sensors

Letter High-Performance Ultra-Thin Optical Design Based on Coddington’s Equations

Zhiwei Feng 1, Guo Xia 2,3,4,*, Rongsheng Lu 1, Xiaobo Cai 1, Hao Cui 1 and Mingyong Hu 2,3,4

1 School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei 230009, China; [email protected] (Z.F.); [email protected] (R.L.); [email protected] (X.C.); [email protected] (H.C.) 2 Academy of Opto-Electric Technology, Hefei University of Technology, Hefei 230009, China; [email protected] 3 Special Display and Imaging Technology Innovation Center of Anhui Province, Hefei 230009, China 4 National Engineering Laboratory of Special Display Technology, Hefei 230009, China * Correspondence: [email protected]

Abstract: A unique method to design a high-throughput and high-resolution ultrathin Czerny–Turner (UTCT) spectrometer is proposed. This paper reveals an infrequent design process of based on Coddington’s equations, which will lead us to develop a high-performance spectrometer from scratch. The spectrometer is composed of cylindrical elements except a planar grating. In the simulation design, spot radius is sub-pixel size, which means that almost all of the energy is collected by the detector. The spectral resolution is 0.4 nm at central and 0.75 nm at edge wavelength when the width of slit is chosen to be 25 µm and the groove density is 900 lines/mm.

Keywords: optical design; Czerny–Turner spectrometer; cylindrical lens

  1. Introduction Citation: Feng, Z.; Xia, G.; Lu, R.; Cai, X.; Cui, H.; Hu, M. Spectrometers play an important role in spatially resolved ultrashort pulse measure- High-Performance Ultra-Thin ment, frequency domain optical coherence tomography, identification of substances and Spectrometer Optical Design Based remote sensing and other fields [1–4]. In general, a spectrometer is composed of a slit, a on Coddington’s Equations. Sensors collimating mirror, a focusing mirror, a dispersing component and a detector. With the 2021, 21, 323. https://doi.org/ diversified developing of application, the miniaturization of spectrometers has been put 10.3390/s21020323 forward. Miniaturization has the advance to increase portability and make it possible to instant measure. Specific applications require miniaturization and miniaturization will Received: 11 December 2020 lead to more applications. Spectrometers are developing rapidly towards miniaturization Accepted: 4 January 2021 with high resolution and high throughput. Published: 6 January 2021 Miniaturization of spectrometers has been studied extensively in recent years and many different approaches have been proposed. Freeform surface is a hot topic in optical Publisher’s Note: MDPI stays neu- design because of its more degrees of freedom. A series of methods based on freeform tral with regard to jurisdictional clai- surface to enhance the compactness of spectrometer have been proposed [5,6]. Jacob ms in published maps and institutio- Reimers et al. introduced freeform surface to eliminate aberrations and designed a freeform nal affiliations. surface spectrometer based on the Offner–Chrisp spectrometer [5]. Freeform design was shown to enable 3× spectral-band or 2× spatial broadening or a 5× reduction in volume when compared to a non-freeform counterpart. Feng et al. also presented a compact Copyright: © 2021 by the authors. Li- imaging spectrometer composed of freeform mirrors [7]. In particular, curved prisms were censee MDPI, Basel, Switzerland. used for dispersing beam. Compared with the conventional system, the volume of the This article is an open access article system was reduced by 60% and covered a wide ranging from 400 nm to 2500 nm. distributed under the terms and con- The grating has been widely used to be the dispersing component due to its ditions of the Creative Commons At- high resolution and good stability. The compactness can also be enhanced by utilizing tribution (CC BY) license (https:// gratings flexibly. Pang demonstrated a compact high-resolution spectrometer by using creativecommons.org/licenses/by/ two plane gratings, a fixed grating and a rotating grating [8]. is first diffracted from 4.0/).

