Influence of wavefront aberration on the imaging performance of the solar grating spectrometer

L. H. Zheng; C. H. Rao; N. T. Gu; L. H. Huang; Q. Qiu

doi:10.1364/OE.24.000153

1. Introduction

Magnetic activity of the Sun plays a dominant role in virtually all processes in the solar atmosphere, and it produces most of the sometimes spectacular visible phenomena, such as the sunspots, prominences, flares and coronal mass ejections [1–3]. They not only have fine structures, but also have manifest highly dynamic features, e.g. the small-scale magnetic fields [4]. From theoretical and numerical computations, it is known that much of the interaction between the solar plasma and the magnetic field occurs on very small spatial scales of about 70 kilometers on the Sun, corresponding to an angle of 0.1 arcsec. Due to the limited instrumental resolution up to now, the fundamental physical processes of these features are still not fully understood, even though they have been observed for a long time [5]. Identifying physical properties (temperature, opticalthickness, velocities, oscillation frequencies and so on) of these fine structures is essential in order to understand formation mechanism of the photospheric features and chromospheric features [6].

The imaging grating spectrometer is one of the most important instruments for the solar observation. Its wide free spectral range allows it to become a straightforward task to co-align a chromospheric feature and a photospheric feature at the same location on the sun [6]. In order to reliably perform the spectrometry of the fine structures, a grating spectrometer with the high spatial resolution, high spectral resolution and high temporal resolution is needed. On the other hand, the spectral resolution and energy utilization are the key parameters of the imaging performance of the grating spectrometer, and they will be influenced by the optical system aberrations [7] and optical elements quality [8]. The influence of the low order Zernike aberrations on the spectral resolution and energy utilization, such as the spherical aberration, coma and astigmatism, was qualitatively analyzed by Wu [7]; The influence of the optical quality on the spectral resolution and energy utilization, such as the manufacturing error of the grating, was qualitatively analyzed by Palmer and Loewen [8]. Their researches are mainly focused on the influence of the low order Zernike aberration on the spectral resolution and energy utilization, and the qualitative analyses are carried out. However, the further research on the quantitative influence of the wavefront aberration on the spectral resolution and energy utilization has not been reported yet. Besides the aberration mentioned above, the higher order Zernike aberrations cannot be ignored, especially for the grating spectrometer installed in a ground-based telescope. Here the atmospheric turbulence cannot be avoided, and it will seriously limit the imaging performance of the grating spectrometer, i.e., it will degrade the spectral resolution and the energy utilization. Therefore, reliable measurements of spectral line in the astronomy observation are very difficult. Fortunately, after development for period of several years, the adaptive optics (AO) technique has made great progress, and it has become an important tool to reduce the influence of the atmospheric aberration and static aberration [9, 10].

In order to quantitatively demonstrate the influence of the wavefront aberration with different types and magnitudes on the imaging performance of the grating spectrometer, the mathematical relation between the wavefront aberration and the imaging performance of the grating spectrometer is derived, and it is fully general for any wavefront aberration. According to the Pinciples of Optics, the narrow slit in the focal plane of the optical system will filter the wavefront aberration to a certain extent [11]. Unfortunately, the influence of the filter slit on the imaging performance of the grating spectrometer is still not understood and has not been reported yet. Our goal is to quantitatively analyze the influence of the wavefront aberration on the imaging performance of the grating spectrometer.

This paper is organized as follows: In section 2, the mathematical relation between the imaging performance of the grating spectrometer and the wavefront aberration is derived and discussed. In section 3, the numerical simulation is performed and is validated experimentally. In section 4, the conclusions are summarized.

