The result of the geometric ray-tracing performed by ZEMAX can be used to predict the appearance of a point image on any plane within the optical system with a reasonable degree of accuracy [11]. This was to be achieved by dividing the entrance
0
)
1
(
2
2
Rx
C
x
y
p Eq. 3.797 pupil of the system into a large number of equal areas and tracing multiple rays from the object point through the centre of each of the small areas [11]. The more rays that are traced, the more accurate the representation of the geometrical image is.
A geometric spot diagram was produced to estimate the shape of the spot at the detector plane with reference to the Airy disk for different fields of view of the optical system (see Figure 3.18) [12]. The incorporation of different fields of view during the analysis is important because aberrations increase at higher angles [6]. This type of diagram shows the effect of aberrations on the focal spot; it does not however take into the account diffraction on the aperture.
Characteristics of a focal spot on the detector plane for the non-scanned system operating with the Gm-array are presented on the ZEMAX-generated spot diagram shown in Figure 3.14. The Airy disk diameter is in good agreement with the specification; it is equal to 66 µm; in addition the focal spot fits well within the Airy disk for FoV = 0 ° (nominal), the FoV = 0.026 ° (detector semi-side) and FoV = 0.036° (detector semi-diagonal).
Figure 3.14. ZEMAX-generated focal spot diagram for the single- element scanned system where (a) represents the spot produced FoVArray = 0 ° (nominal), FoV = 0.026 ° (array semi-side) and
FoVArray = 0.036 ° (array semi-diagonal).
Figure 3.15 shows the ZEMAX-generated focal spot diagram for the single-element scanned system where:
- Figure 3.15(a) represents the spot produced for 0 (0 °), 0.7 (0.0008 °) and 1 (0.0011 °) × semi-FoVPixel for scanning mirrors x, y at 0 ° tilt;
98 - Figure 3.15(b) represents the spot produced for 0 (0 °), 0.7 (0.0008 °) and 1
(0.0011 °) × semi-FoVPixel for scanning mirrors x, y at 0.026 ° tilt;
- Figure 3.15(c) represents the spot produced for 0 (0°), 0.7 (0.0008 °) and 1 (0.0011 °) × semi-FoVPixel for scanning mirrors x, y at 0.036 ° tilt.
Figure 3.15. ZEMAX-generated focal spot diagram for the single- element scanned system where (a) represents the spot produced for 0 (0 °), 0.7 (0.0008 °) and 1 (0.0011 °) × semi-FoVPixel for scanning
mirrors x, y at 0 ° tilt, (b) represents the spot produced for 0 (0 °), 0.7 (0.0008 °) and 1 (0.0011 °) × semi-FoVPixel for scanning mirrors
x, y at 0.026 ° tilt and (c) represents the spot produced for 0 (0 °), 0.7 (0.0008 °) and 1 (0. 0011°) × semi-FoVPixel for scanning mirrors
x, y at 0.036 ° tilt.
In all cases, the Airy disk diameter is equal to 16.5 µm, which is 66 % of the 25 µm detector diameter and demonstrates that the focal spot fits well within the diameter of the Airy disk. This shows a good match with the specifications.
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3.9.2.1
Point Spread Function
The irradiance distribution generated by an optical system that results from a single point source in object space is called a point spread function (PSF) [11]. Although the source is a point, the image is not; it is blurred by the effects of diffraction and aberrations. A method applied by ZEMAX to calculate the PSF is the Fast Fourier Transform (FFT) algorithm which decomposes a spatial distribution into a frequency domain distribution in an efficient manner. Figure 3.16(a) shows a ZEMAX- generated PSF for the 32 × 32 non-scanned system for the nominal (0°) FoV and a graph of relative irradiance while Figure 3.16(b) shows a PSF cross-section in µm. As expected, the FWHM of the FFT PSF is < 66 µm.
Figure 3.16. ZEMAX-generated Fast Fourier Transform point spread function for a 32 × 32 non-scanned configuration at nominal (0°) FoV: (a) surface plot and a cross-section of normalised intensity, (b) PSF cross-section in µm.
Figure 3.17 shows a ZEMAX-generated PSF for the single-element scanned system for the nominal (0 °) FoV. As expected, the FWHM of the FFT PSF is < 17 µm.
100 Figure 3.17. ZEMAX-generated Fast Fourier Transform point
spread function for a single-element scanned sub-system at nominal (0 °) FoV: a cross-section of normalised intensity.
3.9.2.2
Through Focus Depth Diagram
The concept of depth-of-focus is based on the assumption that for a given optical system, there exists a blur due to defocusing of a small enough size that it will not adversely affect the performance of the system [11]. Depth-of-focus is defined by the amount by which the image plane can be shifted along the optical axis and which will introduce no more than an acceptable amount of blur [11].
Modelling depth-of-focus is useful in predicting the degree of allowed defocus on the detector plane before the image starts to degrade beyond the diffraction limit and exceeds the diameter of an Airy disk, thus, imposing tolerances on the mechanical design. Through-focus diagrams of the detector plane generated in ZEMAX are useful in analysing depth-of-focus [12]. The through-focus analysis was performed for the 32 × 32 non-scanned system (shown in Figure 3.19) and for the single-element scanned system (shown in Figure 3.20) for the rays arriving at the optical system at three angles:
- 0°(nominal) to the optical axis (see Figure 3.18(a)),
- For the beam travelling from the side FoVArray, at 0.026 ° to the optical axis (see
Figure 3.18(b));
- For the beam travelling from the diagonal FoVArray, at 0.036 ° to the optical axis
101 Figure 3.18. Diagrams showing rays arriving at the optical system
(a) on axis, (b) from the side FoVArray and (c) from the diagonal
FoVArray.
The array configuration has a long focal length of ~ 3.5 m, which provides a large depth-of-focus of approximately ± 800 µm from the best focus plane. Beyond this region the blur due to the defocus exceeds the size of the Airy disk diameter and thus the system is no longer diffraction-limited. This result suggests that the design provides a reasonable alignment tolerance of the focal spot on the detector plane. The single-element scanned system has a shorter focal length of ~ 0.87 m; this is about four times less than in case of the non-scanned system and therefore the depth-of- focus is shorter. The depth-of-focus is approximately ± 60 µm. Beyond this region the blur due to the defocus exceeds the Airy disk diameter and thus the system is no longer diffraction-limited. This suggests that the alignment tolerance for this system is low and requires the mechanical stage supporting the detector to provide a sufficient degree of adjustment on a micrometre scale.
102 Figure 3.19. Through-focus spot diagram for the 32 × 32 non-
scanned sub-system at nominal (0 °) FOV, side (0.026 °) FOV and diagonal (0.036 °) FOV.
Figure 3.20. Through-focus spot diagram for the single-element scanned sub-system for scanning mirrors pointing at nominal (0 °) FOV, side (0.026 °) FOV and diagonal (0.036 °) FOV.
103