Summary and remarks - Optical transfer functions for the specific-shaped apertures

Chapter 3 Optical transfer functions for the specific-shaped apertures

3.6 Summary and remarks

The pupil function of the specific shaped aperture generated by illumination with a rectangular hollow light pipe has been investigated. The corresponding OTFs were derived in the aberration-free and defocused optical systems, respectively. The semi-analytical results indicate that the OTF’s curve of the optical system can vary with the different shaped apertures that are generated by the illumination with light pipes and light sources of different geometric structures.

In summary, (1) the OTF values of the even-peak frequencies can decrease when the size of the Lambertian light source decreases, (2) if there are a total of n × n individual apertures within the pupil, then there are n near-periodical peaks on the OTF curve, (3) the OTF’s values remain almost unchanged with the different length of the light pipe, (4) the geometric structure of the light pipe does not affect the resolution limit of the optical system, and the case of the defocused system can coincide with that of the aberration-free system under the condition of a larger defocused coefficient ω20.

The semi-analytical method can be extended for any symmetrical cross-section shape of the light pipe, for example, the hexagonal cross-section shape or elliptical one, and any asymmetrical cross-section shape of the light pipe, which can generate the symmetrical and asymmetrical forms of the pupil function, respectively. For these cases, it is difficult to describe the pupil functions of the specific shaped apertures using the photometric method, as we did for rectangular light pipes. However, by utilizing a simulation package, one can easily verify the illumination distribution for the specific light pipes by Monte Carlo non-sequential ray tracing. Then, we can obtain an approximating pupil function which performs a fit for a sequence of ray-tracing data using the interpolation method, and we can calculate and analyze the OTFs of the specific cases finally.

Furthermore, the semi-analytical method can be extended for the lithography optical system with the mercury arc lamp and quartz rectangular light pipe, for the color-LED array with specific light pipe, or for the endoscope optical system with circular light pipe and so on. Also, this method can also be applied to other kinds of light integrator in an optical system, such as for example, the fly-eye integrator.

Because the level of the requirements of the resolution for the object, the light-valve and the light-mask will continue to increase, an investigation into the relationship between the projection system and the illumination system is certainly worthwhile and will be explored in the future.

Chapter 4 Programmable apodizer in incoherent imaging system using a digital micromirror device

4.1 Introduction

This chapter proposes a programmable apodizer using a digital micromirror device and the total-internal-reflection prism subsystem for incoherent imaging systems. It is shown that the proposed programmable apodizer can extend the depth of focus with the specific shaped aperture generated by the digital micromirror device.

With a scale ratio K equal to or less than 0.05, one can achieve almost the same level of imaging quality as provided by the conventional annular apodizer, where K represents the ratio between the integer multiple of the micromirror’s square pixel size and the diameter of the effective aperture stop.

The remainder of this chapter is organized as follows. In Section 2, we revisit on expending the depth of focus (EDOF) and digital micromirror device (DMD). In Section 3, we describe the configuration of the proposed system which consists of a digital micromirror device and a total-internal-reflection prism subsystem is illustrated. In section 4, we derive the pupil functions of the differently shaped apertures which are generated by the digital micromirror device. Then, in section 5, we calculate the optical transfer function and analyze the image quality in such an incoherent imaging system. Furthermore, the corresponding OTFs is evaluated and then to identify the imaging performance for a system of perfect imaging (aberration-free) as well as the defocused one in section 5. Finally, the summary and remark are given in section 6.

4.2 Revisit on extending the depth of focus (EDOF) and digital micromirror device (DMD): technology impact

Extending the depth of focus (EDOF) in an imaging system has been a long-standing issue in optical designs. Enhancing the quality of an image can be achieved and determined not only by the pupil function but also by its amplitude transmittance [1]. Non-uniform amplitude transmission filters can be employed to vary the response of an optical imaging system, for instance, to increase the focal depth and to decrease the influence of spherical aberration. Earlier EDOF investigations and experiments were carried out on annular apodizers [3,4], non-uniform shaped apertures [6,7] and wave-front coding [8,9] in imaging systems, where the nature of light is incoherent. However, none of those are programmable for the amplitude transmission at the aperture stop. From the point of view of potential applications as well as from a purely academic interest perspective, it is worthwhile to explore the possibility of how to realize a programmable apodizer for incoherent imaging systems.

