1. Introduction

It has been more than a half century [1] since diffraction-grating interferometers have been characterized and employed in various applications, including the analysis of gas/fluid flow dynamics [2,3], testing the quality of optical surfaces [1,4], and studying celestial bodies [5]. With the invention of lasers the range of applications has been greatly expanded. Diffraction grating interferometers are now routinely employed in nanometrology [6,7], various ultrafast spectroscopy methods [8–10], and have been suggested as parts of improved gravitational wave detectors [11].

A traditional Michelson interferometer, where a (coated) glass plate plays the role of a beamsplitter, suffers from phase fluctuations in the recorded interferograms due to the vibrations of the beamsplitter that affect the two arms of the interferometer differently. In contrast, diffraction-grating-based interferometers circumvent this drawback as any vibration of the diffraction grating would affect both arms of the interferometers in the same way (i.e. passive phase stabilization). While interferometers based on reflective diffraction gratings offer significant advantages for high-laser-power applications [11], transmission diffraction gratings may be more advantageous in spectroscopy as they can provide high diffraction efficiency over a large spectral range. Previously, a Michelson-type transmission-grating interferometer for the characterisation of XUV attosecond pulses has been reported [8], where the diffraction grating acted as a 50/50 asymmetric beamsplitter, dividing the beam into and recombining back the $0^{th}$ and $1^{st}$ diffraction orders. This lack of symmetry between the active orders makes it challenging to ensure identical spatio-temporal characteristics of the output beams. In addition, the interferometer design described in [8] results in an output beam with angularly diverging spectral components, which can be problematic in some experimental situations. Moreover, the reported efficiency of the interferometer was very low ($\sim$2.5 %) as the grating was not specifically optimized for high diffraction efficiency in the $0^{th}$ and $1^{st}$ orders. In transient grating and two-dimensional spectroscopy experiments, transmission diffraction gratings have been used previously to generate symmetric, phase-locked pulse pairs by selecting the $-1^{st}$ and $+1^{st}$ diffraction order [12–15]. Following this line of development, here we present an implementation of a Michelson interferometer which uses a transmission grating as a beamsplitter and is both fully symmetric and has a throughput efficiency as high as $\sim$25 % owing to the diffraction grating specifically designed to yield an efficient generation of -1$^{st}$ and +1$^{st}$ orders of diffraction with equally distributed intensities between the two. In addition, the presented interferometer is shown to have negligible spatio-temporal dispersion and to be very robust (with phase fluctuations being as low as $\sim \lambda$/400). The interferometer is also based on a simple design and we therefore expect that it will find use in a broad range of applications.

2. Experimental setup

A schematic representation of our transmission-grating Michelson interferometer is shown in Fig. 1(a). A collimated near-infrared (NIR) input pulse ($\sim$750 nm to $\sim 850\,\mathrm {nm}$ center wavelength, $\sim 100\,\mathrm {nm}$ to $\sim 200\,\mathrm {nm}$ bandwith (FWHM) and down to $\sim$10 fs pulse duration) from a tunable lab-built non-collinear optical parametric amplifier (NOPA) is steered towards a spherical mirror (focal length $F'$=25 cm). This mirror focuses the beam to a $\sim$120 $\mu$m (1/e$^{2}$) spot (with a corresponding Rayleigh range of $\sim$1.5 cm) on a 1-mm-thick transmission grating (Holoeye Photonics AG). The grating is a rectangular plate (15$\times$15 mm$^{2}$) of surface relief fused silica subsequently processed by lithography and etching with vertically oriented grooves with a period of 45.8 $\mu$m (corresponding to a groove density of $\sim$21.8 mm$^{-1}$). The grating is optimized for maximum diffraction efficiency in the -1$^{st}$ and +1$^{st}$ order at the wavelength of 800 nm with the intensity distributed equally between these diffraction orders, and with an angular spread of $\sim$2.0 degrees between the diffracted beams. The small separation of the beams makes the setup less susceptible to fluctuations of optical path difference between the two arms induced by air currents. In addition, ray tracing simulations show that a small angular spread of the spectral components in each diffracted beam reduces the dispersion introduced in the variable interferometer arm for nonzero delays. The transmission grating provides high diffraction efficiency over a broad wavelength range, with a maximum diffraction efficiency of $\sim 38\,\%$ and with only small variations in efficiency over more than $200\,\mathrm {nm}$ in the NIR spectral range ($\sim 4\,\%$), as shown in Fig. 1(b). The broad spectral response of the grating makes this interferometer design highly suitable for applications relying on broadband laser pulses.

 

Fig. 1. (a) Schematic of the transmission-grating Michelson interferometer (top view). (b) Normalized spectrum of input beam and diffracted beam ($1^{st}$ order) and corresponding diffraction efficiency.

