ZERO-DISTANCE PULSE FRONTS OF STRETCHER AND ITS OPTICAL SYSTEM

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1 ERODISTANCE PULSE RONTS O STRETCHER AND ITS OPTICAL SYSTEM Author: DOI: /alt.1.70 Corresponding author: agitin@mbiberlin.de

2 erodistance Pulse ronts o a Stretcher and its Optical System Max Born Institute or Nonlinear Optics and Short Pulse Spectroscopy MaxBornStr. 2 A, Berlin, 12489, Germany Abstract A twograting stretcher is a dispersion delay line that is widely used in chirped pulse ampliication or ultraast laser pulses. I the stretcher consists o two relecting diraction gratings with a perectly relective optical between them, its dispersion can be calculated by using the concept o a zerodistance pulse ront. We introduce the concept o a zerodistance pulse ront o the real optical, which characterizes its aberrations. The similarity o these zerodistance pulse ronts allows us to study the inluence o aberrations in an optical on the dispersion o a stretcher. 1. Introduction It is known that the shorter the light pulse is the greater is the inluence o the dispersion on its duration. Thereore, the optics o ultrashort pulses (USPs) has a speciic ield o research, namely dispersion delay lines (DDLs), which allow managing the duration o the pulses in a reversible ashion [1]. DDL is characterized by the group delay o the wavelength λ. I the unction ( ) is strictly increasing, then dispersion is negative and the corresponding DDL is a compressor [2,3]. I the unction ( ) is strictly decreasing, then dispersion is positive and the corresponding DDL is a stretcher [4,5]. The damage threshold or relective optical elements is much greater than that or optical media. Thereore, DDLs or superpowered USPs are usually made rom only relective optical elements, such as mirrors and relection diraction gratings. I DDL is made rom only relective optical elements then inside o it all monochromatic wave components and the USP as a whole run in ree space with the same speed, the speed o light in vacuum c. Thereore, its group delay (λ) equals its phase delay T(λ) which is directly proportional to the path length p o the corresponding monochromatic ray: T(λ) = p(λ)/c. In uture, instead o the phase or the group delay o relection DDL, we will simply speak about its time delay. In natural o units c = 1, so the time delay is just the path length: T(λ) = p(λ). 2. Technique o the zerodistance ront In ree space light rays travel along straight lines but optical elements old the paths o the light rays. A straight line is the simplest geometrical curve. There are a lot o dierent unolding techniques, e.g. the technique o the zerodistance ront erodistance pulse ront o compressor Treacy s consists o a pair o identical relection gratings arranged parallel [2,3]. The irst grating splits an incident polychromatic ray into a diverging homocentric beam o spectrally colored rays. According to the reversibility principle the second grating transorms this beam into a parallel beam o rays, the socalled spatial chirp o Treacy s. The position o a monochromatic ray in the spatial chirp depends on its wavelength, so the cross section o the spatial chirp can be used as a natural nonuniorm wavelengths scale (λ). A ray o the spatial chirp can be continued in its reverse direction to a point *, where points * and are equidistant rom point M: *M = M. The set o points {*} is a zerodistance pulse ront o Treacy s [6] (ig.1). Let the reerence plane o Treacy s be a plane passing through point and perpendicular to the input ray. The time delay T Tr o Treacy s or the monochromatic component is equal to the path length rom point * to point M o the reerence plane: T Tr *M. The zerodistance pulse ront in the coordinate o the time delay T Tr vs the displacement (λ) is the natural characteristic o Treacy s. This characteristic is a strictly

