CC BY
15
UDC 520.6.05
DOI: 10.18698/0236-3933-2021-4-139-151
METHOD FOR CALCULATING THE OFFNER COMPACT-SIZE SPECTROMETER
V.l. Zavarzin I.M. Zaitsev S.V. Yakubovskiy
Bauman Moscow State Technical University, Moscow, Russian Federation
Abstract
Significant advances in development of the optical wavelength range require high-quality optical systems to create optoelectronic equipment on their basis characterized by high speed and information capacity. A method for calculating a compact-size Offher spectrometer was developed having the advantages of its compactness, maintaining high optical characteristics and having relatively low cost in comparison with the large-size equipment. The method is based on using the Rowland circles and the coma and astigmatism correction in the image plane. Analytical expressions were obtained making it possible to calculate design parameters of the spectrometer optical scheme. Two examples of calculating optical systems for visible and infrared ranges were considered. Calculated systems were simulated in the Zemax software program. To evaluate the synthesized optical models image quality, the confusion spot radius in the image plane was used. It is demonstrated that the confusion spot radius value does not exceed the value of the radiation receiver pixel size in the considered spectral ranges. Optimization was carried out for the IR spectrometer according to overall dimensions in order to improve the design manufactur-ability. It is shown that the principles laid down in the method development are effective, and the method itself could be used in design and development of new small-size hyperspectral optoelectronic equipment
Keywords
Compact-size spectrometer, Offher scheme, wide spectral range
Received 11.05.2021 Accepted 08.06.2021 © Author(s), 2021
Introduction. Significant progress was achieved nowadays in the optical wavelength range development. New sources and optical radiation receivers, and high-quality optical systems were developed. Optoelectronic equipment is being created on their basis characterized by high speed and information capacity, which expands and increases the role of optical research in studying
the surrounding world. More and more, optical methods are introduced into the Earth remote sensing. Special place in the Earth remote sensing is occupied by optical equipment and methods of contactless study of the surveyed object characteristics. To obtain not only qualitative, but also quantitative information about the registered object, spectral analysis is used by hyperspectral equipment registering a spectral interval of electromagnetic radiation dozens of times wider than the human eye. Such equipment is often having large dimensions and weight, which technically complicates its use. But, in contrast to large-size optoelectronic equipment, compact-size devices have minor mass and size, due to which design, production and further maintenance cycle are carried out with lower costs in finance and resources once again emphasizing relevance, prospects and efficiency of using the compact-size devices [1].
Reliability of the optoelectronic equipment operation in the Earth remote sensing requires stability in space and time, which originates a number of severe restrictions regarding design and maintenance of technical characteristics at high-level during operation.
Problem statement. General trend in the optoelectronic equipment development lies in increasing optical characteristics and resolution, broadening the spectral range with a decrease in overall dimensions, weight and radiation losses. This places high demands to accuracy and quality of the design work. From the point of view of concept and methods, this currently implies calculation of optical systems of the Earth remote sensing optoelectronic equipment with severe restrictions on dimensions and weight in conditions of import substitution with limited financial and material resources.
Classical schemes and methods could no longer fully meet the ever-increasing requirements to optical system of the Earth remote sensing optoelectronic equipment. Therefore, creation of the compact-size multifunctional optoelectronic equipment became a priority. Creation of such equipment is possible on the basis of new structural and optical materials, fundamentally new optical systems, for example, such as the mirror lens with a minor number of surfaces. The Offner scheme is the most promising among all possible schemes for constructing spectrometers, because of its simplicity, minor dimensions and high optical characteristics in a wide spectral range. Using such schemes would provide possibility to register radiation in a wide spectral range from ultraviolet to the far IR range, including hyperspectral imaging with the number of spectral channels up to several hundred. Possibility to achieve high image quality in a place convenient for installing complex equipment for recording and analyzing images is of particular importance in such systems. Their design should provide reliable image protection from parasitic illumination.