Sensors 2021, 21, 323. https://doi.org/10.3390/s21020323 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 323 2 of 10

the fixed grating to the rotating grating, then the rotating grating diffracts the light back to the fixed grating according to the Littrow diffraction method, and finally the light is diffracted a second time by the fixed grating and focused into the fiber. Triple occurs in the approach and improves resolution and compactness. Li et al. tried to use a plano-concave holographic grating to complete a portable spectrometer design [9]. Planar waveguide is another important method for the miniaturization of spectrometers [10,11]. Faraji-Dana et al. proposed a concept of folded metasurface optics by demonstrating a compact spectrometer. The spectrometer is with a thickness of 1 mm and a volume of 7 cubic millimeters [10]. Using a concave toroidal mirror is also a decent way to improve the miniaturization of spectrometers [12]. Currently, there are microspectrometer products on the market, such as C12880MA and C12666MA developed by Hamamatsu [13]. The overall size is 20.1 × 12.5 × 10.1 mm while the maximum resolution is 15 nm. The methods mentioned above are all aimed at miniaturizing the spectrometer and improving its compactness and portability. These methods are ingenious and remarkable. However, there are still some shortcomings when applied in some situations, such as high resolution but short spectral range, small size but low resolution and good performance but difficult to process. In many spectrometer configurations, tangential image plane and sagittal image plane vary in positions, resulting in astigmatism [14,15]. The astigmatism is inherent in a spectrometer and cannot be corrected by the spherical mirrors themselves. In most cases, the Czerny–Turner spectrometer has the best structure because its structure and adjustment are simple, and its resolution can be improved and the astigmatism can be compensated by modifying the optical system. The various methods to correct astigmatism in Czerny–Turner spectrometers have been investigated, such as using a toroidal lens [16], using an off-the-shelf cylindrical lens [17] and freeform surface [5,7]. As a precursor to the content of this paper, Wu et al. showed how to eliminate astigmatism [18,19]. Unlike conventional Czerny–Turner spectrometers, the spherical collimating mirror was replaced by two cylindrical mirrors. The equations of positions are constructed by adjusting the distance between two cylindrical mirrors so that astigmatism was eliminated. Compared with the conventional Czerny–Turner spectrometers, astigmatism was well corrected and the root-mean-square (RMS) spot radius is reduced to less than 18 µm. In this paper, we propose a high-performance ultra-thin Czerny–Turner (UTCT) spec- trometer. The spectrometer is designed for improving portability significantly. Building on the previous work done by Wu et al., we trace Coddington’s equations from the ground up and present a complete optical design process, which covers the starting point and the ending point. Coddington’s equations are a direct guide to the design of a high- performance spectrometer. This paper is structured in 5 Sections. In Section2, the principle of the method is described, the original Coddington’s equations are given, the complete process of designing is proposed, and the fundamental structure of the spectrometer is Czerny–Turner spec- trometer is described. In Section3, UTCT is designed, the parameters of the optical system are reasonably set and aberrations are corrected in turn (finally, the device F-number is 4 and the spectral range is 670 nm 1130 nm). In Section4, the design objective is verified in relation to high-throughput, high-resolution and optical size. Section5 is the conclusion and future expectations about our work.

2. Materials and Methods Spectrometers are usually composed of light source, a slit, a collimating mirror, a dispersion element, a focusing mirror and a detector. Light beam emitted by light source passes through the slit and reaches the collimating mirror. The beam reflected from the collimating mirror is dispersed and then focused on the detector by the focusing mirror. In a conventional spectrometer, the focal length in sagittal plane of the off-axis spheri- cal mirror is different from that in tangential plane, which is the origin of astigmatism. A practical way to eliminate the difference between tangential and sagittal focus is to split the Sensors 2021, 21, x FOR PEER REVIEW 3 of 10

