1. Introduction

Diffraction phase gratings are traditionally used for light deflection, splitting and scanning and have a number of various applications in optics [1, 2]. Reflective diffraction gratings are widely used in optical spectrometers providing a relatively high resolving power over a wide spectral range due to a large angular dispersion [2–6].

Spectroscopic devices employing a grating as a dispersive element operate either in a spectrometer or a spectrograph mode [3–6]. In the spectrograph mode, spatially separated spectral components of diffracted light are detected simultaneously by an array of photodetectors. To register spectrum in a spectrometer mode, one or more moving elements, e. g. mirrors etc., sweep the spectral components through a photodetector slit.

Earlier we demonstrated that the spatial period and, thus, the angular dispersion of a viscoelastic transmission phase grating could be controlled by mechanical stretching [7]. Due to high elasticity of viscoelastic, the reversible elongation (ΔL/L ₀) of the grating may exceed 20% of its initial length (L ₀). When implemented in a monochromator, the stretchable grating allows scanning of diffracted light over the output slit, converting the monochromator into a scanning spectrometer. The relative spectral range, (λ _max-λ _min)/λ _min, covered by such a spectrometer is proportional to ΔL/L ₀, and thus it can eventually reach 20%.

In this paper, we consider the possibility of using viscoelastic-based diffraction gratings for spectrometric applications. Particular emphasis is given to the deformation-induced aberrations. A demonstrator of a low-cost scanning spectrometer based on a deformable transmissive diffraction grating is proposed and experimentally evaluated.

2. Experimental study

Two transparent 12-mm long gratings were replicated according to the procedure described in [7] in silicone elastomer (Sylgard 184, Dow Corning Corporation) using a reflective echelle grating with a spatial frequency of 600 lines pairs/mm from (Edmund Optics) as a master mold. The samples were 4 mm and 10 mm wide bars with grooves perpendicular to the elongation direction, their thickness in the unstressed state was d ₀=0.3 mm. One end of the first sample was glued to the clamp of an optical mount and the other end was attached to a voice coil. The second grating was glued between two clamps of an optical mount equipped with a micrometer. The diffraction efficiency in the -1st order is measured to be ~35% at a wavelength λ=543.5 nm (green He-Ne laser) for both samples and only weakly dependent on ΔL/L ₀.

To investigate the spectral range and the resolving power of the viscoelastic-grating-based spectrometer we used the experimental setup shown in Fig. 1. Light from a light source (LS) is clipped by a ~50-µm input slit (S₁) and illuminates, after passing a collimating lens (L₁) with a focal length of 35 mm, the grating at 35 mm from the lens.

In the first experiment, the 4-mm wide grating was repeatedly stretched by a voice coil (VC) powered by a function generator (G) at frequencies F~5-40 Hz. The -1st-order diffracted beam from the grating is focused by a 50-mm focal-length lens (L₂), positioned 50 mm behind the grating, on an exit slit (S₂) of ~0.5 mm filtering out the scattered and ambient light, and detected by a photodiode (D). The signal from the photodiode and the voltage applied to the voice coil were monitored simultaneously by a digital oscilloscope with a PC interface.

The inset on the right of Fig. 1 shows the Lissajous figure obtained at a sweep rate F=15 Hz when the input slit of the viscoelastic-grating-based spectrometer was illuminated by a He-Ne laser at λ=633 nm and two semiconductor lasers at 656 nm and 670 nm, respectively. The figure reveals two branches corresponding to the forward and backward scans. This behavior is attributed to the phase shift (φ) between the mechanical force generated by the voice coil at the end of the grating and the local strain that determines the spatial period at the intersection with the laser beam. The amplitude of the local strain and its phase depends on the actuation frequency F. At F≪F ₀, where F ₀=ν/(2L) is the first eigen frequency in the string model and v is the speed of elastic waves in the grating, φ tends to be zero whereas φ=π/2 at F=F ₀. The local strain amplitude increases substantially at frequencies which are integer multiples of F ₀ [8].

 

Fig. 1. Spectrometer with stretchable grating: S₁, S₂, slits; L₁, L₂, lenses; LS, light source; D, photodiode; G, generator; VC, voice coil. Inset shows the Lissajous figure obtained by illuminating the input slit S₁ with three lasers at 633 nm, 656 nm and 670 nm.

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Figure 2 presents the scaled spectrum of the laser sources at 633 nm, 656 nm and 670 nm, correspondingly, obtained with the scanning spectrometer at a sweep rate F=15 Hz. A full-width at half-maximum (FWHM) spectral resolution of ~3 nm was achieved in this experiment.