2. Mathematical relation between imaging performance of the grating spectrometer and wavefront aberration

The spectral resolution is one of the key parameters of the imaging grating spectrometer, and it is influenced by the wavefront aberration, slit width and the spectral sampling of the CCD camera. Among them, the influence of the slit width and the spectral sampling of the CCD camera on the spectral resolution cannot be avoided, and it can be represented by the net spectral resolution [6]. The net spectral resolution of the grating spectrometer [6] is given by

Δ λ = \sqrt{Δ λ_{gr}^{2} + Δ λ_{s p}^{2} + Δ λ_{\det}^{2}}

where

Δ λ_{g r} = \frac{λ}{w_{g} σ m}

Δ λ_{s p} = w_{s} \frac{d \cos α}{m f}

Δ λ_{\det} = 2 l_{p i x} \frac{d \cos β}{m f}

where

Δ λ_{gr}

is the spectral resolution of the grating,

Δ λ_{sp}

is thespectral resolution degradation caused by the slit width,

Δ λ_{det}

is the spectral resolution degradation caused by the spectral sampling of the CCD camera; The w_g and the σ are the illuminated width and the groove density of the grating, respectively; The w_s is the slit width of the grating spectrometer, α and β are the incidence angle and the diffraction angle of the grating respectively; m is the diffraction order, f is the focal length of the collimator, d is grating constant, l_pix is the pixel size of the CCD camera.

Actually, the real spectral resolution $Δ λ_{φ}$ often is determined by the Rayleigh Criterion [8]. That is, the spectral line s1 and s2 with equal intensity can be distinguished, when $| λ_{1} - λ_{2} |$ is equal to the full width at half maximum (FWHM) of the spectral line. On the other hand, the spectral resolution will be degraded by the aberration, and the s1 and s2 cannot be identified as distinct spectral lines again, as depicted in Fig. 1. Hence the instrument spectral resolution is critical in order to carry out spectral analysis and band fitting routines [12].

Fig. 1 The spectral line influenced by the aberration. The red and the blue solid lines are the spectral line s1 and s2, respectively; the red (s1aberration) and blue (s2aberration) dot lines are the spectral lines s1 and the s2 influenced by the aberration, respectively.

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In order to directly measure the relative spectral resolution degradation, the κ is introduced and defined as

κ = \frac{Δ λ_{φ} - Δ λ_{0}}{Δ λ_{φ}} \times 100 %

where the

Δ λ_{φ}

is spectral resolution influenced by aberration, and

Δ λ_{0}

is the spectral resolution without the influence of the aberration. The κ represents a percentual relative error for

Δ λ_{0}

due to the aberration.

A suitable model for the observed spectral line is given by [13]

O (λ) = I (λ) \otimes P S F (λ)

where the O(λ) is the observed spectral line, I(λ) is the real spectral line, and the PSF(λ) denotes the modulation of the system. The

\otimes

is the conlution operator. Hence, the observed spectral line is the conlution product of the real spectral line and the modulation of the modulation of the system.

The imaging grating spectrometer is one of the important post instruments of the telescope, as depicted in Fig. 2.

Fig. 2 The sketch map of imaging grating spectrometer.

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According to the principles of Fourier Optics [14], the image o(x₂,y₂) of the intensity distribution of the point source object i(x₀,y₀) is given by

\begin{matrix} O (x_{2}, y_{2}) = P S F_{t a} (x_{2}, y_{2}) \\ = {| ℱ {A_{0} e^{i φ (x_{_{1}}, y_{_{1}})}} |}^{2} \end{matrix}

where the PSF_ta is the point spread function of the optical system of the telescope and the atmospheric turbulence,

ℱ {•}

is the Fourier transform operator, φ(x₁,y₁) is the equivalent wavefront aberration of the optical system of the telescope and the atmospheric turbulence.

Unlike the general optical system, the slit of the grating spectrometer is narrow enough (e.g. generally 10μm ~100μm in the solar observation). Hence the high frequencies of the wavefront aberration will be filtered by the narrow slit.