In a conventional spectrometer, the focal length in sagittal plane of the off-axis spher- ical mirror is different from that in tangential plane, which is the origin of astigmatism. A practical way to eliminate the difference between tangential and sagittal focus is to split the curvature in the sagittal plane from the curvature in the tangential plane. UTCT is formed with a slit, a sagittal collimating lens, a tangential collimating mirror, a diffraction Sensors 2021, 21, 323 grating, a tangential focusing mirror, a sagittal focusing lens and3 of 10 a linear array charge coupled device (CCD), as is shown in Figure 1. The slit is located at focal plane of the tangentialcurvature collimating in the sagittal plane mirror. from theThe curvature conical in the diverg tangentialent plane. light UTCT is located is formed at a distance from the tangentialwith a slit, collimating a sagittal collimating mirror, lens, so a tangential there will collimating be a certain mirror, a diffractionheight in grating, sagittal plane. In order to a tangential focusing mirror, a sagittal focusing lens and a linear array charge coupled achievedevice the (CCD), ultra-thin as is shown design in Figure intent,1. The slit the is locatedsagittal at focal collimating plane of the lens tangential is placed close to the slit. It meanscollimating that mirror. the height The conical of divergentsagittal light plane is located is getting at a distance smaller from the while tangential the length of tangential collimating mirror, so there will be a certain height in sagittal plane. In order to achieve planethe is ultra-thin no longer design intent,than theconventional sagittal collimating spectr lens isometers. placed close The to the light slit. It meansbeam passes through the sagittalthat thecollimating height of sagittal lens plane and is the getting tangential smaller while co thellimating length of tangential mirror planein turn, is so the beam is col- limatedno longer to be than parallel conventional in the spectrometers. tangential The plane light beam and passes the sagittal through the plane. sagittal Then the beam is dif- collimating lens and the tangential collimating mirror in turn, so the beam is collimated to fractedbe parallel by the in the grating. tangential Finally, plane and thethe sagittal beam plane. is Thenfocused the beam on is the diffracted detector by the by the tangential fo- cusinggrating. mirror Finally, and the beamthe sagittal is focused onfocusing the detector lens, by the which tangential are focusing both mirrorcylindrical and optical elements. the sagittal focusing lens, which are both cylindrical optical elements.

Sagittal Tangential Collimating Lens Collimating Mirror

Tangential Slit Focusing Mirror

Diffraction Grating Linear Array CCD Sagittal Focusing Lens

Figure 1. Schematic diagram of an ultra-thin Czerny–Turner (UTCT) spectrometer, consisting of Figurecylindrical 1. Schematic lenses and diagram mirrors, where of XYan plane ultra-thin is the tangential Czerny–T plane ofurner UTCT while(UTCT) XZ plane spectrometer, is the consisting of cylindricalsagittal plane lenses of UTCT. and mirrors, where XY plane is the tangential plane of UTCT while XZ plane is the sagittalIn an plane optical of system UTCT. a tangential plane and a sagittal plane can be identified. Cod- dington’s equations are an excellent tool in this case. The relation between object distance (OD)In an and optical image distance system (ID) cana tangential be deduced for plane tangential and plane a andsagittal sagittal plane using can be identified. Cod- Coddington’s equations. Through this method, the relative position of each optical element dington’scan be directly equations determined, are an and excellent a high-performance tool in spectrometer this case. canThe be relation designed. Thebetween object distance (OD)theoretical and image schematic distance diagrams (ID) of UTCT can inbe tangential deduced plane for and tangential sagittal plane plane are shown and sagittal plane using in Figure2. T or S represents the optical surface of tangential plane or sagittal plane and Coddington’sthe number of equations. the subscript representsThrough the this sequence method, number ofthe the relative optical plane. position T1(S1) of each optical ele- mentrepresents can be the directly convex surface determined, of the sagittal and collimating a high cylindrical-performance lens. T2(S spectrometer2) represents can be designed. the reflective surface of the tangential collimating mirror. T (S ) represents the working The theoretical schematic diagrams of UTCT in3 3tangential plane and sagittal plane are surface of the planar . T4(S4) represents the reflective surface of the showntangential in Figure focusing 2. mirror.T or S Trepr5(S5)esents represents the the optical convex surface surface of the of sagittal tangential focusing plane or sagittal plane and cylindricalthe number lens. r representsof the subscript the curvature radiusrepresents of optical the surface sequence and d1 and dnumber2 represent of the optical plane. the center thickness of the lens. l1 is the distance between T1(S1) and T2(S2) while l2 is the T1(S1distance) represents between Tthe4(S4 )convex and T5(S5 surface). of the sagittal collimating cylindrical lens. T2(S2) rep- resents the reflective surface of the tangential collimating mirror. T3(S3) represents the working surface of the planar diffraction grating. T4(S4) represents the reflective surface of the tangential focusing mirror. T5(S5) represents the convex surface of the sagittal focus- ing cylindrical lens. r represents the curvature radius of optical surface and d1 and d2 rep- resent the center thickness of the lens. l1 is the distance between T1(S1) and T2(S2) while l2 is the distance between T4(S4) and T5(S5).

