Carrier to envelope and dispersion control in a cavity with prism pairs

Transcription

1 PHYSICAL REVIEW A 75, Carrier to envelope and dispersion ontrol in a avity with prism pairs Ladan Arissian and Jean-Claude Diels Department of Physis and Center for High Tehnology Materials, University of ew Mexio, 1313 Goddard SE, Albuquerque, ew Mexio 87106, USA Reeived 16 August 006; published 5 January 007 Continuously mode-loked lasers ombine a broad bandwidth with a fine omb struture that enables high resolution spetrosopy. Independent ontrol of the pulse duration, enter of gravity of the spetrum, the mode position, and spaing are essential for spetral appliations and metrology. Exat analytial expressions are derived for the first, seond, and third order dispersion in a avity with a pair of intraavity prisms. The influene of avity ontrols suh as tilt and translation of the end mirror on the frequeny omb parameters are analyzed and alulated. DOI: /PhysRevA PACS numbers: 4.60.F, 4.6.Eh I. ITRODUCTIO Continuously mode-loked lasers have been sine the early 70 s the traditional soure of a ontinuous train of short pulses 1. The fat that the output onsists in a train of idential pulses with fixed period lead to the realization that this soure of ultrashort pulses an also beome a tool of high resolution spetrosopy. Coherent interation experiments involving pulse trains, suh as the two-photon spetrosopy pioneered by Ekstein et al., require simultaneous ontrol of the pulse repetition rate, the mode frequeny, the pulse duration, and the enter of gravity of the pulse spetrum. The realization that the spetrum of a modeloked laser an onsist of rigorously equally spaed modes 3,4 led to the expression frequeny omb. The latter an be used as a ruler to ompare and measure different frequenies aross the eletromagneti spetrum. Control and stabilization of frequeny ombs have thus beome inreasingly important in physis and metrology. Intraavity prisms onstitute the most versatile passive ontrol element for mode-loked lasers. Historially, they have played an inreasingly important role for the ontrol of laser output, starting from the ontrol of the pulse entral wavelength, to the pulse duration 5,6 by providing an adjustable dispersion, and lately the pulse repetition rate 7. For the lasers with the shortest pulse duration, there is no requirement for spetral tunability, and dispersion ontrol is often realized with hirped mirrors 8. This all mirror approah laks the flexibility offered by a avity with prisms, where all the parameters ited above an be ontrolled ontinuously. Despite the fat that pairs of prisms have been used for more than two deades, there is to date no omprehensive analytial theory of the ontrol of group delay, phase delay, and spetral bandwidth in avities with pairs of prisms. This paper offers suh an analysis for a standard laser avity with a pair of prisms, and provides expressions for the ontrol of dispersion, group and phase veloity, and spetrum. The essential parameters of a mode-loked omb, and their relation to the avity parameters, are reviewed in Se. II. In Se. III, the equations for phase delay and group veloity dispersion for a pair of prisms are derived. It is shown that some expressions used for the prism dispersion are inorret. umerial examples are given for two ommonly used glasses. Setion IV deals with the variations of phase delay, arrier to envelope phase, and pulse spetrum introdued by a tilt and translation of the end mirror. The relative importane of all these effets will beome lear with the numerial examples given at the end of this last setion. II. PARAMETERS OF A PULSE TRAI The train of pulses emitted by a mode-loked laser results from the leakage of a single pulse traveling bak and forth in a avity of onstant length. The roundtrip time of the avity is thus a onstant, implying a perfetly regular omb of pulses in the time domain. In spetrum, suh a pulse train is a perfet frequeny omb, with teeth equally spaed at frequenies laser,m. By ontrast, the modes of the non-modeloke laser onstitute an unequal omb of frequenies avity,m given by: avity,m = m n i avity,m L i i m n avity,m L, where m is a positive integer and n i m L i is the optial pathlength at the frequeny avity,m of the avity element i of length L i. We formally write nl= i n i avity,m L i, where L is the geometrial avity length and n is an effetive refrative index averaged over the avity, at the frequeny. The mode-spaing avity,m avity,m 1 an be estimated by avity,m avity,m 1 Ln avity,m 1 avity,m n avity,m = v g L = 1. d The time d is thus the group delay through the avity, different from the phase delay /Ln avity,m. The avity modes avity,m and the frequeny omb modes laser,m oinide at an index m=. We will all this entral mode of the mode omb, losest to the enter of gravity of the pulse spetrum. It is desirable to have a small variation of the group delay aross the spetrum. All optial elements having generally a positive group veloity in the visible and near IR, a ombi /007/751/ The Amerian Physial Soiety

