Femtosecond pulse shaper built into a prism compressor

We present a frequency domain, AOM-based pulse shaper that utilizes Brewster prisms rather than the current standard of gratings. In doing so, we demonstrate a three-fold increase in efficiency and the ability to compensate for temporal dispersion created by the acousto-optic modulator that filters the pulse spectrum. The shaper is tested between the wavelengths of 520-660 and 840-&#;nm, creating sub-50 fs pulses for each, and used to collect a 2D white-light spectrum of a thin film of semiconducting carbon nanotubes.

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1. Introduction

Since their inception, pulse shapers have been a useful tool in the field of ultrafast spectroscopy. They are used to compress pulses, transmit information, and alter the yields of photoproducts, to name only a few applications [1&#;9]. Pulse shapers have also become a ubiquitous tool in ultrafast 2D spectroscopy. As first demonstrated by Warren Warren, sequences of pulses with variable time delays and phases can be generated using pulse shapers to collect multidimensional spectra, like 2D IR and 2D Visible spectroscopies or microscopies [10&#;12]. The majority of ultrafast 2D spectroscopy experiments in the pump-probe geometry are now carried out using pulse shapers [13].

Pulse shapers are phase modulators and/or amplitude filters that modify the electric field of an input laser pulse to create the desired time-evolving electric field. This level of control allows the user to define the spectral phase, amplitude, and polarization of the electric field depending on the pulse shaper&#;s configuration [14&#;17]. Of course, there are many static optics that alter the spectral and/or time-dependence of a laser pulse [18,19]. In contrast, pulse shapers are dynamic, permitting computer control over the spectrum and time-dependence of a laser pulse, much like an arbitrary waveform generator for light.

There are many designs of pulse shapers, including ones that utilize acousto-optic programable dispersive filters (AOPDF), spatial light modulator (SLM), micromirrors, and deformable mirrors [4,7,20&#;22]. One common design is the transverse pulse shaper that operates on the pulse in the frequency domain and consists of two gratings, focusing optics, and a programmable filter in a 4-f optical layout as shown in Fig. 1(a). The laser pulses impinge on a grating to spatially resolve the frequencies of the pulse. The light is then focused onto the programmable filter, such as an acousto-optic modulator (AOM) like used in this report, and subsequently exits in a similar fashion, converting the pulse back into the time-domain [23]. The distance between a grating and a focusing optic as well as the distance between a focusing optic and the filter is equal to the focal length of the focusing optic, creating a 4f geometry [6]. By applying a &#;mask&#; to the filter that alters the amplitudes and/or phases of the dispersed frequency components, the spectrum can be altered to create the desired temporally shaped pulse. In its simplest implementation, the mask is the Fourier transform of the desired temporal pulse.

 

Fig. 1. Pulse shaper designs. (a) Grating-based pulse shaper consisting of two gratings and two parabolic mirrors. (b) The presented pulse shaper utilizing four Brewster prisms and two plano-convex lenses. Both pulse shapers depict the 0th order (undiffracted) beam path.

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There are two aspects of the conventional 4-f design that motivated the design reported here: the grating efficiency and the temporal dispersion created by the pulse shaper. There are many options for gratings, but the peak efficiency is typically 70-80% and decreases when moving away from the blazed wavelength, leaving the total efficiency between 60-65% for a 100-200&#;nm pulse. Since two gratings are required, the gratings result in >50% loss in light. Temporal dispersion is caused by transmissive materials within the pulse shaper. Transverse AOMs are made from crystals that are about 1&#;cm thick. The temporal dispersion needs to be removed to create transform limited pulses or to devise a new, well-defined pulse train. The degree to which a pulse can be temporally modulated is set by the frequency resolution of the filter, which depends on the focal spot size at the filter, the number of individual elements the filter can alter, and the pulse&#;s spectral range [24]. A pulse shaper might have 700 individually resolvable frequency elements or &#;pixels&#;. The AOM bandwidth utilized to remove the temporal dispersion from the pulse reduces the amount of bandwidth leftover to create the desired shaped pulse. Indeed, some pulse shapers cannot compensate for the dispersion induced internally, and so chirped mirrors are used to pre-compress the pulses prior to entering the pulse shaper [17].

In what follows, we present an optical design in which the gratings are replaced by a set of Brewster prisms, so the programmable filter resides inside of a prism compressor. By utilizing Brewster prisms rather than gratings, we improve the efficiency of the pulse shaper as well as gain the ability to compress broadband pulses without additional compression optics. In the following sections, we explain the design, demonstrate compression for a series of pulses spanning wavelength ranges of 520&#;660 and 840&#;&#;nm, as well as use it to collect a 2D white-light spectrum of a carbon nanotube thin film.

