1. Introduction

Interferometer-type spectrographs use the only means to resolve the electric field at optical frequencies, interference. Interferometric spectrometers have the highest demonstrated resolving power to date. Besides the well known Fourier and Fabry-Perot Spectrometers, recently - with the advent of high-resolution CCD and CMOS cameras - spectrometers have been developed that use interference in the spatial rather than the time domain. The spectrum is obtained by a Fourier transform as well, from the spatial frequency to wavelength rather than from time to frequency.

A modern version is the “Spatial Heterodyne Spectrometer” (SHS), where a reflective diffraction grating under Littrow angle causes the necessary wavefront tilt for close off-Littrow wavelengths [1,2]. For an SHS instrument, two of these gratings are incorporated into a Michelson interferometer. These spectrometers can be built compact and without moving parts, however, they generally require optical elements of high quality. Other realizations of Fizeau-fringe type spectrometers utilized Wollaston prisms [3, 4] or birefringent liquid crystals [5] to split the incoming wave front into the two that are necessary for interference. The main goal of these research efforts is to eliminate the use of moving parts.

In this paper, we will introduce a spectrometer based on a Sagnac-type interferometer, where one of the mirrors is replaced by a transmission grating. A similar spectrometer benefiting from the inherent stability of common-path interferometers was suggested with two reflection gratings [6]. Another family of devices are multi-/hyper-spectral imagers. They show a very similar structure but serve a different purpose, namely acquiring images in multiple spectral passbands rather than collecting complete spectra [7]. As will be shown here, the inclusion of a transmission grating improves stability and tolerance against misalignment. The weak dependence of the deflection angle on the angle of incidence onto the grating opens the path to measure high-resolution spectra with budget-priced mechanical components. It also enables a calibration procedure based on the radiation under test, i.e. without the need for an auxiliary calibration lamp.

2. Principle

The setup of the Sagnac Fourier Spectrometer (SAFOS) is shown in Fig. 1. A common Sagnac interferometer [8] is modified by inserting a transmission grating into the path. Since the action of a transmission grating is reversible (sin α + sin β = const.), both directions experience the same diffraction under the same angles at a given wavelength λ₀, where the setup works like a standard Sagnac interferometer. For a single wavelength λ₀ (with zero bandwidth) and the device at rest, there is a homogeneous dark field in the output arm. For wavelengths off this center wavelength however, the propagation directions (k-vectors) of the diffracted beams are making a small angle to the optical axis. The resulting tilted wavefronts lead to Fizeau fringes at the output. An odd number of mirrors is necessary to provide the correct number of wavefront inversions that causes the two wavefronts in the output arm to be oppositely tilted.

 

Fig. 1 Principal scheme of the Sagnac Fourier Spectrometer (SAFOS). The green and red lines show the wavefront tilt of the respective directions for a wavelength that is smaller than the design wavelength.

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In the following, we consider operation in the grating’s first diffraction order only. The sign convention for transmission gratings is that angles on the same side of the grating normal have the same sign on both sides of the grating. Hence, in Fig. 1, both angles α and β are positive.

For the wavelength λ₀ + Δλ, the spacing Δx of the fringes is:

Due to the tilt of k-vector into and out of the grating, the energy front experiences a tilt that is different from the tilt of the wavefront [9]. The tilt angle of the energy front γ in the output arm can be calculated according to [10] as tan γ = λ₀ d∊/dλ, where d∊/dλ is the angular dispersion, which in our case is dΔβ/dΔλ, leading to:

We now rotate the grating in an otherwise fixed configuration according to Fig. 1. By design of the spectrometer, the deflection angle γ = π − (α + β) is fixed, defining the optical axis of the spectrometer. The grating equation is sin α + sin β = λ g. When the grating is rotated, the angle of incidence α changes. Since α + β = const. does in general not imply that sin α + sin β = const., the grating equation is fulfilled for a slightly different wavelength after the rotation. This wavelength is given by:

As an example, Fig. 2(a) shows this deviation for a spectrometer configured with g = 2847 mm⁻¹ and several design angles γ. By differentiating Eq. (3) with respect to α, one finds that the maximum of the curve is at α = β = (π − γ)/2. Consequently, the curve has a maximum at λ_D = 2 cos(γ/2)/g, we call this the design wavelength of the spectrometer. Fig. 2(b) shows the same curve for a grating constant of g = 1200 mm⁻¹.

