А possibility of using a diffractive lens as the simplest imaging spectrometer was considered. An experimental construction of the spectral imaging for white, red and green LEDs was conducted. When processing the images the influence of the scattered components of the initial point and nearly points of the image was taken into consideration. There was made a comparison of the received spectral distribution with the spectra of these LEDs, received by means of spectrometer. Root-mean-square error (RMSE) from 8 to 13%.
In the past decade, in applied spectroscopy there have been intensively developing
the new methods, which allow getting and analyzing the spectroscopic information
about an object with a spatial resolution, providing the reception for each small area
(point) the two-dimensional image of the object on the input aperture of the optical
spectrum instrument. In scientific literature such methods are called imaging
spectroscopy, three-dimensional matrix containing the dependence of intensity of
light from the two spatial and spectral coordinates are called spectral images [
In most modern imaging spectrometers optical system forming that imaging and
dispersive element are clearly separated. As a dispersive element in imaging
spectrometer can be used a prism as well as diffraction grating [
To test the possibility of restoring the spectral distribution for every dot of the
image for the objects with a wide spectrum there was conducted an experiment in
which was built a series of white LED images by means of diffraction lens with a
focal length of 50 mm for a wavelength of 500 nm. For the experiment, a four-leveled
diffraction lens was used, described in [
Fig. 1 Shows the optical scheme of the experiment.
Whilst the distance from the LED to the diffraction lens a = 2000 mm was less than the distance from the diffraction lens to a CCD camera (VIDEOSCAN-2-205) with a resolution of 1390*1040. In the experiment, the camera moved from 40 mm to 60 mm with a step of 0.5 mm. Each image corresponds to its own wavelength. For convenience, the diode was located close to the optical axis.
Fig. 2 shows a part of white LED spectrum, obtained by the spectrometer MS7501 (fig. 2). A relative error of spectrum measurement on this spectrometer is not more than 2%.
The spectral distribution in Fig. 2 hereinafter to be used as a standard spectrum for comparison of the spectra of individual image points of the LED obtained by means of diffractive lens.
Fig. 3 shows the examples of the LED images at several distances, which correspond to the wavelengths of 481 nm, 505 nm and 550 nm. The image size of the LED in fig. 3 was nearly 100 micrometer (22 pixels on CCD array).
However, it is not allowed to use the experimental images directly without complementary processing. The matter is that due to the change of distance from lens to image changes the scale. I.e. the LED image shifts in relation to the center of the image pattern and additionally changes its size. It cannot be avoided even if the LED image is positioned at the exact center. In this case what is left is a change if the image’s scale. To overcome this effect, the experimental images were subjected to geometric transformation of tension / compression relative to the center of the picture with a coefficient equal to the ratio of the distances. Fig. 4 shows the LED images at several distances corresponding to different wavelengths (481 nm, 505 nm, 550 nm) after the geometric transformation. After this transformation, the image of any size keeps the size and position for any wavelength. where I x, y, i are the elements of spectral image, bi, bj are distances corresponding to the wavelengths, ki – f-number of the diffractive lens, Ij(x,y) – intensity distributions taken at distances bj, A – square of pixel in CCD array. Fnumber for the diffractive lens is determined by the formula ki d bi , where d is lens diameter. The formula (1) was derived from geometrical considerations. For a precise focus, when the distance bj corresponds to the wavelength we supposed that the size of aberration image is smaller than the size of pixel of the CCD array (4.65 micrometer*4.65 micrometer). It is actually so, the lens diameter is 8 mm at a focal length of 50 mm, i.e. the size of the diffraction aberration image will be about 4 micrometers. At displacement of over a certain distance, we calculated the size of the spot based on the geometrical considerations. For the imaging spectrometer, this system of equations becomes more complicated because it is necessary to take into account the contribution of neighboring dots to the diffuse component on the object
N I x u, y v, i A _____
I j x, y dudv, j 1, N , (2)
i1 u2 v2 R2 A bj bi 2 ki2 / 4
where R bj bi k 1 is the blur size of a dot at focus displacement, when j=i, in the
calculations we shall assume that aberration image coincides in radius with pixel. In
the discrete case, the integral in formula (2) degenerates into a sum over the
aberration image's dot for the appropriate wavelength. The system of equations (2) for
a wide spectral range has a high computational complexity. The number of dots in a
discrete variant according to which the summation is performed can reach several
thousands.
