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Design of airborne imaging spectrometer based on curved prism

Yunfeng NIE*a,b, Bin XIANGLIa , Jinsong ZHOUa, Xiaoxiao WEIc aAcademy of Opto-Electronics, Chinese Academy of Sciences, 9 South Dengzhuang Road, Haidian

District, Beijing, China 100095 ; bGraduate Univeristy of Chinese Academy of Sciences, 19 Yuquan Road, Shijingshan District, Beijing, China 100049 ; cModern Optical Technology

Institute, Soochow University, 1 Shizi Street, Suzhou, China 215006

ABSTRACT

A novel moderate-resolution imaging spectrometer spreading from visible wavelength to near infrared wavelength range with a spectral resolution of 10 nm, which combines curved prisms with the Offner configuration, is introduced. Compared to conventional imaging spectrometers based on dispersive prism or diffractive grating, this design possesses characteristics of small size, compact structure, low mass as well as little spectral line curve (smile) and spectral band curve (keystone or frown). Besides, the usage of compound curved prisms with two or more different materials can greatly reduce the nonlinearity inevitably brought by prismatic dispersion. The utilization ratio of light radiation is much higher than imaging spectrometer of the same type based on combination of diffractive grating and concentric optics. In this paper, the Seidel aberration theory of curved prism and the optical principles of Offner configuration are illuminated firstly. Then the optical design layout of the spectrometer is presented, and the performance evaluation of this design, including spot diagram and MTF, is analyzed. To step further, several types of telescope matching this system are provided. This work provides an innovational perspective upon optical system design of airborne spectral imagers; therefore, it can offer theoretic guide for imaging spectrometer of the same kind.

Keywords: curved prism, Offner configuration, imaging spectroscopy, Seidel aberration theory, spectrometer, smile, keystone, linear dispersion

1. INTRODUCTION Hyper spectral imaging is of growing interest as a novel detection approach since the eighties of the twentieth century, which combines two-dimensional visual picture with one-dimensional spectral information closely related to the biochemical characteristics of the targeted object. This technique was originally developed for remote sensing imaging, including the identification and mapping of minerals and oil resource, monitoring and estimation of water quality as well as air pollution, precision agriculture and forestry, urban investigation and so forth 1.-3. Recently, hyper spectral imaging has spread into fields as extensive as biomedical imaging, anti-counterfeiting and validation of historical manuscript as well as antique, surveillance of food safety, monitoring and treatment of diseases 4-7 . In other words, this technology is continually becoming more available to the public, and has been used in a wide variety of ways, requiring a less expensive and more compact method to design and manufacture imaging spectroscopy, which is the apparatus to capture three-dimensional hyper spectral data cube.

The theory and model of dispersive imaging spectroscopy was first brought up and then realized on various platforms successfully. It contains two fundamental configurations: prismatic dispersion and grating diffraction. Due to its easy principle and low realizing complexity, instruments of this type have been widely applied to many airborne and space borne hyper spectral imagers, such as Hyperion carried by NASA’s satellite EO-18 and medium resolution imaging spectrometer(MERIS) developed by European Space Agency(ESA) for the Envisat-1 polar orbit Earth mission9 . Generally, traditional imaging spectroscopy based on refractive prismatic dispersion is composed of telescopic lenses, collimating lenses and imaging lenses, and considering such a wide spectral band, all of the three components are required to correct chromatic aberrations, especially the conspicuous and inevitable secondary spectrum, leading to more rigorous design requirements for single component and more difficulties on assembling different components, greatly increasing the number of lens, thus reducing image quality and energy utilization efficiency. The usage of reflective design 10 can avoid chromatic aberrations; however, it introduces aspheric mirrors into collimating lenses and imaging lenses as well as difficulty in manufacturing, measuring and integration. The Offner spectrometer based on convex grating dispersion, which is under extensive research at present, possesses some inherent defects such as low diffraction

2011 International Conference on Optical Instruments and Technology: Optical Systems and Modern OptoelectronicInstruments, edited by Y. Wang, Y. Sheng, H.-P. Shieh, K. Tatsuno, Proc. of SPIE Vol. 8197, 81970U

© 2011 SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.904270

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efficiency, low energy efficiency, high-order spectra aliasing, narrow free spectral range, immature manufacturing technique on convex grating and so on, inhibiting further development. Therefore, dispersive prism with merits of wide spectrum range and high light energy transmittance is definitely one of the best options for spectrophotometric elements, thus a good choice for wide-spectrum-range medium resolution imaging spectroscopy.