 

Fig. 2. Spectrum obtained with three lasers at 633 nm, 656 nm and 670 nm.

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In the second experiment, a neon lamp equipped with a diffuser was placed in front of a 100-µm entrance slit S₁, whereas the 10-mm wide grating of the spectrometer was slowly stretched by the screw. By increasing the integration time for each data point in quasi-static measurements of spectra, sufficient signal-to-noise ratio could be obtained with a narrower output slit S₂ of about 20 µm, allowing a higher spectral resolution of the scanning spectrometer. Figure 3 shows the spectrum of the lamp (blue curve with circles) measured by the spectrometer driven by a micrometer screw. For comparison, the lamp spectrum was also measured with a reference spectrometer in the range 580–637 nm using the solid master diffraction grating (red solid curve).

In addition, Figure 4 below compares the spectrum of the neon lamp obtained using the scanning spectrometer (blue curve with circles) with the spectrum measured by the spectrometer employing the unstretched, i. e. ΔL=0, viscoelastic grating (red curve).

 

Fig. 3. Spectrum of a neon lamp measured by the viscoelastic-grating-based spectrometer (blue curve with circles) and reference spectrum of the lamp (red curve).

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As can be seen from the plots in Figs. 3–4, the viscoelastic-grating-based spectrometer provides a FWHM resolution up to ~1.2 nm at the high wavelength side of the spectrum. An average FWHM resolution amounts to ~1.4 nm over a 100-nm spectral range (580–680 nm). In the measurements the grating length was changed gradually from L ₀=12 mm to L ₁≅14 mm, so the relative elongation ΔL/L ₀ reached 17%.

 

Fig. 4. Spectrum of a neon lamp measured by the scanning viscoelastic-based spectrometer (blue curve with circles) and reference spectrum obtained with the untstretched grating (red curve).

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The spectral resolution of the scanning spectrometer as compared to that obtained using the master grating (Fig. 3, red curve) in the same experimental setup is seen to be reduced by approximately a factor of two. This reduction appears to be associated with: (I) inaccuracy in replication of the sample and (II) stress-induced aberrations produced by the viscoelastic grating under stretching. Stress relaxation and creep [9, 10] in the elastomer do not seem to influence the overall precision of the viscoelastic-based spectrometer both in static and in dynamic operation.

3. Spectral resolution

The absolute spectral resolution, estimated from the registered spectra in Figs. 3 and 4 is about 1.35 nm at the half-maximum at λ=633 nm, yielding the resolving power of the spectrometer [3, 4] R=λ/δλ≅470. For the spectrum depicted in Fig. 2, one may obtain R≅210 at the He-Ne wavelength λ=633 nm. These values, however, are below a theoretical spectral resolving power of R_T=λ/δλ=|n|N=1200 of the spectrometer employing an ideal grating with a spatial frequency of 600 line pairs/mm. In the latter expression δλ is the wavelength resolution according to the Rayleigh criterion, n is the diffraction order, and N is the number of grooves within the effective diffraction aperture that is ~2 mm for our experimental setup. Note that the measured resolution obtained with the master echelle grating amounts to R≅800 (see Fig. 3) that is also smaller than the estimate based on the Rayleigh resolution criterion. This can be explained mainly by the effect of the finite slit width. The spectral resolution of the grating spectrometer in the geometry of Fig. 1 imposed by the entrance/exit slit can be estimated as follows [3, 4, 6]:

where Λ is the spatial period of the grating (Λ≅1.67 µm), the subscript q=1 refers to the entrance slit and q=2 to the exit slit, δS_q is the entrance/exit slit width, θ_q and F_q are the light incidence/diffraction angle and the focal distance of the input/output lens, respectively, depending on q.

In the experiment when the viscoelastic grating was repeatedly stretched by the voice coil, the slit-limited resolution is determined by the entrance slit of δS ₁=50 µm. Thus, substituting θ ₁=19.530 and F ₁=35 mm in Eq. 1, we obtain R_S≅280 at λ=633 nm. Analogously, for the experiment with the neon lamp, taking into account that δS ₂=20 µm, θ ₂=0.470 and F ₂=50 mm, the resolving power at λ=633 nm specified by the exit slit becomes R_S≅950. These values are, however, somewhat higher than those obtained experimentally. The discrepancy can be attributed to the aberrations caused by the complex deformation of the viscoelastic diffraction grating subjected to stretching.

4. Aberrations of the viscoelastic grating under stretching

Wave-front aberrations produced by the viscoelastic diffraction grating versus the grating relative elongation ΔL/L ₀ were determined experimentally using a Shack-Hartmann sensor in the transmitted and diffracted beams.

