The majority of the applications of ultrashort laser pulses require a control of its spectral bandwidth. In this paper we show the capability of volume phase holographic gratings recorded in photopolymerizable glasses for spectral pulse reshaping of ultrashort laser pulses originated in an Amplified Ti: Sapphire laser system and its second harmonic. Gratings with high laser induce damage threshold (LIDT) allowing wide spectral bandwidth operability satisfy these demands. We have performed LIDT testing in the photopolymerizable glass showing that the sample remains unaltered after more than 10 million pulses with 0,75 TW/cm2 at 1 KHz repetition rate. Furthermore, it has been developed a theoretical model, as an extension of the Kogelnik’s theory, providing key gratings design for bandwidth operability. The main features of the diffracted beams are in agreement with the model, showing that non-linear effects are negligible in this material up to the fluence threshold for laser induced damage. The high versatility of the grating design along with the excellent LIDT indicates that this material is a promising candidate for ultrashort laser pulses manipulations.
© 2011 OSA
The interest in ultrashort laser pulses began with the invention of the laser and has been one of continuous progress toward shorter time scales . The ultrashort lasers are applied in fundamental areas of physics , chemistry , biology and medicine  and they are on the verge of opening up a large variety of engineering and industrial applications [1,2,5]. The unique physical properties of the high power ultrashort laser pulses rely on the broad power spectral bandwidths and in the well-defined spectral correlation. Both properties can be modified for temporal, spatial and spectral pulse shaping.
Optical elements for ultrashort pulses manipulation can be divided in two groups, according to how they process the spectrum, and therefore, which pulse application are involved. The first group is established by the optical elements for temporal  and spatial pulse shaping [7,8]. They must preserve the complete spectrum and operate as dispersive elements in order to adjust the spectral phase for pulse envelope control. The second group is formed by the optical elements capable to perform wavelengths selectivity in the broad spectral bandwidths of the pulses . These elements are variable spectral filters, which should preserve the correlation of the spectral phase to maintain the pulse character .
Diffraction gratings are ideal candidates to satisfy the demands of the two groups. They are dispersive optical elements that maintain the spectral phase correlation of the pulse and they can be designed to have variable bandwidth operability. The grating bandwidth can be selected to preserve or filtering the pulse spectrum according to the demanded application. Diffraction gratings recorded by holographic methods  exhibit special features compared to standard gratings fabricated by mechanical methods. They are defect-free periodic structures that allow high versatility design together with an easy implementation. Some representative diffractive elements implemented in photosensitive materials for ultrashort laser pulse manipulation have been reported in the literature. The dichromated gelatin holographic gratings  are probably the best-known type of holographic optical elements applied in ultrashort laser. Other elements such as photothermal refractive gratings  and fiber Bragg gratings  have also been proposed. Mirror backed volume phase gratings with coated sol-gel or dichromated gelatin films are applied for telecommunication and pulse compression [14–16], whereas surface relief reflection gratings [17,18] are readily available from commercial sources. However, no report exist treating photosensitive materials which simultaneously satisfy the two groups above mentioned together with a high laser-induced damage threshold. In this paper we demonstrate the capability of volume phase holographic gratings (VPHG) recorded in photopolymerizable glasses to preserve or filtering the incoming pulse spectrum exhibiting an excellent damage threshold.
The paper is organized as follows: in Section 2 we report the properties of the holographic gratings recorded in photopolymerizable glasses; in Section 3 we develop a theoretical model to explain the main features of the diffracted pulses and we find a suitable relation between grating parameters for spectral pulse shaping; in Section 4 is analyzed the diffraction efficiency, as well as the spectral properties of the diffracted laser beams when ultrashort pulses of 50 fs duration centered at 800 nm and 400 nm impinge onto the gratings; in Section 5 we explore the laser induced damage threshold (LIDT) of the photopolymerizable glasses and the work ends with concluding remarks in Section 6.
2. Holographic gratings
The photosensitive material used in this work  is a modified composition of a highly efficient photopolymerizable sol-gel glass synthesized earlier by Cheben et al. , wherein the modification is achieved by incorporating the Zr-based high refractive index species (HRIS) at molecular level [19,21,22]. The sol-gel technique, applied to the synthesis of the material, permits to cast samples with wide thickness design with a low scattering coefficient of 1.2 x 10−3 μm−1 [19,22]. The samples were encapsulated between two coverglasses of 150 μm thickness providing an optical contact between the holographic recording material and the coverglasses.