Sensors 2021, 21, x FOR PEER REVIEW 4 of 10 Sensors 2021, 21, 323 4 of 10

Slit Detector T1 T2 T3 T4 T5

r2 r4 W

d1 l1 l2 ( a )

S1 S2 S3 S4 S5

r1 r5 H

( b )

Figure 2. The theoretical schematic diagram: (a) UTCT in tangential plane (b) UTCT in sagittal plane. Figure 2. The theoretical schematic diagram: (a) UTCT in tangential plane (b) UTCT in sagittal plane. According to Coddington’s equations, the relation of OD and ID in tangential plane and sagittal plane can be expressed as According to Coddington’s equations, the relation of OD and ID in tangential plane n0 n n0 cos I0 − n cos I and sagittal plane can be expressed0 = + as (1) l s ls r ′′′− n0 cos2 I0 n cosnnn2=+I n0 coscosI0 − InIn cos cosI = + (2) (1) l0 l′ r t llsst r 0 where ls and ls represents sagittal object distance (SOD) and sagittal image distance (SID), 0 respectively, lt and lt representsnInInInI tangential′′cos22 object cos distance ′′ cos (TOD)− and cos tangential image distance (TID), respectively, I and I0 represent=+ the angle of incidence and the angle of (2) 0 ′ , n and n represent the refractivelltt index of object space r and image space, and r represents the radius of curvature of the surface. When Equations (1) and (2) are applied to wherea mirror, ls and the l angles’ represents of reflection sagittal could beobject viewed distance as the angle (SOD) of refraction.and sagittal According imageto distance (SID), respectively,Snell’s law, the lt angleand oflt’ refractionrepresents and tangential the angle of reflectionobject distance have the same(TOD) value, and but tangential the image distancedirection (TID), is opposite respectively, (I0 = −I)[20 I]. and I’ represent the angle of incidence and the angle of re- fraction,The n sagittal and n’ collimating represent lens the is cylindrical. refractive To reduceindex theof lossobject of energy, space the and flat surfaceimage space, and r of the lens needs to be aligned with the slit and be cling to the slit. The slit is orthogonal to representsthe tangential the planeradius of theof cylindricalcurvature lens. of the For su convexrface. surface When of Equations the sagittal collimating(1) and (2) are applied to lens,a mirror, the SID the and angle TID can of be reflection expressed ascould be viewed as the angle of refraction. According to Snell’s law, the angle of refraction and the angle of reflection have the same value, but 1 n0 1 − n0 n0 1 − n0 the direction is opposite (I0’ == −I) [20].+ = + (3) l s ls r1 d1 r1 The sagittal collimating1 lens1 is cylindrical. To reduce the loss of energy, the flat sur- 1 n0 face of the lens needs to be aligned with= the slit and be cling to the slit. The(4) slit is orthog- l0 l onal to the tangential plane of the cylindricalt1 t1 lens. For convex surface of the sagittal colli- 0 0 matingwhere lens,l s1 and thel t 1SIDare sagittaland TID image can distance be expressed and tangential as image distance, respectively, d1 is the thickness of sagittal collimating lens. The derivation refers to the principal ray in ′′′′−− sagittal plane and the principal ray hits11 on=+nnnn the cylindrical =+ lens 1 vertically. Therefore, the incidence angle of the convex surface (S1)′ is 0. It is easy to realize that the lens acts as a (3) llss r111 d r plano-convex lens in the sagittal plane and11 acts as a parallel plate in the tangential plane,

as the object distance, ls1 and lt1 are actually equal, and the numerical value is the thickness 1 n′ of the sagittal collimating lens. The slit is placed at the= focus point of sagittal collimating (4) lens and tangential collimating mirror so that thell collimated′ beams can be obtained. tt11

where l′ and l′ are sagittal image distance and tangential image distance, respectively, s1 t1 d1 is the thickness of sagittal collimating lens. The derivation refers to the principal ray in sagittal plane and the principal ray hits on the cylindrical lens vertically. Therefore, the incidence angle of the convex surface (S1) is 0. It is easy to realize that the lens acts as a plano-convex lens in the sagittal plane and acts as a parallel plate in the tangential plane, as the object distance, l and l are actually equal, and the numerical value is the thick- s1 t1

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Similarly, the tangential collimating mirror acts as a spherical mirror in the tangential plane and acts as a flat mirror in sagittal plane. It is easy to derive SID and TID in the following equations: −1 1 1 0 = = 0 (5) l s2 ls2 l s1 −1 1 −2 1 −2 0 = + = 0 + (6) l t2 lt2 r2 cos I2 l t1 + l1 r2 cos I2

where l1 is the distance between the tangential collimating mirror and sagittal collimating lens, I2 represents the incident angle of tangential collimating mirror, and r2 represents the radius of curvature of the tangential collimating mirror. The slit is placed at the focus point so that the collimating beams can be obtained. The third surface is a planar diffraction grating. The grating can be regarded as a flat mirror in sagittal plane while diffracts the beam in tangential plane:

