## Abstract

We propose a new recording system that employs a spherical wave and a toroidal wave to holographically record an aberration-reduced Rowland-circle spherical grating. Two more variables in this recording system are available for the design compared with a conventional symmetric dual-plane-wave recording system, which allows both the astigmatism and sagittal coma of the grating to be corrected to minimum. We derive the aberration coefficients of such grating and give the universal design principles for the recording system. An aberration-reduced grating used in a spectrometer equipped with linear array CCDs is then specifically designed and fabricated. The performances of the aberration-reduced grating and that of the conventional grating recorded by dual plane waves are compared in terms of the ray-tracing spot diagrams, and the photographed spectral images. The experimental results show that the spectral intensity of the aberration-reduced grating is about 4.5 times that of the conventional grating, while the resolution of the aberration-reduced grating is close to that of the conventional grating.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A conventional Rowland-circle spherical grating (CRSG), with straight and equally spaced grooves, is generally recorded by symmetric dual plane waves [1]. CRSG has zero defocus aberration and zero meridional coma, but is afflicted with astigmatism and sagittal coma, resulting in a point light source to be imaged on the Rowland circle as an elongated curved line [2]. The Rowland-circle concave gratings are commonly used in relatively large spectrometers to take advantage of their high resolution.

Three main kinds of detectors are used in the Rowland-circle concave-grating spectrometers. The photomultiplier tubes are most typically adopted, where the spectral lines are detected discretely behind the exit slits on the Rowland circle. The problem of the elongated spectral lines (caused by the astigmatism) can be neglected due to the long exit slits and the large receiving area of the photomultiplier tube. However, since the spectral lines are curved (caused by the sagittal coma) and the exit slits are straight, this inconsistency will cause damage to the spectral resolution.

Another kind of the detectors used is the linear array CCD, which has an advantage of detecting a continuous spectrum, such as the SPECTROBLUE inductively coupled plasma, optical emission spectroscopy (ICP-OES) spectrometer produced by Spectro [3]. Due to the limited pixel height of the linear array CCD, the astigmatism of the grating will be a key disadvantage leading to a poor spectral intensity of the spectrometer. It is noticed that the curvature radius of the Rowland circle should be sufficiently large relative to the length of the linear array CCD to eliminate the spectral resolution errors caused by the incomplete overlap of the linear array CCD and the Rowland circle.

The third kind of the detectors, the 2D CCD is used to detect the image of the imaging spectrometer equipped with Rowland-circle concave grating [4]. At this time, both the astigmatism and the sagittal coma of the grating have great influence on the image quality.

In conclusion, the astigmatism and sagittal coma are the critical shortcomings when different detectors are utilized. To correct these aberrations, various design methods have been reported, which are mainly divided into two types. One of the design ideas is to change the surface profile of the blank. The ellipsoidal grating was first developed by Namioka [5] to reduce the astigmatism using a proper ellipsoidal blank. Masuda et al [2] theoretically studied the design method of an aberration-reduced toroidal grating. The grating was fabricated on a toroidal blank to mount in a Seya-Namioka monochromator. The spectrograms of the Ar 457.93 nm line showed that the astigmatism of the toroidal grating was reduced to about 1/8 compared with that of the spherical grating. Michel Duban [6] proposed a second-generation grating with toroidal blank, recording by aspheric waves, to correct the aberrations up to the fourth degree. However, the high production cost of the aspheric blanks has prevented the widespread use of the aspheric gratings in spectrometers.

The other design idea is to change the recording system for the gratings with spherical blanks, based on the precondition that the defocus aberration and the meridional coma still remain zero for all wavelengths. Brown and Wilson [7,8] proposed a recording system using two point sources lie on the Rowland circle to correct the astigmatism or sagittal coma of the grating at two wavelengths. We [1] proposed recording the Rowland-circle spherical grating by a cylindrical wave and a plane wave to reduce the astigmatism to minimum. We adopted the principle of minimizing the aberration coefficient over a predetermined wavelength range, rather than making aberration coefficient zero at certain wavelengths [7,8]. In both [1] and [7], one more variable in the recording system is available compared with a conventional symmetric dual-plane-wave recording system, which allows either the astigmatism or the sagittal coma to be corrected to minimum. However, these two aberrations cannot be corrected to minimum simultaneously. In all the above-mentioned references about Rowland-circle concave-gratings except [1] and [2], the focus is on the design and simulation of gratings, fabrication and testing of the gratings are not addressed.