2 LADA ARISSIA AD JEA-CLAUDE DIELS PHYSICAL REVIEW A 75, nation of elements suh as a pair of prisms is desirable in order to minimize the variation of group delay aross the spetrum. The proedure will be to alulate the phase shift of the light between an initial and final wave front. The group delay d between these two wave fronts is given by: = d, 3 where = is the angular optial frequeny. The variation of the group delay will be investigated for a pair of prisms. First, the variation of the group delay versus frequeny, whih is the group veloity dispersion, will be derived exatly. ext, it will be shown that the variation of the group delay versus angle of inidene of a beam on the pair of prisms is of the same order of magnitude as the variation of the phase delay. This result applies in partiular to the ontrol of avity roundtrip time by the tilting of an end mirror 7,9. It will be shown also that the tilt of an end mirror results in a shift of the pulse spetrum. III. GROUP VELOCITY COTROL WITH PAIRS OF PRISMS A. Frequeny dependent optial path Cavities with a single prism to ontrol dispersion have been introdued in The adjustable positive dispersion introdued by the material of the prism, ombined with negative phase modulation due to saturation of a dye solution, resulted in intraavity pulse ompression down to 60 fs 5,6. It has been demonstrated that the group veloity dispersion introdued by a single prism an only be positive 10 in a avity 1. In a onfiguration with two or four idential Brewster prisms however, it has been shown by Fork et al. 11 that the group veloity dispersion an be negative as well as positive. umerial urves for the group veloity dispersion of prisms, pairs have been published by Petrov et al. 1. A generalized theory introdued by Duarte yields exat expressions for the angular dispersion and the beam expansion of any pairs of prisms at any inidene. The present work is limited to a pair of prisms in ompensating orientation i.e., oriented suh that the input and output beams are parallel, for any wavelength. However, the interwoven ontributions of angular and material dispersion to the group veloity dispersion are analyzed in detail. Exat expressions for the dispersion under arbitrary inidene are presented below. While the derivation is tedious, the result is seen to have a very simple physial interpretation. The pair of isoseles prisms represented in Fig. 1 are oriented with their faes parallel to eah other. The relative position of these prisms is defined by the two distanes s and t. The position of the beam inident on the pair of prisms is defined by its angle of inidene 0 on the first prism, and the distane a=oa from the apex O of the first prism to the point of inidene A. The optial path ABEFD at a frequeny is represented by the solid line in Fig. 1, while the path for a ray upshifted by is represented by the dashed line. The beam path in FIG. 1. Color online Beam propagation through a pair of parallel fae idential prisms with an arbitrary apex angle. The geometrial setup is defined by only three parameters: 1 OA=a the distane of input beam to the apex of the first prism, not neessary zero, t is the projetion of the distane between the prisms apexes OO on the inner prism fae, and 3 the distane s between the two inner parallel faes of the prisms, so that L=s/os 3 is the beam path between the two prisms. The total propagation path is ABEFD, a funtion of frequeny. All the angles defined on the figure are related by Snell s law, where n is a funtion of. Throughout the paper the index of refration of the air is negleted n air =1, the symbol n is used only for prism material. this onfiguration an be divided into three setions: sin L g = t s tan 3, os 1 BE = L = s os 3, w = j OF sin 0 = j t s tan 3 os + sin tan 1 asin 0, 4 where L g is total path in prisms, BE=L is the propagation between the prisms, and w is the path in air to the end mirror. ote that if the two prisms were brought together merging B and E, they onstitute a slab of glass of thikness g =L g os 1. Sine it is a parallel fae plate of glass, the optial path L g is independent of the distane OA=a. B. Group delay After adding all ontributions to the total phase = nl g + BE + FD, we obtain, after straightforward algebra, the omplete expression for the group delay through the pair of prisms see Appendix A:

3 CARRIER TO EVELOPE AD DISPERSIO COTROL I = nl g + BE + FD + L g. The first term in this equation represents the travel delay at the phase veloity, and the seond part of Eq. 6 is the arrier to envelope delay or CE aused by the pair of prisms. The prisms are onsidered to be in vauum, and the ontribution to the dispersion of air is negleted. C. Group veloity dispersion The rigorous approah to the group veloity dispersion is to take the derivative of Eq. 6, with respet to angular frequeny. In this derivation, all parameters that hange the optial path have to be taken into aount, from the initial point A in Fig. 1 until the final point D, inluding the angular hanges at eah interfae, and the displaement of the beams due to these angular hanges in glass and air. The omplete derivation is rather long and tedious, and is inluded in Appendix A. Thanks to numerous anellations, the general result is surprisingly short and has a simple physial interpretation: d = L g + d n s os 3 3 nl g To understand the physial meaning of eah of these terms, let us reall that the phase shift in a medium of thikness L g and index n is g =nl g /. The seond derivative of that phase shift is the group veloity dispersion for the glass: d g = L g + d n, whih is the first term of the omplete expression 7. Ithas been shown 10 that a group veloity dispersion an be assoiated to any element that reates a wavelength dependent defletion by an angle : d = n 8, 9 where n is the index of refration of the medium in whih the beam propagates a distane from the pivot point to the wave front of observation. In the two prism sequene of Fig. 1, A is a pivot point for the beam propagating a distane =L g in the glass, and B is the pivot point for the beam propagating a distane =BE in air. Adding the ontributions of Eq. 8 to the double ontributions from Eq. 9 leads also to Eq. 7 for the group veloity dispersion of the pair of prisms. ote that expression 7 for the group veloity dispersion applies to any pair of idential isoseles prisms in the parallel fae onfiguration represented in Fig. 1, for an arbitrary angle of inidene. To evaluate the expression 7, it is desirable to write all derivatives in terms of the material dispersion /. The relevant formulas are Eqs. A5 and A6 given in Appendix A. Of most pratial importane is the ase of the minimum deviation angle = 1, 0 = 3 and Brewster angle prism. The simplifiations to be applied in this partiular ase are given in Table III, Appendix A. Taking these simplifiations into aount the group veloity dispersion in terms of wavelength redues to: d = 3 g L d n 4L + L g n3, 10 where is the wavelength of the entral mode laser, for whih the Brewster ondition is satisfied. Here L=s/os 3 is the distane between the two prisms measured at the entral wavelength. In many pratial devies, LL g and the seond term of Eq. 10 redues to L/. The general equation for the third order group veloity dispersion is d 3 3 = 1 d L g + dl g + L g + L g d 3 n 3 s os sin d 3 n L g 1 d 1. L g + n dl g 1 d n s os 3 11 The following derivatives, in addition to the first order angular dispersions given by Eqs. A5 and A6, are needed to determine eah term of this expression: d 1 = tan 1 n 1 os 1 +1 d 3 = d sin os 1 os 3 PHYSICAL REVIEW A 75, tan 1 n d n, 1 sin 1 = 1 os 1 os 3tan + tan d n. 13 In the ase of prisms at Brewster angle in the visible and near infrared, all the signifiant terms of the third order dispersion are ontained in the first line of Eq. 11. The first term is set by seond order dispersion hosen usually to ompensate for positive dispersion of the avity. Even though L g is small, the derivative dl g / is large, beause of the hange in optial path through the seond prism, a hange that is inreasing with the separation between prisms. The last term of the first line of Eq. 11 is among the largest, beause of the generally large third order dispersion of most glasses in the visible part of the spetrum. The reason is that the dispersion of glasses an be approximately modeled as the normal dispersion branh assoiated with a Lorentzian line the ultraviolet resonane. It an easily be shown for a Lorentzian that the ratio