In this paper, the design of curved surface prism integrated with the Offner configuration can considerably relieve the severe problems of spectral line bending (smile) and spectral band bending (keystone or frown) associated with traditional prismatic dispersion. Besides, the combination of curved surface prisms with different materials is able to reduce dispersion nonlinearity. In order to guarantee a compact structure, the distance between those two reflective mirrors of the Offner configuration is strictly controlled, thereby more suitable for aeronautics and astronautics instruments where space resource are scarce. The telecentric structure of the object space is performed in the design of spectrometer, while the image telecentric structure is employed in the telescopic lenses, which can perfectly satisfy the pupil matching principle and not affect the presupposed optical performance of the two components.

2. THEORY OF CURVED SURFACE PRISM 2.1 Review

Curved surface prism is acquired by machining the two flat surfaces of a traditional dispersive prism into sphere surfaces. First prism of such type can be referred to Fery’s invention in 191111, where the prism is described as a superior element for spectrograph or spectroscopy by virtue of its wide available scope both in collimating light path and convergent light path. Miller et al. proposed a new spectrophotometer employing a glass Fery prim in 194812 . In the nineties of last century, Warren et al. established a compact prism spectral imaging system by using Fery prism under aplanatic condition where a refracting surface is free from spherical aberration and coma of all orders13 , and a spherical mirror operating near its center of curvature. Thereafter, Lobb et al. integrated two Fery prisms into a modified Offner relay—a system of two concave mirrors and one convex mirror, resulting in a compact high resolution imaging spectrometer (CHRIS) which provide the main instrument payload on the ESA’s small satellite platform PROBA (Project for On-Board Autonomy) circulating around a 830km altitude near-circular sun-synchronous polar orbit14 . Recently, Korea has also introduced a space borne spectral imager COMIS (Compact Imaging Spectrometer) with curved surface prisms for use in the STSAT3 microsatellite on the sun-synchronous orbit at an altitude of 700km 15.

This paper explores the Seidel aberration principles of curved surface prism as well as the concentric optics theory to achieve a better guidance in the design work of an airborne VIR (visible to near infrared light) imaging spectroscopy. Compared to previous works, this design not only inherits their characteristics of good image quality, little smile and keystone, compact structure and simplicity, but also possesses a wider field of view as well as a better linearity to accommodate applications in more fields.

2.2 Paraxial principles for curved prism

When using as a ray splitting element, traditional flat dispersive prism must be operated in a collimating light path to eliminate most kinds of aberrations. Thus, collimating lenses are required to correct secondary spectrum at a broad spectral band, which means the usage of particular glass materials whose partial dispersion is abnormal and chemical or physical attributes are not satisfying, for example easy corroding and poor processing ability. The system designed with this method is overelaborated, especially in a larger field of view. Thereby a prism used in more flexible light path is recommendable.

The geometry of the converging beam passing through the curved prism is illuminated in Fig.1. The faces of the curved prism of refractive index n are tangent to the equal sides of an isosceles triangle with apex angleα . 1 2,R R are

respectively curvature radiuses of the two light transmitted surfaces. 1S is the objective point, and 2 'S is the final image

point. Additionally the relationship between the incidence angle 1i at the first surface and the incidence angle 2i at the second surface is the same in this thin spherically curved prism as it is in the plane faced prism under a thin-prism approximation:

12

ii an

= − (1)



Each surface has an object distance and an image distance, denoted l and 'l , respectively. The conjugate point 1 'S is defined by the Abbe relation:

01 1 1 1

1 1 1 1'

nR l R l

⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (2)

Figure 1. Geometry of a converging beam passing through a curved prism

According to primary aberration principles, the third-order coefficients for each refracted surface are calculated by formulas:

4 2 21 1 1 1( ) ( )' 'IS h n

r l n l nl= − − (3)

3 2 1 1 1 1( )( )' 'II pS h n i

r l n l nl= − − (4)

2 2 2 1 1( )' 'III pS h n i

n l nl= − (5)

Where IS is the third-order spherical aberration coefficient, IIS is the third-order coma coefficient and IIIS is the third-

order astigmatism coefficient. With the transitional identity 2 1 'l l= and Eq. (1) and (2), we can obtain all the parameters

in Eq. (3), (4) and (5). The special case 1 2R R R= = is chosen to simplify further calculation. Substituting all the

parameters into Eq. (3), (4) and (5), the common solutions for them are 1R l= and ( )

1

0 1lR

n=

+. In other words, when

objective point is positioned in the center of curvature of the first surface or its aplanatic conjugation, the most contributing aberrations will be minimal.