2. Optical design of the prism compressor pulse shaper

In our new design, the first prism disperses the light while the second prism collimates the light before it enters the AOM (Fig. 1(b)). To focus the light at the AOM, we use transmissive lens. We utilize an acousto-optic modulator for the programmable filter, but an SLM or other type of filter might be used instead. After the AOM, the following two prisms and lens recombine the light as it exits the pulse shaper.

The key to the benefits described in the previous section are the prisms themselves. Each Brewster prism is cut so that both the incident and exiting light encounters the prism-air interface at Brewster&#;s angle [25]. The result is that the prisms transmit p-polarized light with near-perfect efficiency across a large range of wavelengths, leading to improved efficiency over gratings. We also arranged the prisms in the pulse shaper in the form of a 4-prism prism compressor. This design allows us to add a controllable amount of anomalous dispersion to the system, a feature not present in grating-based system [25,26]. Transmissive materials in the visible region add normal dispersion, causing shorter wavelengths to lag behind longer wavelengths. Pulse compression optics are intended to counteract this effect, adding anomalous dispersion that allows shorter wavelengths to arrive coincident with longer wavelengths. Prism compressors add anomalous GDD and TOD because longer wavelengths travel through more material than shorter wavelengths. The amount of dispersion depends on the type of prism material, the separation between prism 1 and 2 and prism 3 and 4, the insertion points of each prism (the distance from the center of the beam to the tip of the prism), and angle of the prism relative to the incident light. For this pulse shaper, the insertion point should be as close to the tip as possible to avoid adding normal dispersion, and the prism angle is always set to the Brewster angle for efficiency, so we tune the anomalous dispersion by changing the type of prism material and separation. In the following sections, we discuss how to determine parameters such as choice of programmable filter, prism compressor design, lens focal length and position, and input beam size to build a functional prism compressor pulse shaper.

Programmable filter (AOM)

We chose to utilize an AOM for this pulse shaper design because of its ability to perform shot-to-shot modulation of the time-evolving electric field at &#;100 kHz or higher depending on the method of calibration and data processing [27,28]. This feature enables faster data collection and increased signal-to-noise compared to other programmable filters; however, there are other properties of AOMs that need to be considered other than their ability to operate at higher repetition rates. Discussed in the paragraph below is information regarding AOM material, diffraction efficiency, and dispersion compensation.

An AOM diffracts light by sending a sound wave through a crystal, introducing transient changes in the refractive index along the crystal that diffract the light [15,23]. The diffraction efficiency of the AOM depends both on the polarization of light and the AOM mask. The input light for this pulse shaper needs to be p-polarized to minimize reflections on the Brewster prisms, so we chose a TeO2 AOM over other materials because a TeO2 AOM diffracts p-polarized light. Alternatively, a quartz AOM is designed to diffract s-polarized light and could be used if two waveplates were added to rotate the pulse polarization before and after the AOM. With regards to the AOM mask, the diffraction efficiency can be described by Eq. (1) [15].

$\frac{I}{{{I_0}}}$

is the ratio of the diffracted to the undiffracted light, Q is a parameter that depends on the thickness and acoustic properties of the crystal, ν0 is the frequency of sound wave that diffracts the light such that the Bragg condition is fulfilled, and ν is the frequency of sound wave applied to each wavelength of light. The Bragg condition is necessary for efficient diffraction because it fulfills the requirements of conservation of momentum. We can see from this formula that sending a single frequency sound wave through the crystal will only meet the Bragg condition for a single wavelength of light, leading to less efficient diffraction for other wavelengths. For uniform diffraction, we apply a &#;Bragg mask,&#; in which the instantaneous frequency of the AOM acoustic wave is changed at each pixel such that each wavelength meets the Bragg condition [17,29].

To compensate for dispersion, we multiply the Bragg mask by an additional filter function that creates anomalous GVD and TOD. After dispersion correction, the mask can be further altered to create the desired temporal pulse. Thus, the mask function, M(ω), is given by Eq. (2):

$\psi $

bragg (ω) is the spectral phase necessary to create the Bragg mask, and

$\psi $

M­(ω) is dispersion correction needed to compress the pulses.