4 LADA ARISSIA AD JEA-CLAUDE DIELS PHYSICAL REVIEW A 75, TABLE I. Main parameters for two prism sequenes of different glasses. The index of refration n at =0.8 m, along with the first, seond, and third order derivatives with respet to, are alulated from the Sellmeier equation. The values for B 1,B,B 3,C 1,C,C 3 are taken from the 005 CVI atalog. The separation L between prisms is hosen so that the seond order dispersion with total glass L g =1 m ompensates for 500 fs positive dispersion of half of the avity round trip, from Eq. 7. The last olumn is the third order dispersion of the prisms separated by a distane L. Material n m 1 d n m d3 n 3 m 3 L m d3 3 fs 3 Fused silia Shott -LAK Shott BK Shott F Shott SF Shott -SF Shott -SF CaF d 3 n/ 3 d n/ r = d n/ =6, 14 / demonstrating an inreased influene of the third order dispersion. This model applies only to wavelengths short enough that the influene of the infrared absorption bands annot be felt. An appropriate wavelength for this model is 60 nm for fused silia. Using the values of the indies and their derivatives from Ref. 10, Table.1, one finds the value of this ratio to be r=5.7. The seond and third line of Eq. 11 an be negleted beause the angular third order dispersion terms are very small. The study of the prism dispersion in a linear avity an be applied to the similar ase of four prisms in a ring. In both ases the value for the dispersion has to be multiplied by two due to double passage through the prism pairs. In a linear avity a prism is usually translated along the bisetor of the apex angle to adjust the dispersion through the glass for desired group veloity dispersion in the avity. D. Typial ases Fused silia prisms are most often used for the generation of very short pulses, beause of their low third order dispersion as ompared to other materials. If a high repetition rate is desired, materials suh as LAK1, BK7, SF6 or SF14 glasses are used, allowing for a more ompat design. Table I shows the index of refration and the first, seond, and third derivative of n with respet to wavelength m. The oeffiients of the Sellmeier equation are taken from the CVI atalog 005. The laser is operating at =800 nm. It is assumed that the intraavity elements inluding the gain medium introdue a group veloity dispersion of 500 fs for a half avity round trip, and that the total optial path in the glass is L g =1 m. The minimum separation between prisms L is alulated, for whih the total avity seond order dispersion is zero. Operation of the laser requires a slightly negative dispersion to ompensate for the self-phase modulation Kerr effet in the gain medium. The third order dispersion due to the prism pair is alulated from Eq. 11. E. Evaluation of these results in the ontext of previous work The most ommonly ited equation to evaluate the dispersion of a pair of prisms is that ontained in the artile by Fork et al. 11, whih, limited to the Brewster angle prisms, reads: d P =4L d n +n 1 n3 sin 3 os In this equation P=L os 3 and L is the distane between the two prisms. In this alulation 3 is defined by the assumption that L sin 3 = mm, or twie the beam size. The main differene between Eq. 7 or Eq. 10 and the results of Ref. 11 are: i Equation 15 is only for the ase of tip to tip propagation in the prisms; the positive material dispersion and the angular dispersion of the beam propagating a distane L g in the prisms are negleted. ii In Eq. 15 the only optial path onsidered is L or the path between the two prisms; the beam displaement after the seond prism is not alulated. The angular and transverse displaements of the beam after the seond prism ontribute to a wavelength dependent hange in optial path, as shown in Appendix A. iii Even if it were orret, the term in sin 3 should anel in the four prism onfiguration of a ring avity, or double passage through the pair of prisms in a linear avity, as the beam folds through the third and fourth prism