Download Full Size | PDF

After the diffraction grating, three diffracted beams (0$^{th}$ and $\pm 1^{st}$ orders of diffraction) are allowed to pass through an aperture with two of them ($\pm 1^{st}$ orders of diffraction) hitting a spherical mirror ($F$=7.5 cm) each, while the $0^{th}$ order beam passes between the mirrors and hits a beam block, as shown in Fig. 1(a). The mirrors are placed at a distance of $2F=15$ cm from the grating to re-image the $\pm 1^{st}$ diffraction order back onto the grating. It is the quality of re-imaging of the focused beam spot on the grating back to itself that partly contributes to dispersion characteristics of the interferometer and will be discussed later. Each of the recombined beams results in its own set of diffraction orders, all of which are blocked by an aperture, except the -1$^{st}$ order of the prior +1$^{st}$ order and the +1$^{st}$ order of the prior -1$^{st}$ order, which co-propagate collinearly as a single beam. The two spherical mirrors in the two arms of the interferometer reflect the beams slightly upwards with an angle of $\sim 1^{\circ }$, so that the recombined beam misses the input mirror and continues propagating just above it. One of the spherical mirrors (in the upper arm of the interferometer as shown in Fig. 1(a)) is placed on a high-precision piezoelectric linear translation stage (P-622.1CD, PI Motion), which allows to introduce a variable delay ($250\,\mathrm {\mu m}$ travel range corresponding to a delay range of about $\pm 0.83\,\mathrm {ps}$) between the two pulses in the output beam. For delays different from 0 fs, the condition of perfect re-imaging of the grating onto itself does not hold anymore, and therefore related pulse distortion effects have to be considered.

3. Results and discussion

3.1 Ray tracing

At zero delay between the pulses, the laser pulse propagating through the variable-length arm of the interferometer is perfectly re-imaged onto the diffraction grating, and the setup is fully symmetric and dispersion-free. As soon as delays are introduced by extending or contracting the interferometer arm, however, the re-imaging mirror is no longer at a distance of exactly 2$F$ from the grating, leading to both temporal and spatial distortion of the delayed pulse. We note that the overall travel range of the piezoelectric stage ($250\,\mathrm {\mu m}$) is $\sim$60 times smaller than the Rayleigh range ($\sim 1.5\,\mathrm {cm}$) of the beams, ensuring that the re-focused beams remain well overlapped at the grating even when one of them is delayed by the maximum possible delay of $\pm$0.83 ps. Nonetheless, deviations from the ideal grating-to-mirror distance can introduce pulse distortions. To estimate the magnitude of these effects, we performed geometric ray tracing simulations [16] of our setup for different interferometer delays and evaluated temporal and spatial dispersion of the delayed pulse. In our simulations we considered the planar geometry, where all rays propagate parallel to the optical table. The spherical mirrors were considered ideal and therefore dispersionless, so that any dispersion introduced originates from the geometry of the setup and the angular dispersion of the grating.

First, we focus on the temporal distortions and consider the group delay ($\partial \varphi /\partial \omega ,$where $\varphi$ is the accumulated optical phase and $\omega$ is the optical frequency) introduced by the setup at delays of -1 ps, 0 ps and 1 ps for wavelengths between 700 nm and 900 nm, shown in Fig. 2. No variation of group delay with wavelength can be seen at zero delay, i.e. no temporal dispersion is introduced. This is to be expected since the interferometer arm is by design dispersion free at zero delay. At $\pm$1 ps delay, the extension (contraction) of the interferometer arm leads to variations in optical path length and temporal dispersion because the dispersed spectral components are no longer perfectly refocused onto the grating. However, the resulting difference in group delay between spectral components at 700 nm and 900 nm is only $\sim$80 as, showing that temporal distortions of the pulse are negligible for introduced delays of up to $\pm$1 ps, which encompasses the delay range of the piezo-stage used here. To quantify the spatial chirp introduced by the interferometer, we evaluate the spatial dispersion $dx_{0}/d\lambda$ (variation of centre position $x_{0}$ with wavelength $\lambda$) as a function of propagation distance after the interferometer for a delay of 1 ps. The spatial dispersion increases linearly with distance due to imperfect re-collimation of the spectral components. After 5 m of propagation, we determine a spatial dispersion of $\sim$0.1 $\mu$m/nm, resulting in a separation of $\sim$20 $\mu$m between the centre positions of the 700 nm and 900 nm spectral components. In many experiments, the distance between an interferometer and a sample is considerably less than 5 m, and the spatial dispersion is therefore negligible. Finally, we note that in princinple additional beam distortions may be introduced due to the upward angling of the beams in the two arms of the interferometer, such as comatic aberration or temporal and spatial dispersion due to differences in path length between ingoing and outgoing beam. However, for small angles ($\sim 1^{\circ }$ in our setup) these effects are insignificant for most applications, and path length differences can be readily eliminated by a symmetric beam alignment (i.e. an equal upward angle for ingoing and outgoing beam). In addition, the upward angling is perpendicular to the grating’s plane of diffraction, i.e. the direction of the diffracted beams is not affected.

 

Fig. 2. Temporal dispersion introduced by the interferometer at different pulse delays. The group delay for wavelengths between 700 nm and 900 nm is obtained from ray tracing simulations considering the pulse propagating through the variable arm of the interferometer.

