition given in [2]. In contrst to the constnt indices n 1 nd n 2, the equivlent index shows strong wvelength dependence. Moreover, for wvelengths in the stop-nd regime the equivlent-lyer prmeters ecome complex. It follows from (12) tht the impednce-mtching prolem of chirped mirror trnsltes into the prolem of mtching the equivlent index, s will e discussed in detil in Sect. 2. Additionlly, finding n pproprite chirp lw y djusting the propgtion constnt γ vi the detuning coefficient δ is equivlent to proper choice of the equivlent thickness long the chirped mirror structure ccording to (13). 1.2 Inclusion of the mient medium Now we incorporte the index discontinuity t the mient medium/coting interfce into our coupled-mode description. We define the refrctive-index profile of the unit cell s shown in Fig. 3, i.e., we include the step of the refrctive index from medium 1, n 1,totherefrctiveindexofthemient medium, n, t oth ends of the unit cell. Figure 3 shows the refrctive-index profile of sequence of three unit cells. Inside the mirror structure, the downwrd steps nd susequent upwrd steps cncel ech other. At the ends of the structure, however, n index step from mteril 1 to the respective mient medium remins. This symmetric description of the unit n(m) n(m) n 2 n 1 n n 2 n 1 n A ~ B ~ S u 1_ φ 2 1 A B φ=φ 1 +φ 2 1_ φ 2 φ 2 1 A B A ~ B ~ 1 S d = S ū Fig. 3. Modified refrctive-index profile of the unit cell defined for the inclusion of the mient medium. Trnsfer mtrices S d nd S u convert mplitudes A nd B inside the lyer stck into mplitudes à nd B in the mient medium s shown. Note tht symmetric inclusion my e ssumed without loss of generlity in the cse of high-reflectivity coting, s effectively no light propgtes through the lyer structure. Resulting index profile for sequence of three unit cells. Note tht upwrd nd downwrd index steps cncel except for the ends of the coting structure m m

5 513 cells presupposes tht the refrctive indices of oth mient medi (e.g. sustrte nd ir) re equl, which is generlly not the cse. However, for highly reflecting mirror cotings, s considered in this pper, this is irrelevnt s lmost no light psses through the coting. Mthemticlly, the terminl discontinuity of the refrctive index is descried y Fresnel trnsfer mtrices S u nd S d for the upwrd nd downwrd steps, respectively. These trnsfer mtrices trnsform the mplitudes A nd B into new mplitudes à nd B (see Fig. 3). We write (Ã(m) ) B(m) ( ) A(m) = S u B(m) ( ) (Ã(m) ) A(m) = S d, (14) B(m) B(m) with the Fresnel mtrices [18] ( ) 1 S u = 2 n + n 1 n n 1 = Sd 1, (15) n n 1 n n 1 n + n ( 1 ) 1 S d = 2 n + n 1 (n n 1 ) = Su 1. (16) n n 1 (n n 1 ) n + n 1 Using the trnsformtion (14), we otin the following new coupled-mode equtions for the trnsformed mplitudes (Ã(m) ) d dm B(m) ( )(Ã(m) ) δ(m) κ(m) = i, (17) κ(m) δ(m) B(m) where the trnsformed coefficient mtrix is given y ( ) δ(m) κ(m) κ(m) δ(m) ( ) δ(m) κ(m) = S u Su 1. (18) κ(m) δ(m) Thus, the new coefficients re otined from the originl ones y similrity trnsformtion. Evluting (18) yields the following explicit expressions for the trnsformed exct coupling nd detuning coefficients κ = c 1 κ + c 2 δ, (19) δ = c 1 δ + c 2 κ, (2) with c 1 = 1 2 nd c 2 = 1 2 ( n1 + n ), (21) n n 1 ( n1 n ). (22) n n 1 We cn immeditely see tht the trnsformed coupling nd detuning coefficients reduce to the originl coefficients (2) nd (3) in the cse of n mient medium with refrctive index equl to the index of medium 1, i.e., n = n 1. Tle 1 summrizes exmples of trnsformtion coefficients (21) nd (22) tht will e discussed in Sect. 2. The coefficient c 1 is lwys close to unity, wheres the vlue of c 2 strongly vries. c 2 is direct mesure of the index discontinuity etween the mient medium nd medium 1 nd essentilly descries the difference etween trnsformed nd originl coefficients. Tle 1. Vlues of the trnsformtion coefficients c 1 nd c 2 for the design studies of Sect. 2 c 1 c 2 A. No impednce mtching 8.42 B. Prtil impednce mtching 3.26 C. Perfect impednce mtching Using (11) nd (19) (22) thetrnsformedimpedncefollows s δ(m) κ(m) Z(m) = δ(m) + κ(m) = n Z(m) = n. (23) n 1 N e Finlly, we note tht the equivlent-lyer prmeters re invrint under the trnsformtion (14), i.e., Ñ e = N e nd Γ e = Γ e. 2 Impednce mtching to the mient medium Compring the coupled-mode equtions (1) with (17) we see tht oth equtions re exctly of the sme form. Hence, ll equtions nd conclusions derived in [11 13] still hold, provided the trnsformed coefficients (19) (22) re used insted of (2) nd (3). The conditions for the suppression of dispersion oscilltions Z(m = ) = 1 δ κ(m = ) =, (24) Z (m = ) = δ κ (m = ) =. (25) provide perfect impednce mtching of the chirped mirror structure to n mient medium with the sme refrctive index s medium 1 (see [11]). The prime in (25) denotes the derivtive with respect to m. Condition (24) sttes tht, ccording to (12), the equivlent index must equl the index of medium 1 t the mirror front. Moreover, for perfect mtching, condition (25) requires tht the equivlent index e rmped up or down s slowly nd smoothly s possile. The sme conditions lso hold for the trnsformed coupled-mode prmeters, i.e., Z(m = ) = 1 δ κ(m = ) =, (26) Z (m = ) = δ κ (m = ) =. (27) According to (23), these impednce-mtching conditions re equivlent to mtching of the equivlent index of the unit cell to the refrctive index of the mient medium. Generlly, for n ritrry mient medium, finding rodnd solution stisfying these conditions is very difficult ecuse of the strong wvelength dependence of the equivlent index. In Sects , we investigte mtching condition (26) for three mient medi with different refrctive indices n. Without loss of generlity, we restrict ourselves to the investigtion of (26). If this condition is stisfied, condition (27) cn e lwys fulfilled y sufficiently slow chnge of the design prmeters. We find lrge differences in the qulity of the design, depending on the degree of impednce mtching

6 Group dely (fs) 514 chieved nd indicted y the mgnitude of dispersion oscilltions. In the following exmples, we choose the refrctive indices n 1 = 1.5 nd n 2 = 2.5 for the lyer mterils, leding to r =.25. These vlues re chosen s they re close to the indices of SiO 2 nd TiO 2, commonly used s dielectric coting mterils. 2.1 No impednce mtching In the first exmple, we ssume tht the mient medium is ir, i.e., n =. In terms of dispersion oscilltions we wnt to discuss this s the worst possile cse. We refrin from ny ttempts to improve impednce mtching t the interfce to ir nd consider plin simple-chirped mirror structure. A simple-chirped mirror refers to multilyer structure with grdully incresing Brgg wvelength ut constnt 5% duty cycle, i.e., φ 1 (m) = φ 2 (m) m. Figure 4 shows the response chrcteristics of simplechirped mirror. In this exmple, the Brgg wvelength is chirped over the initil 2 unit cells nd then kept constnt for nother five unit cells to increse the reflectnce for long wvelengths. The reflectnce is very high over rod wvelength rnge nd covers most of the Ti:spphire gin spectrum. However, the group dely (GD) shows extremely lrge oscilltions, similr to Gires Tournois interferometer (GTI, [28]), with pek-to-pek mplitude on the order of 1 fs, which mkes such mirror useless for ultr shortpulse genertion. Oviously, the simple-chirped mirror structure does not provide ny impednce mtching. To further investigte this, we explore the trnsformed coupled-mode design prmeters ( κ, δ) nd the equivlent-lyer prmeters (N e, Γ e ) s two-dimensionl functions of the unit cell nd wvelength, depicted s contour plots. In these plots the x-xis gives the unit cell nd the y-xis gives the incident wvelength. It is ssumed tht the light is incident from the right. In Fig. 5 the contour lines of the trnsformed coupling coefficient (19) re lmost horizontl lines with vlues fr Reflectnce Fig. 4. Clculted mplitude (left xis) nd phse (right xis) properties of stndrd front-coted simple-chirped mirror s function of wvelength. Shown re the reflectnce nd the group dely upon reflection, respectively. In this exmple the Brgg wvenumer is linerly decresed over the first 2 unit cells from the mximum vlue kb mx = 2π/(65 nm) to the minimum vlue kb min = 2π/(95 nm) ccording to k B(m) = kb mx ( m 1) ( kb mx ) kmin B /19. For the lst five unit cells the Brgg wvenumer is kept constnt t its minimum vlue kb min.airhseenssumeds the mient medium. The simple-chirped mirror provides rod highreflectnce rnge, ut exhiits strong dispersion oscilltions over most of this rnge c Unit Cell Coupling Coefficient κ ~ ~ Detuning Coefficient δ Equivlent Index N e Equivlent Thickness Γ e /π d Unit Cell Fig. 5. Contour plots of the trnsformed coupling coefficient κ, the detuning coefficient δ, c the equivlent index N e, d nd the equivlent thickness Γ e (in units of π) ofthesimple-chirpedmirroroffig.4sfunctions of unit cell m nd wvelength from zero t the coting surfce. Condition (26) is not fulfilled for ny given wvelength in the