CHAPTER FIVE
5.5 Comparison and discussions
The assumptions made in the last two sections are :
1. A red LED is used for the illumination,, with central wavelength, X, of 660 nm and
linewidth (AÀ) of 50 nm.
2. The diameters of the microlens and the pinhole (2a) are both set to match the pixel pitch of the spatial light modulator, 165 pm.
3. The distance between the input (zi) and filter plane and that between the filter plane and the output plane (z?) are equal and are given by:
4.
Correlator system : theoretical A n a ly sis 118
The magnification (M = zi/zi) is therefore 1 and both systems have the same length.
It follows that the focal length for the shared microlens system (/) and the effective focal length for the pinhole system (Zo) is :
/ = Zo = Z2/2 = 1.03 cm
The results obtained are given in table 5.1. The working point (z?) chosen in the pinhole system is in the Fraunhofer diffraction region and since Fraunhofer diffraction always occurs shared microlens system, the resolution limit of the pinhole system is same as that in the shared microlens system (Fig. 5.10). However, the lens system is about twice as efficient as the pinhole system since more light is collected within the central lobe. For the shadow casting correlator, the encircled energy within the ‘shadow’ spot increases linearly until the full spot size is reached and is then 100% efficient. At q = 100 pm, the efficiency is 60% which is lower than the shared microlens system but larger than the pinhole system (fig. 5 .10b). This value is chosen to maximise the number of elements and yet still maintain a relatively high efficiency. If fewer elements could be tolerated than the highest efficiency could be achieved using the shadow casting system at q =165 pm. So why not use the shadow casting correlator instead? The benefit of going through the trouble of using long focal length microlenses is that the number of pixels in 1-D that are allowed for the shared microlens system is N = 2.11 cm 100 pm = 211 whereas that in the shadow casting system is N = 2.11 cm 4- 165 pm = 128; so we get 2.7 times improvement in total number of pixels.
System Resolution limit (diameter) Diameter of -3dB point
Encircled energy at
formula figures Res. limit -3dB point
Microlens 1.22 Z2X/a 201 pm 85 pm 83.8 % 16.2 %
Pinhole 1.22 a (1+M) 201 pm 90 pm 48% 23%
Shadow casting 2 a (l+ M ) 330 pm - 100% -
Table 5.1 Comparison of resolution and efficiency for the different systems
The distance z? chosen for the pinhole system is a realistic value as a smaller size would imply large off-axis angles since the size of the input and mask planes are both equal
to 2.11
c m
(128X
165)Lim).
This gives an off-axis angle of about ( tan ' f — — j = ) 27°. V 2 X 2.06 JThe illumination system cannot provide larger off-axis angles than this (see chapter 6).
I
0.8 S 0.6 "S = 0.4I
o Z 0.2 X = 660 nm 2a = 165 p.m z,= 2.06 cm microlens — Pinhole • • - Shadow casting 100 200 300Radius of the output spot (q) [pm] (a)
c 0.48 .u 0.4
Shared lens system Pinhole system • • • - shadow casting
100 200 300
Radius of the output spot (q) [pm] (b)
Fig. 5.10 Comparison of shared microlens and pinhole systems in (a) resolution and (b) efficiency
If the focal length of the microlens system is allowed to be variable so that the system plane separation ratio is always kept at 1:1, the spot size (diameter) varies with the distance in the following relationship and is depicted in fig. 5.11:
qidiameter) = 2a (5.27) E zL
a
c/2 400 350 300 250 Geometric limit 200 150 microlens + pinhole shadow casting 100 50 0 1 2 3 4Distance between the filter and output plane , (cm) Fig. 5.11 Focal spot size variation with distances for each system
C orrelator system : theoretical A n a lysis 120 The result is a linear relationship. Similarly, if the pinhole system is placed under the same constraints, the spot size varies in a very similar way to the shared microlens system (shown by the ‘+’ in the graph which are calculated values). The data points correspond to the distances shown in fig. 5.5c-f. The reason for discrete points is that each spot size have to be numerically determined. The data point at z i= 1.4 pm lies outside the line because it is not at the one of the maxima shown in fig. 5.5b in the Fresnel diffraction region. Although it may suggest a smaller spot size can be obtained in the Fresnel region if a maxima is not chosen, this usually accompanied with the penalty of low efficiency and high crosstalk as the spot has large sidelobes. For the shadow casting system, the spot size remains the same with distance unless the geometry is changed, i.e. M < 1 for smaller spot size. We conclude that for medium size (3.5-4 cm) systems the shadow casting arrangement is best as it gives high efficiency and smaller spot size. For small systems (< 3.5 cm), the shared microlens system is best as it gives smaller spot size (higher SBWP) and higher efficiency than the pinhole system. The spot size and SBWP improves as the system shrinks but the maximum off-axis angle increases. The minimum size of the system is limited by the range of angles that is possible to generate by the illumination system. This is the reason why devote chapter 6 to this issue.