 

Fig. 2 Tuning curve of the SAFOS, showing the wavelength that propagates on the optical axis (i.e. the heterodyne wavelength) in dependence on the angle of incidence α on the grating. Curves are shown for different design angles γ, which is constant for a given configuration. The dotted line connects the maxima of the curves, where α = β.

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Note that the curvatures of the tuning curves are different for different design angles γ. Since the equations for the Littrow angle and the grating equation for α = β are identical, the dotted lines in Fig. 2 are the same as the tuning curve for a SHS, when the two reflections gratings are rotated by the exact same amount.

3. Two modes of operation - theory

The fact that the wavelength range under investigation must not overlap with the wavelength propagating along the optical axis dictates the operational conditions of the SAFOS. One could operate the device in the same way the SHS is usually operated [2]: the grating is locked in position under a fixed angle, covering a wavelength range to either side of the design wavelength. This configuration, for α = β, is shown in Fig. 3(a), the hatched regions denominating the principal operation range of the spectrometer (not to scale).

 

Fig. 3 Two modes of operating the SAFOS.

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The inner limits of these regions (i.e. the lower limit of A and the upper limit of B) are dictated by the mathematical processing. If the difference to the design wavelength is too small, the wavelength to be measured overlaps with the zero-frequency spike of the Fourier transform and cannot be sensibly extracted anymore. The outer limits represent the usable bandwidth of the SAFOS. The further away a spectral component is from the design wavelength, the larger the spatial frequency of the Fizeau fringe pattern that it generates as it is evident from Eq. (1). Hence, the bandwidth is limited by the spatial resolution of the detector arrangement. According to the Nyquist limit, at least two pixels are necessary to resolve one fringe. With a pixel size of xp, the largest |Δλ| that can be resolved is determined by the following equation:

One also has to guarantee that higher diffraction orders are not propagating along the optical axis. Since the next wavelength fulfilling this condition is half the wavelength under test, this can be easily achieved with edge filters.

For any fixed angle, the standard Fourier transform as known from the SHS processing, involving the scaling parameter

Using the weak dependence of the center wavelength on the grating angle, one can tune the spectrometer, opening two ranges of usability, as shown in Fig. 3(b). The range B however, is located further away from the design wavelength. In this configuration, the ambiguity concerning the sign of Δλ can be resolved. By scanning the angle of incidence, the spectrum will move one way or the other, the direction of which tells the sign. As noted, the curvature of the tuning curve depends on the design wavelength. This facilitates the determination of the absolute position of this curve with no prior knowledge of the input spectrum.

4. Two modes of operation - experiment

An immediate experimental verification of the curve shown in Fig. 2 would require a tunable, narrow-band light source, which had to be tuned for each grating angle until a uniform dark field in the output would be detected. Because such a light source was not available, we modified this experiment, using a blue laser pointer with a nominal wavelength of 405 nm. As determined by measurement with an Echelle spectrograph (DEMON, Lasertechnik Berlin GmbH), the output of this laser consists of several narrow lines (FWHM < 6 pm), equally spaced by 72 pm, with the strongest line at 404.000 nm. The laser pointer was operated with an external power supply; care had to be taken to precisely control the bias voltage V_B, since the absolute position and the number of lines (not the spacing) depends on the bias voltage. The number of lines within the FWHM of the envelope varied between 30 (V_B = 1.5 V) and 3 (V_B = 3.0 V).

