High-resolution stigmatic spectrograph for a wavelength range of 12.5&#x2013;30 nm

A. N. Shatokhin; A. O. Kolesnikov; P. V. Sasorov; E. A. Vishnyakov; E. N. Ragozin

doi:10.1364/OE.26.019009

1. Introduction

The need to record space-resolved spectra and, more generally, record spectral images arises in the investigation of laboratory and (extra) solar plasmas and laboratory XUV radiation sources. Among the recently emerged laboratory objects that invite the use of imaging (stigmatic) instruments are the source of high-order (even and odd) harmonics in a relativistic helium plasma produced by a multiterawatt femtosecond laser [1,2] and a relativistic “flying mirror” – the XUV source arising in the reflection of Ti:sapphire laser radiation from the relativistic plasma wave driven by multiterawatt laser pulses [3]. These objects call for a spatial resolution on a micrometer scale.

Classical Rowland-type grazing incidence spectrographs, which have long been used below ~300 Å, are inherently astigmatic. This entails a loss in the irradiance in spectral lines and the loss of spatial information. Furthermore, diffracted rays are incident on the focal curve (Rowland circle) at small grazing angles, hindering the use of CCD detectors. These drawbacks are eliminated by using plane and concave varied line-space (VLS) gratings.

Hettrick and Bowyer [4] came up with the idea to obtain a stigmatic XUV spectral image by directing a homocentric converging beam to a plane VLS grating. In this case, its local line density has to obey the local grating equation that forces the diffracted rays in the principal plane to arrive at the common spectral (horizontal) focus, which coincides with the vertical one. This concept was brought to fruition in the implementation of the Extreme Ultraviolet Explorer spectrometer [5], which was intended for the observation of extrasolar objects in the 70 – 760 Å range and comprised a Wolter–Schwarzschild Type II collecting optics and three plane VLS gratings with different central line densities.

Hettrick and Underwood [6] introduced the concept of a high-resolution scanning plane VLS-grating spectrometer/monochromator with a constant deflection angle and immobile entrance and exit slits. A plane grazing-incidence VLS grating is illuminated with a converging beam, and wavelength scanning is effected by grating rotation. An ultrahigh-resolution scanning spectrometer of this type was demonstrated by Hettrick et al. [7] with a Penning discharge. An impressive demonstration of its capabilities was the measurement of the linewidth of the Ne-like Se-ion soft X-ray laser (206.38 Å) [8]. Since then, this configuration enjoys numerous applications as a monochromator with laser plasma sources, synchrotron radiation and free-electron lasers (see, for instance Ref [9].).

Recent years have seen the development of spectrographs with a plane VLS grating intended for resonant inelastic X-ray scattering (RIXS) research with synchrotron radiation (see Fuchs et al. [10] and Warwick et al. [11]). The imaging spectrograph described in Ref [11] is intended to produce a 10-fold magnification in the dispersion (horizontal) plane and about a 5-fold one vertically.

The above instruments with plane VLS gratings are of two types: broadband spectrographs with immobile elements and narrowband scanning spectrometers (or, optionally, monochromators) with a constant focal length over a broad spectral region. Our recent numerical ray-tracing simulations show that the focal length of a scanning VLS spectrometer/monochromator may remain practically constant throughout a range of two octaves in wavelength (e.g., 80–320 Å) (Fig. 3 in Ref [12].). In this case, the ray-traced point spread function does not exceed the diffraction limited linewidth, thereby maintaining the theoretical resolving power at its limit defined by the number of grating lines.

We emphasize that scanning VLS spectrometers/monochromators with a constant deflection angle and invariable focal length of the Hettrick–Underwood type [6] are beyond the scope of our present work.

Harada and Kita [13] and Kita et al. [14] introduced a grazing-incidence spherical VLS-grating spectrograph with a flat portion of the focal surface in a wavelength range of 50–200 Å, in which the diffracted rays made relatively small angles (7°–13°) with the detector normal. The spectrograph became commercially available and came to be known as the Harada spectrograph. More recently, several flat-field VLS spectrographs of this kind with different configuration parameters were designed. For instance, Beiersdorfer et al. [15] reported a flat-field spectrograph with a VLS grating of radius 44.3 m intended for recording the spectra of multiply charged ions in an electron beam ion trap (EBIT) in a range of 10–50 Å and Dunn et al. [16] employed a similar instrument for laser plasma studies.