−1 1 1 0 = = 0 (7) l s3 ls3 l s2

−1 cos2 i 1 cos2 i 1 = = 0 2 2 0 (8) l t3 cos θ lt3 cos θ l t2 where i and θ are the incident and diffraction angles of the grating. The tangential focusing mirror is cylindrical, so the following equations can be ob- tained: −1 1 1 0 = = 0 (9) l s4 ls4 l s3 −1 1 −2 1 −2 0 = + = 0 + (10) l t4 lt4 r4 cos I4 l t3 r4 cos I4

where I4 represents the incident angle, it is also the angle of inclination of the tangential focusing mirror. r4 represents the radius of curvature of the mirror. For the last surface, the sagittal focusing lens, the distance can be expressed as

n0 1 n0 − 1 1 n0 − 1 0 = + = 0 + (11) l s5 ls5 r5 l s4 r5

n0 1 1 0 = = 0 (12) l t5 lt5 l t4 − l2

where l2 is the distance between the tangential focusing mirror and sagittal focusing lens. The tangential and sagittal image distances of the optical system can be derived as

n0r r d = 0 = 1 5 1 LS l s5 0 0 (13) (n − 1)(r1 + r5)d1 − r1r5n

0 0 0 n r2 cos I2r4 cos I4(n l1 + d1) 0 LT = l t5 = 2 − n l2 (14) r I (n0l + d ) + cos i r I ( n0l + d − r I ) 2 2 cos 2 1 1 cos2 θ 4 cos 4 2 1 2 1 2 cos 2

Astigmatism can be eliminated if LS = LT, so l2 can be derived as

r cos I r cos I (n0l + d ) r r d = 2 2 4 4 1 1 − 1 5 1 l2 2 0 0 (15) r I (n0l + d ) + cos i r I ( n0l + d − r I ) (n − 1)(r1 + r5)d1 − r1r5n 2 2 cos 2 1 1 cos2 θ 4 cos 4 2 1 2 1 2 cos 2 Thus the distance relationship between the optical elements of the spectrometer is derived.

3. Results An UTCT is designed according to the principle in Section2. The principle is based on Coddington’s equations. In this section, the parameters that were chosen for the Sensors 2021, 21, 323 6 of 10

proposed spectrometer are further elaborated according to the previously established UTCT optical model. The tangential plane of an optical system is more familiar than its sagittal plane. For UTCT, the tangential collimating mirror determines the collection of the incident light and the tangential focusing mirror determines the linear dispersion of the system. In the design case, the F number is chosen to be 4 (the option can completely collect beams with NA of 0.12). f 2 can be inferred to be 40 mm when entrance pupil is 10 mm. In the design, it is desirable to cover the spectrum all over the photosensitive surface of the detector. The detector we choose is a linear array charge-coupled device (CCD), of which the size is 28.67 mm × 0.2 mm (pixel size is 14 µm × 200 µm and pixel number is 2048). If the spectrum covers the CCD photosensitive surface, f 4 can be derived from the formula

Ld cos θ f4 = (16) λ2 − λ1 where L represents the sum of all pixel lengths of the CCD, d represents the groove spacing of the grating, λ1 and λ2 represent the edge at each edge. In the case, the spectral range is 670–1130 nm and the groove spacing is 1.111 µm/line. Diffraction beam of the grating can be determined by the grating equation:

d(sin i + sin θ) = mλ (17)

where m is the diffraction order and is chosen as 1. In order to restrain the stray light, a certain interval should be kept between the collimating mirror and the focusing mirror in the tangential plane. For this reason, the incident angle i is set to 45◦, which means the ◦ diffraction angle θ would be 5.906 . After substituting the values into Equation (16), f 4 is approximated to 68 mm. Aberration will affect the performance of the spectrometer, and the design method mentioned in Section2 can effectively avoid the loss of energy caused by astigmatism. If the incident angle of tangential collimating mirror (I2) is too small, the grating will be very close ◦ to the sagittal collimating lens. To avoid this trouble, the I2 is set to be 14.5 . According to Shafer [21], the coma at the central wavelength can be eliminated by the formula:

sin I r2 cos3 I cos3 i 4 = 4 4 2 3 3 (18) sin I2 r2 cos I2 cos θ ◦ So I4 can be obtained, I4 = 14.968 . The spherical aberration is inherent in a spherical mirror. The max wavefront aberra- tion (WS(max)) due to the spherical aberration can be expressed as Equation (19):

y 4 W = max (19) S(max) 8r3

where ymax is the half-aperture of the spherical mirror and r represents the spherical radius of curvature of the spherical mirror [22]. Rayleigh principle can be used to determine the effect of the spherical aberration: When the wave aberration produced by the spherical aberration is less than λ/4, the effect of the spherical aberration on the imaging can be ignored. So the limitation can be expressed as Equation (20):

y 4 λ max ≤ (20) 8r3 4 When using the method described in this paper, the initial parameters of the system can be derived more accurately. After the radius of curvature is determined, the optimized variables are only the distance between mirrors. The fixed parameters by means of the Coddington’s Equations are shown in Table1. The optimized parameters are shown in Table2. The parameter α represents the tilt angle of the detector. The tilt angle of the Sensors 2021, 21, 323 7 of 10

detector α is set to 0◦ for the convenience of derivation when the optical system model is built in Section2. It is changed to be a variable during optimization to compensate for the aberration caused by wide spectral region.

Table 1. Parameters set by means of the Coddington s Equations (fixed parameters).

Fixed Parameter Value

r1 (mm) 2.00 r2 (mm) 80.00 r4 (mm) 136.00 r5 (mm) 2.00 d (µm/line) 1.111 d1 (mm) 5.93 d2 (mm) 5.93 ◦ I2 ( ) 14.500 ◦ I4 ( ) 14.968 i (◦) 45.000 θ (◦) 5.906

Table 2. Parameters set by means of the optimization routine.

Parameter Initial Value Optimized Value ◦ Sensors 2021, 21, x FOR PEER REVIEW α( ) 0.000 0.900 8 of 10 l1 (mm) 34.82 34.80 l2 (mm) 61.78 61.57

4. Discussion results show that all of the light beam can be collected by the detector, demonstrating that In this section we present a design case of UTCT. The initial parameters are calculated UTCT is with high inthroughput. Section3. The optical system is modeled by using an optical tracing software. Figure3 shows the layout of UTCT. The height of optical elements in sagittal plane is under 2 mm.

Figure 3. Layout of UTCT, of which the height in sagittal plane is under 2 mm. Figure 3. Layout of UTCT, of which the height in sagittal plane is under 2 mm. The spot diagram is one of the main evaluation indexes. The performance of UTCT can be evaluated by analyzing the size of spot diagrams. The spot diagrams of several specific wavelengths are shown in Figure4. It shows that the imaging effect of the center wavelength is excellent as well as the imaging size of the edge wavelength on both sides is kept within 200 µm. As mentioned in Section3, the height of a pixel is 200 µm. The results show that all of the light beam can be collected by the detector, demonstrating that UTCT is with high throughput.

Figure 4. Spot diagrams at several specific wavelengths.

For better comparison, we also design a Czerny–Turner spectrometer with a cylin- drical focusing lens (CCT). The sagittal collimating lens is removed and the tangential collimating mirror is replaced by a spherical mirror with the same radius of curvature. Similarly, the tangential collimating mirror is replaced by a spherical mirror with the same radius of curvature and an identical sagittal focusing lens is placed in front of the focal plane. In order to ensure the reasonableness of the comparison, the parameters are opti- mized in the same way as UTCT. The result of the comparison is shown in Figure 5.

Sagittal-direcetion of CCT 180 Tangential-direcetion of CCT Sagittal-direcetion of UTCT 170 Tangential-direcetion of UTCT 160 150 140 130 120

30

RMS Spot Radius in μm in Radius Spot RMS 20

10

0 0.670 0.716 0.762 0.808 0.854 0.900 0.946 0.992 1.038 1.084 1.130 Wavelength in μm Figure 5. RMS spot radius versus wavelength for UTCT and CCT.