The VPHGs formation is obtained by the interference of two mutually coherent s-polarized writing beams, using a continuous wave (cw) frequency-doubled Nd:YAG laser at 532 nm. The writing beams were incident on the sample in an unslanted configuration, intersecting the latter at different angles with respect to the sample normal, as is described in [21,23]. Thus, interference fringes in this work are formed with various spatial frequencies ranging from 500 to 4000 lines/mm. Properties of the gratings has been characterized extensively for monochromatic light demonstrating an excellent optical performance and a negligible surface modulation, as it is shown in references [19–23]. The grating thickness is measured directly on the sample and the refractive index modulation is calculated fitting the diffraction efficiency of one of the writing beams under Bragg angle to Kogelnik’s coupled wave theory . In this paper we extend the grating characterization to ultrashort laser pulses. The value of the Klein-Cook parameter Q 
3. Theoretical model
In this section we generalize the one dimensional Kogelnik’s coupled wave theory  to the case of polychromatic light, by including pulses with broad spectral bandwidths. We analyze the particular case of transmission VPHGs composed by unslanted planes perpendicular to the grating interphase. We start from the following assumptions:
- 1) The gratings are encapsulated between two dispersive coverglasses (no(λ)) providing optical contact, as indicated in Section 2. The optical element (grating and coverglasses) is embedded in a non-dispersive medium with refraction index n = 1 (in the air).
- 2) The light dispersion in the optical element mainly occurs at the interface air-coverglass since the refractive index of the grating and the coverglasess are similar (~1.52). Therefore, once the light is dispersed it maintains the direction of propagation along the optical element without noticeable change. As a first approximation we consider an unique background refractive index for the coverglasess and the gratings, equal to no(λ).
- 3) The refractive index modulation Δn fulfills Δn << no(λ). Hence, Δn can be considered as a constant for the wavelength range of interest.
- 4) The ultrashort pulsed plane wave can be described as a superposition of monochromatic planes waves propagating in the same direction in the air. The complex amplitude of each component is obtained by the Fourier Transform of its time-dependent electric field .
- 5) The monochromatic components of the pulse impinge onto the grating under the same incidence angle θi. Snell’s refraction law applied to the interface air-coverglass sinθi = no(λ)sinθ(θi,λ), defines the incidence angle θ(θi,λ) inside the grating for each spectral component according to 2).
The coupled wave theory analyzes the diffraction efficiency originated by a grating when an incoming monochromatic plane wave impinges onto the sample under-Bragg angle as well as off-Bragg angle. The detuning from Bragg condition was analyzed only for two individual cases. First, monochromatic light with angle of incidence mismatch and second, collinear polychromatic light propagation inside the grating, with light wavelength mismatch. We extend this theory to a general case in which we place ourselves in the regime of point 5): the angle and wavelength mismatch occurs simultaneously. We refer to this case as the generalized mismatch condition, and it can be defined through the function :
This equation depends on the grating parameters, on the incident angle outside the grating θi and on the illumination wavelength λ. Notice that a monochromatic plane wave under Bragg angle reduces to zero the function . Off-Bragg incidences give rise to and the diffracted complex field is generated with a detuning proportional to and the grating depth. The expression of the diffraction efficiency, , is :Eq. (2) can be written as ; 2). The diffraction efficiency under Bragg incidence reaches the maximum value (), thus . According to Eq. (3), the first minimum in the spectral selectivity curve occurs when and the grating bandwidth is:
The main features of the diffracted beams are determinate by the grating properties as well as the incoming pulse characteristics. We can define the diffracted spectral intensity per unit wavelength, through the expression:Eq. (8), gratings that satisfy ΔλG >> ΔλP preserve the pulse spectrum, meanwhile ΔλG < ΔλP perform a wavelengths selectivity. This criterion will be used in the next section for spectral pulse shaping.
The total diffraction efficiency of the ultrashort pulse, , takes into account the contribution of the whole spectrum through this expression:
The representation of as a function of the incidence angle θi outside the grating is denominated the angular selectivity curve (ASC). When ΔλG >> ΔλP, can be regarded as constant in the spectral interval in which g(λ) is defined and Eq. (9) can be written as . The ASC is identical to that for monochromatic light. Hence the pulse characteristics are not detectable and the ASC shape is defined only by the grating properties. The opposite case occurs when ΔλG <<ΔλP. In this case, only for the wavelength λi that satisfies the Bragg angle θi: . can be regarded as a delta-like function centered in λi and the total diffraction efficiency is:Eqs. (8) and (9) to fit and interpret the experimental results.