The PSF_ta(x₂,y₂) after the filter slit is:

\begin{matrix} O (x_{2}, y_{2}) = P S F_{t a} (x_{2}, y_{2}) r e c t^{2} (x_{_{2}} / w_{_{s}}) r e c t^{2} (y_{_{2}} / L_{_{s}}) \\ = {| ℱ {A_{_{0}} e^{i φ (x_{_{1}}, y_{_{1}})}} |}^{2} r e c t^{2} (x_{_{2}} / w_{s}) r e c t^{2} (y_{_{2}} / L_{s}) \end{matrix}

where

r e c t (x_{_{2}} / w_{_{s}}) = {\begin{cases} 1, - \frac{1}{2} w_{s} \leq x_{_{2}} \leq \frac{1}{2} w_{s} \\ 0, o t h e r w i s e \end{cases}

r e c t (y_{_{2}} / L_{_{s}}) = {\begin{cases} 1, - \frac{1}{2} L_{s} \leq x_{_{2}} \leq \frac{1}{2} L_{s} \\ 0, o t h e r w i s e \end{cases}

And the complex amplitude E_s after the filter slit is given by

E_{s} = ℱ {A_{0} e^{i φ (x_{1}, y_{1})}} r e c t (x_{2} / w_{s}) r e c t (y_{2} / L_{s})

Ignoring the magnification of the system and image inversion, the complex amplitude φ_s(x₃,y₃) after the slit is given by

φ_{s} (x_{3}, y_{3}) = ℱ^{- 1} {ℱ {A_{0} e^{i φ (x_{1}, y_{1})}} r e c t (x_{2} / w_{s}) r e c t (y_{2} / L_{s})}

where the

ℱ^{- 1} {•}

is the inverse Fourier transform operator.

It can be inferred from Eqs. (7) that, the wavefront fluctuation φ_s(x₃,y₃) decreases with the slit width and the slit length. Considering the extreme case, when the w_s and L_s are approaching zero, the $r e c t (x_{2} / w_{s}) r e c t (y_{2} / L_{s})$ is approaching 1 and the φ_s(x₃,y₃) is a plane wave here.

After the filter slit, the light is collimated by the collimator and then is dispersed by the grating. That is, the diffraction angles of different wavelengths are different. Then the light with the different wavelengths is converged by the imager to the different positions along the dispersion direction,as in the sketch map depicted in Fig. 3.

Fig. 3 The grating dispersion model of imaging grating spectrometer. After the grating dispersion, the 2-spatial dimensions are converted to the spectral dimension (along the x direction) and spatial dimension (along the y direction). The o(x₀,y₀) is set as the reference point, corresponding to the diffraction angle β. “o” is any point along the grating dispersion direction, Δθ_x is the deviation angle from the β, Δx_θ is the distance “oo”, f' is the focal length of the imager.

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The grating equation is given by

d (\sin α + \sin β) = m λ

where the λ₀ is the central wavelength, the α and β are the incidence angle and diffraction angle, respectively. The m is the diffraction order, and d is the grating constant.

The relation between the pixels along the grating dispersion and the diffraction angles of the different wavelengths is given by

d [\sin α + \sin (β + Δ θ_{x})] = m (λ_{0} + Δ λ_{θ})

The relation between the Δx_θ and the Δθ_x is given by

Δ θ_{x} = Δ x_{θ} f^{'}

Substituting Eqs. (8) and (10) into Eqs. (9), the simplest form of Eqs. (9) is:

\begin{matrix} Δ x_{θ} = Δ λ_{θ} (\frac{m f^{'}}{d \cos β}) \\ = Δ λ_{θ} D_{l} \end{matrix}

where the D_l is the linear dispersion of the grating spectrometer. A band-pass filter is placed in front of the image focal plane which is used to achieve order sorting.

The relation between the position along the grating dispersion direction and the wavelength is given by

\begin{matrix} x = x_{0} + Δ x_{θ} \\ = (λ_{0} + Δ λ_{θ}) D_{l} \\ = λ D_{l} \end{matrix}

where the λ = λ₀ + Δλ_θ.