Flat-field Harada-type spectrographs themselves are astigmatic, and obtaining space-resolved information invites either a narrow slit placed in the dispersion plane like in Ref [16] (which decreases the radiation flux) or additional focusing optics to endow the Harada grating with the imaging capability in the perpendicular (“vertical”) direction.

To record space-resolved plasma spectra, Fan et al. [17] employed the combination of a Harada spectrograph with two grazing-incidence gold-coated mirrors: a cylindrical mirror, which formed a 1-D focus of the plasma on the entrance slit to illuminate the grating, and a crossed spherical mirror to focus the source on the focal plane in the perpendicular direction. The solid acceptance angle was estimated at ~10⁻⁵ sr. One ~800-ps long laser pulse with an energy of 40 J focused on a slab silicon or copper target was sufficient to produce a well-exposed spectrum on X-ray photographic film.

Neely et al. [18] designed a slitless three-channel flat-field VLS spectrometer for a range of 5–90 nm intended to record the point-like source of high-order laser harmonics. A Harada grating produces the spectral source images focused in one direction, and three grazing-incidence mirrors, which are tapered in width to acquire elliptical profiles under the application of bending moments, focus the source on the detector in the perpendicular direction. The instrument is therefore a stigmatic (imaging) flat-field spectrometer. The measured spatial resolution of the system was estimated at ~0.1 mm and the spectral resolution at ~1 Å for a wavelength of 350 Å.

Poletto et al. [19] reported a stigmatic spectroscopic system (2.5–40 nm) for laser-plasma research. A grazing-incidence toroidal mirror focuses the radiation source onto the entrance slit of a flat-filed spectrograph, which used VLS gratings with central line densities of 1800 mm⁻¹ (2.5 – 20 nm) and 1200 mm⁻¹ (10 – 40 nm). A grazing-incidence spherical mirror crossed relative to the grating is used to compensate for the astigmatism. Regions up to ± 1 mm off axis in the direction parallel to the entrance slit could be analyzed.

Poletto et al. [20] designed and implemented a spectrometer/monochromator for the study of high-order harmonics of laser radiation. The spectrometer spans the 5–75-nm region with a 600 mm⁻¹ plane grazing-incidence VLS grating. The grating in illuminated by the beam from a grazing-incidence toroidal mirror, which produces the stigmatic image of the entrance slit behind the grating. The configuration is stigmatic at one selected wavelength and is essentially similar to that of Hettrick and Bowyer [4], with the difference that the converging homocentric beam is produced by a toroidal mirror rather than a telescope.

Poletto and Tondello [21] designed an imaging spectrograph for solar spectroscopy, which comprised a flat-field spectrograph with a spherical VLS grating and a sophisticated three-component telescope, with focusing decoupled in two orthogonal directions. This concept was implemented by Frassetto et al. [22]. The instrument combines a flat-field Harada spectrograph, a 1-D parabolic grazing-incidence mirror to one-dimensionally focus a distant source on the entrance slit of the spectrograph, and a 1-D two-component Walter–Schwarzschild type-2 grazing-incidence telescope, which focuses the distant source on the detector plane in the perpendicular direction. The field of view is sufficient to image the entire solar disk.

Our intention is to make a high-resolution imaging spectrograph for laboratory applications which is stigmatic throughout a broad wavelength range. Normal-incidence focusing optics possesses small aberrations, a large field of view, and a large solid acceptance angle in comparison with grazing-incidence optics. Its use in the XUV has become possible with the advent of multilayer mirrors (MMs) and, in particular, broadband normal-incidence MMs based on aperiodic multilayer structures [23–26], which offer reasonable normal-incidence reflectivity.

When a homocentric beam is incident on a plane VLS grating [4], the stigmatic condition is fulfilled at a single wavelength $λ_{1}$ (Fig. 1). As the wavelength recedes from $λ_{1}$ , the horizontal focus drifts away from the vertical one.