In this article, we propose a new recording system that records an aberration-reduced Rowland-circle spherical grating (ARSG) on a spherical blank by a spherical wave and a toroidal wave. This recording system provides two more variables than a conventional symmetric dual-plane-wave recording system. To the best of our knowledge, this is the first time that the astigmatism and the sagittal coma of the Rowland circle spherical grating can be simultaneously corrected to minimum, while the defocus aberration and the meridional coma of the grating remain at zero. Thus, the spectral lines of ARSG will become shorter and less curved than those of CRSG and the gratings designed according to [1] and [7]. We derive the aberration coefficients of ARSG through the optical path function and give the design principle of the ARSG. An ARSG and a CRSG used in an actual spectrometer equipped with linear array CCDs were designed and fabricated. The spectral resolution and intensity of ARSG and CRSG were measured and compared. Comparing with CRSG, the spectral intensity of ARSG is effectively increased, meanwhile the spectral resolution keeps unchanged.

## 2. Theory of the design

#### 2.1 Calculation of the aberrations

We derive the aberration coefficients by analyzing the optical-path function *F* in Eq. (1) to characterize the aberrations of ARSG in Eq. (4). Figure 1 gives the geometry of ARSG. The Cartesian coordinate systems of Figs. 1(a)-1(c) share a same definition. The coordinate origin *O* is the grating center, the *x* axis is the normal of the grating at *O,* and the *z* axis is the groove's direction at *O*. *R* is the curvature radius of the grating and is equal to the diameter of the Rowland circle. The Rowland circle is on the *x*-*O*-*y* plane. Point $P(e,w,l)$ is on the grating surface. Figure 1(a) gives the schematic diagram of the Rowland-circle mounting. The object point $A({x}_{A},{y}_{A},0)$ and the image point $B({x}_{B},{y}_{B},0)$ are on the Rowland circle.

Figure 1(b) gives the idealized geometry of one of the recording waves, which is actually generated by a spherical wave passing through a cylindrical lens. According to the imaging characteristics of the cylindrical lens, the source of this recording wave is approximate to an arc. For the convenience of calculation, we idealize this recording wave source as arc ${E}_{1}{E}_{2}$ in Fig. 1(b). Arc ${E}_{1}{E}_{2}$ located in the *x*-*O*-*y* plane with the center *D* and the midpoint *E*. Line ${D}_{1}{D}_{2}$ is parallel to the z-axis. Points *O*, *D*, and *E* are collinear and *EO* is the chief ray of the recording wave. The lights emitted from point *E*_{1} pass through points *D*_{1} and *D*_{2}, respectively, to *F*_{1} and *F*_{2}, and ${E}_{1}{F}_{1}={E}_{1}{F}_{2}$, that is, the center of the arc ${F}_{1}{F}_{2}$ is *E*_{1}. The surface ${F}_{1}{F}_{2}{F}_{3}{F}_{4}$ is defined by rotating the arc ${F}_{1}{F}_{2}$ about the axis ${D}_{1}{D}_{2}$. At this time, the light emitted from any point of the arc source ${E}_{1}{E}_{2}$ through ${D}_{1}{D}_{2}$ to the surface ${F}_{1}{F}_{2}{F}_{3}{F}_{4}$ has the same optical path length as equal to ${E}_{1}{F}_{1}$, which satisfies the Huygens–Fresnel principle. That is to say, ${F}_{1}{F}_{2}{F}_{3}{F}_{4}$ is the wavefront of this recording wave. According to the definition of the toroidal surface, ${F}_{1}{F}_{2}{F}_{3}{F}_{4}$is a toroidal surface. Thus, we name this ideal recording wave as toroidal wave.

In actual exposure, the toroidal wave is approximated by a spherical wave passing through a cylindrical lens. The actual recording structure and error analysis are shown in Chapters 4 and 5 respectively. In the principle and formula derivation, we use the ideal toroidal wave for the convenience of calculation.