high-reflectnce region. In fct, the trnsformed coupling coefficient in Fig. 5 is even frther from eing mtched to zero thn the originl coupling coefficient. The originl coupling coefficient is lwys negtive nd pproximtely constnt due to the simple-chirped mirror structure (κ 2r =.5, [22]). According to (19), mtching cn only e chieved y positive vlues of c 2 δ. For wvelengths ove 65 nm, however, this is not fulfilled t the coting surfce ecuse of the negtive detuning coefficient δ. Consequently, solutely no impednce mtching is chieved in this simple-chirped structure. Figure 5 shows the contour lines of the trnsformed detuning coefficient. These contour lines re nerly liner functions over the rnge of chirped Brgg wvenumers. According to (2), the trnsformed detuning coefficient essentilly

7 515 follows the originl detuning coefficient with n offset of c 2 κ.2. The qulittive ehvior of the detuning coefficient is not chnged y the trnsformtion. The zero contour line of the originl detuning coefficient indictes the Brgg condition, i.e., the dependence of Brgg wvelength on penetrtion depth or chirp lw. Compred to the originl detuning coefficient, the zero contour line of the trnsformed detuning coefficient is shifted y 5 nm to shorter wvelengths (dshed line). This plot clerly illustrtes the pproximtely liner increse of the GD with wvelength (negtive GDD). This scenrio cn e lso interpreted in the picture of equivlent lyers. According to Sect. 1, impednce mtching requires tht the equivlent index t the mirror front equls the refrctive index of the mient medium (n = ). However, Fig. 5c shows tht the equivlent index is extremely fr from eing mtched to unity t the mirror front, illustrting the lrge impednce mismtch. The rod re surrounded y contour lines with the vlues zero nd infinity mrks the stop-nd region, where the equivlent index ecomes complex. For this region the unit cells represent potentil rrier with n exponentil ehvior of the electromgnetic field. In the pssnd regions outside the evnescent region, the equivlent index is rel nd the unit cells re trnsprent for the light. With respect to the quntum-mechnicl scttering prolem s introduced in [11], the upper line (N e = ) corresponds to right turning points nd the lower line N e = toleftturningpointsofthecorrespondingclssicl motion. Figure 5d shows tht, for fixed wvelength, the equivlent thickness of the unit cells monotoniclly increses with penetrtion depth, illustrting the chirp of the Brgg wvelength long the mirror structure. As mentioned ove, custom-tilored dispersion properties cn e designed y proper control of the equivlent thickness. For pssnd regions the equivlent thickness of unit cell m pproximtely corresponds to the totl opticl phse shift (5) evluted t wvelength λ [22]. Similrly to Fig. 5c, the evnescent region is surrounded on oth sides y the dshed unity contour line (given in units of π). 2.2 Prtil impednce mtching In the second exmple we gin consider the cse of simplechirped mirror, ut now we serch for the refrctive index of the mient medium with est possile impednce mtching without dditionl mtching lyers (e.g., n AR coting). It might e suspected tht est mtching is chieved if the index of the mient medium is the geometric verge of the indices of medi 1 nd 2. Structures mtched to the geometric verge of the refrctive indices hve een used efore s strting design for chirped mirrors (see, e.g., [23]). We will exemplify, however, tht this method does not yet led to the optimum solution for the impednce-mtching prolem considered nd only provides prtil impednce mtching. For simple-chirped mirror with φ(m) = m, using (12) nd (23) the impednce-mtching condition (26) is rewritten s Z(m) = n n 1 sin(φ) 2r sin(φ/2) sin(φ) + 2r sin(φ/2)! = 1. (28) From this the mient refrctive index cn e resolved s cos(φ/2) +r n = n 1 cos(φ/2) r. (29) The right-hnd side of (29) is the equivlent index s function of the phse vrile φ. Impedncemtchingrequires tht (29) e fulfilled over rod wvelength rnge, i.e., the equivlent index should equl the index of the mient medium n independent of the totl phse shift φ. The refrctive index of the mient medium is rel quntity. This requires φ < 2rccos( r ) s necessry condition if we restrict ourselves to phse shifts φ π. The equivlent index is pproximtely constnt only for sufficiently smll phse shifts φ close to zero, see Fig. 6. In prticulr, this figure clerly illustrtes the strong wvelength dependence of the equivlent index, s mentioned in Sect. 