3 increasing unction. Thereore, Treacy s has a negative dispersion. erodistance pulse ront Tr ( ) T Tr * 0 Axis o the spatial chirp (transorm limited pulse) M ig. 1. Treacy's and its zerodistance pulse ront T Тr ( ) in the coordinate o the time delay T Tr vs the displacement (λ). The red ray is shown by the double dashanddot line and the blue ray is shown by the dashanddot line. According to the reversibility principle i two Treacy s s are arranged backtoback (ig. 2 then this turns an input polychromatic ray caring a transormlimited pulse into an output polychromatic ray caring a pulse with a negative dispersion, the socalled downchirp. This is a compressor. Its relative time delay is a double relative time delay o a single Theacy s. Because o the mirror symmetry this compressor is equal to a single Theacy s with a retromirror (ig. 2. Natural nonuniorm wavelengths scale (λ) M RP Tr 2.2. erodistance pulse ront o stretcher with perect optical Martinez s consists o a pair o identical relection gratings and a mirror optical with unit magniication between them (ig. 3 [4,5]. Let points and be optical conjugate onaxis points o the optical. The irst grating splits an input polychromatic ray into a diverging homocentric beam o spectrally colored rays. The perect optical turns this beam into a mirror symmetrical converging homocentric beam. According to the principle o reversibility, the second grating turns the convergence beam into an output polychromatic ray which is mirror symmetrical to the input polychromatic ray [7,8]. Let us shit the second grating along the optical axis (ig. 3. The incident angle is preserved, so the second grating transorms the converging homocentric beam into a parallel beam o monochromatic rays, the socalled spatial chirp o Martinez s. The position o the monochromatic ray in this spatial chirp depends on its wavelength, so the cross section o the spatial chirp ( ) can be used as a natural nonuniorm wavelengths scale. (а) ourierplane T in Input pulse ront T in ( ) (the transorm limited pulse) Theacy s T Tr erodistant pulse ront T Tr ( ) o Treacy erodistant pulse ront T Com ( ) = 2T Tr ( ) o compressor G Tr3 G Tr4 Retromirror T Com Theacy s ( Virtual Treacy s ourierplane l = erodistance pulse ront T Mar ( ) o Martinez RP Mar l 0 * M M * erodistance pulse ront T vtr ( ) o the virtual Treacy output ray ig. 3. Mirrorsymmetric optical (l = ) (, Martinez s as an imaging perect optical with a minus unit linear magniication (l < ) and its zerodistance pulse ront (. ig. 2. ourgrating compressor. Twograting compressor with the retro mirror. The ray o the spatial chirp can be continued in its reverse direction to a point *,

4 where points and * are equidistant rom point M (ig. 3: *M = M. The set o points {*} can be called a zerodistance pulse ront o Martinez s [9]. In the image space o a perect optical the second grating and the image o the irst grating G Mar1 orm a virtual Treacy s with the reerence plane RP vtr, the zerodistant pulse ront, and the time delay which is the path length rom the point * to the point M o the reerence plane (ig. 3: T vtr *M. I the reerence plane o the virtual Treacy s is accepted or a reerence plane RP Mar o Martinez s, then the time delay o Martinez s T Mar is directly proportional to the path length * M ). The zerodistance pulse ront in the coordinate o the time delay T Mar vs. the displacement (λ) is the natural characteristic o Martinez s. Note that the time delay o Martinez s T Mar is equal to the time delay o the perect optical T o =R/c minus the time delay o the virtual Treacy s T vtr. ( ) (the path length *M ) [9]: T Mar ( ) = T o T vtr. ( ). So Martinez s has a positive dispersion. erodistant pulse ront T Str ( )=2T Mar ( ) o stretcher (the upchirp pulse) G Mar4 G Mar3 TStr Martinez Optical with m =1 Input pulse ront T in ( ) (the transorm limited pulse) T in T Mar Martinez Optical with m =1 Optical with m=1 ero distance pulse ront T Mar ( ) o Martinez s ig. 4. ourgrating stretcher. Twograting stretcher with the retro mirror. According to the reversibility principle i two Martinez s s are arranged backtoback (ig. 4 then this turns an input polychromatic ray caring a transormlimited pulse into an output polychromatic ray caring a pulse with a positive dispersion, the socalled upchirp. This is a stretcher. Its relative time delay is a double relative time delay o a single Martinez s. Because o the mirror symmetry this stretcher is equal to a single Martinez s with a retromirror (ig erodistance pulse ront o a real optical o a stretcher Let us consider a real optical (ig.5). The real optical transorms a broad diverging homocentric beam o rays rom the object point into a converging nonhomocentric beam having a breakshaped envelope with the top at the image point. Continue the straight ray rom a point A o the caustic in its reverse direction to the point q, where points and q are equidistant rom the point A (ig. 5: [q, M](u )= OPL[,A](u ). The set o points {q } is a zerodistance phase ront o the optical. Note that the zerodistance phase ront o a perect optical { o } is a circle with the radius R = ct o, the socalled a reerence sphere. The zerodistance phase ront{ q } q Reerence sphere { o } o y Caustic {A} y n Point source y Real optical The zerodistance pulse ront{ } o Reerence sphere { o } G n Real optical 1 1 u R Caustic {A} u R A y A Retromirror