The recording geometry is shown in Fig. 1(c). *C* is the point source of the spherical wave. Points *C* and *D* are on the Rowland circle. The chief rays of the recording beams trace from $C({x}_{C},{y}_{C},0)$ and $E({x}_{E},{y}_{E},0)$ to *O*. The distances of *OC*, *OD* and *OE* are ${r}_{C}$, ${r}_{D}$ and ${r}_{E}$, respectively. The recording light emitted from *E*_{p} on the arc source ${E}_{1}{E}_{2}$propagates through point *D*_{p} on${D}_{1}{D}_{2}$ to point *P*. All the angles are measured counterclockwise from the positive *x* axis. Since point C and arc ${E}_{1}{E}_{2}$ are the wave sources of the two recording beams, the optical-path function F of ray path APB is calculated:

*m*is the diffraction order. In Eq. (1),

*AP*+

*PB*is the geometrical optical path from the mounting system, as shown in Fig. 1(a);

*PC*-

*PE*

_{P}is the holographic contribution from the recording system, as shown in Fig. 1(c). The expression of $P{E}_{P}$ is obtained based on the spatial geometry. We expand

*F*as a Taylor series in combined power of

*w*and

*l*up to the third order:

We write the aberration coefficient ${F}_{ij}$as the sum of a geometrical contribution ${M}_{ij}$ and a holographic contribution proportional to ${H}_{ij}$:

Considering the geometrical relations ${r}_{A}=R\mathrm{cos}{\theta}_{A}$, ${r}_{B}=R\mathrm{cos}{\theta}_{B}$, ${r}_{C}=R\mathrm{cos}{\theta}_{C}$, and ${r}_{D}=R\mathrm{cos}{\theta}_{D}$, we derive the expressions of ${M}_{ij}$ and ${H}_{ij}$ for ARSG from Eq. (1):When the parameters of the Rowland circle mounting are fixed, the coefficients ${F}_{\text{02}}$ and ${F}_{\text{12}}$ of CRSG are also fixed and normally so large that leads to a bad astigmatism and a bending of the spectral lines. However, in an ARSG, the extra variables ${\theta}_{C}$ and ${r}_{\text{E}}$ can be used to reduce the absolute values of ${F}_{\text{02}}$ and ${F}_{\text{12}}$. Thus, the astigmatism and sagittal coma of ARSG are corrected, which makes the spectral lines of ARSG shorter and less curved.

#### 2.2 Design principle

We fixed the mounting parameters of ARSG first. Then two variables ${\theta}_{C}$ and ${r}_{\text{E}}$ of the recording parameters are available for design. The aberrations of ARSG can be expressed as the objective function of ${\theta}_{C}$ and ${r}_{\text{E}}$:

In some particular situations, we consider trade-offs between the aberrations at different wavelengths and add a weight coefficient $k\left(\lambda \right)$ to the objective function:

The design method of the recording parameters described above is executed when the mounting parameters of the Rowland-circle grating are fixed. If we need to optimize the mounting parameters, ${\left[{I}_{20}\right]}_{\mathrm{min}}$or${\left[{I}_{12}\right]}_{\mathrm{min}}$ of Eq. (8) or the sum of them can be employed as the objective function. Since Eq. (8) is independent of the recording parameters, the calculation of the mounting and recording parameters can be divided into two processes: first optimizing the mounting parameters and then the recording parameters, thereby avoiding putting all the recording and mounting variables together for multi-parameter optimization. The algorithm for calculating the mounting parameters should be chosen according to the degrees of freedom of the mounting system.

As mentioned above, the number of the variables in the recording system determines how many terms of the aberrations can be corrected to minimum according to Eq. (9). In [1] and [7], since only one variable is available in the recording system, the astigmatism and the sagittal coma cannot be corrected to minimum simultaneously. For example, when the astigmatism of the grating is corrected to minimum and the spectral lines become very short, the damage of the sagittal coma will stand out which leads to a severe spectral line bending and a serious decline in spectral resolution. Due to the two variables ${\theta}_{C}$ and ${r}_{\text{E}}$, our method can solve this problem well.