1. In the limit φ (λ ) we get the long-known result [21]: n = n 1 1 +r 1 r = n 1 n 2. (3) This is exctly the geometric verge refrctive index of oth lyer mterils. The limit φ ppers to e in contrdiction to the ssumption of qurter-wve lyers, where φ π for wvelengths ner the Brgg wvelength. In this relevnt cse, the impednce-mtching condition prcticlly requires us to strt chirping t Brgg wvelengths much smller thn the minimum design wvelength of the coting. In other words, the impednce-mtching condition φ isonlyfulfilled for wvelengths much lrger thn the Brgg wvelength of the unit cell considered. Let us consider the exmple of simple-chirped mirror structure in Figs. 7 nd 8. Unlike in the previous exmple Equivlent Index Phse Shift φ/π Fig. 6. Equivlent index s function of phse shift s given y (29). As in the other exmples, n 1 = 1.5, n 2 = 2.5 hseenssumed.forsmllvlues of φ, theequivlentindexpprochesthegeometricvergeofn 1 nd n 2, i.e., n = The equivlent index diverges for φ 2rccos( r ) nd is complex-vlued for the fundmentl stop-nd region t Brgg resonnce φ B = π

8 Group dely (fs) 516 Reflectnce Fig. 7. Clculted reflectnce (left xis) nd group dely (right xis) of ck-side-coted simple-chirped mirror s function of wvelength. The Brgg wvenumer is linerly decresed over 5 unit cells from kb mx = 2π/(3 nm) to kb min = 2π/( nm). Amterilwiththegeometric verge of the coting mterils (n 1.94) hs een ssumed s the mient medium. The ck-side-coted simple-chirped mirror provides prtil impednce mtching, s indicted y the reduced dispersion oscilltions over most of this rnge (Figs. 4 nd 5) we now ssume n mient medium with n index equl to the geometricl verge of the two lyer mterils. For coting providing dispersion compenstion from nm to nm, we strt chirping the Brgg wvelength t 3 nm. The high-reflectnce rnge of the coting is extremely wide s shown in Fig. 7, ut most of this rnge lies well out of the nd with phse properties tht might e useful for dispersion compenstion. The GD oscilltions with pek-to-pek mplitude of out 1 fs re considerly reduced compred to Fig. 4. The dispersion oscilltions decrese with n incresing wvelength ut never reduce to negligile vlue. The curve is fr from eing smooth nd such oscilltions re not tolerle for n ultr-short-pulse lser source. Reducing the numer of lyers nd incresing the initil Brgg wvelength of the design further deteriortes the phse properties of the coting. On the other hnd, dispersion properties t nm my e improved y strting to chirp t even lower Brgg wvelengths < 3 nm. In ny cse, only smll prt of the spectrum is covered, providing high reflectnce nd decent dispersion properties simultneously. The contour plots shown in Fig. 8 further illustrte the prtilly chieved impednce mtching. Still, the contour lines t the mirror front re essentilly horizontl lines similr to the contour lines shown in Fig. 5. In contrst, mtching condition (26) is now prtilly fulfilled, s indicted y the contour plots for the trnsformed coupling coefficient nd for the equivlent index (see Fig. 8 nd c). These contour plots confirm tht mtching is improved t longer wvelengths. With incresing wvelength the coupling coefficient pproximtes zero, Fig. 8, nd the equivlent index in Fig. 8c pproches n given y (3). In conclusion, ssuming qurter-wve lyers nd n mient medium with geometric verge of the indices of medi 1 nd 2 provides prtil reduction of dispersion oscilltions. The qulity of impednce mtching, however, is directly relted to the excess ndwidth, with excellent reflectnce ut poor dispersion properties. Therefore, this design pproch mkes very uneconomic use of the ville c Equivlent Thickness Γ /π e d Unit Cell Fig. 8. Contour plots of the trnsformed coupling coefficient κ, the detuning coefficient δ, c the equivlent index N e, d nd the equivlent thickness Γ e (in units of π) oftheck-side-cotedsimple-chirpedmirrorof Fig. 7 s functions of unit cell m nd wvelength ~ Detuning Coefficient δ Equivlent Index N e Unit Cell Coupling Coefficient ~ κ numer of lyers, which renders it imprcticle for mny pplictions. 2.3 Perfect impednce mtching In the finl exmple, we ssume tht the mient medium hs the sme refrctive index s medium 1, i.e., n = n 1 = 1.5. As mentioned in Sect. 1, this reduces the trnsformed coupledmode prmeters to the originl ones nd mtching condition (26) reduces to the old mtching condition (24). According to [11], the DCM design technique now llows for perfect impednce mtching. This mens using very thin lyers of mteril 2 for the initil unit cells in the mirror structure, i.e., n 2 d 2,m λ B (m)/4, nd then slowly rmping up the duty cycle of the coting until 5% is reched, i.e., n 2 d 2,m = λ B (m)/4. With the DCM technique, the coupling coefficient vnishes ccording to (2) nd (4 8), nd the impednce equls unity ccording to (11). Other thn in stndrd DCM,

