1. Introduction

Interferometric spectrographs have always been a strong competition to purely dispersive ones, yielding the highest resolving powers to date. A modern version of the spatial type of interferometers (i.e. without moving parts) is the spatial heterodyne spectrometer (SHS) [1], where a reflective diffraction grating under Littrow angle causes the wavefront tilt for near off-Littrow wavelengths that is necessary to obtain an interferogram containing the spectral information. These gratings are placed in the arms of a Michelson interferometer as shown in Fig. 1. The two wave fronts of an infinitely narrow emission line at Littrow wavelength returning from the two arms are parallel, leading to a bright or dark field in the output arm, depending on the difference in length of the two arms. For such a line at an off-Littrow wavelength however, the wave fronts are tilted in opposite directions as shown in Fig. 1. To ensure this opposite tilt, the two gratings have to be operated in the same diffraction order (1^st or − 1^st). One of the returning wave fronts is reversed by the reflection at the beam splitter while the other one is transmitted and keeps its orientation.

 

Fig. 1 Principal scheme of the Spatial Heterodyne Spectrometer.

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The principle of interferometric detection of spectra in a Michelson interferometer (e.g. with one of the mirrors tilted) was known before [2, 3]. The major breakthrough, however, was only achieved with the availability of sufficiently highly resolving fast readout detectors as CCD or CMOS cameras. A specific version of this device, where the beam splitting is achieved by utilizing the 1^st and −1^st order of a symmetric diffraction grating [4, 5] is sometimes called self-compensating, all-reflection interferometer (SCARI) [6]. Often, these devices are designed to operate in a higher diffraction order of the usually blazed grating [7, 8]. The principal advantage of this is merely technical, that blazed gratings with a lower groove density can be manufactured more easily. Otherwise, all configurations with a constant product of diffraction order and groove density are equivalent. Hence, the following considerations will be made for an SHS that operates in first diffraction order of the gratings.

To acquire the spectrum from the interferogram, one has to know the fringe spacing in dependence on the wavelength. Fizeau fringes of two plane wavefronts of wavelenght λ crossing each other under an angle δ have a spacing of λ/2sin δ. We assume a small departure ∆λ from λ₀ (λ = λ₀ + ∆λ), resulting in a small angle ∆α, which is the angle (with respect to the optical axis of each arm) under which the wavefront at λ is diffracted. Note that the wavefronts λ = λ₀ + ∆λ and λ = λ₀ − ∆λ generate the exact same Fizeau fringe pattern, causing an ambiguity of the heterodyned spatial spectrum. Most notations are explained in Figs. 2 and 3. The small angle approximations cos∆α = 1 and sin∆α = ∆α lead to the following expression:

 

Fig. 2 Diffraction at Littrow wavelength λ₀: (a) Incident wave packet with center wavelength λ₀ and coherence length cτ_coh. (b) Interfering wavepackets diffracted under Littrow angle Θ₀, the resulting diffracted wavefront is still perpendicular to the direction of incidence.

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Fig. 3 Diffraction at a wavelength λ₀ + ∆α: (a) Incident wave packet with center wavelength λ₀ + ∆λ and coherence length cτ_coh. (b) Interfering wavepackets diffracted under angle ∆α, the resulting diffracted wavefront is perpendicular to the diffraction direction, hence tilted by ∆α with respect to the direction of incidence.

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The fringe spacing s now amounts to (still assuming ∆λ ≪ λ₀):

2. Bandwidth (Wavelength coverage)

It is evident, that in this configuration, the product of bandwidth of the device (which is the wavelength range that it covers without turning the gratings) and resolving power is limited. The further away a spectral component is from the Littrow wavelength, the higher the spatial frequency of the Fizeau fringe pattern that it generates as it is evident from Eq. (2). 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 x_p, the largest ∆λ that can be resolved is determined by the following equation:

Here we see one of the compromises that have to be made in this kind of spectrometer: Increasing the resolving power (decreasing d) severely limits the bandwidth ∆_Bλ. One can easily increase the bandwidth by changing the magnification of the imaging optics (zooming into the interferogram) but this will be at the expense of resolving power.

