In application of ultra-short laser pulses the pulse parameters have to be controlled accurately. Hence the manipulation of the propagation behavior of ultra-short pulses requires for specially designed optics. We have developed a tool for the simulation of ultra-short laser pulse propagation through complex real optical systems based on a combination of ray-tracing and wave optical propagation methods. For the practical implementation of the approach two commercially available software packages have been linked together, which are ZEMAX and Virtual Optics Lab. The focussing properties of different lenses will be analyzed and the results are demonstrated.
©2005 Optical Society of America
An emerging number of real world applications in science and industry require the use of ultra-short laser pulses. Prominent examples are multi-photon laser scanning microscopy and ultra fast laser micro-machining . Several applications require for a tight focussing of the laser beam in order to realize a high spatial resolution and high local field strengths. However, focussing optics usually have significant influence on the spatial and temporal characteristics of the laser pulse in the focal region. During the last fifteen years several analytical approaches have been made in order to model different effects caused by material dispersion or chromatic and spherical aberrations [2–12]. However, all these investigations are limited to idealized single lenses or achromatic doublets. Furthermore, also the description of the incident pulses is idealized in these models. The treatment of ultra short pulse interactions with real lenses or even complex optical systems as well as optical design is impossible with these analytical approaches.
In the current paper we report on a general approach for the treatment of the propagation of arbitrary laser pulses through complex optical systems. The calculations are based on a combination of ray-tracing and wave optical propagation methods. Therefore all kinds of aberrations are considered. On top of that not only the analysis of occurring effects but also the optimization of system parameters with respect to the behavior of the focussed pulses is possible.
The electrical feild of an ultrashort laser pulse can be decomposed into its spectral components by applying a fourier transform. Therefore it can be written as
with the spectral amplitude described by A(v, x, y, z) and ϕ(v, x, y, z) being the spectral phase of the laser pulse. For the computations the incident laser pulse is separated into its spectral components (Eq. (1)) which are traced separately through an optical system applying geometrical optics methods. Propagating the laser pulse through an optical system modifies both the spectral phase ϕ(v, x, y, z) and the spectral amplitude A(v, x, y, z). The latter one is influenced by diffraction effects during propagation, absorption, reflection, scattering effects, pupil aberrations and nonlinear material interaction. In addition dispersion and aberrations within the optical system affect the spectral phase ϕ(v, x, y, z) and its spatial distribution. In general it is possible to consider all these effects within our calculation. However, in the present paper we turn our attention especially to imaging systems because of their broad field of application. Therefore, it is sufficient in most cases of practical relevance to limit the effects of the propagation between entrance and exit pupil of the optical system to the modification of the spectral phase ϕ (n, x, y, z) by linear material interaction and the lateral scaling of the beam size.
If an optical system involves only collimated propagation with no focussing of the laser pulse in the observation plane, e.g. propagation through a simple stretcher-compressor-system, the electrical field is simply calculated by a superposition of all spectral components in that plane applying Eq. (1). Calculating the electrical field in the vicinity of the focus of an imaging system is more complex. Since the geometrical optical theory may not be valid in the focal region wave optical methods have to be induced. For this purpose we extended the combination of geometrical optics and wave optics firstly proposed by Stamnes  in 1986 to the treatment of ultra-short laser pulse propagation.
In the special case of imaging systems (Fig. 1) the propagation of the laser pulse between the entrance and the exit pupil can be treated very accurately with ray-tracing methods. The propagation from the exit pupil into the focal region is then performed by solving the diffraction integral for each spectral component. Thus the electrical field E(t, x, y, zf) of the laser pulse in the focal plane is given by the weighted superposition of the diffraction pattern from all spectral components. Since the entrance and exit pupil are both images of the same limiting aperture within the imaging system diffraction effects can be treated as occurring from the exit pupil only . There are several ways of calculating the wave optical propagation from the exit pupil into the focal region . However, for the applicability it is important to have a fast way of computation because each spectral component needs to be propagated separately. Due to its convenient handling we have chosen a method based on fourier transformation. If the spectrum E(v, px, py, zex) in the exit pupil is defined relative to a reference sphere, which is centered in the focal region, these calculations are no longer limited to paraxial beams . The coordinates (px, py) then refere to the reference sphere. The approximations made even hold for spherical aberrations up to about 50λ.