Work objective is to elaborate methodology for designing the compact-size spectrometer optical system making it possible to optimize design solution and obtain high-quality image close to the diffraction. The confusion spot radius in the image plane is used as a criterion in the image quality evaluation. With ideal optical system and dispersing device, it is the confusion spot size that determines spectral and spatial resolution.
Spectrometer optical scheme calculation. Optical system of the spectrometer consists of lens and dispersive component, which role could be played by various elements in the form of a prism/set of prisms and a diffraction grating. In order to reduce the system overall size and minimize the chromatic aberration effect on image quality in selecting the lens components, mirrors are generally preferred. Spectrometer design built according to the Offner scheme (Fig. 1) is, in the general case, a three-mirror system with a convex diffraction grating on the second mirror, where the aperture diaphragm is also installed.
Calculation is based on using the Rowland circles [2-8] and the corresponding condition, under which it is sufficient to obtain image without a coma that the object point О (entrance slit) lies on the first Rowland circle, and at the same time the I final and intermediate images of the О point should lay on the Rowland circles of each scheme component (Fig. 2). Replacing the first concave mirror by the two separate ones with all the spherical elements installed on the Rowland circles with the z axis intersection point leads to a decrease in astigmatism. Point С is the circles' intersection point at the z-axis.
In Fig. 2 point I and angle cp' are the points of meridional image and the meridional angle.
Slit diaphragm
Fig. 1. Offner spectrometer optical scheme
Fig. 2. Rowland circles formed by three spectrometer scheme components
The designed spectral device should operate in the wavelength range from Xi flo where the main wavelength is selected as the average for this range or being correspondent to the maximum sensitivity of a radiation receiver. Then, the difference between extreme wavelengths of the operating spectral range is
&k = X2 - h- (1)
Let the required spectral resolution of the dk device be given. Hence the number of channels operating in the radiation receiver is determined as
AX
Nc =-. (2)
dX
Resolution element (pixel) size of the radiation receiver is Ô. And the size of the spectral image is
/Zsp =Ncd. (3)
Calculation is carried out for the k - -1 diffraction order, since in this case the dispersed radiation wave contains most of the light energy [2]. The main beam incidence angle on the third surface is S3. It is recommended to consider the angle value of 0 < 63 < 25° [3], so that the grating could possess up to 90 % of energy concentration in a certain spectrum order (-1). The diffraction grating groove frequency is denoted by m (lines/mm).
Let us use the anastigmatism condition to find the angle of image position deviation (meridional angle) from the original 7-axis (object plane), thereby ensuring that the astigmatism value is at least of the second order of the wavelength [3]:
sin Б3 , . , 1 + 2 sin2 e3 , ,
-+ tg ф - tg 83 cos 2б3 tgz ф +-tgj ф = 0. (4)
cos 83
Having solved the cubic equation for cp', desired value of the image position deviation angle is obtained from the original axis.
Main beam reflection angle from the second surface is in the following form:
82 = cp' + 2s3. (5)
The second mirror radius is determined by the formula
km AX
The third surface (third mirror) radius is found as
= (7)
sin 83
Distance CI from the circles' intersection point to the image point could be determined in two ways from formula (7), and due to the cp' angle small-ness, it could be considered equal to the y' image position on the y-axis:
CI = R2 sin 82 = £3 sin s3 s y'. (8)
Main beam incidence angle on the second surface is defined as
82 = kXm - sin82. (9)
Object position deviation angle from the original axis is equal to
^ ~ sin 82tg cp' ^ sin 82
Ф = arctg
(10)
Main beam incidence angle on the first surface is calculated as
(id
2
First surface (first mirror) radius is in the form
^ = _K2sin82 (12)
sin Si
Object position by analogy with (8) could be determined in two ways from formula (12), and it could be considered equal to the CO distance from the circles' intersection point to the object point:
CO = -Ri sin si = R2 sin s2 = y. (13)
Next let us determine position of mirrors relative to each other through the axial distances. Distance from the object plane (y-axis) to the first mirror taking into account the cp inclination angle is calculated by the formula
do = |i?i cos Si | - |CO tg cp|. (14)
Distance between the first and the second mirrors is the radii difference:
di=Ri~ R2. (15)
Distance between the second and the third mirrors by analogy with (15) is
d2=R2~ Rs. (16)
Calculation according to formulas (1)—(16) completely determines the Offner spectrometer design parameters.