$\psi $

M(ω) is typically written in the form of a Taylor series expansion, where different terms correspond to different orders of dispersion (GVD, TOD, etc.). Finally, S(ω, ϕ) is the mask for creating the desired pulse shape. For example, S(ω, ϕ) for a double pulse with a time delay of τ is

The limit to the complexity of any mask is set by the bandwidth of the AOM (100&#;MHz centered at 200&#;MHz for TeO2). One can estimate the amount of AOM bandwidth used from dispersion compensation alone from the following formula:

n is the amount of AOM bandwidth required for dispersion correction up to the nth order, TM is the mask length in time,

$\psi $

i represents the ith order of dispersion compensation (GVD, TOD, etc.), and Δω is the spectral width of light being shaped. Thus, any bandwidth remaining after the Bragg mask and dispersion compensation can be utilized to make a more complex pulse or pulse train. As an example, we show in Fig. 

 

Fig. 2. Depicts the AOM bandwidth necessary to produce a (blue) single frequency mask, (red) a Bragg mask, a mask with the dispersion compensation used to compress the (yellow) shaped pulse (840&#;&#;nm) below 50 fs, (purple) and a mask with dispersion compensation and double pulse time delay of 600 fs.

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Prism compressor design

The prism compressor&#;s material and spacings determine the extent to which the pulse can be compressed. Most materials add more normal GDD than TOD; however, prism compressors add more anomalous TOD than GDD, lengthening the pulse even if one compensates for the system&#;s GDD. To minimize the problem, we chose prism materials with low TOD-to-GDD ratios in the wavelength regions of interest: LAK21 for 520&#;660&#;nm and SF11 for 840 - &#;nm. Next, we set the distance of the prisms such that they compensate for the normal dispersion introduced by the lenses and AOM, which were 150&#;cm for 520&#;660&#;nm and 50&#;cm for 840&#;&#;nm. The anomalous TOD from the prisms will be greater than the normal TOD introduced by the transmissive optics in the apparatus, meaning that the remainder of the pulse compression will need to be performed by the AOM.

Lens position and focal length

The choice of lens position and focal length impacts the frequency resolution of the pulse shaper. The resolution to which the AOM can alter the frequency components within the bandwidth is set by size of the AOM mask and the spot size of the focused beam at the AOM. The bandwidth multiplied by the length of the AOM mask in time sets an upper limit on the number of independently programmable pixels and corresponds to the intrinsic pixel size of the AOM (&#;45 microns for a TeO2 AOM). To design the pulse shaper, we utilized the Light Tools ray tracing software to simulate the spot size and mask length that is utilized for a series of lens focal lengths and positions, from which we calculated the number of pixels.

We simulated a total of eight different lenses with focal lengths ranging from 150 mm to 700 mm with each lens focused at the AOM aperture (Fig. 3(a)). To determine the number of pixels for each lens, we simulated the focused spot sizes of two, 3 mm diameter, monochromatic input beams (840&#;nm and &#;nm) that represent the extremes of the bandwidth of our white light. From these simulations, we calculated the FWHMs of the focused beams at the AOM aperture and took the average of the two values as the pixel size (Fig. 3(b), left axis). The lenses also reduce the distance over which the light is spatially dispersed, and so we calculated the spatial separation between the two colors of light to estimate the amount of the aperture spanned by the white-light and thus the length of mask (Fig. 3(b), right axis). By taking the ratio of the mask length to pixel size, we calculate the number of pixels of the pulse shaper (Fig. 3(c)) [24]. According to Fig. 3(c), the position with the largest number of pixels corresponds to a lens placed immediately after prism 1 for a shaper built using SF11 prisms and operating between 840&#;&#;nm. That position also maximized the number of pixels for the other two wavelength ranges, 970&#;&#;nm and 520&#;660&#;nm, with SF11 and LAK21 prisms, respectively. We predict 88 pixels (2.2&#;nm resolution) and 74 pixels (2.7&#;nm resolution) for the 840&#; and 970&#;&#;nm ranges, respectively, when using 500 mm lenses. We predict 116 pixels (1.2&#;nm resolution) for 520&#;660&#;nm light using mm lenses (the longer focal length corresponds to the larger prism separation). We note that the AOM is placed as close as physically possible to prism 2 to minimize the focal length and thus provide as small a beam width as possible at the AOM.

 

Fig. 3. Results of ray tracing simulations for finding the optimal lens placement in the prism compressor pulse shaper using SF11 prisms and 840 &#; &#;nm light. (a) Diagram of the ray tracing simulations with the various lens positions at different points in the beam path. The dotted outlines of the lenses represent tested positions, and the filled lens corresponds to the position the ray tracing simulation predicts gives the highest resolution. (b) Effective pixel size, mask length, and lens focal length. (c) Pixels achievable with each focal length lens.