Hence the propagation from the exit pupil into the focal plane is equivalent to a fourier transformation of the spectrum E(n, px, py, zex) with
containing the spectral amplitude A(v, px, py, zen) and phase ϕ(v, px, py, zen) of the incident laser pulse plus an additional part of the spectral phase ϕab(v, px, py, zex) caused by dispersion and aberrations of the system. The pulse spectrum E(v, x, y, zf) in the focal plane can be expressed as
The quadratic phase factor in front of the fourier transformation has been neglected because its spatial variation found to be very weak in regions near the center of the focus where the major part of the energy is concentrated if aberrations are not too large.
Applying Eq. (1) on (3) yields a phase correct superposition of all spectral components by fourier transform and therefore the space-time-distribution in the focal plane. If another z-plane is of special interest the field distribution in the exit pupil needs to be manipulated with an additional spherical phase factor to perform the propagation with a fourier transform as well . Calculating the field distribution for a sufficient amount of z-planes and rearranging those numerically leads to an x-z-distribution at a given time.
For the practical implementation of this approach any software package for optical design could be used. The calculations in this paper were performed with the ray-tracing software ZEMAX  and the wave optical simulation software Virtual Optics Lab . The ray-tracing of all spectral components from the entrance to the exit pupil of the optics under consideration was done with ZEMAX. Whereas the wave optical propagation from the exit pupil into the focal region was accomplished with Virtual Optics Lab. The special method used here is not limited to paraxial focussing optics as will be seen in the examples presented in Section 3. Although not shown here systems like pulse compressors or focussing into materials can be analyzed with this method as well. Limitations are up to now the exclusion of nonlinear material interactions of the pulses. For the consideration of such effects the propagation algorithm has to be substantially extended.
3. Numerical results
For a comparison of various effects occurring while focussing ultra-short laser pulses we have chosen three different optical systems. They offer similar parameters such as a numerical aperture (NA) of about 0.45 and a focal length of about 9mm. All optics are real world examples. Hence the distortion caused by dispersion and aberrations is not reduced to that of model systems and in addition all effects appear in a realistic mixture. However, to be able to study the impact of major effects such as dispersion, chromatic and spherical aberrations the optics chosen exhibit dominantly one of them each. The first optic is a planoconvex lens from Linos (312011) made of BK7 having severe spherical aberrations. The second one is a Geltech asphere (Thorlabs 350240) showing chromatic aberrations. The third optical system is a microscope objective from Linos (038723) made of several different sorts of glass, therefore giving rise to a great amount of material dispersion. Even though it is possible to compute with an arbitrary incident laser pulse using the described method we have chosen a 24fs-pulse centered at 800nm with a gaussian shaped spectrum to keep possible effects well separated and as simple as possible.
3.1. Spectral phase in the exit pupil
Ray-tracing for each spectral component through an optical system gives us the spectral phase ϕ (v, px, py, zex) in the exit pupil relative to a reference sphere. Since all considered optical systems are rotationally symmetrical the analysis of the spectral phase along one coordinate axis is sufficient. Depicting this phase for each spectral component as a function of the normalized exit pupil coordinate py gives a first insight to the focussing behavior for ultra-short laser pulses. The spectral phase in the exit pupil of the three optical systems discussed here are depicted in Fig. 2. The horizontal axis shows the variation of the phase as a function of py which is therefore the aberration for each frequency v. The vertical dependence in phase with the frequency shows the influence of material dispersion and therefore the group velocity dispersion (GVD) and higher orders. Just from this illustration one can estimate the amount of distortion of the laser pulse. For an ideal lens the spectral phase on the reference sphere is constant for all frequencies and pupil coordinates.