With the analysis above, we can find that the additional variables 1R and 2R provide potentials for correcting primary aberrations originating from dispersive prism in a converging light path. It’s a conspicuous improvement, because the adding of curved prism excludes the necessity for collimating lenses, which is a great simplification for optical design of spectrometer instruments. What’s more, the two spherical surfaces can also provide focal power for imaging, playing both dispersion and imaging roles in a spectral imager.

3. PRINCIPALS OF OFFNER RELAY Offner relay has one concave spherical mirror and two convex spherical mirrors. All spherical surfaces have a common center of curvature, and the object and image planes (or slit and detector planes in a spectrometer) are close to the common center, as shown in Fig. 2. When using in a pushbroom imaging spectroscopy, Offner relay has excellent



properties such as no chromatic aberrations over quite broad spectral ranges, very compact configuration, little distortion and no aspheric surface 16 .

Figure 2. Optical layout of Offner relay

The meridional image of an off-axis object point produced by a reflective mirror satisfies:

1 1 2' cosMr r R θ

+ = (6)

And the sagittal image verifies:

1 1 2cos'Sr r R

θ+ = (7)

In these Eqs. R is the curvature radius of mirror, , ', 'M Sr r r are respectively the distances from the incidence point of the reference ray on the mirror to the point and to both the meridional and the sagittal image point. Given a particular condition cosr R θ= , substituting the parameter in Eq. (6) we get ' cosMr R θ= , which means meridional image point is on the circle with its diameter OA, just like the Rowland’s Condition. And the image point isn’t function of aperture angle, so the meridional image of spherical mirror is absence of spherical aberration and coma. Similarly in Eq.

(7) we get 'cos 2S

rrθ

= . It’s clear that the sagittal image doesn’t lie on the Rowland circle in general, which makes

astigmatism the most important aberration.

Repeating the same calculation and analysis process onto the second reflective mirror and the third one as well, we finally deduce the conclusion that: 1) the third-order and fifth-order spherical aberration and coma coefficients of the Offner relay are zero, and 2) absolute value of the primary astigmatism introduced by the first and third mirror are exactly the same with opposite sign, yet leaving little high-order astigmatism after counteracting, and the second mirror produces no astigmatism at all, and 3) field curvatures from the first and third mirror are negative while the one from the second is positive, all the three added together resulting in a minimal field curvature, and 4) distortion is zero under a complete symmetric configuration. In a word, both images can be matched by a suitable combination of the three mirrors.

A quick conclusion can be made by the theoretic analysis above that the combination of curved prism and Offner relay is a great choice for their excellent competence in correcting spherical aberration, coma and astigmatism. Besides, when curved prism is used as an imaging element, the object and image point are aligned in the same direction of the prism itself, not in the same optical axis though, but it still perfectly matches the Offner configuration and guarantees a smaller size for the spectrometer.



PARAMETERS Optics Spectral ranges 450nm~1050nm Spectral resolution 10nm Numerical aperture in object space 0.125 Slit elevation 20mm Detector pixel size 14um

4. SIMULATION AND EVALUATION 4.1 Assumptive optical performance

This spectrometer is realized by inserting two achromatic curved prisms into arms of the Offner relay configuration. Achromatic curved prism is a composite prism made of two different glass materials—one is a flint glass and the other a crown one—in order to decrease dispersive nonlinearity. By reflecting, light beam passes through the curved prism twice, which promotes the dispersive ratio. The Offner relay is performed at near unit-magnification to achieve an approximately symmetric structure whose distortion is little.

The proposed performance parameters are listed in Table 1.

Table 1. Performance parameters for spectrometer

4.2 Optical design of spectrometer

The initial structure plays an extremely important role in optical design. By employing the conclusions in section 2, we make the curvature radius R of the first surface of the curved prism and the objective distance 1l satisfy the relation:

1

0 1lR

n=

+ (8)

Additionally, the radii of the three mirrors verify:

3 2 1: : 2 :1: 2R R R = (9)

Where 1 2 3, ,R R R represents the radius of the first, second and third mirror respectively.

After inserting the initial parameters and editing merit function carefully, we optimize the spectrometer system by the optical design software ZEMAX. During the optimization process, the apex angle of the curved prism should be adjusted manually so as to ensure a sufficient dispersion length. The eventual layout of the spectrometer is shown in Fig.3.

Figure 3. Optical layout of spectrometer



4.3 Evaluation of image quality

Generally, basic criteria for image quality of camera include Point Spread Function (PSF), Modulation Transfer Function (MTF), visual resolution, distortion and etc. PSF and MTF are interrelated metrics that indicate the performance of imaging systems. PSF is in spatial domain and MTF is its counterpart in frequency domain. MTF is a non-subjective diagnostic tool to describe the characterizations of an optical system, and has been widely adopted.