5 CARRIER TO EVELOPE AD DISPERSIO COTROL I iv The orret expression for group veloity dispersion is d /, where =n/p, and not d P/. v A numerial example might help to omplete the evaluation of the simplifiations made in Referene 11. Equation 15 applied to a pair of fused silia prisms for 800 nm pulses leads to a minimum separation between prisms of L=138 mm, even in the absene of any positive dispersion element. The alulation from Eq. 10 for a similar ase of zero passage through glass, L g =0, will result in negative dispersion for any positive separation L. ote that the separation L between the prisms an only ontribute in negative dispersion, as shown by Eq. 7. Equation 15 instead implies that the separation between the prisms L has both a positive and negative effet. IV. EFFECTS OF A TILT AD TRASLATIO OF THE ED MIRROR Sine the publiations of Reihert 7, it seems to be an aepted onept that a tilt of the end mirror modifies exlusively the group veloity of the pulse. This onfidene expressed in numerous artiles see for instane Refs. 16,17, stems from a ommonly referened artile 9. This partiular paper does not relate to a pair of prisms, but to a ombination of a single grating with a lens the grating being plaed in the foal plane of the ahromati lens, faing a pivoting mirror in the opposite foal plane. In the stabilization loop of a laser, it is desirable to have independent ontrol of group and phase delays. As pointed out in Ref. 18, suh a goal an be rigorously ahieved in a synhronously pumped optial parametri osillator OPO, where the group delay is uniquely determined by the pump avity length, and the phase delay solely by the OPO signal avity length. The ommon tehnique to irumvent the absene of orthogonal ontrol parameters is to apply two linear ombinations of the error signals for the phase and group delays, to two different but not orthogonal ontrol elements of the laser 19,0. The two ontrol elements hosen in the two ited referenes 16,17 are the tilt of an end mirror following a prism sequene, and the avity length. The alulations that follow lead to a determination of the response of these two atuators tilt and translation, in terms of a hange in phase and group delays, and also spetral shift. It will be shown that, ontrary to widespread belief, the tilt of the end mirror of a two prism sequene is not equivalent to the mehanial sanning delay line 9. While there is a hange in group delay, there is also an important hange in phase delay and a shift of the spetrum, as the end mirror after a prism pair is tilted. Understanding this optial arrangement is important, onsidering the high preision that is required for today s metrology. A. The avity onfiguration and effets of the end mirror tilt The avity under onsideration is a standard arrangement onsisting in a flat output oupler, a gain setion omprising the Brewster angle rystal between two urved mirrors of equal foal distane, followed by a pair of prisms, and finally an end avity mirror M 1, as shown in Fig. a. The ombination of output oupler and the two urved mirrors an be replaed by a urved end avity mirror M of radius of R avity loated at a distane L avity from the end mirror Fig. b.a prism sequene not shown in the figure is generally loated between the tilting flat end mirror M 1 and the gain setion. If the end mirror M 1 is tilted about an axis not loated at the point of inidene of the beam, rotation by an angle leads, to first order, to a avity elongation L as skethed in Fig.. This effet is only of seond order in if the axis of rotation is at the point of inidene, as in Fig.. A displaement h of the beam parallel to itself does not hange the optial path through the prisms, hene will not have any effet on phase or group delay. However, after a tilt of mirror M 1, the beam moves to remain along the ommon normal to both M 1 and the equivalent mirror M, hene along a line through the enter of urvature C of mirror M. As shown in Fig. e, the intraavity laser beam is not only rotated by on mirror M 1, but also translated by h=r avity L avity. ote that there is a orresponding avity elongation h=r avity L avity, whih is seond order with respet to. In a typial example of R avity = m and L avity =1 m, the additional delay is h/=3.3 fs, for expressed in milliradian, and h in mm. B. Spetral shift It is lear that a translation of an end mirror displaes the avity modes, without affeting the overall spetral envelope of the laser. In a solid state laser, the alignment of the avity mode inside the gain rod, with respet to the foal volume of the pump beam, provides the wavelength seletion Fig. f. The gain line inside the rystal is oriented at the fixe internal Brewster angle Br =artan1/n. The external angle i, however, has a variation due to the tilt of the end mirror M 1. Snell s law applied to the Brewster interfae is: Differentiating: Therefore: PHYSICAL REVIEW A 75, sin i = n sin Br. os i = sin Br. = The spetral shift due to 1 mrad rotation of the end mirror, for the Ti:sapphire /= m 1 at 0.8 m, is 55 nm. This effet is thus quite large, and annot be negleted if the laser is to be tuned to an atomi or moleular transition. C. Change in group and phase delay due to tilt and translation To a tilt of the end mirror M 1 by an angle orresponds a displaement of the beam that an be deomposed into a