Fig. 1 Compensation of astigmatism at a non-zero wavelength $λ_{1}$ in the incidence of a converging homocentric beam on a plane VLS grating (m is the order of diffraction). The vertical focal curve is a circle drawn about the grating center, its radius being equal to the distance between the converging-beam focus and the grating center (Ref [27], Fig. 1).

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In the irradiation of a VLS grating by a slightly astigmatic beam it is possible to satisfy the condition of strict stigmatism simultaneously at two wavelengths, $λ_{1}$ and $λ_{2}$ , with the understanding that this strictly applies to the paraxial rays lying in the principal plane. When $λ_{1}$ and $λ_{2}$ are properly spaced, the condition of practical stigmatism is fulfilled over about two octaves in wavelength. Here, the term “practical stigmatism” implies that the point spread function calculated by numerical ray tracing is mostly confined to the size of one CCD detector pixel (13 μm).

In this paper we describe the implementation of a broadband stigmatic high-resolution plane-VLS-grating spectrograph, in which the function of producing a slightly astigmatic beam is imposed on a broadband aperiodic Mo/Si multilayer mirror operating at near-normal-incidence. The spectrograph is used to record the line spectra of multiply charged ions in plasmas produced by 0.5-J, 8-ns, 1.06-μm focused laser pulses, and its applicability to the characterization of plasma or other radiation sources is thereby demonstrated.

2. Spectrograph configuration

Let the VLS-grating line density be described by a polynomial

p (w) = p_{0} + p_{1} w + p_{2} w^{2} + p_{3} w^{3},

where

p_{0}

is the line density at the grating center. As is well known, factor

p_{1}

modifies the spectral focal curve, while

p_{2}

and

p_{3}

may be used to cancel meridional coma and spherical aberration at fixed (not necessarily the same) wavelengths. We assume that a slightly astigmatic beam is incident on the plane VLS grating,

L_{h}

and

L_{v}

denote the distance of its horizontal and vertical foci from the grating center,

φ

and

ψ

are the grazing angles of incidence and diffraction of the central ray,

{r^{'}}_{h}

and

{r^{'}}_{v}

are the distances to the horizontal and vertical (paraxial) foci upon diffraction by the grating. We impose the requirement

{r^{'}}_{h} = {r^{'}}_{v}

to obtain

\frac{L_{h} \sin^{2} ψ}{\sin^{2} φ + m p_{1} λ L_{h}} = L_{v},

In combination with the grating equation

\cos φ - \cos ψ = m λ p_{0}

for the central ray, Eq. (2) defines the stigmatic condition at a wavelength λ. At this stage, the incidence angle φ and the central line density

p_{0}

are free parameters. Next we impose the requirement that the stigmatic condition is fulfilled simultaneously at two wavelengths,

λ_{1}

and

λ_{2}

, and express

p_{1}

from Eq. (2):

p_{1} = \frac{1}{m λ_{1}} [- \frac{\sin^{2} φ}{L_{h}} + \frac{\sin^{2} ψ_{1}}{L_{v}}] = \frac{1}{m λ_{2}} [- \frac{\sin^{2} φ}{L_{h}} + \frac{\sin^{2} ψ_{2}}{L_{v}}] .

On rearrangement we arrive at a constraint

φ = \arcsin (m p_{0} \sqrt{λ_{1} λ_{2}} / \sqrt{L_{v} / L_{h} - 1})

, which binds together

φ

and

p_{0}

. This signifies that astigmatism can be eliminated simultaneously at two wavelengths at a sacrifice of one of the free parameters (

φ

or

p_{0}

). For the first implementation of the spectrograph we assumed a customary value

p_{0} = 600

mm⁻¹.

Here, a remark is in order. The grazing angle φ has to be sufficiently small to ensure a high grating reflectivity. This requirement is fulfilled even when the operating range measures about two octaves in wavelength, because $p_{0} \sqrt{λ_{1} λ_{2}} ~ 2 p_{0} λ_{1} ~ 1.7 \cdot 10^{- 2} < < 1$ , and $L_{v} / L_{h} - 1 < 1$ is a controllable figure.