5 ig. 5. ( The zerodistance phase ront o an optical {q } and its wave aberration o q q R. ( The correspondent zerodistance pulse o the optical { } and its wave aberration o R. The dierence between the zerodistance phase ront o the real optical and the reerence sphere is its wave aberrations unction W(u ): W(u ) [q, A](u ) R. Note that a diraction grating G illuminated by a thin polychromatic light ray can act as a point source o radiation (ig. 5, but diracted rays o this point source are not coherent and have dierent requencies, so they cannot interere. In this case, the term zerodistance phase ront o the optical {q } should be replaced with the term zerodistance pulse ront o the optical { }. In a relective optical, the zerodistant pulse ront and the zerodistant phase ront are the same erodistance pulse ront o stretcher with real optical Note that a real optical o the stretcher can be replaced by its zerodistance phase ront (ig. 6). The time delay o Martinez s T Mar (the path length M ) is equal to the time delay o the perect optical T o =R/c plus the wave aberration unction o the real optical W( )/c minus the time delay o the virtual Treacy s T vtr. ( ) (the path length *M ): T Mar ( ) = T o W( )/c T vtr. ( ). ( ) erodistance pulse ront o Martinez s stretcher {*} T vtr s erodistance wave ront o real optical { } RP Mar * T Mar * M ( ) M( ) erodistance pulse ront o virtual Treacy s { *} o compressor, stretcher and the aberrations o the stretcher optical in a pictorial manner. The similarity o the zerodistance pulse ronts o the stretcher and its optical allows us to take into account the aberrations o the optical while calculating the dispersion o the stretcher. Reerences [1] D. Strickland, G. Mourou (1985), Compression o ampliied chirped optical pulses, Optics communications, 56, [2] E.B. Treacy (1968), Compression o picosecond light pulse, Physics Letters, 28A (1), 3435 [3] E.B. Treacy (1969), Optical pulse compression with diraction gratings, IEEE Jornal o uantum Electronics, E5 (9), [4] O.E. Martinez (1987), 3000 Times Gratig Compressor with Positive Group Veloсity Dispers ion: Applicatiom to iber Compensation in m Region, IEEE Journal o uantum Electronics, E23 (1), 5964 [5] O.E. Martinez (1987), Design o highpower ultrashort pulse ampliiers by Expansion and Recompression, IEEE Journal o uantum Electronics, E23 (8), [6] A.V. Gitin (2010), erodistance pulse ront as a group delay characteristic o the twograting compressor, Optics Communication, 283, [7] A.V. Gitin (2008), Geometrical method o calculation o a stretcher s dispersion, which allows to consider inluence o parametres o the optical, uantum Electronics, 38 (11), [8] A.V. Gitin (2012), Tautochronism principle and grating dispersive delay lines, Applied Optics, 51 (1), 2732 [9] A.V. Gitin (2012), Using the unolding method or dispersion calculations o relective grating delay lines in the chirped pulses ampliier, Optics Communication, 285, ig. 6. Relationship between the zerodistance pulse ront o Martinez s {*}, the zerodistance pulse ront o its optical { }, and the zerodistance pulse ront o virtual Treacy s { *}.Conclusion The concept o zerodistance pulse ront provides a new way o representing the dispersion