6 LADA ARISSIA AD JEA-CLAUDE DIELS PHYSICAL REVIEW A 75, M 1 (a) (b) R avity L θ L avity M () Displaement h = (R-L) θ M 1 θ (e) (d) L avity R avity M 1 (f) θ ι θ θ Br FIG.. Color online Representation of a standard avity of a femtoseond solid state laser. In a, overall avity onfiguration, exluding the prisms, whih is equivalent to the two mirror avity shown in b. If the axis of rotation of the tilting mirror is not positioned exatly at the point of inidene of the intraavity beam, the effet of a mirror tilt results also in a avity length hange L. This avity length hange is zero to first order when the rotation axis passes through the point of inidene. Ine, the equivalent two mirror avity is used to illustrate the displaement h of the beam on mirror M 1, due to a tilt by an angle. Inf, only a portion of the avity extending from the tilting mirror M 1 to the Brewster angle laser rod represented enlarged for larity is skethed. The envelope of the Gaussian mode is shown. The lasing mode has a waist at M 1 and inside the gain rod. The axis of the tilted mode is represented by the dotted line. rotation and a translation by h of the inidene point on the mirror. In evaluating the effet of this motion of the beam on the prism sequene, we an eliminate the effet of translation. Indeed, it an easily be seen that the optial phase of the beam passing a two prism sequene is independent of a beam translation normal to the wave front. It is therefore suffiient to study the hange in optial path, for a beam tilted at the pivot point by an angle, as shown in Fig. 3. D. Tilting of the end mirror The path lengths are as mentioned in Eq. 4 with the addition of an initial path in air: u = i OA sin 0 = i a sin ote that A is no longer a pivot point. The proper pivot axis is in the plane of the rotating mirror. The total hange in optial phase, per unit angle, is: 0 = t s tan 3 os 0 os os 1. 0 ote that neither a nor the variation in a do ontribute to the hange in optial phase, as these variations anel out Appendix B. For the partiular ase of minimum deviation 3 = 0 and 1 = =/ and Brewster inidene tan 0 =n, the variation of optial phase due to tilt redues to: or = sn 0 t 1 n 1+n 1+n 1 n 1+n = 0 L g, where L g is the total length of propagation in the glass. For a laser at 800 nm propagating to total amount of glass of 1 m, the phase hange is 6.5/mrad. Let us next proeed to a alulation of the variation in group delay due to the tilt. Adding the air-path ontribution from the pivot point to the first prism to Eq. 6 leads to the following expression for the group delay: = OPLu + ABEF + w + L g. 3 As in the ase of Eq. 6, the group delay onsists in two parts: the left term being the phase delay, and the right term the arrier to envelope delay. The group delay variation, by

7 CARRIER TO EVELOPE AD DISPERSIO COTROL I PHYSICAL REVIEW A 75, FIG. 3. Color online The figure shows the beam propagation through prisms and air. The dashed trajetory is the initial beam and the dotted one is the shifted beam due to a negative tilt of the mirror at far left. u and w are the path in air before and after passing through the prism pairs, respetively. BE is the path in air between the two parallel fae prisms. L g =AB+EF is the total travel through two prisms. t is the apex to apex distane between prisms projeted on an internal fae, and s is the parallel internal fae distanes. OA=a is the distane of the beam input trajetory to the apex of the prism. tilting the end mirror in a linear avity with prism pairs, is d 0 = 1 s tan 3 t os 0 os + os 1 s os os 0 sin os 3 os 1 os 3 os 1 + t s tan 3 sin sin 1 os 1 os 0 n os 1. 4 This expression simplifies in the ase of minimum deviation and Brewster prism: d 0 = L g + + L L g n 3, 5 where the total variation with angle of the group delay is expressed in two terms, the left term being the angular variation of phase delay, and the seond the angular variation of arrier to envelope CEO delay. The material parameters were given in Table I, for the ase of a laser operating at =800 nm. Table II lists the values of phase delay, CEO delay, and group delay, due to the rotation of an end mirror, for the distane L alulated in Table I. The phase delay does not inlude the seond order ontribution R avity L avity mentioned at the beginning of Se. IV A. E. Translation of the end mirror For a pulse propagating in a homogeneous medium of index n, the group delay is defined by: dk = 1 = 1 v g n. 6 If the end mirror of the avity is translated perpendiular to the phase front by L, the hange in length ours only in air, and the hange in group delay is simply the phase delay: L/. 