4. Experimental results
The VPHGs recorded in the photopolymerizable glass were tested with ultrashort pulses from an Amplified Ti: sapphire laser system (ATLS) delivering 50fs, 3.6mJ pulses centered at 800nm with 1KHz repetition rate. The second harmonic generation of the laser is achieved by using a non-linear crystal BBO to obtain pulses centered at 400 nm, which also have been used to analyze the diffraction features originated by the gratings.
Two types of experiments were carried out to demonstrate the pulse spectrum preservation or filtering. In the first one, the diffracted and the incident intensity per unit wavelength was measured using a portable spectrophotometer. In the second experiment, the total diffraction efficiency was obtained by normalizing the diffracted power spectrum by the addition of the transmitted and diffracted power spectra. This normalization is commonly used in holography and it does not take into account losses by Fresnel reflection. The measurements were carried out with a two-channel photodetector to collect the total transmitted and diffracted power spectra. The sample was placed on a high precision rotation stage (0.001 degrees) controlled by a computer that allow as to change the incident angle of the incoming pulses and to perform the angular selective curves. In the following, pulses centered at 800 and 400 nm will be denominated pulses a and b, respectively.
4.1 Gratings for pulse spectral conservation
In this section we design holographic gratings for pulse spectral conservation of the fundamental emission of the ATLS and its second harmonic, operating with maximum diffraction efficiency. To make an appropriated grating design it is necessary to know the spectral intensity per unit wavelength of the incoming pulses, , and their spectral bandwidths ΔλP. The black lines in Figs. 1(a) and 1(b) provide the normalized experimental results for pulses a and b, delivering and . The Gratings 1 and 2, shown in Table 1 , have been used to perform the experiments with pulses a and b, respectively. To know the gratings bandwidth operability we use Eq. (7) and the parameters Q displayed in Table 1, obtaining and . According to the theory, these values assure pulse spectral conservation.
The diffracted spectral intensities per unit wavelength are shown in Fig. 1(a) and 1(b), dotted curves, and the red lines represent the theoretical fit using Eq. (8) and the grating parameters. We confirmed the spectral conservation together with the high diffraction efficiencies close to ~92% and ~95% for pulses centered at 800nm and 400nm, respectively. In the experiments, the incidence angle has been selected to satisfy the Bragg condition for the central wavelengths of pulses a and b. The total diffraction efficiency as a function of the incidence angle θi has been used to double check the theoretical model. In Figs. 2(a) and 2(b) are displayed the ASC for Grating 1 and 2.
The shape of the ASCs resemble to the one associated to a quasi-monochromatic light, for which it is possible to resolve secondary maxima and minima. The pulse characteristics are not detectable and the ASC shape is defined only by the grating properties. This result is in accordance with the behavior predicted by the model when the condition is satisfied. The red lines in the figures represent the fitting using Eq. (9) and parameters in Table 1, demonstrating an excellent correspondence.
The gratings analyzed are ideal candidates for ultrashort pulses manipulation when the complete spectrum must be preserved. They involve diffractive optical elements for temporal and spatial pulse shaping. A typical arrangement for temporal pulse shaping is composed by four or pair identical gratings, as it is discussed in  that would be suitable to bring up the out-going pulses to its transform-limited duration. The estimated efficiency of the optical system performed with these gratings (without considering losses by Fresnel reflections) is ~72% and ~82% for the fundamental and SH pulses of the ATLS, respectively. These estimated values show an excellent potential of the optical set-up by using VPHGs in the photopolymerizable glass.
4.2 Gratings for pulse spectral filtering
VPHGs with narrow bandwidth operability were also recorded in the photopolymerizable glass. In order to provide an illustrative example, we have designed two spectral filters, one for the fundamental and another one for the first harmonic of the ATLS, capable to reduce four times the bandwidth of the incident spectrum. This condition is achieved if . These spectral filters are variable ones in the sense that the central wavelength λi of the diffracted beam is selected changing the angle of incidence θi.. Specifically, λi must satisfies the Bragg condition. Notice that, for pulses centered at 800 nm, , while for pulses centered in 400 nm, . The Gratings 3 and 4 shown in Table 1 have been used to perform the experiments with pulses a and b, respectively. According to Eq. (7), and .