The remaining parameters $p_{i}$ (i = 2, 3,…) of the plane VLS grating are derived by expanding the following expression into a Taylor series:

m p (w) λ_{o p t} = \cos [arccot (\cot φ - \frac{w}{L_{h} \sin φ})] - \cos [arccot (\cot ψ - \frac{w}{L_{v} \sin ψ})] .

Meridional coma is compensated at a wavelength $λ_{o p t}$ when coefficient $p_{2}$ satisfies Eq. (5):

m λ_{o p t} p_{2} = \frac{3}{2} (- \frac{\sin^{2} φ \cos φ}{L_{h}^{2}} + \frac{\sin^{2} ψ \cos ψ}{L_{v}^{2}}),

and spherical aberration is compensated at a wavelength

λ_{o p t}

when coefficient

p_{3}

satisfies Eq. (6):

m λ_{o p t} p_{3} = \frac{\sin^{2} φ}{L_{h}^{3}} (- 2 \cos^{2} φ + \frac{\sin^{2} φ}{2}) + \frac{\sin^{2} ψ}{L_{v}^{3}} (2 \cos^{2} ψ - \frac{\sin^{2} ψ}{2}) .

The angles φ and ψ are related to

λ_{o p t}

by the grating equation. Note that wavelengths

λ_{o p t}

in Eq. (5) and Eq. (6) are not necessarily the same. In our case, we put

λ_{o p t} = λ_{1}

.

Figure 2 is a schematic of the spectrograph. The mirror is a broadband aperiodic Mo/Si MM [23–26] of radius 1 m. Proceeding from the spectral range of the MM (approximately 125–300 Å), in the instrument design we adopted $λ_{1} = 144$ Å and $λ_{2} = 270$ Å, which minimized the geometrical defocusing on a plane detector surface in the spectral range of the MM. The VLS grating is placed approximately half way from the MM to the grating.

Fig. 2 Schematic of the imaging VLS spectrograph (not to scale). “Vertical focus” is a circle of radius $L_{v}$ drawn about the grating center and is the vertical focus of the image of the source at all wavelengths. “Horizontal focus” is the horizontal focus of the image of the entrance slit produced by the MM. “Spectral focus” is the spectral (horizontal) focal curve upon diffraction by the VLS grating, which intersects the “Vertical focus” at points $λ_{1}$ and $λ_{2}$ .

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Two experimental uses of the spectrograph are possible: (i) the source is imaged onto the entrance slit using auxiliary optics, the entrance slit is considered as the source and the instrument produces stigmatic spectral images of the entrance slit; (ii) the source may be placed several centimeters in front of the entrance slit to produce vertically space-resolved spectra of the plasma. The difference between the two configurations consists in a minor difference in the angle of incidence on the MM. Several versions of the former configuration were analyzed and extensively ray-traced in Ref [28]. In one of them, for an average plate scale of ~6 Å/mm, a length of 28 mm of a 120–300 Å broad spectrum, and an acceptance angle of 4.5⋅10⁻⁴ sr, the ray-trace simulations confirmed that the point-spread function (the image of a point source on the entrance slit) was primarily confined, both spectrally and vertically, to one detector pixel (13 μm) throughout the wavelength range. Furthermore, the vertical field of view was ± 1 cm. The spectral resolving power corresponding to one detector pixel was roughly equal to 2.5⋅10³ (at a wavelength $λ_{1} = 144$ Å).

The latter configuration (with the source in front of the entrance slit) was experimentally realized in the present work, and its parameters are collected in Table 1. The relatively low reflectivity of the aperiodic MM is overcompensated for by the large acceptance angle of the instrument.

Table 1. Parameters of the imaging spectrograph

View Table

In both cases the spectral focal curve intersects at points $λ_{1}$ and $λ_{2}$ the vertical focus of the image, be it the point of the entrance slit or the point radiation source located ~30 mm in front of the entrance slit (Fig. 3).

Fig. 3 Behavior of the vertical and horizontal (spectral) focal curves in the wavelength range of interest. Plotted on the abscissa is the distance to the VLS-grating center (in the former case). The straight lines show the paths of the diffracted central rays with different wavelengths. The two focal curves intersect at wavelengths $λ_{1} = 144$ Å and $λ_{2} = 270 Å$ . The range of practical stigmatism spans two octaves in wavelength, approximately from 90 to 360 Å.