Example 1. Design of Offner spectrometer operating in the visible range. Initial design data are as follows: spectral range is - X2 (0.4-0.8 (J,m); radiation receiver resolution element size 5 = 6.5 jam; required spectral resolution dX = 5 nm; F-number K- 10.
According to formulas (1)-(16), let us calculate design parameters of the Offner spectrometer optical scheme (Table 1). Design parameters obtained are used for simulation in the Zemax analysis and optical systems design software package [9]. Note that the number of m lines of the diffraction grating in the Zemax program is set in lines/[o.m. In this case m = 0.1 lines/(am.
Simulation results shown in Fig. 3 make it possible to estimate the root-mean-square value of the confusion spot points' radius in the image plane for the fundamental wavelength Xq - 0.6 p,m and at the following field points: -3.745, -1.872 mm and 0. Coordinates are set in such a way as to ensure possi-
bility to evaluate the confusion spot points' size along the entire length of the radiation receiver line equal to 7.488 mm. Slit width is taken equal to the pixel size. Object position is y = 5.513 mm. The size of the Airy disc shown in Fig. 3 in the form of a circle is 7.329 |j,m.
Table 1
Design parameters of a spectrometer operating in the visible range
Surface Radius R, mm Thickness R, mm Optical material
Object plane 00 26.009 -
Concave mirror 1 -26.575 -13.575 Mirror
Aperture diaphragm. Diffraction grating -13.000 11.330
Concave mirror 3 -24.330 -24.897
Image plane 00 - -
№
/
abc Fig. 3. Confusion spot points in the image plane of the 0.6 |iim wavelength: a are O (0; 5.513), I (0; -4.732 mm), RMSV-rad (0.454 (¿m); b are O (-1.872; 5.513), I (1.872 mm; -4.731 mm), RMSV-rad (0.85 jam); c are O (-3.745; 5.513), I (3.741 mm; -4.726 mm); (RMSV-rad (2.451 |im) O is object (slit); I is image (on the matrix); RMSV-rad is root-mean-square value of the confusion spot radius; reference grid scale 40 (am)
It follows from Fig. 3 that the maximum root-mean-square value of the confusion spot radius is 2.451 p.m. It may be concluded from this that the confusion spot point size in the image plane is not exceeding the pixel size of the radiation receiver matrix (2 • 2.451 fj,m < 6.5 |J,m), i.e., there is no loss in the useful light signal (excluding diffraction phenomena), which confirms correct design of the Offner spectrometer operating in the visible spectral range in accordance with the initial requirements.
After simulating the spectrometer, let us evaluate its overall dimensions. To do this, the Zemax program will be used. Based on Zemax data obtained, the total system length is 27.1422 mm; maximum light diameter is 16.3543 mm.
Calculated spectrometer demonstrated satisfactory compliance with the specified conditions for correcting coma and astigmatism, which values are -0.0001211 mm and -0.02575 mm (maximum field).
Example 2. Design of Offner spectrometer operating in the near IR range. Initial design data are as follows: near infrared range X\-%2 (0.85-1.70 fom), where the fundamental wavelength is selected equal to Xq - 1.275 |im; required spectral resolution of the device dX = 10 nm; receiver resolution element (matrix pixel) size 8-20 [am; line frequency m - 0.1 lines/pm; F-number K- 10.