In the focal plane of the grating spectrometer, the image of the PSF_s(x₂,y₂) is given by

P S F (x, y) = P S F_{s} (x_{2}, y_{2}) \otimes P S F_{s p} {(x, y)}_{| x = D_{l} λ}

where the PSF_s(x,y) is the image of the PSF_s(x₂,y₂), the PSF_sp(x,y) is the point spread function of the imaging grating spectrometer, the φ^’(x₃,y₃) is equivalent wavefront aberration of the grating spectrometer. The influence of the PSF_sp can be eliminated by precise adjustment of the optical system, and the influence of the φ^’(x₃,y₃) will not be discussed anymore in this paper [15,16]. The simplest form of Eqs. (13) is given by

\begin{matrix} P S F (x, y) = P S F_{s} (_{x_{2}, y_{2}) | x = D_{l} λ} \\ = i m g {_{P S F_{t a} (x, y) r e c t^{2} (x_{2} / w_{s}) r e c t^{2} (y_{2} / L_{s})} | x = D_{l} λ} \\ = i m g {_{{| ℱ {A_{0} e^{i φ (x_{1}, y_{1})}} |}^{2} r e c t^{2} (x_{2} / w_{s}) r e c t^{2} (y_{2} / L_{s})} | x = D_{l} λ} \end{matrix}

where the img{·} denotes the image process from the focal plane of the solar telescope to that of the solar grating spectrometer.

The observed spectral line O(λ) [13] is then given by

\begin{matrix} O (λ) = I (λ) \otimes P S F (λ) \\ = I (λ) \otimes \int i m g {{| ℱ {A_{0} e^{i φ (x_{1}, y_{1})}} |}^{2} r e c t^{2} (x_{2} / w_{s}) r e c t^{2} (y_{2} / L_{s})} d y_{| x = D_{l} λ} \end{matrix}

Supposing the value P is the maximum of the O(λ), the FWHM Δλ can be obtained by

O (λ) = 0.5 P

The real solutions of Eqs. (16) is given by

λ_{+} = ℑ {O (λ_{+}) = 0.5 P}

λ_{-} = ℑ {O (λ_{-}) = 0.5 P}

where the λ₊>λ_-, and the notation

ℑ {•}

is to return the real solution of equation.

The real FWHM Δλ_φ are given by

\begin{array}{l} Δ λ_{φ} = λ_{+ φ} - λ_{- φ} \\ = ℑ {_{O (λ_{+}) = 0.5 P} φ} - ℑ {_{O (λ_{-}) = 0.5 P} φ} \end{array}

when the φ = 0, the FWHM Δλ_φ are given by

Δ λ_{φ = 0} = ℑ {O (λ_{+}) = 0.5 P}_{φ = 0} - ℑ {O (λ_{-}) = 0.5 P}_{φ = 0}

when the φ≠0, the FWHM Δλ_real are given by

Δ λ_{φ \neq 0} = ℑ {O (λ_{+}) = 0.5 P}_{φ \neq 0} - ℑ {O (λ_{-}) = 0.5 P}_{φ \neq 0}

Substituting Eqs. (20) and (21) into Eqs. (2), the κ is given by

κ = (\frac{ℑ {O (λ_{+}) = 0.5 P}_{φ \neq 0} - ℑ {O (λ_{-}) = 0.5 P}_{φ \neq 0}}{ℑ {O (λ_{+}) = 0.5 P}_{φ = 0} - ℑ {O (λ_{-}) = 0.5 P}_{φ = 0}} - 1) \times 100 %

On the other hand, the energy utilization η of the grating spectrometer also is an important factor, and it is defined as

η = \frac{I_{φ \neq 0}}{I_{φ = 0}} \times 100 %

where the I_φ≠0 is intensity with the influence of the aberration, the I_{φ = 0} is the intensity without the influence of the aberration.