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Shown in Fig. 4 are the ray-trace simulations of several spectral images of the plasma source. To illuminate the entire grating width, the source is a section of a line arranged 30 mm in front of the entrance slit parallel to the principal plane of the spectrograph. All ray-trace 120–300 Å spectral images of the source spaced at ± 5 mm from the principal plane are primarily confined to one detector pixel (13 μm). In the ray tracing the entrance slit width was equal to 10 μm.

Fig. 4 The upper boxes of size 13 by 13 μm show the ray-trace spectral images of the source (140 Å, left box; 210 Å, right box) lying in the principal plane. The lower boxes of size 26 by 26 μm show the spectral images of the source (140 Å, left box; 210 Å, right box) spaced at 5 mm from the principal plane. The 140 Å wavelength is close to the best focusing point and the 210 Å wavelength is close to the point of maximum geometrical spectral defocusing. The rectangles show the halfwidths (FWHM) of the ray-trace images in the dispersion and vertical directions.

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3. Experiment

The aperiodic Mo/Si multilayer structure of the MM was numerically designed for maximum uniform reflectivity throughout a range of 125 – 250 Å [23–26]. The substrate of the MM was a superpolished fused silica substrate with an rms roughness of under 3 Å. The MM was synthesized by magnetron sputtering in the Laboratory directed by V. V. Kondratenko in the National Technical University “Kharkov Polytechnic Institute,” Kharkov, Ukraine. In the 125–250 Å range the measured MM reflectivity varies between 18 and 13%, and decreases to ~8% at 304 Å [25]. The plane Au-coated VLS-grating with a ruled area of 55 × 25 mm was made by 532-nm interference lithography in the State Institute of Applied Optics, Kazan, Russia. It had a quasi-sinusoidal groove profile with a depth of about 30 nm. The detector was a backside-illuminated CCD with 2048 × 1024 square pixels of size 13 μm each. The instrument was assembled on a rigid duralumin plate of size 1.1 × 0.6 m (Fig. 5). In view of a possible misalignment of the optical configuration and departure of coefficients $p_{i}$ from the design values, the grating was mounted on a precision motorized rotary stage and the CCD detector on a motorized translation stage. The spectrograph was accommodated in a 3.8-m long vacuum chamber 0.9 m in diameter pumped to a residual pressure of 5⋅10⁻⁵ Torr. The stages were computer-controlled to enable a fine alignment of the spectrograph in the evacuated chamber in the course of experiments.

Fig. 5 Layout of the spectrograph. Several auxiliary elements (blend, aperture stop) are removed.

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We recorded the vertically resolved line spectra of the multiply charged Li, C, F, and Mg ions, which were excited by 0.5 J, 8 ns, 1.06 μm laser pulses focused with an $f = 110$ mm lens. The spectra were recorded in one laser shot. The slit width was equal to 30 μm. Its geometrical image width on the detector was equal to 20 μm at 140 Å and to 15 μm at 270 Å. The targets were ground LiF, Mg and polyethylene discs, which could be rotated about the vertical axis. The target surface lay in the principal plane of the spectrograph to provide spatial resolution of the spectra in the normal direction to the target. Figure 6 shows the line spectrum of the Mg target, in which we indicated several lines of Mg III–Mg X as well as of O V. The height of the bracket at the left corresponds to an altitude of 0.5 mm above the target. The sharpness of the light–shadow boundary testifies to a spatial resolution of two detector pixels. The spectral resolving power also corresponds to two detector pixels and amounts to 10³.

Fig. 6 Vertically space-resolved (stigmatic) spectrum of Mg plasma recorded in one laser shot. The target surface lies in the horizontal plane, which coincides with the principal (dispersion) plane of the instrument and gives rise to a sharp light–shadow boundary. The bracket at the left indicates a 0.5 mm vertical scale. The spatial resolution is equal to two detector pixels (26 μm), as evidenced by the sharp light–shadow boundary.