As illustrative example, in Fig. 3(a) and 3(b) we display the normalized spectral intensity of the incidence pulses a and b (black lines) and the diffracted beams (dotted lines) at different angles of incidence θi. The angle values select the central wavelength λi of the diffracted beams according to Bragg’s law. The (θi, λi) correspondence is shown in the figure caption, while the gratings parameters are displayed on Table 1. We stress the high diffraction efficiency delivered for a single wavelength since the spectral intensity associated with λi approximates the spectrum envelope at this point. The diffracted beams FWHM for pulses centered at 800 nm and 400 nm are ~6 nm and ~2 nm, respectively. These values correspond to ~25% of the and , in accordance to the gratings design. The color lines in Figs. 3(a) and 3(b) are the theoretical results using Eq. (8) and parameters in Table 1. Notice that secondary maxima and minima are predicted by the theory, but they are partially resolved in the experiments. The slightly smoothness occurs due to small thickness gradient (~2%) in the sample. Nevertheless, there is a well correspondence between theory and experiments.
In Fig. 4(a) and 4(b) it is shown the angular selectivity curves associated to pulses a and b, respectively, and the fitting to the experimental results. The shapes of the curves are alike to the spectral intensity profiles of the corresponding incoming pulses. This fact has been predicted by the theory and it becomes more appreciable for gratings that satisfy ΔλG <<ΔλP. Notice that the slightly asymmetry of Fig. 4(b) is in accordance to the spectrum shape of pulse b. The maximum value of the total diffraction efficiency for gratings that perform spectral filtering is always smaller than one, even if the gratings are high efficient for a single wavelength (see Eq. (9)), as the cases shown in Fig. 3. Finally, the small thickness gradient does not change appreciably the ASC´s shape since each single point on the curve is the contribution of the whole diffracted spectra at one angle of incidence.
According to Figs. 1–4, we conclude that the theoretical model predicts satisfactory the experimental results for both cases, gratings with wide and narrow spectral bandwidth operation, when the spectral distribution of the diffracted beam and the total diffracted power are analyzed. This indicates that undesired effects non-covered by the Kogelnik's theory like nonlinear effects are negligible in the gratings recorded in the photopolymerizable glass, up to the fluencies proven in these experiments.
We would like to mention that Grating 4 has been recorded using an alternative arrangement arrangement. The narrow grating period, Λ = 0.25 μm, induces high values of Q capable to filter out the incoming pulse spectrum of the SH of the ATLS. The incidence angle of the writing beams to record such a grating can only be achieved if the photopolymerizable glass material is sandwiched between two right-angle prisms, as explained in . Bragg angle for the incoming pulses with central wavelength λc = 400nm corresponds to a 19° incidence angle on the interface air-prism. The dispersion originated in the interface air-prism has been considered in the theoretical model. Hence, the angles θi displayed on Fig. 3 and 4, are measured inside the recording photomaterial.
5. Femtosecond laser induced damage threshold in photopolymerizable glasses
One of the core issues in development VPHGs is their laser-induced damage threshold [14–16]. In this section we analyze the LIDT for the photopolymerizable glasses in order to define the applicability range when it is used to manipulate high power ultrashort laser pulses. For this purpose, we have used VPHGs with thickness of 145 μm and a single period of Λ = 2 μm. The LIDT experiments have been performed with an Amplified Ti: sapphire laser system delivering 50fs pulses centered at 800nm with 1 KHz repetition rate.
The incident pulses on the sample have spatial Gaussian-distribution of 2.0 mm FWHM, reached after passing by plano-convex lens with a 100cm focal-length. A variable neutral density filter was installed in the beam’s path to provide energy control. The LIDT was tested with the temporal evolution of the first order diffraction efficiency of a nonactinic He–Ne laser (λ = 632nm) probe beam during sample irradiation. The angle of incidence of the probe beam as well as the ultrashort pulses were chosen to fulfill Bragg`s condition on the same volume of the grating, see Fig. 5 . In all experiments, an appropriate air absorption mechanism was activated in order to avoid breading molecules generated by evaporation and by a possible laser ablation.