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Beam size

The input beam size is another important parameter since it affects both frequency resolution and the amount of dispersion compensation needed to compress the pulse. Increasing the input beam size leads to tighter focusing at the AOM aperture and a corresponding increase in the number of pixels. A larger beam diameter, however, adds normal dispersion because it increases the insertion point with prism 1 and 4, causing the light to travel though more material. One can compensate for the dispersion with the AOM mask, although that reduces the AOM bandwidth available for further shaping the pulse (Eq. (4)) as well as diffraction efficiency. To characterize this tradeoff, we calculated the effects of beam sizes from 1 to 6 mm diameter on the maximum time delay between two pulses as well as diffraction efficiency.

Using Light Tools ray tracing, we simulated the focused spot size at the AOM aperture for each input beam diameter. We then simulated the acoustic wave necessary for dispersion compensation as well as generating a max double pulse time delay based on the resolution predicted by the ray tracing simulations. Upon Fourier transforming the mask, we examine the amount of AOM bandwidth it requires. One such calculation is shown in Fig. 4(a) for a 5 mm input beam where we compare the amount of AOM bandwidth used by masks with and without dispersion compensation. Shown in Fig. 4(b), are the max double pulse delays calculated for 1 to 6 mm beam sizes with and without dispersion compensation. The largest time delay that can be achieved with dispersion compensation is about 650 fs using a 5 mm input beam even though the number of pixels permits larger delays. We also calculated the diffraction efficiencies for each input beam size based on the mask using Eq. (1) (Fig. 4(c)). The diffraction efficiency decreases on the blue side of the white-light due to deviations away from the Bragg mask, but it can still support a&#;>&#;100&#;nm spectral range.

 

Fig. 4. Calculations of mask frequencies, max double pulse time delay, and diffraction efficiency based on the additional dispersion needed to compensate when using different input beam sizes. (a) Amount of the AOM bandwidth required to produce (blue) the max double pulse time delay based on number of pixels and (red) the max double pulse time delay while compensating for dispersion originating from the larger insertion point. The AOM is unable to reproduce the latter waveform since it requires frequencies below 150&#;MHz (b) The maximum double pulse time delay as a function of input beam size for both dispersion compensation and no dispersion (c) The simulated wavelength dependent diffraction efficiencies with the dispersion necessary to compensate for various input beam sizes.

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Based on the calculations above, we built a pulse shaper using the following parameters: For the 840&#;&#;nm and 970&#;&#;nm wavelengths, we used SF11 prisms separated by 50&#;cm, a 500 mm focal length lens placed directly after the first prism, and a 3 mm input beam size; For the 520&#;660&#;nm wavelengths, we used LAK21 prisms separated by 150&#;cm, a mm lens placed directly after the first prism, and a 3 mm input beam size.

3. Methods

Pulse shaper

To construct the prism compressor pulse shaper for wavelengths 840&#;&#;nm, we used four SF11 Brewster prisms cut for 800&#;nm from Newlight Photonics (IPS-59.0), and two 500 mm focal length, AR coated plano-convex lenses from Thorlabs (LJRM-B). For the prism compressor pulse shaper spanning 520&#;660&#;nm, we used two one-inch LAK21 prisms from Edmund Optics (#89-843), two custom two-inch prisms LAK21 prisms from Hangzhou Shalom Electro-Optics, and two mm focal length, AR coated plano-convex lenses from Thorlabs (LA-AB). All experiments we conducted with an AR coated (500&#;nm &#; &#;nm) TeO2 AOM from Phasetech. We also used the Phasetech RFA-VIS amplifier and Quick Control software with a Signatec PXDAC AWG to control the AOM at a 0.25 duty cycle for the 840&#;&#;nm measurements and a 0.75 duty cycle for the 520&#;660&#;nm measurements.

Spectra measurements

For both near-IR and visible frequency domain measurements, we used the Princeton Instruments Acton SP spectrometer and OMA V InGaAs detector. For the near-IR time domain measurements, we used a Thorlabs biased InGaAs photodiode (DET10A2) and a Zurich Instruments UHFLI 600&#;MHz with the boxcar integrator option.

White light generation

To generate NIR white light (840&#;&#;nm or 970&#;&#;nm), we used 200&#;mW of 800&#;nm light (2.1 µJ/ pulse at 95 kHz) from the Light Conversion Pharos 1 with an iOPA-FW-HP focused through a 4 mm YAG crystal. To generate visible white light (520&#;660&#;nm), we used 300&#;mW of &#;nm light (3.1 µJ/pulse at 95 kHz) from the Light Conversion Pharos 1 focused through a 4 mm YAG crystal.

Pulse compression measurements

We used polarization-gated frequency resolved optical gating to measure the spectral phase of each region of light with 150&#;mW of 800&#;nm light (1.67 µJ/per pulse at 95 kHz, FWHM&#;=&#;195fs +/- 2fs) as the gate pulse.