## 3. Application of ARSG to a spectrometer

#### 3.1 Design of ARSG

We name the grating proposed in [1] IRSG (improved Rowland-circle spherical grating), which is recorded by a cylindrical wave and a plane wave. In order to evaluate our method, the performance of an ARSG was compared with that of a CRSG and that of an IRSG by spot diagrams. The gratings were designed to be used in a commercial spark optical emission spectroscopy metal analyzer equipped with linear array CCDs. The mounting parameters of the three gratings were obtained in common based on the spectrometer structure, including the curvature radius of the grating *R* = 750 mm, the incident angle ${\theta}_{A}=37.5\xb0$, the diffraction order *m* = 1, the groove density at the center of the grating *GD* = 2400 gr/mm, and the spectral range from 200 nm to 500 nm. The pixel size of the CCD is 5.3μm × 200 μm (width × height).

We calculated the recording parameters of ARSG by solving Eq. (9), obtaining ${\theta}_{C}=-8.41\xb0$ and ${r}_{\text{E}}=595.6\text{\hspace{0.17em}}\text{mm}$. The CRSG was designed to recorded by symmetric dual plane waves with the exposure angle of 29.72°. The IRSG was designed, obeying the method described in [1], to correct the astigmatism to minimum, obtaining that the optical length of the chief ray between the cylindrical wave source and the grating center equals to 1067 mm. A krypton ion laser with wavelength of ${\lambda}_{0}=413.1\text{\hspace{0.17em}}\text{nm}$ was used to record the gratings. The diameter of the exposed area on the grating is 20 mm.

Figure 2 gives the spot diagrams obtained by ray tracing for the three types of the gratings. When comparing Fig. 2(a) with Fig. 2(c), it is evident that the spectral image heights of ARSG are much smaller than those of CRSG, which means that the spectral intensity of ARSG will be much higher when the linear array CCDs are used as the detectors. Meanwhile, the spectral images of ARSG are less curved than those of the CRSG, which will be a desirable property in the imaging spectrometers or the spectrometers equipped with the photomultiplier tubes. In fact, it is consistent with our expectation that the spectral lines of ARSG become shorter as the astigmatism term ${I}_{\text{02}}$ is corrected and become less curved as the sagittal coma term ${I}_{12}$ is corrected.

When comparing Fig. 2(b) with Fig. 2(c), we obtained that the spectral image of IRSG is about the same height as that of ARSG but much more curved than that of ARSG. The spectral image of IRSG is so curved that the spectral resolution will be deteriorated to a significant extent and the entrance slit height will be limited to a small range. As only one variable is available in the recording system of IRSG, the astigmatism and the sagittal coma cannot be corrected to minimum simultaneously. In actual use, the astigmatism and the sagittal coma of IRSG are usually sacrificed to each other and both corrected to non-minimum. The comparison shows a great superiority of ARSG with two variables versus IRSG with only one variable in recording system. The grating proposed in [7] recorded by two spherical waves also has only one variable in the recording system. When this grating is designed to a minimum astigmatism, the simulated spot diagrams are similar with Fig. 2(b). Therefore, we did not give a detailed design and simulation of the grating recorded by two spherical waves.

So far, we have used the ideal toroidal wave for the design and simulation. When a cylindrical lens is placed behind a spherical wave to generate an approximate toroidal wave for actual fabrication, error occurs, which will be analyzed in Sec. 4.

#### 3.2 Experimental result of ARSG and CRSG

To evaluate our grating experimentally, an ARSG and a CRSG were fabricated and their performances were compared by the spectral images taken by CCD.

As shown in Fig. 3(a), we set up an exposure system to record ARSG. The laser beam was split into two by a polarization beam splitter (PBS) and the resulting two beams entered the spatial filters, generating two spherical waves. One of the spherical waves passed through a cylindrical lens and became an approximate toroidal wave. The shape and location parameters of the cylindrical lens were shown in Fig. 3(b). Other parameters of the exposure system were given in Sec. 3.1. The CRSG was recorded by a common exposure system of symmetric dual plane waves.