The η is given by

η = \frac{\iint P S F {(x, y)}_{| φ \neq 0} d x d y}{\iint P S F {(x, y)}_{| φ = 0} d x d y} \times 100 %

Any wavefrontaberrationscan be expressed by the Zernike polynomials, including the atmospheric turbulence fluctuation [17]. An arbitrary wavefront aberration φ(x,y) over a unit circular aperture is written in polar coordinates as

φ (r, θ) = \sum_{j = 0}^{N} a_{j} Z_{j} (r, θ)

where, 0≤r≤1, x = rcosθ, y = rsinθ, θ is the angle some point makes with respect to the x-axis, the θ = tan⁻¹(y/x), the subscript j is the ordering number, the amplitudes a_j of the Zernike components are given by

a_{j} = \int W (r) φ (r, θ) Z_{j} (r, θ) d^{2} r

where

\begin{array}{l} \begin{array}{l} Z_{e v e n j} (r, θ) = \sqrt{n+1} R_{n}^{m} (r) \sqrt{2} \cos (m θ) \\ Z_{odd j} (r, θ) = \sqrt{n+1} R_{n}^{m} (r) \sqrt{2} \sin (m θ) \end{array}} m \neq 0 \\ Z_{j} (r, θ) = \sqrt{n+1} R_{n}^{0} (r) m=0 \\ W (r) = {\begin{cases} 1 / π r \leq 1 \\ 0 r >1 \end{cases} \end{array}

where j is the function of n and m, the values of n and m are always integral and satisfy m≤n, n-|m| = even. R_n^m(r) is theradial function,

R_{n}^{m} (r) = \sum_{s = 0}^{(n - m) / 2} \frac{{(- 1)}^{s} (n - s)!}{s! [(n + m) / 2 - s]! [(n - m) / 2 - s]} r^{n - 2 s}

The phase structure function is given by

D_{φ} (ρ, θ) = 〈 {| φ (r, θ) - φ (r + ρ, θ) |}^{2} 〉

where the ρ is the displacement.

The long exposure OTF [16] is then

〈 O T F (ρ, θ) 〉 = τ_{0} (ρ, θ) e^{- 0.5 D_{φ} (ρ, θ)}

where τ₀ is the OTF of the perfect imaging system, and it is given by

τ_{0} (ρ, θ) = \frac{2}{π} {\cos^{- 1} (\frac{ρ}{D}) - \frac{ρ}{D} {[1 - {(\frac{ρ}{D})}^{2}]}^{0.5}}

where D is the diameter of the aperture, ρ₀/D is the normalized aperture radius.

The point spread function (PSF) of the system is given by

P S F = ℱ {〈 O T F (ρ, θ) 〉}

Substituting Eqs. (30) into Eqs. (15), the observed spectral line O(λ) is given by

\begin{matrix} O (λ) = I (λ) \otimes \int {| ℱ {\frac{2}{π} {\cos^{- 1} (\frac{ρ}{D}) - \frac{ρ}{D} {[1 - {(\frac{ρ}{D})}^{2}]}^{0.5}} e^{- 0.5 D_{φ} (ρ, θ)}} |}^{2} \\ r e c t^{2} (x_{2} / w_{s}) r e c t^{2} (y_{2} / L_{s}) d y_{| x = λ D_{l}} \end{matrix}

where ρ = [(x/λf')² + (y/λf')²]^0.⁵.

After the influence of the atmospheric turbulence, the κ and η are given by

κ = (\frac{ℑ {O (λ_{+}) = 0.5 P}_{φ \neq 0} - ℑ {O (λ_{-}) = 0.5 P}_{φ \neq 0}}{ℑ {O (λ_{+}) = 0.5 P}_{φ = 0} - ℑ {O (λ_{-}) = 0.5 P}_{φ = 0}} - 1) \times 100 %

η = \frac{\iint P S F {(x, y)}_{| φ \neq 0} d x d y}{\iint P S F {(x, y)}_{| φ = 0} d x d y} \times 100 %

3. Numerical and experimental validation

To prove the validity of the derived mathematical relation between the imaging performance of the grating spectrometer and the wavefront aberration, the numerical simulation and experiment have been carried out.