For the case where the object for which the spectral image is formed is relatively
compact, the computational complexity reduces. Thus, for the LED (fig. 3) it is
necessary to take into account just a few tens of dots. Besides, for the spectrum
recovery process there was considered a different diffractive effectiveness of the
diffractive lens for different wavelengths, which is calculated according to the
formula [
4 n 1h 2 n 1h 4 2 where η is spectral sensitivity, h is relief depth, n is a refractive index of the lens material.
The spectral sensitivity of CCD array [
As a result of the solution of equations (2), the following spectral distribution was obtained for one of the dots for the diode image (fig. 6).
RMSE of recovery range was 11%. However, for such compact objects as LED it is possible to use a simpler means of obtaining a spectral image with an insignificant loss of accuracy. After obtaining a geometrically correct image (fig. 4), a scattered component can be removed from all images. To do that, there was determined an average number of the intensity of the scattered light outside the LED image for each image individually, then this number was subtracted from every experimental image. Fig. 5 shows an example of the thus processed images for the wavelengths: 481 nm, 505 nm, 550 nm.
Based on the obtained images, a spectral distribution for each dot of the image can be obtained (fig. 7).
Fig. 7 shows a certain deterioration (in comparison with fig. 6) of the correspondence of a spectrum obtained on the spectrometer (dark line) and a spectrum obtained after processing the image sequence, formed by the diffraction lens. RMSE in this case was 13%. Which is 2% more than RMSE of the exact solution. However, the processing time here significantly reduces and in certain cases, it is possible to use a simplified algorithm to obtain spectral images of compact objects.
Spectra obtained for the different LED dots are a little bit different (fig. 8). Various dots at which the spectrum was measured were marked in fig. 8a. The spectrum for a point marked with an arrow is shown in Fig. 7. The spectra for the points marked with a triangle and square, are shown in Fig. 6bc, respectively.
For fig. 8b and 8c graphics differ slightly from the graphic in Fig. 7, and RMSE value is about 13%.
Let us also consider the spectral image of the structure of the green, red and white LEDs in a cross shape, a white diode is situated in the center, green diodes are in vertical direction, and red diodes are in horizontal direction (Fig. 9). The spectra of these diodes are quite different from each other, and obtaining the spectral image of such object, we hereby confirm that the spectrometer based on diffractive lens is indeed imaging.
In Fig. 9, it is clearly seen that images of the red and green diodes are formed at different distances from the lens. Fig. 10 shows the spectra obtained for red and green diodes in comparison with their spectra measured on MS7501. b) Fig. 10. – Spectrum of the green diode obtained in the experiment in comparison with its spectrum measured on MS7501 (a), the spectrum of the red diode obtained in the experiment in comparison with its spectrum measured on MS7501 (b)
The value of RMSE in fig.10a - 8%, in fig. 10b – 11%. The RMSE value obtained
in the experiment is rather large 8-13%, but we must consider three things:
1. spectrum of a single dot on the spectral image should not be identical with the
spectrum of the entire diode (fig. 8b and 8c) (spectrometer measures the integral
characteristics);
2. only a part of the spectrum was studied, but CCD array is also sensitive to
components outside the measured range, therefore, they also introduce distortions
in the spectrum reconstruction. In the future this can be avoided by installing the
cutoff light filters;
3. the usual error level of imaging spectrometers is quite large compared to the errors
of non-imaging spectrometers (1-2%). As a rule, manufacturers of hyperspectral
cameras indicate an error in the measurement of the spectrum in the range of
510% [
When working with more complex images in the future, a precise method based on system should be used (2).
However, based on the obtained results, we can talk about the working efficiency of this simple imaging spectrometer, consisting of a single diffractive lens.
The error in operation of the simplest imaging spectrometer, where the imaging system and the dispersion element are integrated, is investigated – spectrometer is based on diffractive lens.
The experimental results on the formation of the spectral image from the series of images formed by the diffraction lens, allow us to speak about the error of forming spectral images using diffractive lens with an error of 8-13%, which roughly corresponds to the error of the existing samples of hyperspectral cameras.
The work was funded RSF grant 14-19-00114.