Before exposure, we coated a 100nm layer of photoresist (Shipley 9912) on the concave blanks. The intensities of the two exposure beams were measured by a photodiode power sensor and adjusted to be basically the same. After 25 seconds of exposure and about 20 seconds of development, we obtained a mask grating with the groove depth about 100 nm and the duty cycle about 0.45. At last, a 200-nm-thick aluminum layer was coated by evaporation technique on the mask grating. We selected an ARSG and a CRSG with similar groove shape through atomic force microscopy (AFM) images for the experiment, which guaranteed that the diffraction efficiencies of the gratings are substantially the same.

In order to compare the spectral resolution and spectral intensity of the two gratings, we set up a detection system to take the photos of the spectral images. The geometry of the detection system is similar as Fig. 1(a). A width adjustable entrance slit was put at point *A*. The aperture height of the slit was 1 mm, and the slit width was adjustable from 5 μm to 6 mm. The light emitted from a mercury lamp passed through a light collecting system and focused on the slit, becoming the light source of the detection system. Characteristic spectral lines 365.0, 365.4, and 366.3 nm of the mercury lamps were chosen for detection. The detector was a 1280 × 1024 (horizontal pixels × vertical pixels) 2D CCD, with pixel size 5.3 μm × 5.3 μm. To accurately place the 2D CCD to the position tangent to the Rowland circle, we needed to fine-tune the position and the attitude of the CCD, until the best spectral resolution was observed.

Figures 4(a), 4(b) and 4(c) respectively give the photographed spectral images of ARSG with the slit width of 107 μm, denoted as ARGS_{1}, of CRSG with the same slit width, and of ARSG with the slit width of 28 μm, denoted as ARSG_{2}. To show the energy distribution clearly, we used the sum of the gray values of each pixel column within a 200-μm-wide vertical strip (this corresponds to the use of a linear array CCD with the pixel height of 200 μm) to describe the spectral energy. The spectral energy distribution is shown as Fig. 4(d). The full width at half maximum (FWHM) of the spectral energy distribution for given spectral line is used to represent the spectral resolution, and the peak value of this curve to represent the spectral intensity.

In order to compare the spectral intensities of ARSG and CRSG under the similar spectral resolutions, the slit widths were adjusted to be both 107 μm in Figs. 4(a) and 4(b). As shown in Fig. 4(d), the FWHMs of ARSG_{1} and CRSG at *λ* = 365.4 nm are 0.075 nm and 0.071, respectively, almost the same. However, the spectral intensities of ARSG_{1} and CRSG at *λ* = 365.4 nm are 8108 and 1729, respectively. In another words, the spectral intensity of ARSG_{1} reaches about 4.5 times that of CRSG at *λ* = 365.4 nm, and the FWHM of ARSG_{1} is just larger than that of CRSG by about 5%.

To compare the spectral resolutions of ARSG and CRSG with similar spectral intensities, we reduced the slit width of ARSG from 107 μm to 28 μm, obtaining Fig. 4(c). Now for Hg 365.4 nm line, the spectral intensities of CRSG and ARSG_{2} are roughly the same, are 1729 and 1672, respectively. Meanwhile, as shown by the red and blue curves in Fig. 4(d), the spectral resolutions of CRSG and ARSG_{2} are 0.071 nm and 0.050 nm, respectively, which means that the spectral resolution of ARSG_{2} is about 1.4 time better than that of CRSG. The spectral doublet in Fig. 4(d) can be clearly distinguished by ARSG_{2}.

The experimental comparison above was given mostly to ensure the feasibility of the fabrication. The contrast values between the properties of ARSG and CRSG will change when the area of the grating or the exposure time of the CCD is changed. As the experimental results are consistent with the simulation results, we recommend to evaluate our method by using the simulation result in Sec. 3.1.

In conclusion, with the same slit width, ARSG has a higher spectral intensity than CRSG and their spectral resolutions are similar, which helps the spectrometer with an ARSG to obtain lower spectral detection limits. Moreover, as the spectral intensity can be sacrificed to obtain a better spectral resolution by reducing the slit width, ARSG has a larger spectral intensity margin than CRSG for sacrifice to improve the spectral resolution.

## 4. Error analysis

When the gratings are applied to the spectrometers equipped with liner array CCD, considering that the diameter of the Rowland circle is 750mm and the length of the linear array CCD is 19.3 mm, the spectral resolution error caused by the incomplete overlap of the linear array CCD and the Rowland circle is calculated to be less than 0.5 pm and can be negligible.