In Figure 5, we associate RMS spot radius with wavelengths to evaluate and analyze the effect of energy . Obviously, the RMS spot radius is reduced to less than 25 μm in the wavelength range 670 nm–1130 nm in tangential plane and the RMS spot

Sensors 2021, 21, x FOR PEER REVIEW 8 of 10

Sensors 2021, 21, x FOR PEER REVIEWresults show that all of the light beam can be collected by the detector, demonstrating8 of 10 that

UTCT is with high throughput.

results show that all of the light beam can be collected by the detector, demonstrating that UTCT is with high throughput.

Figure 3. Layout of UTCT, of which the height in sagittal plane is under 2 mm. Sensors 2021, 21, 323 8 of 10 Figure 3. Layout of UTCT, of which the height in sagittal plane is under 2 mm.

Figure 4. Spot diagrams at several specific wavelengths.

FigureFor 4. 4.Spot Spotbetter diagrams diagrams comparison, at severalat several specific we specific also wavelengths. wavelengths.design a Czerny–Turner spectrometer with a cylin- drical focusing lens (CCT). The sagittal collimating lens is removed and the tangential ForFor better better comparison, comparison, we also we design also design a Czerny–Turner a Czerny–Turner spectrometer spectrometer with a cylindrical with a cylin- collimatingfocusing lens (CCT).mirror The is sagittalreplaced collimating by a spherical lens is removed mirror and with the tangential the same collimating radius of curvature. drical focusing lens (CCT). The sagittal collimating lens is removed and the tangential Similarly,mirror is replaced the tangential by a spherical collimating mirror with mirror the same is replaced radius of by curvature. a spherical Similarly, mirror the with the same collimating mirror is replaced by a spherical mirror with the same radius of curvature. radiustangential of collimatingcurvature mirrorand an is replacedidentical by sagittal a spherical focusing mirror withlens theis placed same radius in front of of the focal Similarly,curvature and the antangential identical collimating sagittal focusing mirror lens is is replaced placed in by front a spherical of the focal mirror plane. with In the same plane. In order to ensure the reasonableness of the comparison, the parameters are opti- radiusorder to of ensure curvature the reasonableness and an identical of the comparison, sagittal focusing the parameters lens is areplaced optimized in front in the of the focal mizedplane.same way inIn the asorder UTCT. same to Theensure way result as the ofUTCT. reasonableness the comparison The result is of shownof the the comparison, in comparison Figure5. the is shownparameters in Figure are opti- 5. mized in the same way as UTCT. The result of the comparison is shown in Figure 5. Sagittal-direcetion of CCT 180 Tangential-direcetion of CCT Sagittal-direcetion Sagittal-direcetion of CCTof UTCT 170180 Tangential-direcetion Tangential-direcetion of CCTof UTCT Sagittal-direcetion of UTCT 160170 Tangential-direcetion of UTCT 150160 140150 130140 120130 120 30 30

RMS Spot Radius in μm in Radius Spot RMS 20

RMS Spot Radius in μm in Radius Spot RMS 20 10 10

0 0 0.670 0.716 0.762 0.808 0.854 0.900 0.946 0.992 1.038 1.084 1.130 0.670 0.716 0.762 0.808 0.854 0.900 0.946 0.992 1.038 1.084 1.130 Wavelength in μm Wavelength in μm FigureFigure 5. 5.RMS RMS spot spotspot radius radiusradius versus versus versus wavelength wavelength wavelength for UTCT for for UTCT and UTCT CCT. and and CCT. CCT.

InIn FigureFigure5, we5, we associate associate RMS spotRMS radius spot with radius wavelengths with wavelengths to evaluate and to analyzeevaluate and analyze the effectIn Figure of energy 5, we concentration. associate RMS Obviously, spot radius the RMS with spot wavelengths radius is reduced to evaluate to less than and analyze thethe25 µ effecteffectm in the of wavelengthenergyenergy concentration. concentration. range 670 nm–1130 Obviously, Obviously, nm in the tangential the RMS RMS spot plane spot radius andradius theis reduced RMSis reduced spot to less to thanless than 2525radius μμmm is in less the than wavelengthwavelength 5 µm in the range sagittalrange 670 plane670 nm–1130 nm–1130 as shown. nm Fromnm in intangential the tangential contrast plane of theplane RMSand and spotthe RMSthe RMS spot spot radius between UTCT and CCT in Figure5, the effect of astigmatism on energy loss is almost eliminated. The results demonstrate that the energy collected by the detector of UTCT will be more concentrated while the resolution remains the same level as CCT. What is more, the thickness of UTCT is obviously get thinner. Spectral resolution is another important index to evaluate the performance of a spec- trometer. Although spectral resolution is defined as the spectral bandwidth detected by a pixel, the effect of slit width should also be considered. In view of this, we calculate the line spread function (LSF) of the ultrathin spectrometer, and convolute the slit and pixel function with LSF. In the whole broadband range, we calculated the LSF of five typical wavelengths and completed the convolution. All results are shown in Figure6. Sensors 2021, 21, x FOR PEER REVIEW 9 of 10