Generally, the real spectral resolution of the grating spectrometer is determined by the narrow bandwidth light source, hence a polarized HeNe laser with 632.8nm wavelength and 1pm FWHM is used as the light source to calibrate the spectral resolution precisely in the experiments. To study the influence of the wavefront aberration with different types and magnitudes on the spectral resolution and energy utilization, the AO system is integrated in the grating spectrometer system. To better assess the energy utilization of the grating spectrometer, the Littrow configuration is adopted.

3.1 System description

The optical layout of the echelle grating spectrometer with AO correction is built, as depicted in Fig. 4. The AO system is based on the Hartmann Shack Wavefront sensor (HS WFS) and the electrically addressed phase-only liquid-crystal spatial light modulator(LC SLM). A polarized HeNe laser with 632.8nm wavelength and 1pm FWHM is used as the light source for all the experiments. The linearly polarized collimated laser output passes through a non-polarizing beam splitter, such that the beam is divided into two beams, with nearly equal intensity. One of the two beams incidents on the LC SLM, while the other incidents on the reference mirror, serving as the reference of the optical adjustment. Each of these reflected beams is then recombined by the beam-splitter, and then incidents into the grating spectrometer. The phase-only modulation in the LC SLM can be achieved by adjusting the polarizer.

Fig. 4 The optical layout of the grating spectrometer. L1~L3: lenses, BS1~BS2: 50/50 non-polarizing beam splitters, OAP: off axis parabola mirror.

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In the grating spectrometer, the incoming beam is firstly filtered by the diffraction from the tiny slit. The tiny slit widths we used are 50μm and 100μm, respectively, which are roughly equal to 1.22λf/D and 2.44λf/D, respectively. Where f is the focal length of L1, λ is the wavelength, D is the input aperture diameter. In this case, the high frequency of the wavefront aberration will be blocked along the slit width direction. After the filter slit, the beam illustrates the off axis parabola mirror (OAP) and is collimated into the plane wave. Then the plane wave illuminates the grating. Groove density of the grating is 79gr/mm, size is 220mm × 110mm × 30mm, purchased from the Newport Rechardson Grating. As a result, the plane wave is dispersed. That is: different wavelengths are deflected to different positions. Besides, the different diffraction orders of the same wavelength are also deflected to different positions, and the bandpass filter is used to achieve order sorting in the experiment. The beam with the maximum intensity is reflected to the OAP. Finally, the beam converged by the OAP is divided into two beams, one of the two beams is used to image in the far-field CCD camera, while the other is collimated by the L2 and illuminates the HS WFS.

The HS WFS is used to measure the aberratedwavefront and the data is sent to the control computer to calculate the conjugated wavefront. Each coordinate position of the calculated conjugated wavefront is used to drive the corresponding pixel of the LC-SLM to perform the wavefront correction. The RMS value of the initial wavefront aberration generated by the adjusting error and the optical quality is 0.15λ, measured by the HS WFS with 28 × 28 sub-apertures. After the AO correction, the RMS value of the residual wavefront aberration is roughly 0.025λ, and a nearly diffration-limited focal spot is obtained, the result of simulation and experiment is as depicted in Fig. 5. The real spatial resolution of the solar grating spectrometer is 55μm, which is close to the diffraction-limited spatial resolution (50μm). After the spectral line calibration, the real spectral resolution of the solar grating spectrometer is 6pm, which is close to the net spectral resolution (5.6pm).

Fig. 5 The focal spots after the AO correction.(a)The numerical simulation,(b)CCD camera captured.