 

Fig. 5. Wave-front aberrations measured in (a) the transmitted and (b) the diffracted beams of the viscoelastic grating (solid figures). Dashed lines with open figures in panel b are calculations based on the grating deformation model described in the text.

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A 3.78-mm aperture wave-front sensor from OKO Technologies (The Netherlands) [11] employing a hexagonal 127-subaperture lenslet array and a CCD camera was placed in a plane conjugate to the grating. The registered images, i. e. hartmanngrames, were then acquired by an 8-bit frame grabber and processed by the FrontSurfer software (OKO Technologies).

Figure 5(a) and (b) above show the most significant wave-front aberrations produced by the viscoelastic grating subjected to stretching in the 0th and -1st orders of diffraction, respectively. In the experiment, the micrometer-driven 10-mm wide grating was illuminated by a collimated beam of a green He-Ne laser at λ=543.5 nm. The aberration coefficients a_i in Fig. 5 represent the wave-front expansion in terms of Zernike polynomials Z_i normalized and numbered according to Noll [12] over a circular aperture of the sensor with a radius r ₀=1.89 mm.

As seen in Fig. 5(a) tilts Z ₂, Z ₃ and astigmatism Z ₆ dominate in the transmitted beam, whereas astigmatisms Z ₅, Z ₆, comma Z ₈ and trefoil Z ₁₀ give a major contribution to the wave-front aberrations of the diffracted beam. The corresponding modal coefficients a_i reveal an almost linear dependence on the relative grating elongation ΔL/L ₀.

The aberrations observed experimentally in the diffraction order can be attributed to: (i) nonuniform deformation of the viscoelastic grating in the thickness direction along the Z-axis, as shown in Fig. 6 below, and (ii) nonuniform stretching over the grating aperture that results in the spatial frequency chirp, i. e. the local spatial period of the grating Λ(x, y) becomes a function of the transverse coordinates x, y.

 

Fig. 6. Geometry of light diffraction by the transmission viscoelastic grating subjected to stretching. The origin O of the reference frame coincides with center of the grating.

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Assuming that the grating experiences elastic deformation linearly depending on the applied force (at least, in the experimental range of ΔL/L ₀), its deformation can be found from the measured wave-front optical path difference u(x, y) of the transmitted beam. The thickness change Δd=d(x, y)-d ₀, where d(x, y) is the thickness of the grating stretched by ΔL and d ₀ is the initial grating thickness, takes the form:

where ν is the Poisson’s ratio and n is the refractive index of the viscoelastic material. For the elastomer Sylgard 184, we have ν≅0.5 and n=1.43, respectively. Supposing that the local spatial period Λ(x, y) is proportional to the grating deformation Δd(x, y):

one may deduce the following expression for the spatial period of the viscoelastic diffraction grating:

Λ₀ being the spatial period of the unstretched grating (for our samples Λ₀≅1.67 µm). The second term in Eq. 4 is a linear change of the spatial period due to stretching by ΔL, whereas the third term originates from the complex volume deformation of the sample. As discussed above, u(x, y) can be expressed in a series of Zernike polynomials:

where r={x, y} is the transverse radius-vector and a ^(t) _i a is the ith modal coefficient measured in wavelength units (λ), the superscript t denotes aberrations of the transmitted beam. Recalling from Fig. 5(a) that only tilts Z ₂, Z ₃ and astigmatism Z ₆ contribute to the wave-front aberrations of the transmitted beam, Eq. 5 results in a simplified form:

Substituting the general expression Eq. 5 into Eq. 4, we obtain:

here γ=λ/[νd ₀(n-1)(1+ΔL/L) is a small parameter γ≪1.

Following Harada and Kita [13] and Mahajan [14], in the geometry depicted in Fig. 6 for light coming from point A, falling upon B on the grating, outgoing from C, and diffracting to D, the light-path function becomes:

where m is the order of diffraction and q is the groove number (grooves are parallel to the Y-axis). The origin O of the Cartesian coordinate system depicted in Fig. 6 is in the center of the grating and q is calculated from x=0.

If the viscoelastic grating is illuminated by a collimated beam and the wave-front of the diffracted beam is measured in a plane conjugate to the grating, then the resulting aberrations are determined only by the combination S′=〈BC〉-mλq. The first contribution to the light-path function S′ yields 〈BC〉≅nd(x, y) which coincides, with accuracy to a constant, with u(x, y) specified by Eq. 5. The second diffraction term mλq depends on the spatial period Λ(x, y) [13]. By using Eq. 7 and recalling that γ≪1, the diffraction term takes the form:

here r′={x′, y}.