radius is less than 5 μm in the sagittal plane as shown. From the contrast of the RMS spot radius between UTCT and CCT in Figure 5, the effect of astigmatism on energy loss is almost eliminated. The results demonstrate that the energy collected by the detector of UTCT will be more concentrated while the resolution remains the same level as CCT. What is more, the thickness of UTCT is obviously get thinner. Spectral resolution is another important index to evaluate the performance of a spec- trometer. Although spectral resolution is defined as the spectral bandwidth detected by a pixel, the effect of slit width should also be considered. In view of this, we calculate the line spread function (LSF) of the ultrathin spectrometer, and convolute the slit and pixel Sensorsfunction2021, 21, 323 with LSF. In the whole broadband range, we calculated the LSF of five typical9 of 10 wavelengths and completed the convolution. All results are shown in Figure 6.

Figure 6. The spectralFigure resolution 6. The spectral of the resolution UTCT at of typical the UTCT wavelengths. at typical wavelengths.

The full width at half maximum (FWHM) is used to characterize the spectral resolution The full width ofat ultrathin half maximum spectrometer. (FWHM) In Figure is6, weused calculate to characterize the FWHM of the five spectral typical wavelengths resolu- tion of ultrathin spectrometer.after convolution, In Figure and the 6, results we calculate show that thethe resolution FWHM ofof thefive whole typical band wave- is within lengths after convolution,0.75 nm, and the the resolution resultsof show the central that wavelengththe resolution is even of reach the 0.4 whole nm. band is within 0.75 nm, and5. the Conclusions resolution of the central wavelength is even reach 0.4 nm. Based on Coddington’s equations, a high-throughput and high-resolution spectrome- 5. Conclusions ter, UTCT, is proposed in this paper. The theoretical analysis of throughput and resolution Based on Coddington’sare detailed equations, in sagittal plane a high and tangential-throughput plane, and respectively. high-resolution The results of spectrom- the case show that UTCT achieves a full spectral resolution better than 0.8 nm with a minimum thickness eter, UTCT, is proposedof 2 mm. in Thethis resolution paper. The of the th centraleoretical wavelength analysis (900 of nm) throughput is 0.4 nm. Compared and resolu- with a tion are detailed in sagittalconventional plane spectrometer, and tangential UTCT notplane, only respectively. achieves the level The of results spectral of resolution, the case but show that UTCT achievesalso obtains a full more spectral concentrated resolution energy atbetter the detector. than 0.8 Although nm with there a are minimum more optical elements in UTCT, UTCT is much easier to process than the spectrometer composed by thickness of 2 mm. Thecomplex resolution surfaces. of When the compared central wavelength to a waveguide (900 spectrometer, nm) is 0.4 the nm. size ofCompared UTCT may be with a conventionallarger spectrometer, but the resolution UTCT may not be only better achieves and the spectral the level region of may spectral be wider. resolution, UTCT reduces but also obtains morethe concentrated difficulty of processing energy andat the promotes detector. the developmentAlthough there of compact are more spectrometer optical to elements in UTCT, portableUTCT devices.is much easier to process than the spectrometer composed by complex surfaces. WhenAuthor Contributions:compared toConceptualization, a waveguide Z.F. spectrometer, and G.X.; methodology, the Z.F.size and of G.X.; UTCT validation, may R.L. be larger but the resolutionand M.H.; data may curation, be better X.C. and and H.C.; the writing—original spectral region draft preparation, may be Z.F.;wider. writing—review UTCT and editing, Z.F.; funding acquisition, G.X. All authors have read and agreed to the published version of the manuscript.

Sensors 2021, 21, 323 10 of 10

Funding: This research was funded by Key Research and Development Program of Anhui Province, grant number 1804d08020310. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data is contained within the article. Conflicts of Interest: The authors declare no conflict of interest.

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