9 Group dely (fs) 517 however, no rodnd AR coting is required to mtch the index to the mient medium. Figure 9 shows reflectnce nd GD of DCM consisting of 25 unit cells. The Brgg wvenumer is linerly chirped over the first 2 unit cells nd then kept constnt for the remining five unit cells in the sme wy s in Sect In Sect. 2.3, however, mteril with n index identicl to the low-index lyer mteril hs een ssumed s the mient medium. Also, we independently chirp the thickness of the mteril 2 over the first 12 unit cells (doule-chirping). The very smooth dispersion over the entire high-reflectnce rnge in Fig. 9 clerly illustrtes the resulting effect of impednce mtching. This comes t the expense of slight reduction of the high-reflectnce ndwidth on the short-wvelength side, s comprison of Figs. 4 nd 9 revels [11]. In Fig. 1 the (trnsformed) coupling coefficient of the DCM is plotted. At the eginning of the mirror structure, the contour lines re lmost verticlly oriented. Ech individul unit cell provides n lmost constnt coupling coefficient for ny given wvelength. The coupling coefficient vnishes t the eginning, indicting excellent rodnd impednce mtching. The solute vlue of the coupling coefficient increses long the grting structure until it reches its mximum vlue. The ehvior of the (trnsformed) detuning coefficient shown in Fig. 1 generlly resemles the first exmple shown in Fig. 5. In contrst, however, the detuning vlues of Fig. 1 re shifted towrds lrger vlues. The zero contour line pinpoints Brgg resonnce, yielding the chirp lw [13]. The high degree of impednce mtching chieved is lso clerly illustrted y the contour plot of the equivlent index. For the pssnd region t the mirror front, the contour lines re essentilly verticl lines with vlue close to 1.5 over rod wvelength rnge. This demonstrtes gin the lmost perfect mtching to the refrctive index of the mient medium. The equivlent thickness plotted in Fig. 1d shows the sme qulittive ehvior s the equivlent thickness in the first exmple (see Fig. 5d). However, one devition e- Reflectnce Fig. 9. Clculted reflectnce (left xis) nd group dely (right xis) of BASIC doule-chirped mirror s function of wvelength. In this exmple the Brgg wvenumer is linerly decresed over the first 2 unit cells from the mximum vlue kb mx = 2π/(65 nm) to the minimum vlue kb min = 2π/(95 nm) ccording to k B(m) = kb mx ( m 1)(kB mx kb min )/19. For the lst five unit cells the Brgg wvenumer is kept constnt t its minimum vlue kb min.thethicknessofthemteril2isvried with penetrtion depth ccording to d 2,m = π/(2k B (12)n 2 )( m /12) 1.2 over the first 12 unit cells (doule-chirp section in Fig. 1). A mteril with n index equl to the low-index mteril of the coting hs een ssumed s the mient medium (n = n 1 = 1.5). The BASIC doule-chirped mirror provides rod high-reflectnce rnge nd smooth dispersion chrcteristics, simultneously c Unit Cell Coupling Coefficient κ ~ ~ Detuning Coefficient δ 2.5 Equivlent Index N e 1.1 Equivlent Thickness Γ e /π d Unit Cell Fig. 1. Contour plots of the coupling coefficient κ, the detuning coefficient δ, c the equivlent index N e, d nd the equivlent thickness Γ e (in units of π) ofthebasicdoule-chirpedmirrorinfig.9sfunctionsof unit cell m nd wvelength. Note tht in nd trnsformed nd originl vlues for the coupling coefficient nd the detuning coefficient re identicl tween the equivlent-lyer prmeters of Figs. 5 nd 1 is very ovious. For short wvelengths t the mirror front the dshed contour lines, which surround the stop-nd region, re nrrower, indicting the reduction of the high-reflection ndwidth for short wvelengths (compre Figs. 4 nd 9). 3 BASIC mirrordesign 3.1 Sustrte properties The theoreticl findings of Sects. 1 nd 2 now offer convenient method to void dispersion oscilltions of chirped mirrors y proper design. Choosing n index of refrction of the mient medium identicl to the index of one of the coting mterils, perfect impednce mtching t this interfce is chieved y doule-chirping of the chirped mirror structure. Prcticlly, this mens inversion of the lyer sequence 1.57