At this point we want to infer on the focussing behavior from the spectral phase of the three optical systems discussed here. The phase in the exit pupil of the planoconvex lens (Fig. 2(a)) shows strong variations with py for each wavelength which means that there are severe aberrations (spherical aberrations in this case). On the other hand the variation of the spectral phase with the frequency is small. Hence material dispersion just causes small changes in the pulse duration as expected for thin lenses. The variation of the spectral phase in the exit pupil of the Geltech asphere (Fig. 2(b)) is quite weak compared to the planoconvex lens. The curvature of the phase can be identified as chromatic aberrations. The spectral phase also shows a modification with the wavelength which can not be neglected and is caused by the highly dispersive material the lens is made of. The spectral phase in the exit pupil of the microscope objective (Fig. 2(c)) is nearly constant for all py which is due to the well corrected achromatic system. Whereas the high amount of material dispersion caused by the different sorts of glasses and the long glass path lead to a strong variation of the spectral phase with the frequency which means a strong GVD. One can now expect good focussing attributes but an expansion of the pulse duration for the microscope objective. The other two optical systems will cause pulse front distortions because of their aberration. These effects will be discussed next.
3.2. Propagation into the focal region
For a comparison of the influence of material dispersion and aberrations on the focussing behavior we behold the focussing of an ultra-short laser pulse with an ideal lens (NA=0.45 and f=9mm) first (Fig. 3). There is no pulse front distortion and a perfectly spherical pulse front is propagating into the focus. Since the aperture of the lens is fully illuminated typical diffraction patterns on the pulse front can be observed.
We will initiate our exploration with the influence of material dispersion on the focussing behavior of ultra-short laser pulses starting with the microscope objective. From the analysis of the spectral phase (Fig. 2(c)) one can expect an almost undistorted pulse front but an increased pulse duration which is proportional to the amount of material dispersion. Exactly this behavior can be found from Fig. 4(a), which shows the focussing of the laser pulse in the vicinity of the focus (±200µm). After propagating through the microscope objective the pulse duration is 116fs being five times larger than the original duration. The also occurring intensity pattern on the optical axis will be discussed in Section 3.3.
Secondly we turn our attention to the focussing behavior of the Geltech asphere. As noted before this lens will cause an increase of the pulse duration as well. But the major effect here are the chromatic aberrations which cause a distortion of the pulse front. The typical horseshoe shape  can be observed in Fig. 4(b). Notice that the pulse duration near the optical axis is longer than at the edge due to the radially varying thickness of the lens. An additional pulse occurs on the optical axis which is refered to as boundary wave pulse in  and further described in . We will discuss this effect in Section 3.3.
Finally the focussing behavior of the planoconvex lens will be analyzed. Due to the severe spherical aberrations not even monochromatic light can be focussed well with this lens. In Fig. 5 one can observe what happens to an ultrashort laser pulse after propagating through that planoconvex lens. As one can see the spherical aberrations lead to the formation of a Bessel-like pulse  with its typical x-shape. Indeed there are three different intensity peaks propagating along the optical axis between the marginal and paraxial focus. This behavior has been observed before in , but the explanation given there is in contrast to our findings. The first intensity peak (max1) is originating from the paraxial part of the lens being only weakly influenced by aberrations and behaving similar to the pulse focused by the microscope objective. This intensity peak is directly correlated to the original incoming pulse and is focused in the paraxial focal plane. The second intensity peak (max2) is a Bessel-like pulse caused by the spherical aberrations. It is existing between the marginal and paraxial focal plane only. The third intensity peak (max3) known as boundary wave pulse exhibits very special attributes and will be discussed in the following Section 3.3. The formation of max2 and max3 is very similar, so the properties of max2 are not explained in detail. We just want to point out that the radial intensity distribution and the velocity of propagation of max2 depend on its position along the z-axis. At the marginal focal plane max2 and max3 coincide both propagating with the velocity vb, which will be given below in Eq. (4). While propagating towards the paraxial focal plane the velocity of max2 decreases because the effective aperture causing max2 shrinks and therefore the angle α decreases. At the paraxial focal plane max1 and max2 coincide both propagating with the vacuum velocity of light c.
3.3. Side effects
There are two major side effects observed for the focussing of ultra-short laser pulses with real lenses which are of special interest being the intensity distribution along the optical axis and the additional pulse known as boundary wave pulse. Due to our general approach and the higher flexibility in calculating the propagation in the whole vicinity of the focus we are able to study these effects in more detail.