Ray Tracing simulations

We used Synopsys&#;s Light Tools to perform the ray tracing simulations. Each monochromatic simulation was coherent with the input light following a Gaussian intensity profile.

4. Results

Efficiency

We find the efficiency of the undiffracted, transmitted light through the prism compressor pulse shaper to quantify the losses coming from reflections from lenses, prisms, and the AOM. The transmission efficiency of these optics is approximately 90% in each wavelength region, an improvement over the >50% loss seen in grating-based pulse shapers. Since the efficiency is so high with prisms, the overall efficiency is largely set by the AOM (or other type of programmable filter). For a TeO2 AOM, we find the overall efficiency of the pulse shaper to be 32% for 840-&#;nm light and 35% in the 520-660&#;nm light by taking the ratio of the diffracted output light power to the input light power. For comparison, a pulse shaper using a TeO2 AOM and gratings typically has around 10% efficiency. Thus, replacing the gratings with prisms triples the efficiency, which is valuable when using low intensity pulses such as those generated with white-light continuum. We also note that most grating pulse shapers would also require chirped mirrors, which are not necessary with the prism compressor pulse shaper at these wavelengths. Removing these optics gives an additional enhancement of roughly 10-20% depending on the number of bounces.

Resolution

To experimentally quantify the frequency resolution, we measured the FWHM of the narrowest linewidths we could create with an AOM mask [24]. By placing the 500 mm lens ( mm for 520&#;660&#;nm) close to the first prism while remaining focused at the AOM, we find the prism compressor pulse shaper has approximately 80 pixels (2.5&#;nm resolution) with 840&#;&#;nm light, 70 pixels (3&#;nm resolution) with 970&#;&#;nm light, and 108 pixels (1.3&#;nm resolution) with 520&#;660&#;nm light.

Pulse compression

We measured the pulse compression of the shaped light from 840 - &#;nm, 970 - &#;nm, and 520 - 660&#;nm using polarization-gated frequency resolved optical gating (PG-FROG) as shown in Fig. 5(a)-(c) [30]. We only compress up to 200&#;nm of the white-light at one time because, for broader spectral regions, the AOM does not have sufficient bandwidth to apply a Bragg mask, dispersion compensation, and still generate a max double pulse delay based on the apparatus&#;s frequency resolution.

 

Fig. 5. Cross correlation of compressed pulses. (a-c) PG-FROG traces, (d-f) their corresponding time marginals, (g-i) and the (blue) amount of AOM bandwidth used to compress each pulse as well as the (red) AOM bandwidth utilized to achieve the max double pulse time delay based on the number of pixels (g-i). The arrows on each of the FROG traces correspond FWHMs of the traces along the wavelength axis and represent the bandwidth of the compressed light. The time marginals represent the 3rd order autocorrelation of the gate pulse with the white-light and are used to convolute the measurement to obtain the duration of the shaped pulse.

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We chose PG-FROG because autocorrelation techniques are difficult with white-light pulse energies (&#;500 pJ) [31&#;33]. PG-FROG, on the other hand, is a cross-correlation technique that utilizes a strong gate pulse to induce a time dependent birefringence in a X3 medium (Kerr effect). This phenomenon rotates the polarization of white-light and allows us to characterize our pulse with low white-light pulse energy. We characterized our gate using a 2-photon autocorrelation and measured a FWHM of 195 +/- 2 fs, where the uncertainty is calculated using the standard error of the mean of the FWHM.

Given our limited temporal resolution with PG-FROG, our goal was to compress our white-light pulses to sub 50 fs while remaining as close as possible to the Bragg mask to maintain the benefits of low loss [17]. We were able to compress the regions of light spanning 840&#;&#;nm and 970&#;&#;nm of light below 50 fs without attenuating any given wavelength more than 3&#;db. To determine the bandwidth of our compressed light, we calculated the frequency marginal of the PG-FROG scans integrating over the time axis [34]. From these calculations we find that the shaped light spanning 840&#;&#;nm and 970&#;&#;nm have FWHM of 115&#;nm and 130&#;nm, respectively. These white-light bandwidths are represented by the red arrows in Fig. 5(b)-(c).

To determine the white-light pulse duration, we calculate the time marginal of the PG-FROG scan by integrating over the wavelength axis, fitting to a Gaussian, and deconvoluting using the gate pulse (Fig. 5(e)-(f)). We also used error propagation based on the standard error of the mean of the gate pulse and PG-FROG time marginal to determine the uncertainty of our results [34]. After deconvolution, we found that the pulse durations shaped light spanning 840&#;&#;nm, and 970&#;&#;nm are 39 +/- 10 and 37&#;+&#;/- 7 fs, respectively.

