Chapter 3 Optical transfer functions for the specific-shaped apertures
3.5 Optical transfer function and image performance evaluation
The optical transfer function (OTF) is derived from the autocorrelation of the pupil function [37] by using the Hopkins canonical coordinates [48,49] and is given by
( ) ( )
f/# is the F-number of the projection lens system , λ is the wavelength and N is the number of cycles per unit length in the image plane. The denominator of Eq. (3-14) is the normalizing factor to make D0 (0) = 1. Then the g (s, 0) and g(0, 0) in OTF for the exp[iKx] can be reduced to cos[iKx] in the integral of Eq. (3-15). Then g(s, 0) is rewritten asIn order to have the numerical computation in the Mathematica software [41], we need to modify the Eq. (3-16) and Eq. (3-17) to the following convenient forms, i.e.,
( )
[( )
]where
(3-16) and Eq. (3-17) with y-axis for the summation in Eq. (3-18) and (3-19), an initial setting of p=100 is made for the number of the interval used to find the value of
[ ( ) ]
The OTF of different pupil functions is numerically calculated using the Mathematica software based on the Eq. (3-11)-(3-13), (3-18) and (3-19). The OTF of the aberration-free and defocused system with the clear aperture T0(x, y) =1 (x2+y2≦ 1) and 0 (x2+y2 >1), already investigated by Hopkins [37], was calculated as a comparison. We calculated and analyzed the OTFs of the four cases that are with specific shaped aperture generated by the illumination with light pipe as follows, (1) Case-1: the OTF in the aberration-free system with specific shaped apertures of
different geometric structure of the light pipe: a = b= 2.5 is fixed, L= 60 is fixed, but c varies from 0.5, 1.0, 1.5 to 2.0, as shown in Fig. 3-7. The OTF’s calculation results are shown in Fig. 3-10. The general tendency of the OTF curve versus the number of the individual aperture within the pupil in readily evident. There are a total of 9 × 9 =81 individual apertures within the pupil. Meanwhile, there are 9 peaks on the OTF curve at the near-periodically spatial frequencies around 0, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.70 and 1.90. The OTF values of the odd-peak
frequencies for different sizes of the Lambertian light sources (i.e. value c) remain very similar to the values of the OTF’s of the corresponding frequencies for the clear aperture. However, the OTF values of the even-peak frequencies can decrease with the decrease in the light-source size. Therefore, if the size of an individual aperture is much less than that of the pupil, then OTF values of the even-peak frequencies will be almost zero.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
s
OTF
c=2.0
c=1.5
c=1.0 c=0.5
T0
Figure 3-10 Optical transfer functions in aberration-free system with a clear aperture T0 and specific apertures generated by different light pipe’s geometric structure with fixed a=b=2.5 and fixed L= 60 and different conditions of c=0.5, c=1.0, c=1.5 and c=2.0.
(2) Case-2: the OTF in the aberration-free system with specific shaped apertures of different geometric structure of the light pipe: c = 2.0 is fixed, L= 60 is fixed, but a = b varies with 2.5, 3.5, 5.0, 7.5 and 10.0, as shown in Fig. 3-8. The results of the OTF’s calculation are shown in Fig. 3-11. For a=b=2.5, there are a total of 9
× 9 = 81 individual apertures within the pupil, and there are 9 peaks on the OTF curve. For a=b=3.5, there are a total of 7 × 7 = 63 individual apertures within the pupil, and there are 7 peaks on the OTF curve. Together with the other cases,
we can conclude that if there are a total of n × n individual apertures within the pupil then there will be n peaks on the OTF curve. Furthermore, the peaks show at nearly periodic responses to the OTF curve.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
s
OTF
a=b=2.5
3.5
5.0 10.0 7.5
T0
Figure 3-11 Optical transfer functions in aberration-free system with a clear aperture T0 and specific apertures generated by different light pipe’s geometric structure with fixed c=2.0 and fixed L= 60 and different conditions of a=b=2.5, a=b=3.5, a=b=5.0, a=b=7.5 and a=b=10.0.
(3) Case-3: the OTF in the aberration-free system with specific shaped apertures of different geometric structure of the light pipe: a = b= 2.5 is fixed, c= 2.0 in fixed but L varies with 20, 30, 60 and 120, as shown in Fig. 3-9. The OTF’s calculation results are shown in Fig. 3-12. We can see that the values of the OTF remain almost unchanged with the different lengths of the light pipe (i.e. the value of L.) even though the amplitude of the individual aperture decreases from the center to the peripheral when the length of the light pipe is increases.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 s
OTF
L=20 L=30
L=60
L=120
T0
Figure 3-12 Optical transfer functions in aberration-free system with a clear aperture T0 and specific apertures generated by different light pipe’s geometric structure with fixed a=b=2.5 and fixed c= 2.0 and different conditions of L=20, L=30, L=60 and L=120.
(4) Case-4: the OTF in the defocused system with the clear shape and the fixed shaped apertures and with a geometric structure of the fixed light pipe: a = b= 2.5 is fixed, c= 2.0 is fixed, and L =60 is fixed too, as shown in Fig. 3-8 (a). The OTFs are calculated for the different defocused coefficients for ω20= 0, λ/π, 2λ/π, 3λ/π, 5λ/π and 10λ/π, as shown in Fig. 3-13 (a) and Fig. 3-13 (b), respectively. The spatial frequency of the first zero is commonly defined as the resolution limit of the optical system with aberration, and we can assume that the first corresponding zero is the degree of the defocused system. When we compare Fig. 3-13 (a) to Fig. 3-13 (b), it is evident that the degree of defocus for the optical system with a clear aperture is equal to that of the optical system with specific shaped apertures generated by the light pipe. Therefore, we can conclude that the geometric structure of the light pipe does not affect the resolution limit of
the optical system. We also find that the OTFs of the defocused system can coincide with those of the aberration-free system under the condition of a larger ω20.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
s
OTF
ω20=0 λ/π
2λ/π 3λ/π 5λ/π 10λ/π
(a)
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
s
OTF
w20=0
λ/π
2λ/π 5λ/π 3λ/π
10λ/π
(b)
Figure 3-13 Optical transfer functions in defocused system with (a) a clear aperture and (b) one specific aperture generated by different light pipe’s geometric structure with a=b=2.5, c= 2.0 and L=60 for the different defocused coefficients ω20= 0, λ/π, 2λ/π, 3λ/π, 5λ/π and 10λ/π.