Analogously to Eq. 5, the wave-front w(x, y) of the diffracted beam specified by the light-path function S′ can be expressed as a Zernike series:

where a ^(d) _i is the ith modal coefficient measured in waves. Substituting Eqs. 9 and 5 into the light-path function S′ and taking into account Eq. 10, we finally obtain:

where C is the constant.

Thus, it can be concluded from Eq. 11 that the aberration coefficients of the diffracted and transmitted waves are intrinsically interrelated. In particular, by measuring aberrations a ^(t) _i in the transmitted beam, the wave-front aberrations a ^(d) _i in the diffracted beam can be found.

As an illustration, we consider the wave-front of the transmitted beam specified by Eq. 6 with the modal coefficients shown in Fig. 5(a). In this case the number of significant modes N=10 in Eq. 10. From Eq. 11 the coefficients of the most pronounced aberration terms, i. e. astigmatisms Z ₅, Z ₆, comma Z ₈ and trefoil Z ₁₀, for the above specified parameters of the viscoelastic grating, become: a ^(d) ₅≅-3.08a ^(t) ₃+a ^(t) ₅, a ^(d) ₆≅-1.54a ^(t) ₂+a ^(t) ₆, a ^(d) ₈≅0.54a ^(t) ₆+a ^(t) ₈, a ^(d) ₁₀≅-1.63a ^(t) ₆+a ^(t) ₁₀. The results of the calculation of the Zernike coefficients are plotted in Fig. 5(b) with dashed lines and open figures corresponding to those for the measured aberrations depicted with solid lines and filled figures.

We have already noted in section 3 that some reduction in the resolving power of the viscoelastic-based spectrometer may be attributed to the wave-front aberrations in the diffracted beam caused by the grating deformation. Now, let us evaluate the effect of the deformation-induced aberrations by imposing Eq. 10 with the coefficients presented in Fig. 5(b) on the spectrum measured using the untstretched grating, i. e. ΔL=0 (see Fig. 4, red curve).

 

Fig. 7. Comparison of the neon lamp spectrum measured by the scanning viscoelastic-based spectrometer (blue curve with circles) with convoluted reference spectrum obtained with the untstretched grating (red curve).

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By substituting the measured coefficients a ^(d) _i a into Eq. 10, the aberration-degraded spectrum S_A(λ) can be calculated from the spectrum S ₀(λ) obtained with the unstretched grating:

where U is the aberration-dependent smearing function that can be expressed in terms of the system point-spread function H(x, λ) [15, 16]:

where:

The function H(x,λ) specifies the intensity distribution in the plane of the exit slit S₂ produced by a line-source located in the plane S₁ at a given wavelength λ. In Eq. 13 the relation (that follows from Eq. 1) δλ=δxΛcosθ ₂/F ₂ between the width δx of the line image (in the plane S₂) and the corresponding wavelength spread δλ was employed.

The point-spread function on H(x,λ) of the system with aberrations a ^(d) _i was calculated at every λ point using Zemax ray-tracing software (Zemax Development Corp., USA). Figure 7 (red solid curve) shows the final spectrum obtained by convoluting the measured spectrum S ₀(λ) (viscoelastic grating at ΔL=0) with the smearing function U specified by Eq. 13. As seen from the comparison in Fig. 7, the calculated spectrum S_A(λ) agrees well with the measured data and, thus, the limitation of the spectral resolution of the viscoelastic-based spectrometer can be attributed mainly to the deformation induced aberrations.

5. Conclusion

In conclusion, we have presented an inexpensive scanning spectrometer employing a stretchable viscoelastic diffraction grating as a tunable dispersive element. A WFHM resolution of ~3 nm was demonstrated in dynamic scanning when the grating was repeatedly stretched by a voice coil at a sweep rate of 15 Hz. In quasi-static operation (with a micrometer screw) a WFHM resolution of ~1.4 nm was reached over a 100-nm spectral range (580–680 nm). At the high wavelength side of the spectrum (~680 nm) a FWHM resolution of ~1.2 nm was achieved.

We developed a model relating the deformation of the viscoelastic grating subjected to stretching and the resultant wave-front aberrations of the diffracted beam. The model suggests that the aberrations of the diffracted beam originate mostly from the chirp in the spatial period of the grating and these aberrations can be calculated by measuring the grating deformation or wave-front aberrations in the transmitted beam. Comparison with the case of the untstretched grating confirms that the spectral resolution of the viscoelastic-based spectrometer is limited mainly by the deformation-induced aberrations in the diffracted beam.