10 518 nd coting the lyers on the ck side of sustrte. Consequently, the light em psses through the sustrte efore reflection y the chirped mirror coting. Therefore, we nme this method ck-side-coted (BASIC) chirped mirrors. It might pper difficult to choose sustrte mteril with exctly mtching index ecuse the refrctive index of sputtered or evported mterils my significntly devite from the refrctive index of ulk mterils. A refrctive index n SiO2 1.49, e.g., for sputtered silic compres to ulk vlue n FS 1.45 t nm. Such n index discontinuity t the interfce etween coting nd sustrte, however, merely cuses reflection of t norml incidence. A further reduction is chieved y suitle choice of n opticl glss with refrctive index closer to sputtered SiO 2 (e.g., BK7 with n BK t nm). In ny cse, the prolem of residul mismtch etween sustrte nd coting index is fr less stringent thn the prolem of mtching to ir. A residul mismtch of the order discussed ove is esily resolved y numericl optimiztion of the lyer sequence. One might rgue tht the BASIC pproch only trnsfers the impednce-mtching prolem to the opposite interfce of the sustrte. In fct, if plne-prllel sustrtes were used, interference with the reflection from the interfce to ir gives rise to pronounced stellite pulses. With the high reflection from the ck nd the prtil Fresnel reflection from the top, such mirror would form GTI, cusing strong spectrl vrition of the mirror phse. However, these detrimentl interference effects re suitly suppressed y geometricl mismtch of the two surfces. This cn e chieved y mking the two surfces non-prllel nd therefore noninterfering, i.e., y wedging or y choosing different centers of curvture for the two surfces. Residul interference is clculted from the overlp of the reflections in the fr field. For typicl pplictions in lser cvity with em divergences of fewmrd,wedgenglesontheorderofonedegreeredily reduce interference to negligile mount for plne mirror. To replce the concve focusing mirrors in typicl lser cvity, thin plno-convex lens sustrtes my e used to diminish interference effects. The geometricl mismtch of the sustrte surfces llows for suppression of detrimentl interference effects y ny given degree, ut the power losses cused y the Fresnel reflection t the top surfce pper prohiitive for use inside lser cvity. Of course, n AR coting on the front surfce of the BASIC mirror redily reduces this prolem. Additionlly, n AR coting further suppresses dispersion oscilltions cused y smll residul overlp of the ems in the fr field nd llows for smller geometricl mismtch of the two surfces. Hence, even though it is not strictly required, the AR coting lso improves the dispersion properties of the BASIC mirror. The totl dispersion of BASIC mirror is given y the dispersion of the chirped mirror structure plus twice the dispersion of sustrte nd AR coting upon trnsmission. Typicl chirped mirror cotings for use in Ti:spphire lsers llow for the compenstion of mteril dispersion equivlent to pth length of out 2-mm fused silic per ounce [29]. Therefore, it is desirle to use sustrtes of few 1-µm thicknessto permit the genertion of net negtive GDD with BASIC mirror. On the other hnd, however, multilyer cotings of 6 or more lyers re known to generte tensile or compressive stress on the sustrte, cusing deformtion of extremely thin sustrtes. In our experiments we found sustrte thickness of out.5 mm vile compromise etween surfce deformtion nd tolerle mteril pth. 3.2 Optimized design for ultr-short-pulse Ti:spphire lsers For BASIC chirped mirrors, one hs to design two independent cotings, n AR coting for the top surfce nd the chirped mirror coting for the ck surfce. We use n 18- lyer AR coting nd 6-lyer BASIC coting for the coverge of the gin ndwidth of Ti:spphire (65 nm) with oth high net reflectnce nd smooth dispersion properties. The AR coting is designed using commercil softwre [3]. Figure 11 depicts the residul reflectivity of out 1 3 for the given wvelength rnge. Note tht this coting lso provides high trnsmission for the pump wvelengths 488 nd 514 nm (rgon-ion lser). Figure 11 shows the resulting GDD of the BASIC AR coting upon trnsmission with nerly negligile dispersion over the entire gin ndwidth of Ti:spphire. The 6-lyer chirped mirror coting on the ck surfce is designed ccording to [13] nd is susequently computeroptimized with locl grdient lgorithm [31]. The design gols re high reflectnce nd smooth GDD over the 45-nm wvelength rnge of the AR coting. Additionlly, the mirror hs to e highly trnsprent for the pump lser. The mirror is designed for p-polrized