7 A hange in avity length of L=5 m orresponds to the phase hange due to 1 mrad rotation 6.5. V. COCLUSIO Complete expressions were derived for the seond and third order dispersion introdued by pairs of isoseles prisms at any angle of inidene. It is shown that the short path in glass, often negleted, leads to the dominant terms in the TABLE II. Phase, CEO delay, and group delay due to rotation of an end mirror. Prism pair onfigurations are as mentioned in Table I. Material Phase delay fs/mra CEO delay fs/mra Group delay fs/mra Fused silia Shott -LAK Shott BK Shott F Shott SF Shott -SF Shott -SF CaF

8 LADA ARISSIA AD JEA-CLAUDE DIELS third order dispersion. Contrary to aepted belief, tilt of the mirrors do not provide ontrol of group delay, independent of phase delay. In addition, the tilt results in a large spetral shift, as we have observed in several Ti:sapphire laser avities. Fused silia prisms are the best solution for dispersion smallest third order and tilt largest ratio of group to phase delay. Unfortunately, the use of fused silia requires longer avities. For avities with very small dispersion, it may be advantageous to use small apex angle prisms rather than Brewster prisms, whih result in a smaller third order dispersion, and a smaller phase delay upon tilt. TABLE III. Simplifiations for Brewster prisms at minimum deviation angle. tan 0 =n tan 1 = 1 n sin = n n +1 PHYSICAL REVIEW A 75, n sin 0 =os 1 = 1+n 1 os 0 =sin 1 = 1+n ACKOWLEDGMET This work was supported by the ational Siene Foundation under Grant umber ECS APPEDIX A: DISPERSIO FOR A PAIR OF PRISMS This appendix is to show and share the detailed alulation of the group delay and first and seond order dispersion for a pair of idential parallel prisms. The alulations are made for the general ase, unless speified in Brewster or minimum deviation. 1. Group delay From Fig. 1 the group delay is: = d nl g + d BE + d FD = nl g + + BE + FD + + L g ns sin os + 3 os 1 ns + nl g s os 3 tan 3 os 3os sin 1 + sin sin 1 os 1 n t s tan 3 sin sin 1 1 os 1 = OPLABEFD + L g + s n sin os 1 + os 3 tan 3 + n os sin 1 3, os 3 tan A1 where we have defined the optial path length OPLABEFD=nL g +BE+FD. The fator preeding 3 / anels, sine = 1 + the term os sin 1 sin os 1 + sin 3 n = sin 1 + sin A vanishes. The omplete expression for the group delay through the pair of prisms redues to: = OPLABEFD + L g. First order group veloity dispersion A3 The seond derivative of the phase, obtained by taking the derivative of Eq. A3, is: d = L g + d n + L g tan 1 1. os 1 s sin 3 os 3 A4 The derivatives with respet to are related. By differentiating Snell s law for the first interfae: 1 = tan 1 n =. A5 For the seond interfae, taking the previous relation into aount, we find: 3 os 3 = n os + sin, 3 = sin os 1 os 3. A6 In these equations, one an easily make the substitution = 1, and 1 =arsinn 1 sin 0. The seond order dispersion Eq. A4 redues to an easily interpretable form: d = L g + d n s os 3 3 g n L This expression an be simplified for the ase of inidene at the minimum deviation angle symmetri beam path through

9 CARRIER TO EVELOPE AD DISPERSIO COTROL I PHYSICAL REVIEW A 75, R avity is the effetive urvature of the avity mirrors. In the ase of a onentri avity at the limit of stability, the tilting mirror is loated at the enter of urvature and is equal to R avity, that is the only ase where the laser spot will not move on the mirror with rotation. In Fig. 4, the pivot point is the initial laser beam spot at M, the laser spots move to M and the beam path to the prism hanges from u to u. This hange an be deomposed to hange due to the rotation or hange in the inident angle and a hange due to translation of the spot on the mirror. The differential path hange before the prism negleting seond order terms is: FIG. 4. The figure shows the hange in angles and optial paths due to the rotation of the end mirror. The figure shows how the spot will displae on the mirror with rotation as well. The solid line trajetory MA is the beam path in initial position, and the solid line MA is the modified beam path after the rotation. The dotted line trajetory MA is the intermediate step ignoring the beam displaement on the mirror to alulate the hange in OA to OA. the prism for =. In addition, for minimum intraavity loss, the prisms will generally be ut with an apex angle suh that the minimum deviation