We were unable to compress the 520&#;660&#;nm pulse with the pulse shaper configuration described above, so for this wavelength range we decreased the material dispersion in the system by removing the focusing lenses and adjusting the position of the white-light collimation optic (Fig. 3(a)) such that the light focused at the AOM aperture. Doing so degraded the shaper frequency resolution (85 pixels and 1.7&#;nm resolution), but less dispersion compensation enabled sub-50 fs pulse compression and double pulses up to a delay of 460 fs. Removing he lenses did not improve the results for the other two regions of light because it disproportionately impacted the frequency resolution relative to dispersion compensation (simulations predict 43 and 29 pixels for 840&#; and 970&#;&#;nm light, respectively).

With the new configuration for the 520&#;660&#;nm light, we were able to compress this region of light below 50 fs without attenuating any given wavelength more than 3&#;db. The cross correlation in Fig. 5(a) has a FWHM of 55&#;nm and a pulse duration of 35 +/- 7 fs. We also note that because the prisms are much further spaced for the 520&#;660&#;nm light versus the two near-infrared regions (150&#;cm vs. 50&#;cm) that the same deviation away from the Bragg mask creates &#;3-times the spatial chirp. Since each region of light required a different amount of dispersion correction to compress the pulse below 50 fs, pulse compression required a different amount of the AOM bandwidth for each wavelength (Fig. 5(h)-(i)). We also plot the amount of AOM bandwidth utilized that creates the largest delay possible for a transform limited pulse pair, which is 460 fs, 430 fs, and 475 fs, for the three wavelength ranges, respectively.

2D Spectroscopy with pulse shaping

We also demonstrate the prism compressor pulse shaper&#;s ability to measure multidimensional spectra of a thin film of 6,5 carbon nanotubes in Fig. 6. Semiconducting carbon nanotubes are strong absorbers with narrow line shapes making them a good standard to test a new apparatus. The spectrum shown in Fig. 6 was measured in the time domain using an InGaAs photodiode with a 4-frame phase cycle where the pulse shaper was used to generate the pump-pulse pairs.

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Fig. 6. 6,5 carbon nanotube 2D white-light spectra measured in the time domain with a 4-frame phase cycle. The spectrum was measured at t2&#;=&#;200 fs and exhibits a strong inhomogeneous lineshape and a characteristic excited state absorption

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The ground state bleach appears at &#;nm, which is expected for 6,5 carbon nanotubes, and there is also an excited state absorption peak which appears at a redder frequency representing the S11 to S22 transition, as expected from prior publications that did not use pulse shaping [35]. The measured homogenous linewidth is &#;16&#;nm while photoluminescence experiments measure a 14&#;nm homogenous linewidth [36]. Thus, the resolution of this shaper design is sufficient to measure the electronic transitions of many molecules and materials.

5. Conclusion

With the implementation of the prism compressor pulse shaper, we have demonstrated improved efficiency over grating-based pulse shapers and the ability to compress the shaped light below 50 fs with no additional pulse compression optics. The improvement in efficiency will be useful in spectrometers that have low pulse energies, such as those utilizing white-light. Furthermore, the prism compressor compensates for dispersion and can create sub 50 fs pulse pairs with delays up to 430-475 fs, corresponding to a 2.5&#;nm frequency resolution at 840&#;nm when doing Fourier transform spectroscopy. While grating pulse shapers are able to obtain higher resolutions because the grating dispersion is independent of pulse compression, they are unable to achieve the same level of dispersion compensation without additional optics.

Another variation in the design reported here would be to use a quartz AOM instead of a TeO2 AOM thereby permitting pulse shaping below 370&#;nm. Quartz has much smaller GVD and TOD coefficients than TeO2 and so a larger fraction of AOM bandwidth would remain for pulse compression and the prisms would require a smaller separation. The diffraction efficiency will be about 35%, as compared to 40% for TeO2, and will require waveplates before and after the AOM to rotate the pulse polarization [37,38]. Regardless, the design presented here capitalizes on the spatial dispersion inherent to a prism compressor and, as we have shown, has sufficient resolution to measure linewidths of typical molecules and materials with 3-times the transmission efficiency of grating-based pulse shapers.

Funding

National Science Foundation (CHE-, DGE-).

Acknowledgements

This work was supported by the National Science Foundation via the Center for Adapting Flaws in Features, and the NSF Center for Chemical Innovation supported by grant CHE-. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-.

Disclosures

I: Martin T. Zanni is a co-owner of PhaseTech Spectroscopy, which sells ultrafast pulse shapers and multidimensional spectrometers.