In connection with complexity of the IR devices control and adjustment operations [10,11] caused by peculiarities of the light rays invisible to the human eye passage through optical elements, it makes sense to simplify the optical circuit, which includes reducing the number of its components, eliminating all possible tilts thereof and maintaining symmetry relative to the optical axis. This becomes possible, if the first and the third mirrors of the Offner spectrometer are made in the form of a common element, i.e., the main mirror, which is ensured by equality of the R\ and R3 radii. Then, positions of the spectrometer first and third mirrors would coincide. It follows from formulas (7) and (12) that this equality is attained under the following condition:
Rx = i?3 о
sin 82
sin 83
sin 82
sin Si
(17)
Analyzing formulas (4), (5), (9), (11), conclusion is made that the cp', s2, 82 and si angles depend on the E3, i.e., when choosing a certain value of the S3 angle, condition (17) is satisfied.
Value of the cp' angle in practice is less than 0.1 rad, and then equation (4) could be written in the form of an iterative expression:
sin3S3 , , l + 2sin2s3 ,
tg9n =--+ tg S3 cos 2&3 tg <Pn—1----tg cpn_i (18)
cos S3 2
at cpo = 0. Let us write expression (18) up to the fifth iteration order to obtain high calculation accuracy. Using the MathCAD mathematical software package [12], approximation will be as follows:
<pi = arctg
• 3 N Sirr 83
COS S3
Ф2 = arctg
sin3 S3
COS S3
1 -
sur S3
COS 83
tg S3 cos 2S3 +
(1 + 2 sin2 S3) sin3 S3
2 cos S3
(19)
q>3 = arctg Cp4 = arctg Cp5 = arctg
' • 3 Slir £3
^ COS 83
f ■ 3 sin5 83
COS £3 sin3 £3
, 1 + 2 sin2 83 , , + tg 83 COS 283 tg2 Ф2----tgj Cp2
tg3 Фз
, 1 + 2 sin2 83 , + tg 83 cos 2e3tgz фз--
(19)
COS 83
, , 1 + 2 sin2 83 , , + tg 83 COS 2s3 tgZ Ф4----tg* Ф4
Let us write formulas (5) and (9)—(11) taking into account (19):
82 = 95 + 2s3; 82 = kXm - sin82; f sin 82 tg 95 ^
Ф = arctg
V
81 =
(20)
sin 82 у ф - 82
Let us denote the left and the right sides of the equality in condition (17) as new functions of the 83 angle dependence; then, taking into account (20), the following is obtained:
Sin 83
/2(63) =
- Sin 82 sin Si
(21)
(22)
Having solved the system of equations (21) and (22), let us find the 83 angle value. It should be noted that finding the absolute value analytically is tedious, so graphical solution is an easier way. The 83 angle value given in radians is the abscissa of the intersection point of the function graphs (21) and (22) shown in Fig. 4.
According to the data in Fig. 4, approximate value of the main beam incidence angle on the third surface (which is also the first) is: 83 « 0.113528 rad = = 6°30'17". Function values for this argument are equal to: /1 (0.113528) = = 1.9745402208; f2 (0.113528) = 1.9745402332. The difference is observed only in the eighth decimal place. The 95 angle value is: 95 (0.113528) = = -0.0014629936.
Spectrometer representation (Table 2) was simulated in the Zemax automated analysis and optical systems design program. In accordance with it, the number of diffraction grating lines is set as m - 0.1 lines/¿im.
1.9745410
Ш)
,, ч 1.9745405
/гО з)
1.9745400 --
0.113526 0.113528 0.113530
E3
Fig. 4. Graphic determination of the main beam incidence angle on the third (first) surface
Design parameters of the IR spectrometer
Table 2
Surface Radius R, mm Thickness d, mm Material
Object plane 00 38.880 -
Concave mirror 1 -39.495 -19.491 Mirror
Aperture diaphragm. Diffraction grating -20.000 -19.491
Concave mirror 3(1) -39.495 -40.119
Image plane 00 - -
It is necessary to estimate the confusion spot size over the entire width of the entrance slit corresponding to the line length of the selected radiation receiver. Its maximum line length in our case is 13.00 mm. Field split points in the sagittal section: -6.50 mm; -3.25 mm; 0; 3.25 and 6.50 mm. Object position jy = 9.5 mm.