In the experimental setup (Fig. 1), we used a commercially available transmission grating of size 51 × 51 mm² with a groove density of 1200/mm (Edmund Optics), superpolished 100-mm square mirrors with a UV-enhanced aluminum coating and a custom-made beam splitter (Layertec GmbH, Germany) with a reflectivity of (50 ± 8)% for unpolarized light over the wavelength range 230–530 nm. The resulting clear aperture of the device was ≈ 25 mm. Using a 150-mm focal length UV fused silica lens or a multi-lens camera objective, the grating surface was imaged onto a full-frame 16-MPixel CCD camera (Thorlabs) while magnifying the image by a factor of 1.6 to fit the 36-mm wide CCD chip. The incoming light was collimated by using a fiber with a N.A. of 0.22 and an off-axis parabolic mirror (100 mm, 90°, UV-enhanced aluminum).

We obtained interferograms by scanning a large range of input angles α and calculated spectra by way of Fourier Transform after “cleaning” the interferogram using some of the techniques described in [11]. These were flat-fielding (correcting for pixel sensitivity fluctuations of the used camera), removing a sloped DC background, and normalization to spatial variations in the illumination (by taking a background image with a screen in place of the grating). Although apodization is widely used in processing this kind of interferogram, we decided against it, since it reduces the resolving power without yielding any substantive benefit for our application. The Fourier transform was computed with a fixed design wavelength λ_D. Actually, this procedure yields a ’wrong’ wavelength scale for the investigated peak. However, the Fourier transformed spectrum is heterodyned around the design wavelength, i.e. it only depends on (λ₀ − λ_p), where λ_p is the wavelength of the emission peak. Therefore, the so acquired experimental curve yields exactly the deviation of the design wavelength from the value at α = β as shown in Fig. 2(b). In other words, a wavelength λ₁ in a SAFOS with λ_D = λ₂ yields the same spatial frequency of the interferogram as a wavelength λ₂ in a SAFOS with λ_D = λ₁.

The results for a configuration with a design wavelength of 403.513 nm is shown in Fig. 4. Obviously, the scan curve can easily be fitted to determine the design angle with a precision of better than 0.01°. At the given configuration, this corresponds to an error in the design wavelength of ±0.133 nm. Note that the wavelength scale has an offset which also has to be determined by the fitting process. A detailed procedure to accomplish this is given in [12]. In addition to fitting equation (3), one has to determine the offsets in α and λ between experimental data and the formula. It turned out that the most robust fitting procedure is to separate the two. The angular offset is calculated by moving the maximum of the measured curve to Δα = 0, while the wavelength offset Δλ is simply added as a free parameter, transforming equation (3) to:

The dependence for a configuration with a design wavelength of > 404 nm is shown in Fig. 5(a). For two points of the resulting graph (marked by arrows) the corresponding spectra are displayed. Fig. 5(b) shows that the resolving power of the SAFOS depends on Δλ, as expected. The major peak of the red curve here is < 0.01 nm wide.

 

Fig. 4 Experimental results for a SAFOS configuration in which the wavelength to be measured (horizontal line) is outside the scan range (corresponding to range A in Fig. 3(a)). The red line is a fit using a design angle of 151.978°. The green lines show fits with a deviation of γ of only ±0.01°.

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Fig. 5 Experimental data for a SAFOS configuration in which the wavelength to be measured is within the scan range (corresponding to range B in Fig. 3(b)).

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5. Conclusion

We introduced a spectrometer based on a common-path Sagnac-type interferometer. For a given wavelength range, this spectrometer can work at a large resolving power (R=40,000 demonstrated) without any moving parts. The high resolution relies on the fact that the spectrum is heterodyned around one single wavelength, which the spectrometer is designed for. This wavelength has to be larger or smaller than the wavelength of the radiation under test. It can be determined in the common way, by a calibrated light source or by using an unknown line source, rotating the grating, and measuring the tuning curve as shown in Fig. 2. A fit to the grating equation yields the design angle and wavelength. The resolving power of the device depends on the difference between illumination and design wavelength.

This spectrometer is much easier to align than the SHS, as in general, a Sagnac interferometer is much easier to align than a Michelson interferometer. The sensitivity against grating angle changes is much lower than in an interferometric spectrometer using reflection gratings. There is no need to search for an elusive balanced condition (equal arms in lengths and dispersion). There is no need for synchronized alignment of two gratings. Fig. 4 indicates increased tolerance against misalignment.