The conditions were unaltered during the experiments and different tests have been performed varying the incident energy over the gratings from 0.6mJ to 1mJ per pulse. In Fig. 6 we report the evolution of the first order diffraction efficiency of the probe beam for different pulse energies. The diffraction efficiency of the gratings was obtained dividing the diffracted power by the addition of the transmitted and diffracted power. The measurements were carried out with a two-channel photodetector to collect the transmitted and diffracted power.
In all cases analyzed, but the corresponding to 0.6 mJ, the diffraction efficiency keeps constant until a critical time in which the efficiency decline and the sample exhibits damage. Data corresponding to the first 10 seconds for each curve in Fig. 6 were taking before irradiation, showing that the diffraction efficiency is constant until damage occurs. Similarly, the last 10 seconds for each experiment shows the efficiency after irradiation. It also takes a constant value equal to the one reached at the last pulse. Notice that incident pulses of 1 mJ energy (~1.25 TW/cm2 per pulse) at 1 KHz repetition rate induce damage after more than 106 pulses. On the contrary, sample properties remain unaltered after 107 pulses of 0.6 mJ (~0.75 TW/cm2 per pulse). This energy represents the upper limit of the LIDT for the volume phase holographic gratings recorded in the photopolymerizable glasses.
When damage occurs, the irradiated area of the sample is gradually changed from transparent to a highly absorbing circular spot whose diameter increases with exposition. At the final stage the diameter is 2 mm. Figure 7(a) is a photograph of the damaged area and Fig. 7(b) is a magnified photograph using an optical microscope (10x), showing the damage in more detail. The spot shape and dimension coincide with the spatial distribution of ~2 mm FWHM pulses, in accordance with intensity shape measurements on the sample. The dark spot absorb light generating a heat expansion that induces cracking in much more extended areas; see Figs. 7(a) and 7(b). This behavior has also been reported for femtosecond laser induce damage in highly transparent materials [27,28]. Accordingly, the photopolymerizable glass shows an excellent femtosecond LIDT and it is particularly promising for ultraintense and ultrashort pulse applications.
In this paper, we explore the capability of the photopolymerizable glass to be applied in spectral manipulation of femtosecond laser pulses. For this purpose, we used VPHGs with both, wide and narrow bandwidth operations. Diffraction efficiency as well as spectral composition of the diffracted beams were analyzed for femtosecond laser pulses illumination centered at 800 nm and 400 nm, respectively. Gratings with high diffraction efficiency and high performance were implemented, showing the versatility of the photopolymerizable glass that can be exploited to design diffractive elements for temporal and spatial pulse shaping, as well as for the generation of variable spectral filters with tailored desired bandwidth operation.
We have performed experiments to determine the femtosecond laser induce damage threshold of the VPHGs implemented in the photopolymerizable glass. We have shown that the holographic material remain unaltered after more than 10 million Gaussian-shaped 50 fs pulses at 1 KHz repetition rate with a spatial FWHM of 2mm centered at 800nm and with 0.6mJ per pulse. LIDT occurs for values up to 0.75 TW/cm2 indicating an excellent damage threshold. This holographic material is promising for intense-fields excitations. Meanwhile, we developed a theoretical model that takes into account the grating bandwidth operability together with the spectral bandwidth of the incoming ultrashort pulses. This model interprets the main features of the diffracted beams by VPHGs with wide and narrow spectral bandwidths. It was shown that the model fits satisfactorily the spectral composition of the diffracted pulses and the total diffraction efficiency as a function of the incidence angle, as well. We also found a suitable relation between the Klein-Cook parameter Q and the grating bandwidth operation (Eq. (7)) which has been used to design gratings to perform the spectral pulse shaping. As the main Kogelnik's hypotheses hold in the whole excitation range, undesired effects limiting the holographic performance seems to be negligible in this material up to fluencies close to the damage threshold.
We thank J. A. Rodrigo and R. Weigand for helpful discussions. The financial support of the Spanish Ministry of Innovation and Science, under grant TEC2008-1045 is acknowledged. One of us (M.P. Hernandez-Garay) acknowledges the MAEC-AECID (Spain) and CONACyT (Mexico) grants, P. Vaveliuk is grateful to CNPq (Brazilian federal grant agency) for financial support. We thank to Chemical Reaction Dynamics and Femtochemistry Group: J. G. Izquierdo and L. Bañares (Spanish Ministry of Science and Innovation (MICINN) through Grant CTQ2008-02578/BQU, and Consolider SAUUL CSD2007-00013). Finally, the authors gratefully the reviewers’s suggestions.
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