The stationary intensity distribution along the optical axis (Fig. 4(a)) is of special interest for the application of fs-pulses for micro-structuring of materials. Calculations show that they always appear when aberrations are not to high. In addition their appearance and structure are not much influenced by the spectral width of a laser pulse. They even occur for monochromatic illumination. The spacing of two maximums can be explained using fresnel numbers and therefore is depending on the NA of an optical system. The intensity pattern along the optical axis is caused by diffraction of the pulse front at the system aperture and is therefore an inherent attribute of the optical system. The ratio of the intensity along the optical axis compared to the total pulse intensity is dependent on the NA of the system. With increasing NA the distribution (Fig. 6) becomes narrower and steeper. Also the spacing of two intensity maximums decreases with an increasing NA because the Fresnel number is increasing too. It can also be seen that for a gaussian shaped illumination of the aperture apart from a homogeneous illumination the oscillation of the intensity distribution along the optical axis disappears but the envelope stays unchanged.
While processing transparent material with ultra-short laser pulses streak formation in front and after the focus can occur . Our calculations show that especially for focussing with low NA optics there is a non neglectable amount of the pulse intensity distributed along the optical axis (Fig. 6). In addition this intensity distribution is laterally very narrow (about 1µm) and could cause a higher local free-electron density which might lead to permanent modifications of the material. Our assumption is supported by the dimensions of the streaks observed in  which are about 1µm wide and more than 50µm long in front and after the position where the optical breakdown occurs. In this region there is more than 70% of the total pulse intensity on the optical axis. To avoid the occurrence of streaks we suggest focussing with higher NA optics.
The second side effect to be discussed here is the additional pulse occurring while focussing ultra-short pulses. This effect has been observed before and is known as forerunner pulse or the so called boundary wave pulse [5, 10]. Our calculations show that its attributes such as velocity of propagation and radial intensity distribution are described well by assuming the additional pulse being a Bessel-like pulse  originating from the system aperture. Its formation as an interference pattern along the optical axis is illustrated in Fig. 7. Notice that at each point along the optical axis different parts of the pulse front interfere. Of special interest is its superluminal velocity vb which exceeds the vacuum velocity of light c depending on the NA of the optical system and is given by
Since the additional pulse is no classical wave package this is not contradictory to the theory of relativity.
Our calculations also show that for a fully illuminated aperture this additional pulse is always appearing no matter how strong aberrations are. Having a gaussian distribution of the field in the aperture attenuates the intensity of the additional pulse. At a certain width of the gaussian distribution the additional pulse even disappears. Furthermore, our calculations show that for an ideal lens the main pulse is overtaken by the additional pulse exactly in the focus. Chromatic aberrations lead to a delay of the pulse front close to the optical axis which causes a shift of the passing point towards the optical system. Therefore it seemed to be a forerunner pulse as reported in earlier papers which is exact only behind the focus. Knowing this is of special interest for experimental work where the additional pulse could corrupt the results, e.g. in pump-probe-experiments. Using a well corrected achromatic system for focussing sets the passing point of both pulses into the focus. Doing so should prevent from having a forerunner pulse in the focal plane.
We demonstrated a powerful method for the calculation and extensive analysis of the propagation of ultra-short laser pulses. This approach significantly extends the analytical calculations of pulse propagation through idealized lenses presented in the literature up to now. Due to the fact that all aberrations and linear propagations effects are taken into account and the calculations are not limited to paraxial focussing the focussing behavior of real world imaging systems could be investigated for the first time. In addition our approach is not limited to idealized field distributions and laser pulse spectra. The implementation of the propagation algorithm into two commercially available optical design packages makes it very flexible and easy to use for optical design tasks. The required effort for the analysis of a specific system is nearly independent from the number of optical components the system is made of.
As a result of our investigations two major side effects occurring while focussing ultra-short laser pulses have been analyzed in detail. The stationary intensity distribution along the optical axis results in a streak formation when focussing with low NA optics. Secondly, the attributes of the additional accompanying pulse could be described with the model of a Bessel-like pulse. It was shown that chromatic aberrations determine the passing point of this pulse and the major pulse front.
The authors like to thank B. Wilhelmi and M. Kempe for fruitful discussions.
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