3. Resolving power

The fringe spacing s_i for a monochromatic wave at λ₀ + ∆λ_i amounts to (from Eq. (2)):

We can derive a resolution criterion by assuming that for two wavelengths (λ₀ + ∆λ₁) and (λ₀ + ∆λ₂) the number of fringes over the width of the aperture has to differ by at least 1/2. This is based on the Rayleigh criterion: When the fringes line up on one side of the aperture, on the other side, the intensity minimum of one fringe pattern (s₁) is collocated with the intensity maximum of the other fringe pattern (s₂). The number of fringes is given by the aperture width divided by the fringe spacing. W.l.o.g. we write the criterion:

The aperture width w and the width of the grating W are related by the Littrow angle: w = W cosθ₀. With W = Nd, the resolving power turns out to be four times the number of illuminated grooves: R = 4N. This is twice the resolution that we would generally expect of a grating spectrometer with the two gratings combined.

On the other hand, if we consider the path difference ∆x = wtanθ₀ (cf. Appendix) and insert this into Eq. (4), we get R = 8∆x/λ₀, which is twice the maximum path difference expressed in wavelengths. Also here, the resolution turns out to be twice the value that we would expect from a Fourier Transform Spectrometer. As we will show in the following, Eq. (4) overestimates the resolving power, since it does not take into account the lateral confinement of the interference pattern, generated by diffraction of the wave under Littrow’s angle.

4. Considering the tilt of the energy front

Figure 2 shows a footprint of the SHS, in which the optical paths of the two arms are projected onto a single optical axis, depicting the wavefront situation in the output arm. The incoming beam incident on the two gratings at Littrow angle Θ₀ is shown in Fig. 2(a). Since we are dealing with linear optics, it is easier to represent incoherent light with a coherence length ℓ_coh by a pulse of duration τ_coh = ℓ_coh/c, where c is the speed of light. The side of the rectangle in the propagation direction (Fig. 2(a)) is then cτ_coh.

Figure 2(b) shows the situation in the output arm of the SHS for the wave packets centered at Littrow wavelength λ₀. Here, we neglect the fact that Fourier components which are off the Littrow wavelength (as definitely present in wave packets of finite length) are actually diffracted under a slightly different angle. Due to diffraction under Littrow angle and the resulting optical path difference between the two beam edges, the energy front is tilted with respect to the incoming energy front by an angle γ. This angle is approximately given by (see Appendix):

On a side note, Bor et al. [9] used the exact SHS configuration already in 1985 to construct a single-shot autocorrelator for ultrashort laser pulses. Here, the energy front is called pulse front for obvious reasons and actually used to laterally map a fringe-resolved autocorrelation function on detector with a single laser pulse. In this paper, a general expression for the energy front tilt in terms of angular dispersion dε/dλ was derived:

Applying this to the present case (i.e. dε/dλ = d∆α/d∆λ) shows that this general expression applies: If one uses Eq. (1) to calculate the derivative in Eq. (6) and replaces λ₀ with 2d sinθ₀ (Littrow condition), one gets Eq. (5).

It is often stated that for the exact Littrow wavelength, the SHS is equivalent to a Michelson interferometer. From Fig. 2(b) it becomes obvious that this equivalency only applies for waves of infinite extent.

The actual disadvantage for the resolving power of the SHS is now that this tilt of the energy front causes a lateral confinement of the overlap region, i.e. the region in which Fizeau fringes can be observed. As mentioned, this very feature of the two-grating Michelson interferometer and its variations was used earlier to determine the duration of ultrashort laser pulses in a single shot [10–12] or to intentionally generate an energy front tilt to construct a traveling-wave amplifier [13].

For incident radiation at λ₀ + ∆λ, the diffracted beam is deflected from the optical axis by a small angle ∆α (with respect to the incident direction). The wavefront tilt (with respect to the direction of incidence) is ∆α, the energy front is tilted by a larger angle γ as given by Eq. (5). Although γ slightly changes for ∆α ≠ 0 (see AAppendix), this change can be neglected for high-resolution versions of the SHS. The gratings are oriented such that the energy front tilt (as well as the wave front tilt) is in opposite directions for the two wave fronts (see Fig. 3(b)).