When the ARSG is recorded, the cylindrical lens shown in Fig. 3(b) will introduce aberrations into the toroidal recording wave, which leads to errors of the spectral images. We simulated the actual exposure system by Zemax and drew the spectral images of ARSG as shown in Fig. 5. The toroidal recording wave was simulated obeying the mounting shown in Fig. 3(b). Other parameters used to draw Fig. 5 are the same as those for drawing Fig. 2(b). Comparing the spot diagrams shown in Figs. 5 and 2(c), we can safely conclude that the shapes and sizes of the spectral lines of Fig. 5 are almost the same as those of Fig. 2(c). Therefore, the errors of the spectral images caused by the cylindrical lens can be ignored.

In addition, errors will be introduced if the cylindrical lens deviates from the ideal position. We also simulated the effect of the position errors of the cylindrical lens. When the cylindrical lens moves within ± 2 mm from its ideal position, the best spectral image shifts but can be regained by fine-tuning the position of the CCD. The resulting relative errors of the spectral resolution is less than 3%, which does not have a big impact on practical applications.

## 5. Conclusion

We proposed to record the Rowland-circle spherical grating by a spherical wave and a toroidal wave. In [1] and [7], only one variable is available for design, so, inferred from Eq. (9), only one of the astigmatism term or sagittal coma term can be corrected to minimum. However, in our method, use of the toroidal wave in recording system provides one more variable to correct both the astigmatism term and the sagittal coma term to minimum. Thus, the spectral lines of ARSG will be shorter and less curved than those of CRSG. When designing an ARSG, we recommend to employ Eq. (8) as the objective function to firstly optimize the mounting parameters, then the recording parameters of ARSG can be calculate by solving Eq. (9).

We designed and fabricated an ARSG and a CRSG for the Rowland-circle spectrometer equipped with linear array CCD. As the ARSG was designed to upgrade a commercial spectrometer, we ignored the first design step and used the origin mounting parameters of the spectrometer. We used a 2D CCD to photographed the spectral images and a virtual linear array CCD to calculate the spectral resolutions and intensities. The experimental results show that the spectral intensity of the ARSG_{1} is about 4.5 times that of the CRSG, while the differences of spectral resolutions between ARSG_{1} and CRSG are less than 0.005 nm. On the other hand, when the spectral intensity of the ARSG_{2} is comparable to that of the CRSG by varying the width of the slit, the spectral resolution of the ARSG is about 1.4 times better than that of the CRSG.

## References

**1. **X. Chen and L. Zeng, “Astigmatism-reduced spherical concave grating holographically recorded by a cylindrical wave and a plane wave for Rowland circle mounting,” Appl. Opt. **57**(25), 7281–7286 (2018). [CrossRef] [PubMed]

**2. **F. Masuda, H. Noda, and T. Namioka, “Design and performance of toroidal holographic gratings,” J. Spectrosc. Soc. Jpn. **27**(3), 211–223 (1978). [CrossRef]

**3. **https://www.spectro.com/products/icp-oes-aes-spectrometers/spectroblue-icp-oes.

**4. **C. Montero-Orille, X. Prieto-Blanco, H. González-Núñez, and R. de la Fuente, “Design of Dyson imaging spectrometers based on the Rowland circle concept,” Appl. Opt. **50**(35), 6487–6494 (2011). [CrossRef] [PubMed]

**5. **T. Namioka, “Theory of the ellipsoidal concave grating. I,” J. Opt. Soc. Am. **51**(1), 4–12 (1961). [CrossRef]

**6. **M. Duban, “Holographic aspheric gratings printed with aberrant waves,” Appl. Opt. **26**(19), 4263–4273 (1987). [CrossRef] [PubMed]

**7. **B. J. Brown and I. J. Wilson, “Holographic grating aberration correction for a Rowland circle mount I,” Opt. Acta (Lond.) **28**(12), 1587–1599 (1981). [CrossRef]

**8. **B. J. Brown and I. J. Wilson, “Holographic grating aberration correction for a Rowland circle mount II,” Opt. Acta (Lond.) **28**(12), 1601–1610 (1981). [CrossRef]