angle is also the Brewster angle. The partiular values of the angles are given in Table III. We an also make the substitutions 1 /= 1/n /, and 3 /=/. With these simplifiations, the total seond order dispersion in this ase beomes: d = L g + d n 4L + L g n3, A7 where we have introdued the distane between the two prisms measured along the entral wavelength L=s/os 3. In terms of wavelength: d = 3 g L d n 4L + L g n3. A8 APPEDIX B: TILTIG THE ED MIRROR The hanges in paths and angles an be monitored by appliation of the Snell s law. Let us look at the hange of the optial path when the end mirror is rotated by angle 0 from the pivot point positioned at a distane h i from the prism fae. i The hange in path length to the first prism surfae u: ote that due to the rotation of the mirror the laser spot will move on the surfae of the mirror by m =R avity L avity 0, where L avity is the avity length and du = u h + u R avity L avity + h i sin 0 0 h 0 0 os 0 os. 0 B1 In the above expression, the first term, representing a translation of the beam, does not ontribute to any optial path hange through the prism ombination. The seond one is due to the rotation or hange in angle of inidene on the mirror. The spot shift due to the mirror tilt will ontribute to the variation of parameter a. ii Change in angles: The hange in angles due to the mirror tilt starts from 0, and is transferred to all the other angles as follows: 1 = os 0 n os 1 0, = 1, 3 = n os = os os 0 0. B os 3 os 1 os 3 iii Propagation in air between the two parallel faes of the prisms BB: BE = s sin 3 os os 0 0 os. B3 3 os 1 os 3 iv The total propagation in glass prisms L g : dl g s os os 0 sin = 0 os 3 os 1 os 3 os 1 + t s tan 3 sin sin 1 os 0 os. B4 1 n os 1 v The hange in path in air outside the two prisms or u+w: Using Eqs. 4 and B u + w = os 0 t s tan 3 os + sin tan s sin 0 os 3 os os 0 os 1 os 3 os + sin tan 1 sin os 0 sin 0 t s tan 3 os. B5 1 n os 1 There is no dependene on a or the position of the beam on the first prism in the ombination of u+w. In the parallel

10 LADA ARISSIA AD JEA-CLAUDE DIELS prism pair onfiguration, if the beam moves away from the apex of the first prism, it will result in the beam spot moving loser to the apex of the seond one. The same opposite variation orresponds to the path before the first and after the seond prism Fig. 4. The total hange in phase due to the mirror rotation is: n sin sin 1 0 = t s tan 3 os 0 os + sin tan 1 sin 0 sin os 1 os 0 n os 1 + os 0 os 1 n os 1 s os 3 3 os 0 os os 1 PHYSICAL REVIEW A 75, sin 0 os + sin tan 1 sin 3 + n sin os 1. B6 The first two lines an be simplified, using the equalities n sin 1 =sin 0 and = + 1. The same onsiderations will result in a vanishing third line of Eq. B6. The group delay due to the mirror tilt is thus: 0 = t s tan 3 os 0 os os 1. B7 1 I. S. Ruddok and D. J. Bradley, Appl. Phys. Lett. 9, J.. Ekstein, A. I. Ferguson, and T. W. Hänsh, Phys. Rev. Lett. 40, R. J. Jones, J. C. Diels, J. Jasapara, and W. Rudolph, Opt. Commun. 175, T. Udem, J. Reihert, R. Holzwarth, and T. Hänsh, Opt. Lett. 4, W. Dietel, J. J. Fontaine, and J.-C. Diels, Opt. Lett. 8, J. J. Fontaine, W. Dietel, and J.-C. Diels, IEEE J. Quantum Eletron. QE-19, J. Reihert, R. Holzwarth, T. Udem, and T. W. Hänsh, Opt. Commun. 17, F. X. Kärtner,. Matushek, T. Shibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Sheuer, M. Tilsh, and T. Tshudi, Opt. Lett., K. F. Kwong, D. Yankelevih, K. C. Chu, J. P. Heritage, and A. Dienes, Opt. Lett. 18, J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena Elsevier, Boston, R. L. Fork, O. E. Martinez, and J. P. Gordon, Opt. Lett. 9, V. Petrov, F. oak, W. Rudolph, and C. Rempel, Exp. Teh. Phys. Berlin 36, F. J. Duarte and J. A. Piper, Opt. Commun. 43, F. J. Duarte, Opt. Quantum Eletron. 19, F. J. Duarte, Opt. Quantum Eletron. 4, J. Ye and S. Cundiff, Femtoseond Optial Frequeny Comb: Priniple, Operation and Appliations Springer, ew York, F. W. Helbing, G. Steinmeyer, and U. Keller, Laser Phys. 13, J.-C. Diels, J. Jones, and L. Arissian, in Femtoseond Optial Frequeny Comb: Priniple, Operation and Appliations, edited by J. Ye and S. Cundiff Springer, ew York, 005, Chap. 3, pp K. W. Holman, D. J. Jones, J. Ye, and E. Ippen, Opt. Lett. 8, T. H. Yoon, S. T. Park, E. B. Kim, and J. Y. Yeom, IEEE J. Sel. Top. Quantum Eletron. 9, In performing suh an analysis, are must be taken to define the optial path as onneting two real optial phase fronts, and not from a phase front to an arbitrary surfae. CVI, 00 Dorado Plae, Albuquerque, P.O. Box 1108, ABQ, M