The projection p of the coherence width on the detecting image sensor, which is perpendicular to the optical axis, is (from Fig. 3):

From this, we can deduce a criterion for the resolution of the SHS that is connected to the aperture of the whole system. If the width of the emission line is so small (i.e. the coherence time is so large) that the width of the fringe pattern p exceeds the aperture of the system A, the resolution limit is reached. This is, when:

This leads to the following expression for the resolving power:

Presuming that we cannot resolve better than the linewidth anyway, we can write ∆λ = ∆λ_coh. Taking into account Eq. (5), the Littrow condition and the fact that the total number of grating grooves N within the aperture A is given by N = Ag/cosθ₀, we get

Given the fact that γ > 0 is necessary to have angular dispersion in the first place, this is less than a quarter of the value given by Eq. (4) and decreases further with increasing Littrow angle which is tantamount to increasing resolution. Here again, we see the compromise that has to be made for finding the optimum resolving power of the SHS. A resolving power in the order of N was observed in [7], but was attributed to the low quality of the utilized optics.

5. Experimental results

For the experiments, we used a slightly modified version of the original SHS [14], shown in Fig. 4. Mounting the two gratings back to back on a common rotation stage eliminates the problem of synchronizing their Littrow angles. Also, this configuration mitigates the angular alignment (horizontal and vertical) of the two arms, since not the two gratings mounted on the rotation stage but only the two more stably mounted mirrors have to be aligned.

 

Fig. 4 Modified version of the Spatial Heterodyne Spectrometer.

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The holographic symmetric reflection gratings were of size 34 × 34 mm² with a groove density of 3000/mm. We used 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 SHS was 25 mm up to a grating angle of θ₀ = 42°.

Of course, the two arms of the interferometer have to be of equal length, for which purpose the grating including the rotation stage were placed on a translation mount that moves parallel to the optical axis. On this mount, a motor-controlled rotation stage (Thorlabs, CR1-Z7) was placed with the two gratings on top of it. This stage allowed precise positioning (compared to the resolving power shown below) within in an angular range that corresponds to a few nanometers of Littrow wavelength. Thus, the SHS has to be calibrated with a known emission line that is within a few nanometers of the targeted range. For a more precise absolute positioning, a rotation stage with much better repeatability is necessary, as used e.g. in [4]. Care has to be taken to align the two gratings so that their grooves are parallel. A tilt of the gratings however, can easily be compensated with the two mirrors, as can a horizontal wedge between them.

Using a 50-mm focal length UV fused silica lens, the grating surface was imaged onto a UV-sensitive CCD camera (JAI Inc. CM-140 GE-UV) while demagnifying the image by a factor of ≈ 4 to fit the 6-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).

Figure 5 shows the interferogram recorded by the CCD camera for 4 slightly different angles of the two gratings. The light source is a Hg calibration lamp (Stellarnet Inc.). From the Littrow condition, we know that the spectral line that forms the interferogram is the Hg line at 253.65 nm. The angles and fringe spacings extracted from the interferograms agree with the values calculated by Eq. (2) within the error margins of the utilized rotation stage.

 

Fig. 5 Left: Four interferograms obtained with the folded SHS at grating angles 22.344°, 22.351°, 22.363°, and 22.375°, respectively. The numbers in the pictures are the Littrow angle difference to the top image. The light source is a Hg lamp, the angles correspond to the emission line at 253.65 nm. The slight tilt is due to imperfections in the gratings’ rotation. Right: Line scan along the vertical center of the corresponding interferograms. The grey vertical lines show the edges of the fringes’ envelope, corresponding to the width of the interference pattern as shown in Fig. 3(b). The abscissa shows the horizontal position on the CCD camera in mm.

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The most obvious feature is the envelope of the fringe pattern as predicted by Fig. 3(b), the width of which does not change when the gratings are rotated. The total width of the image is 25 mm (taking into account the magnification before the camera), the width of the envelope is 7 mm. Using Eq. (7), we can estimate the coherence time of the emission line to be τ_coh = 15 ps, which corresponds to a linewidth of 14 pm.