We select three representative wavelengths (central wavelength 750nm, two extreme wavelengths 450nm and 1050nm) to evaluate optical performance of this system, as illustrated in Fig.4. Generally, the image quality at the central wavelength is the best and at the other two wavelengths are among the worst. Correspondingly, traversing all the wavelengths across all the fields, RMS (root mean square) radii of the spectrometer are smaller than half pixel size, and MTF values are more than 0.65, approaching the diffraction limit, which admirably meet the qualification.

Figure 4. Spot diagram and MTF curve of three representative wavelengths

4.4 Evaluation of spectral attributes

Table 2 illustrates Monte-Carlo ray-tracing results at three normalized field of view (0, 0.7 and 1) across seven wavelengths (450nm, 550nm, 650nm, 750nm, 850nm, 950nm and 1050nm). Statistics show that for all wavelengths the maximum departure from a straight line parallel to a row is less than +/- 0.023 pixels, which is the maximal value of smile. Frown is also indicated in Table 2 that at any field position the image of slit lies within +/- 0.14 of a column (pixel) width over the full wavelength range.

The ultimate spectral resolution is measured at 64 wavelengths, and the results for on-axis measurements are plotted in figure 5. Off-axis measurements give similar results. The plot shows that the spectral resolution of the spectrometer varies from approximately 8.5 to 11.5 nm across the spectrum while the highest dispersion at 600nm and the lowest in the short wavelength at 450nm. The average spectral resolution is about 10nm.

Compared to most designs of this type 17-19 , not only the smile and frown is corrected to a minimal level, but also the spectral linearity is much better. For example, CHRIS varies from 1.25nm to 11nm whose spectral band is almost the same as our design.



Table 2. Ray-tracing results at different fields over different wavelengths

Deviations of slit images from a straight line for different wavelengths(um) Wavelengths 450nm 550nm 650nm 750nm 850nm 950nm 1050nm

0 0 0 0 0 0 0 0 0.7 +0.26 +0.3 +0.31 +0.3 +0.3 +0.28 +0.271 +0.54 +0.62 +0.64 +0.63 +0.61 +0.59 +0.56

Changes of position in column for different field points(um) Fields 450nm 550nm 650nm 750nm 850nm 950nm 1050nm

0.7 -0.26 -0.03 +0.01 0 -0.03 -0.08 -0.13 1 -0.38 -0.05 +0.02 0 -0.05 -0.11 -0.19

500 600 700 800 900 10008

8.5

9

9.5

10

10.5

11

11.5

12

Wavelength/nm

Spe

ctra

l res

olut

ion/

nm

Figure 5. Spectral resolution of the spectrometer

5. CONCLUSIONS When integrating telescopic lenses with the spectrometer, the pupil matching rule must be complied with. Usually, spectrometer is object-space telecentric, while telescopic lens meets image-space telecentricity requirement. However, in some complicated systems, for example, a traditional prismatic dispersion spectral imager composed of telescopic lenses, collimating lenses and imaging lenses, the feasible method is to assemble two lenses first, calculating the exit pupil position to match the third’s entrance pupil position.

The choice of telescopic lenses determines the achievable spatial resolution, and further the platform to carry this payload. There are three most frequent configurations—the three mirror anastigmatic telescope, catadioptric telescope and refractive fore-optics. The former two are priorities in long-focal-length narrow-field optical systems for absence of color aberration, which is a most devastating factor in refractive lens. Consider an airborne imaging spectroscopy with moderate length focal, a refractive fore-optics is preferable, due to its cheapness of manufacture, ease of alignment, convenience of assembly, fast focal ratio and so on. Two options of this kind are recommended, modified double gauss with field lens group and modified Huygens eyepiece with field lens20 , and making the final decision needs more comprehensive research.

In this paper, a VIR moderate resolution imaging spectroscopy is presented in details, covering theoretic analysis and calculation, simulation testing, and image evaluation. Correspondingly, Monte-Carlo ray-tracing results show that the image quality is excellent, and registration error by smile and frown is less than 5% of the pixel, which means spectral resolution is limited essentially by the detector pixel size. Compared to previous works, the field of view of this design is larger, and the linearity is fairly excellent from the perspective of dispersive prism. To summarize, with properties of compact configuration, light weight, and high image quality, linear dispersion, high throughput and stabilization, this design can be a good choice for airborne moderate resolution imaging spectroscopy system.



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