P: The described design for the prism compressor pulse shaper is in the patent application process.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Therefore, in this paper, focusing on the demand of mass production and engineering prototype, facing these challenges in current technologies, we design a new lens module for this kind of grating rulers, a single prism called a quadrangular frustum pyramid, enabling simultaneously deflecting the four diffraction beams without any power loss and easy alignment. To test the lens module, a grating ruler system is designed, constructed and experimental verification are introduced in this manuscript.

As illustrated in these prototype MDOF grating rulers, grating diffraction beams interferometry is the key technology, which means that the interference is generated by using four beams of ±1 order diffracted light of a two-dimensional grating so that to obtain the displacement distance by detecting the interference signal [ 6 8 ]. Compared with the one-dimensional interferometric grating ruler, the two-dimensional grating ruler is equivalent to integrating two sets of one-dimensional grating rulers whose working direction is orthogonal to each other, however this makes complexity of the two-dimensional grating ruler light path increase greatly. The complexity is mainly manifested in the need of the interference of four beams both from the reference grating and the scale grating parallel or perpendicular to the incident light in the system in order to produce an ideal interference signal. This means that the above eight diffracted beams need to be finely deflected, which determines the quality of the interference signals. In previous research, there are three proposed methods, collimation lens represented by Reference [ 6 ], separate prisms represented by Reference [ 5 ], and a home fabricated diffraction device by combining four separate one-axis line gratings in a glass substrate [ 13 ]. These methods meet the demand of the light path design and can be manufactured under the lab conditions. However, due to the complex structure and alignment, loss of signal intensity and the low sensitivity, these solutions are difficult to meet the condition of mass production.

In order to eliminate the abbe error in multi-axis measurement and improve the integration of the measurement system, measurement devices with MDOF measurement capabilities have been paid more attention. Among them, the technique schemes of two-dimensional grating rulers and the MDOF grating rulers are of great application potential [ 6 9 ]. These MDOF grating rulers, generally with usage of a two-axis planar scale grating, are able to measure not only the translational motions but the small order rotational error motions, and it can be also applied to six-DOF measurement. Compared with the traditional single-axis measuring devices, the MDOF grating ruler has a higher level integration and the abbe error could be avoided Thus, the reliability and precision of the multi-axis precision stage and measurement system can be improved effectively [ 10 12 ].

With the development of industrial automation and intelligence, the popularization and demand of multi-axis computer numerical control (CNC) equipment in industrial production are increasing. Therefore, the multi-degree-of-freedom (MDOF) measurement technology is a key basic technology. For instance, in a MDOF CNC machine tool, the current common measurement scheme is to equip each motion axis with a separate single-axis measuring device, including linear grating ruler, linear photoelectric encoder, and so forth [ 1 3 ]. The installation of these measuring devices is usually stacked along the motion axis. This brings about the abbe error problem in the assembly process, which will have an adverse effect on the measurement accuracy [ 4 5 ].

The design and calculation of parameters are described in Figure 2 b. The grating diffraction angle is denoted as. The refractive angle of diffraction light is denoted as. The reflection angle of diffraction light is denoted as. The angle between the normal side of the prism and the top plane is denoted as. The design requires that the diffraction light emitted is parallel to the incident light, then α andshould satisfy Equation (1).

In order to overcome the above problems, an integrated grating diffractive light deflection prism design is proposed. The design takes the advantages of prism deflection that does not reduce the beam intensity and transmission grating high integration, providing a high precision, high brightness and easy to assemble and debug solution for reference grating and scale grating diffractive light deflection. The effect of the integrated prism and the deflection of four ±1 order diffractive beams is shown in Figure 2 . The key parameters in the design are the integration of the prism angle, performance parameters for the diffraction light horizontal/vertical emergent spacing, and variable parameter to the top of the prism and grating spacing and the beam radius.