Another experiment was performed using a laser pointer that was labeled with an emission wavelength of 405 nm. The resulting interferogram is shown in Fig. 6, the beating between at least two neighboring emission lines can be clearly recognized. With a Fourier Transform and several image processing and averaging procedures, the spectrum shown in Fig. 7 was obtained. The red line denominates the Littrow angle that the SHS was set to. The spacing between the emission lines corresponds to a frequency difference of 133 GHz at this wavelength. Consequently, these lines are the longitudinal resonator modes of the InGaN laser diode with a length of about 1.1 mm.

 

Fig. 6 SHS interferogram of a nominal 405-nm diode laser showing the beating of neighboring emission lines.

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Fig. 7 Spectrum of a nominal 405-nm diode laser derived from the interferogram in Fig. 6.

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To relate Eq. (7) to this specific experiment, Fig. 8 shows the calculated width of the interference pattern for the major emission line from Fig. 7 in dependence on the spectral width of that emission line. In the experiment presented here, we used a grating with a groove density of 1200/mm. The data for a grating with 3000 lines per mm are shown for comparison, demonstrating that - although the resolving power of the SHS increases with this grating, the reasonable range of use for this device decreases. For a standard detector (full frame CCD or CMOS camera) for instance, the Fizeau fringe pattern utilizes the full width of the detector only for a width of the emission lines below 5 pm. For the case described here (18 pm line width, 20 mm detector width) we still occupy the range where one can use the simple mathematical treatment (Fourier transform according to [1]) for the retrieval of the spectrum. By a measurement with a high-resolution Echelle spectrometer (EMU-120/65 VIS-NIR, Catalina Scientific Inc.), we determined the width of the laser lines to be less than 10 pm. Hence, the 18 pm, that we see in Fig. 7 represent the resolving power of the SHS being about 22,400. This agrees well with the value of 21,500 that we get from Eq. (8)

 

Fig. 8 Interference pattern width p of the major emission line of the nominal 405-nm diode laser (402.995 nm) in dependence on the width of that line for two different gratings used in the SHS. The spectrum of Fig. 7 was acquired using a grating with 1200 lines per mm, the data for the 3000/mm-grating are for comparison.

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6. Conclusion

The Spatial Heterodyne Spectrometer can be used in high-resolution spectroscopy, however, it is not as versatile as described so far. Due to the fact that an interferogram is formed by wave packets with a tilted energy front, there is a severe limitation to the resolving power when the coherence length of the radiation under investigation exceeds a certain limit, determined by the aperture of the device and the resolving power itself. Another obstacle for a wide dissemination of this device as a high-resolution spectrometer is the requirement for very precise angle tuning. However, wavelength differences (as for instance the separation of isotopic lines) can be precisely determined since this value is given by a low beat frequency to be resolved by the utilized CCD. The feasibility of the SHS to detect spectra in Laser-Induced Breakdown Spectroscopy (LIBS) was recently demonstrated [15], however, the achieved resolution was far below the necessities for this method. More details about the mathematical treatment of SHS interferograms can be found in [16].

Appendix: Angle of energy front tilt

Figure 9 shows an incoming wave packet at Littrow angle θ₀ having the corresponding Littrow wavelength λ₀ = 2sinθ₀/g. The diffracted wavefront with an effective aperture of w is parallel to the incoming wavefront while the diffracted energy front travels twice the distance ∆x more on the right side than on the left side. From triangle ABC, we get: tanθ₀ = ∆x/w, from the triangle ABD: tan γ = 2∆x/w resulting in

$$\tan \gamma =2\tan {\theta }_0$$

 

Fig. 9 Obtaining the energy front tilt from the Littrow angle.

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This equation is strictly valid only for the wavefront under Littrows angle but can be used for small angles ∆α as well. The exact expression for any diffraction angle would be:

$$\tan \gamma =\frac{\sin {\theta }_0+\sin \left({\theta }_0-\mathrm{\Delta }\alpha \right)}{\cos \left({\theta }_0-\mathrm{\Delta }\alpha \right)},$$

yielding the former equation for ∆α = 0.

Acknowledgments

We thank Burt Beardsley from Catalina Scientific Inc. for help with the retrieval of the spectrum. This work was funded in part by the US Department of Energy grant number DE-SC0011446.

References and links

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16. http://www.lenzner.us/shs