Download Full Size | PDF

3.2 Phase stability

One of the requirements of any interferometric measurement is the low level of relative phase fluctuations $\Delta \phi$ between the two arms of an interferometer. These fluctuations can be caused by the relative variation of the optical path length or dispersion (e.g. caused by the air currents) in both arms. They can have both a rapidly varying component, originating from air currents, and a slowly varying component, originating, for example, from relaxation of the translation stage or instabilities in the optical setup. Therefore, we carried out long-term (Figs. 3(a) and (b)) and short-term (Figs. 3(c) and (d)) measurements of phase fluctuations in the setup presented above.

 

Fig. 3. (a,b) Long-term and (c,d) short-term phase fluctuations with the piezoelectric stage in (a,c) closed-loop and (b,d) switched off. (e) Second-order interferometric autocorrelation trace (blue) and the corresponding intensity autocorrelation (orange) with FWHM of $\sim$16.5 fs corresponding to $\sim$12 fs pulse duration (FWHM).

Download Full Size | PDF

Figure 3(a) shows phase fluctuations with the piezo-electric stage locked in position using closed-loop operation. The measurements were carried out over an interval of $\sim$60 min, using input pulses at 760 nm center wavelength ($\sim$100 nm FWHM bandwidth and $\sim$12 fs pulse duration). The phase’s standard deviation (STD) in this case was estimated to be as low as $\sim \lambda$/170, which corresponds to a timing error $\Delta t$ of $\sim$15 as at 760 nm wavelength. These measurements, however, are affected by the efficiency and accuracy of the closed-loop control of the piezo-stage controller and the properties of the piezo-stage itself. Therefore, we carried out the measurements of phase fluctuations when the translation stage was switched off and allowed to relax from residual strain for $\sim$12 h. The result of these measurements is shown in Fig. 3(b). In this passive regime the phase fluctuations are reduced with a STD of $\sim \lambda$/240 which corresponds to time fluctuations $\Delta t\sim$11 as at 760 nm. This high stability of the interferometer is achieved due to the use of transmission grating as a beamsplitter, the compact design, and by having the beams in both arms of the interferometer propagating close to each other, thus experiencing similar air currents. Additionally, the setup is shielded by a box enclosure. In both measurements, the sampling interval was $\sim$2 sec and interferograms were recorded with an integration time of 300 ms.

Long-term measurements of phase fluctuations described above do not capture the effects of rapid variations of air currents and vibrations of optical and optomechanical components, which affect the relative pathlengths in the two arms of the interferometer. To test for the quick variations, we increased the sampling frequency of the measurements up to $\sim$100 Hz. The results of these fast measurements are shown in Figs. 3(c) and (d) for the cases when piezo-stage was actively locked (Fig. 3(c)) and when it was completely switched off (Fig. 3(d)). The measurement interval was $\sim$10 ms, and integration time was set to 2 ms. The STD-value of phase fluctuations was estimated to be $\sim \lambda$/370 ($\Delta t\sim$7 as at 760 nm) in the case of actively locked translation stage, and $\sim \lambda$/410 ($\Delta t\sim$6 as at 760 nm) in the case of a switched off and fully relaxed stage, once again confirming a high phase stability of the presented interferometer. This is not surprising as our interferometer is expected to be insensitive to vibrations of the beamsplitter (transmission grating), in contrast to plate-beamsplitter-based interferometers.

Time errors of the order of $\sim$10 as achieved here fulfill well the requirements posed on Fourier-transform-based ultrafast spectroscopy experiments, in which phase stability of at least $\sim \lambda$/100 fs is desired in order to obtain meaningful spectra and clean time traces [17]. Indeed, in fully non-collinear two-dimensional spectroscopy setups transmission grating beamsplitter and beam co-propagation through the same optical elements have been successfully employed for creating phase-locked pairs of pulses [14,15].

In order to confirm the absence of substantial noise in the time-resolved measurements using the transmission-grating interferometer under consideration, we have retrieved an interferometric autocorrelation (IAC) trace (Fig. 3(e)) measured using $\sim$12 fs pulses focused onto a beta barium borate (BBO) crystal. Clear and prominent interferometric fringes can be easily observed with overall shape having a bell-like envelope as expected for the case of close to Fourier-transform limited ultrashort pulses. The ratio between the background (the level of IAC signal far from zero delay) and the peak intensity is close to $\sim$1:8, as expected for an IAC trace [18], confirming that pulse distortions are minor. The deviation from the perfect 1:8 ratio could be due to possible leakage of IR radiation through the blue filter in front of the detector. We also note that time fluctuations could be introduced while scanning piezoelectric stage. However, judging from the retrieved IAC trace, we estimate that these fluctuations in our experiments do not contribute significantly.

4. Conclusion

In conclusion, we have presented an easy to build transmission grating-based fully symmetric Michelson interferometer, which is essentially dispersion-free, has a comparatively high throughput, and exhibits a high level of phase stability. The device is promising to be useful for applications in ultrafast spectroscopy, astronomy, and metrology. A nearly dispersionless operation of the device is especially important in the applications involving X-Ray and attosecond UV pulses, whereas having a high phase stability is significant in Fourier-transform-based ultrafast spectroscopy techniques.