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As would be expected, different lines behave differently with altitude. Some of them exhibit apparent broadening close to the target surface. The Mg III line intensities peak at an altitude of ~1 mm, unlike the lines of higher charge ions. Figure 7 shows the altitude dependent Stark broadening of the H_β (135 Å, 4→2) line of hydrogen-like C VI recorded in polyethylene plasma. The linewidth (FWHM) averaged over an altitude of 0 – 39 μm (three-pixel vertical binning) is equal to 0.53 Å. Assuming that the ion positions are independent, which is equivalent to the assumption that the thermal energies are appreciably higher than Coulomb interaction energies, we arrive at a density $N_{e} \approx 5.4 \cdot 10^{19}$ cm⁻³.

Fig. 7 Stark-broadened H_β line of C VI.

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4. Summary

We have introduced a way of broadband high-resolution stigmatic spectral imaging in the XUV, which involves the use of a slightly astigmatic beam incident on a grazing-incidence plane VLS grating, the astigmatic beam being formed by a broadband aperiodic near-normal incidence multilayer mirror.

A 1-m spectrograph was made, which makes use of an aperiodic Mo/Si MM efficient in a range of approximately 125–300 Å. The spectral resolving power ( $λ / δ λ \approx 10^{3}$ ) and the spatial resolution (26 μm) of the instrument correspond to two detector pixels, in accord with the predictions of numerical ray tracing (see Fig. 4 and Ref [28].). By recording the 1-D (vertically) space-resolved line spectra of multiply charged ions and, in particular, the altitude-dependent density-induced Stark broadening of the H_β line of C VI, the spectrograph was shown to be a useful tool for plasma diagnostics.

There are firm grounds to believe that the high-resolution stigmatic spectral imaging will be extended down to 111 Å with the use of Be-based aperiodic MMs, which yield a reflectivity plateau of ~0.2 throughout a range of 111 – 135 Å in numerical simulations [26] (in this case, the present optical configuration does not call for modifications, see Fig. 3). And it is not unlikely that this kind of imaging will be extended below 111 Å with the use of, say, La-based aperiodic MMs, which show (in simulations) reasonable reflectivities of ~4% throughout a range of 66 – 110 Å and reflectivities of ~8% in a range of 88 – 110 Å [24], or may be with the use of aperiodic Pd/Y MMs with B₄C barrier layers: as experimentally shown by Windt and Gullikson [29], Pd/B₄C/Y exhibit a reflectivity of ~5% throughout a range of 89 – 112 Å.

Funding

Russian Science Foundation (Grant No. 14-12-00506).

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Angle of incidence on the MM (central ray), deg	7.59
Radius of curvature of the MM, mm	1000
MM diameter, mm	50
Reflection band of the MM, Å	125 – 300
Grazing incidence angle $φ$ , deg	6.44
Stigmatic wavelength $λ_{1}$ , Å	144
Grazing diffraction angle at a wavelength $λ_{1}$ , deg	9.92
Stigmatic wavelength $λ_{2}$ , Å	272
Grazing diffraction angle at a wavelength $λ_{2}$ , deg	12.21
Plasma source–entrance slit distance, mm	30
Entrance slit–MM distance, mm	991.2
MM–grating distance, mm	461.2
Distance $L_{h}$ of the horizontal MM focus from the grating, mm	530.0
Distance $L_{v}$ of the vertical MM focus from the grating, mm	535.5
$p_{0}$ , mm⁻¹	600
$p_{1}$ , mm⁻²	2.2
$p_{2}$ , 10⁻³ mm⁻³	6.0
Plate scale at a wavelength $λ_{1}$ , Å/mm	5.36
Resolving power at $λ_{1}$ corresponding to one detector pixel	2070
Plate scale at a wavelength $λ_{2}$ , Å/mm	6.5
Resolving power at $λ_{2}$ corresponding to one detector pixel	3200
Illuminated grating area, mm	40 (w) × 18 (h)
Acceptance angle, sr	2.9⋅10⁻⁴
Field of view (vertically), mm	≥10

High-resolution stigmatic spectrograph for a wavelength range of 12.5–30 nm

Abstract

1. Introduction

2. Spectrograph configuration

3. Experiment

4. Summary

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (6)

Optics Express