As illustrated in previous research [ 5 13 ], further phase delay and amplitude division of these signals are required so that the direct current components of the interference signals can be eliminated and the motion direction can be distinguished. Thus, more optics are added into this optical layout and this results the light path increasing to more than 100 mm [ 14 ]. This makes paralleling these diffraction beams necessary. As mentioned in the first section of this paper, Figure 1 b&#;d illustrate these three present optical designs. As shown in Figure 1 b, a collimation lens was employed. The diffraction beams were refracted. This is suitable for large grating period, generally a 10 micrometer order. Under such a condition, the diffraction angle from gratings are relatively small, and the collimation effect can be ensured [ 6 ]. However, a large grating period will result in a relatively poor resolution. A high resolution requires a small grating period. Correspondingly a large diffraction angle makes using the collimation difficult. The second trial is that of the triangular prism refraction method, as shown in Figure 1 c. Similarly, a triangular prism can be replaced with a mirror. Both methods need the diffractive light deflection mirror to be precisely calibrated so that the reference and the scale grating diffractive light will be able to interfere after transmission to the detector. It was found in the trial production of the actual test prototype [ 5 12 ], the process of assembly and adjusting the diffractive light deflection prism is extremely complicated and difficult for the mechanical assembly stress and the influence of such factors as poor precision stability of the optical path. In order to avoid the problem that diffractive light deflection prisms need to be adjusted separately, a structure using transmission grating for diffraction light deflection is proposed, as shown in Figure 1 d [ 13 16 ]. In this structure, the transmission grating is actually four one-dimensional transmission gratings combined. This scheme greatly reduces the difficulty of assembling and debugging optical circuits by employing the integrating transmission grating, which is used as a diffractive light deflection device. However, limiting the transmission grating diffraction efficiency, the plan will greatly reduce the intensity of the diffractive light. And it is not conducive to improve the signal-to-noise ratio (SNR) in subsequent signal processing, which leads to reduce the measuring accuracy of the grating ruler system.

3. Results

Based on the above geometric optical design, a batch of proposed QFP prisms were fabricated by a lens manufacturing company (Union Optic Incorporated, Wuhan, China) and the parameters were measured and given as shown in Figure 3 a. It should be noted that a cuboid base was added onto the bottom of the QFP prism so that the prism can be mounted onto the plate base of the grating ruler, which also benefits the prism manufacture process.

In order to test the effectiveness of the QFP prism, a fundamental grating ruler optical path using interferometric measuring principle is constructed, as shown in Figure 3 b,c. This optical layout is the same as the optical path principle shown in Figure 1 c. The incident beam with a calibrated wavelength of 660 nm was divided into two beams and are projected to the reference grating and the scale grating after passing through the QFP prisms, respectively. The scale grating is mounted onto a moving table and the reference grating is stable.

First of all, performance of the QFP prism in diffraction beams propagation directions modulation was evaluated by observing the beam spots on the screen. The observation as shown in Figure 3 c was placed at a distance about 150 mm, a similar distance with that of a real grating ruler. Figure 4 a illustrated the beam spots on the screen. It can be seen that the four diffraction beams from the reference grating coincided well with those from the scale grating. The distance between these four diffraction beams from the central beam (reflective beam, also called zero-order diffraction beam) are consistent with the designed values. These results verified the geometry accuracy of the fabrication process. It should be noted that there are many undemand spots on the screen which are caused by the non-orthogonal diffraction beams from the grating and can be easily blocked by setting a physical holes aperture.

In order to further verify the effectiveness of the proposal in terms of optical property, such as polarization, the interference signal was tested. As shown in Figure 4 b, clear diffraction stripe structure can be observed in all four interference spots and these represent that they have almost same order of interference intensity, while in previous separated prism layouts this was always the most challenging task.

The results can preliminarily prove that the QFP prisms based interferometric grating ruler optical configuration is feasible and effective. To further test the optical path, the scale grating was driven by the micron motion platform (PI-M112.1 DG, Physik Instrumente (PI) GmbH & Co. KG, Karlsruhe, Germany), as shown in Figure 3 c.

I

X+1 and

I

X&#;1, were recorded by a photodiode. The interference signals are given in

As the principle of grating interferometry, the movement of the scale grating brings a periodic change of the interference signal. In this demonstration, two interference signals,and, were recorded by a photodiode. The interference signals are given in Figure 5 . It can be seen that the visibility of the interference signal almost remains stable, which is essential for the grating ruler application. The direct current components in the interference signal can be eliminated optically. The results show that the given QFP prisms based optical layout can form a good interference signal, the signal is in good sinusoidal shape under constant speed drive and can be used for bit shifting calculation.

+1 is slight less than X&#;1, because the rear surface of the QFP prism is not fully parallel to the grating surface, which is inevitable and will have no or little influence on the result after calibrating the system and further processing the subdivision signals.

The motion velocity along the direction X and direction Y of the measured grating was 0.025 mm/s, the motion stroke was 7 mm, and the sampling rate was Hz. A total of four repeated experiments were conducted. The calculation results of bit shifting of each test are shown in Table 1 . As can be seen from the data in the table, the repeatability of the bit transfer calculation is good, and the relative error is no more than 2.5%. The measuring error can be reduced by calibrating the measure system for the machining error in fabricating the grating and the QFP prism and assembly error. In each group, the result of Xis slight less than X, because the rear surface of the QFP prism is not fully parallel to the grating surface, which is inevitable and will have no or little influence on the result after calibrating the system and further processing the subdivision signals.

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