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3.2 Influence of the single order Zernike aberration on the κ and η

To investigate the influence of the single order Zernike aberration on the κ and η, the initial wavefront aberration generated by the adjusting error and the optical quality should be firstly compensated by the AO. After the AO correction, the RMS value of the residual wavefront aberration is 0.025λ, and the nearly diffraction-limited focal spot is attained. Secondly, the validation of the influence of the wavefront aberrations described by the Zernike polynomials with order number from 3 to 36 are performed. RMS values of the wavefront aberrations are 0.1λ and 0.2λ, which is generated by the LC SLM. Zernike tip and tilt terms have been eliminated. Examples of the influence of the 4th Zernike aberrations with the RMS = 0.2λ are illustrated in [Fig. 6(a)].

Fig. 6 The influence of wavefront aberration on the spectral resolution. The Figs from left to right represent the influence of the 4th Zernike order aberration with the RMS = 0.2λ and kolmogorov phase with the D/ro = 5, respectively. The Figs from top to bottom represent the wavefront aberration, the simulation farfield image, the simulation spectral profile, the farfield image captured by the CCD camera and the observed spectral profile.

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It is apparent that the spectral line is broadened by the Zernike order aberration, and the spectral resolution is definitely degraded. The results of the influence of the different types and magnitudes Zernike aberrations with the order number from 3 to 36 on the κ and η are illustrated in Fig. 7.

Fig. 7 The influence of different types and magnitudes of aberrations on the κ with 1dA (diameter of of Airy disk) of the slit width. The Figs from top to bottom represent the influence of the different types and magnitudes on the κ and η, respectively. From left to right represent the results of the simulation and the experiment.

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We have demonstrated that the influence of the different order Zernike aberrations on the κ is different, and it increases with the magnitudes of the Zernike aberration. The larger the magnitude of the Zernike aberration is, the larger the κ is, and the spectral resolution is worse. On the other hand, the influence of the different order Zernike aberrations on the η are different, and it increases with the magnitudes of the Zernike aberrations. The larger the magnitude of the Zernike aberration is, the smaller the η is, and the energy utilization degradation is worse, which adversely affects the signal to noise ratio and the temporal resolution.

In actuality, the narrow slit will filter the wavefront aberration to some extent. The validations of the influence of the different slits on the κ and η have been carried out. The slit widths we used in the experiments are 1dA (50μm) and 2dA (100μm), respectively. The Zernike order is from 3 to 36, and the RMS value of each Zernike order aberration is 0.2λ. The result is illustrated in Fig. 8.

Fig. 8 The influence of different slit widths on the κ and η. The Figs from top to bottom represent the influence of the different slits widths on the κ and η, respectively. From left to right represent the results of the simulation and the experiment.

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We have demonstrated that the influence of the different slit on the κ is different. Generally, the smaller the slit is, the better the spectral resolution is. However, this is not true for the low-frequency dominated aberration, e.g. the defocus and astigmatism, since its low frequencies will also be blocked by the narrow slit. If the influence of the filter slit on the wavefront aberration cannot be taken into account, it will lead to adaptive optics over-compensation. On the other hand, we have demonstrated that the η increases with the slit width, the wider the slit is, the higher the η is. Thus it is possible to improve the signal to noise ratio of the spectral data and the temporal resolution, which is important for the observation of the highly dynamic features of the solar atmosphere.

3.3 Influence of the Kolmogorov turbulence on the κ and η

Next, the validation experiments of the influence of an atmospheric phase screen consistent with Kolmogorov's theory on the κ and η are performed. The phase screen consists of 3 to 36 Zernike orders,and the D/ro = 5, D/ro = 7 and D/ro = 10 are used, where D is the input aperture diameter, and ro is the Fried parameter. An example of the influence of the Kolmogorov turbulence with D/ro = 5 on the κ is illustrated in the [Fig. 6(b)]. It is apparent that the spectral line is broadened by the atmospheric turbulence, and the spectral resolution is definitely degraded. The result of the influence of the wavefront aberration generated by the atmospheric turbulence with the D/ro = 5, D/ro = 7 and D/ro = 10 on the κ and η is illustrated in Fig. 9.