Radiation receiver transverse position is determined from the main beam tracing, it is equal to y' = -6.944 mm. Length of the resulting system is 40.12 mm, and the maximum mirrors' diameter is about 30 mm.
Let us demonstrate that the confusion spot size fits into the receiver matrix pixel size. Fig. 5 presents diagrams of the confusion spot for the fundamental wavelength (1.275 (J,m) at all field split points, as well as the corresponding values of the confusion spot radius root-mean-square values. The Airy disk (shown as a circle) for the fundamental wavelength in this case is 15.570 p,m.
a b c
Fig. 5. Confusion spot points in the image plane for the wavelength of 1.275 jixm:
a are O (0; 9.500), I (0; -6.944 mm), RMSV-rad (2.417 (am); b are O (-3.250; 9.500), I (3.248 mm; -6.943 mm), RMSV-rad (3.614 jxm); c are O (-6.500; 9.500), I (6.493 mm;
-6.940 mm), RMSV-rad (8.897 |xm); reference grid scale 40 |im
Data in Fig. 5 make it possible to determine the confusion spot size in the image plane: it is 2 • 8.9 jam. In this case, the receiver matrix pixel size is 20 jim. This means that the pixel is illuminated completely, there is no loss of the light signal, except due to diffraction phenomena, which satisfies the design assignment. Coma and astigmatism values for the maximum field are -0.00015 and 0.00786 mm.
Results obtained and discussion. The developed method for calculating the compact-size Offher spectrometer optical system is based on the Rowland circles and made it possible to calculate two optical schemes for the visible and the IR ranges. Calculation algorithm obtained from the condition of eliminating coma and astigmatism aberrations demonstrated its possibility to create compact optical systems with improved optical, technical and operational characteristics for promising and highly efficient Earth remote sensing optoelectronic equipment [13-15]. It was found that the confusion spot size of the simulated systems differs from the Airy disk size by not more than 2.5 jam, residual coma value for the maximum field is on the order of tenths of a micrometer, and residual astigmatism value is on the order of several micrometers.
Design solution optimization for the IR spectrum range made it possible to reduce the number of mirror surfaces in the optical system to two, which simplifies design and improves manufacturability. Simplicity, small overall dimensions, high optical characteristics and image quality of the Offner scheme in a wide spectral range close to the diffraction range were confirmed by simulating the developed optical schemes in the Zemax software program.
Effectiveness of the methodology is ensured by its simplicity and minimization of the computing resources in design and optimization of the design solutions.
Conclusion. A method for calculating a compact-size Offner spectrometer based on the Rowland circles was developed making it possible to optimize design solution based on the conditions of improving design, technological and optical characteristics. It is shown that theoretical assumptions of the promising capabilities of the Offner scheme underlying the study, i.e., simplicity, small overall dimensions and high optical characteristics in a wide spectral range, are confirmed by simulating the developed optical schemes in the Zemax program. Obtained algorithms, methods of optical systems calculation and design, on the one hand, reduce the development time, and on the other hand, increase quality of design.
Application of this method in the practice of calculating optical systems for the hyperspectral devices would make it possible to improve elemental base and precision optics technology up to a new higher level, which would contribute to creation of modern high-speed information-intensive tools for spectral analysis of the remote objects, including compact-size Earth remote sensing optoelectronic equipment.
Translated by K. Zykova
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Zavarzin V.I. — Dr. Sc. (Eng.), Professor, Department of Laser and Optoelectronic Systems, Dean of the Department of Optoelectronic Instrument Engineering, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Zaitsev I.M. — Student, Department of Laser and Optoelectronic Systems, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Yakubovskiy S.V. — Student, Department of Laser and Optoelectronic Systems, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Please cite this article as:
Zavarzin V.I., Zaitsev I.M., Yakubovskiy S.V. Method for calculating the Offner compact-size spectrometer. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2021, no. 4 (137), pp. 139-151. DOI: https://doi.org/10.18698/0236-3933-2021-4-139-151