light with n incidence ngle of 5 in ir. This corresponds to n incidence ngle of 3.44 in the fused silic sustrte. Figure 12 depicts the spectrl response chrcteristics of the resulting BASIC DCM structure, clculted for fused silic s the incident Reflectnce GDD (fs 2 ) BASIC-DCM Stndrd DCM -6 5 Fig. 11. Reflectnce of two AR cotings with different ndwidth. The dshed line shows 14-lyer AR coting tht provides out 24-nm ndwidth with less thn 1 4 reflectnce s used in stndrd DCMs [11]. The solid line refers to n ultr-rodnd coting (18 lyers) with more thn 4-nm ndwidth nd out 1 3 residul reflectnce s used on the front surfce of the BASIC chirped mirror. As the AR coting is pssed twice per ounce, the effective losses mount to doule the residul reflectnce of the coting. The ultr-rodnd coting lso provides high trnsmission t the pump wvelengths 488 nd 514 nm. Group dely dispersion of the ultr-rodnd AR coting upon trnsmission

11 519 8 Reflectnce Group dely (fs) Designed Desired GDD (fs 2 ) Designed Desired c -8 Fig. 12. Clculted reflectnce of BASIC doule-chirped mirror structure. The structure is designed for use in n ultr-short-pulse Ti:spphire lser. In computer optimiztion process, highly trnsmissive window for the pump lser is introduced.mesured refrctive-index dt for the coting mterils re used. In these clcultions, the influence of AR coting nd sustrte on spectrl properties is not considered. The reference plne is chosen in the sustrte close to the coting, isolting the effect of the DCM coting. The reflectnce is plotted on two different scles to show the high reflectnce in the 62- to 15-nm rnge nd the trnsmissive region t 5 nm. Designed (solid line) nd desired (dshed line) group dely of the BASIC doule-chirped mirror structure upon reflection. Note tht oth curves prcticlly coincide except for the short-wvelength end of the rnge shown. c Designed (solid line) nd desired (dshed line) GDD of the BASIC doule-chirped mirror structure upon reflection medium. The chirped mirror structure provides 99.8% reflectnce from 61 nm (19 THz ndwidth). Accounting for the dditionl.2% losses due to the doule pss through the AR coting, this results in net reflectivity of 99.6% over the sme 4-nm ndwidth. The wide reflectnce rnge fvorly mtches the unprecedented ndwidth of extremely smooth dispersion of 22 THz. In contrst, the BASIC pproch tolertes residul reflectnce of 1 3 for use inside Ti:spphire lser, nd this cn e immeditely trded for 2 nm of extr ndwidth. The BASIC design technique is minly limited y the chievle net reflectivity rther thn y the mgnitude of dispersion oscilltions s in conventionl DCM designs. The insensi- 3.3 Comprison to stndrd DCMs Comprison of the structure of BASIC mirror nd stndrd DCM redily revels three identicl functionl sections ut in different order: sustrte, chirped mirror structure, nd n AR coting (Fig. 1). In the cse of the BASIC mirror, however, the two coting sections re non-interfering, seprted y the sustrte in etween. The non-interference of the two coting sections decouples their design, nd imperfections of the AR sections do not spoil the dispersion properties of the mirror. Figure 11 illustrtes this y compring the reflectnce of the AR coting used in the BASIC design pproch with similr 14-lyer cotings used in the design of previous stndrd DCMs [11]. Due to the high sensitivity of the DCM dispersion towrds residul reflection from the AR section, the residul reflectivity hs to e kept t 1 4 or elow, which limits the ndwidth of such coting to out 25 nm t -nm center wvelength. Given the lyer mterils nd the 45-nm ndwidth of this design, the residul reflectivity generlly sturtes t out 1 3 with only mrginl improvement from n incresed numer of lyers [32]. Consequently, ny ttempt to further increse the ndwidth of stndrd DCM cotings cuses drmtic increse of dispersion oscilltions. (Group dely) (fs) (GDD) (fs 2 ) BASIC-DCM Stndrd DCM 18 THz 22 THz BASIC-DCM Stndrd DCM -4 Fig. 13,. Dispersion oscilltions of BASIC doule-chirped mirrors (solid line) nd stndrd DCMs (dshed line). DtfromFig.12nd[1]hveeen used for this comprison. Shown re the differences etween desired nd designed vlues of group dely nd GDD