Fig. 9 The influence of the wavefront aberration generated by the Kolmogorov turbulence with D/ro = 5, D/ro = 7 and D/ro = 10 on the κ and η, the slit width is 2dA (100μm). (a)The influence of the wavefront aberration on the κ. (b)The influence of the wavefront aberration on the η.

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We have demonstrated that the influence of the wavefront aberration generated by the Kolmogov turbulence on the κ is different. Generally, κ increases with the D/ro. The worse the seeing ro is, the larger the κ is, and the spectral resolution is worse, which is consistent with the conclusions of the single Zernike aberration. On the other hand, the influence of the wavefront aberration generated by the Kolmogorov turbulence on the η is also different. The η increases with the seeing parameter ro, which is also consistent with the conclusions of the single order Zernike aberration.

3.4 White light experimental results

The Fianium white light laser, SC400, purchased from Fianium Ltd, is adopted in the experiment. The spectral range is between 400nm to 2400nm. To attain the free spectral range, the narrow bandpass filter is placed in front of the CCD camera. The central line is 632nm, FWHM is 1nm. The white light experiment results are illustrated in Fig. 10. It is evident that the influence of the wavefront aberration will degrade the spectral resolution and the energy utilization. When the wavefront aberration of the optical system is corrected by the AO system, the spectral resolution and energy utilization are improved obviously.

Fig. 10 The spectrum images of the white light without and with AO correction. The aberration is the10th Zernike aberration with the RMS = 0.2λ.

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Conclusions

High spectral resolution and high energy utilization are very important to a solar grating spectrometer. However, the spectral resolution and the energy utilization are influenced by the wavefront aberrations generated by the atmospheric turbulence, the optical system adjustment errors and the optical elements manufacturing errors and so on. On the other hand, when the slit width of the grating spectrometer is narrow enough, it naturally will filter the wavefront aberration to some extent. Therefore, the quantitative analysis of the influence of the wavefront aberration on the spectral resolution and energy utilization becomes a necessity. Drawing upon our theoretical analysis and the validations of the simulations and experiments, our conclusions can be summarized as follows:

The influence of the wavefront aberration with different types on the κ and η is different, and the influence increases with the magnitudes of the Zernike aberrations, the larger the magnitudes Zernike aberrations are, the worse the spectral resolution and energy utilization are. On the other hand, the influence of the different slit widths on the wavefront aberrations is different. We have demonstrated that the influence of the wavefront aberrations, except the low-frequency dominated aberration (e.g. the defocus) on the κ and η is generally increased with the slit width. That is because the narrower the slit width is, the more the low frequency is blocked by the slit, and this will lower the spectral resolution. The energy utilization increases with the slit width, the larger the slit width is, the higher the utilization is. If the influence of the filter slit on the wavefront aberrations are not taken into account, it will therefore lead to adaptive optics over-compensation.

Acknowledgments

This work is funded by the National Natural Science Foundation of China, Grant No.11178004. A special acknowledgment should be shown to Prof. Wenhan Jiang from the Institute of Optics and Electronics, Chinese Academy of Sciences, for his revision we benefited greatly. We also would like to express our gratitude to the reviewers for their valuable advice.

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Influence of wavefront aberration on the imaging performance of the solar grating spectrometer

Abstract

1. Introduction

2. Mathematical relation between imaging performance of the grating spectrometer and wavefront aberration

3. Numerical and experimental validation

3.1 System description

3.2 Influence of the single order Zernike aberration on the κ and η

3.3 Influence of the Kolmogorov turbulence on the κ and η

3.4 White light experimental results

Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Equations (40)

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