12 52 tivity towrds dispersion oscilltions is illustrted in Fig. 13, compring the dispersion properties of BASIC DCM nd conventionl DCM. Both cotings shown use 6 lyers, of which 14 re used in the AR section of the stndrd DCM. This exmple clerly shows tht the BASIC design pproch reduces dispersion oscilltions to pek-to-pek vlue of 5fs 2 (2 fs 2 rms), wheres the stndrd DCM of significntly less ndwidth exhiits pek-to-pek oscilltions of 5 fs 2 (15 fs 2 rms). Some dditionl dvntges of the BASIC pproch should e pointed out. With the wek remining impedncemtching prolem, computer optimiztion of the coting structure is less demnding thn in the conventionl DCM design pproch. Using the sme numer of lyers in BASIC mirror, wider wvelength rnge cn e covered ecuse no lyers hve to e scrificed for impednce mtching to ir. Finlly, the AR coting of BASIC mirror my e grown independently, using mterils with higher index contrst to further increse the net reflectnce. The design in Fig. 12 my serve s n illustrtion of how to chieve optimum performnce given the ndwidth nd other constrints of mode-locked Ti:spphire lser. Giving up the highly trnsmissive region round 5 nm used for pumping the lser llows for n dditionl reduction of dispersion oscilltions in the design. A further extension of the dispersion-compenstion ndwidth, e.g., for the compression of white-light supercontinu, is possile y llowing for reduced net reflectnce or using more lyers. For pplictions outside lser cvity, slightly incresed losses typiclly pose no prolem. In principle, support of single-cycle spectr with BASIC chirped mirrors is fesile in externl compression schemes. 4 Experimentl results 4.1 Bck-side-coted chirped mirrors The design presented in Sect. 3 ws grown using highprecision ion-em sputtering [33]. Active control of lyer deposition in the few-angstrom rnge [34] is indispensle ecuse of the high sensitivity of the GDD to deposition errors [12, 23]. We used.45-mm-center-thickness plnoconvex lens sustrtes s replcement for the 1-cm-rdius concve focusing mirrors previously used in our lser. The convex side of the lens sustrtes hs rdius of curvture of 15 mm (focl length f = R/2n 5 cm) nd is coted with the BASIC chirped mirror structure. The AR coting is pplied to the plne surfce. During polishing, the thin lens sustrtes re opticlly contcted to plne crrier sustrtes. With the support of the crrier sustrtes, surfce deformtions of the sustrtes cn e kept elow λ/5 (pekto-vlley vlues over 1-cm dimeter) in the mnufcturing process. For the coting process, the sustrtes re removed from the crrier. With the chirped mirror coting pplied, the thin lens sustrtes re finlly cemented into mting concve 1-mm-thick crrier sustrtes, similr to the construction of n chromtic lens. Cementing to crrier sustrte prevents mechnicl stress to the thin mirrors nd lso protects the chirped mirror cotings from environmentl influences. The internl stress of the coting is decresed y the reduced sustrte curvture. Since the BASIC DCM is uried ehind the lens, the front surfce cn e clened with no dnger of mechnicl dmge to the chirped mirror coting. The finl structure consisting of the thin BASIC DCM lens nd the thick glss support forms mechniclly rugged sl with two flt surfces. The ck side of the thick supporting glss sustrte cn e dditionlly AR-coted to improve trnsmission of the pump light. We use white-light interferometry to chrcterize the dispersion of the mnufctured mirrors. Figure 14 shows the mesured GDD of BASIC chirped mirror in comprison with the trget GDD of the design. Note tht these curves now include twice the GDD of.45-mm sustrte nd twice the GDD of the AR coting upon trnsmission. We GDD (fs 2 ) (GDD) (fs 2 ) (GDD) (fs 2 ) c Trget 3 x BASIC 2 x BASIC + 1 x conventionl BASIC-DCM Stndrd DCM Error simultions Mesured GDD Fig. 14. Desired round-trip GDD (single pss through fused silic prism sequence with 5-cm pex seprtion nd 2.3mm of Ti:spphire) used s trgetfunctionforthemirrordesign(dotted line). ThemesuredGDDof the BASIC chirped mirror is shown s dots. Notethenetpositivedispersion of the BASIC mirror t nm. In comprison with Figs. 12 nd 13, dispersion of the AR coting (Fig. 11) nd the sustrte with mesured thickness of.45 mm re now included. The differences etween designed nd mnufctured GDD re shown (BASIC chirped mirror, solid line; stndrd DCM[1], dshed line). The stndrd DCM exhiits similr dispersion oscilltions, which re, however, shifted such tht they prtilly cncel out dispersion oscilltions of the BASIC mirror. A comintion of one stndrd DCM nd two BASIC mirrors yields smooth dispersion compenstion from 65 nm to 95 nm, shown s the solid line in. The mgnitude of the dispersion oscilltions of the BASIC mirror coting grees with numericl simultions, indicting 2-Å (rms) lyer-deposition ccurcy (c)