In the literature, amplitude-transmitting filters for apodizing and hyper-resolving can be implemented with a programmable liquid-crystal spatial light modulator operating in a transmission-only mode in a coherent imaging system with the laser light source, polarizers and quarter-wavelength plates [50,51]. In this chapter, we proposed a programmable apodizer using the digital micromirror device (DMD™; Texas Instrument, Dallas, Tex.) [52] and the total-internal-reflection (TIR) prism subsystem in a polarization-free mode in an incoherent imaging system. We evaluated the imaging properties of the incoherent imaging system with a specific shaped aperture generated by DMD™ by calculating the optical transfer function (OTF) using the Hopkins method [37]. We also included the OTF of the specific shaped aperture for the conventional annular apodizer, which has been demonstrated,

both theoretically and experimentally, by Mino and Okano [6] to show that the proposed programmable apodizer can not only extend the depth of focus but can also achieve almost the same level of imaging quality as the conventional annular apodizer in an incoherent imaging system.

4.3 Configuration of the optical system with digital micromirror device

The schematic sketch of an incoherent imaging system using one DMD™ and a charge-coupled device (CCD) imager is illustrated in Fig. 4-1. The system is formed by an image-taking lens module and a prism module. By following the paths of the axial rays as indicated by the solid lines in Fig. 4-1, the rays starting from a point in the object pass through Lens #1 and a prism module. The size of the axial cone of energy from the object is limited by the aperture stop on the DMD™. The DMD™

consists of hundreds of thousands of moving micromirrors that are made to rotate to either +12° or -12° positions depending on the binary state, i.e. on-state or off-state, of the underlying Complementary Metal Oxide Semi-conductor Synchronized Dynamic Random Access Memory (CMOS SRAM) cells below each micromirror [52].

Figure 4-1 Schematic diagram of the incoherent imaging system with the DMD™ and the total-internal-reflection (TIR) prism subsystem.

The DMD™ array size is 1024×768, and the pixel micromirrors measure

~13.7 µm square to form a matrix having a high fill factor of around 90%. The prism system comprises two transparent prisms, with an air gap between them. Total internal reflection (TIR) at the interface between the prism and the air gap is utilized to separate the rays by their angle. Total-internal-reflection (TIR) prism has been applied into the DMD™-based projection display in practice [14]. The prism system can guide the rays onto and away from the DMD™ simultaneously. The rays indicated by the dotted lines in Fig. 4-1 from the object are imaged and focused onto the CCD by Lens #2 when the configuration of the DMD™ is the on-state. When the configuration of the DMD™ is the off-state, the rays indicated by the dashed lines in Fig. 4-1 are steered away in the opposite direction, and the rays from the object are not imaged on the CCD. The DMD™ performs a spatial light modulation to rapidly generate a specific shaped aperture with either uniform or non-uniform illumination distribution at the aperture stop in an imaging system within the limited exposure time.

This digital micromirror device can provide a programmable apodizer with a specific binary transmission for the incoherent imaging system. This implementation is not limited by this practical device. The TIR prism performs light separation to manage the illuminations and also make the normal vectors of the object, aperture stop and image planes, respectively, coincide with the optical axis of the optical imaging system with the most compact volume.

4.4 Optical computation for pupil function

The pupil function of a defocused optical system with a circular symmetrical aperture is referring to Eq. (2-2) and given by [6]

( ) ( ) [ ( ) ]

where ω20 is the wave aberration of the defocused coefficient, (x ,y ) are the normalized Cartesian coordinates, and k = 2π/λ, where λ is the wavelength of the light.

Function ^{T ′}

( )

^x,^y in Eq. (4-1) represents the binary amplitude distribution over the normalized pupil coordinate that is scaled and normalized to make the outer periphery the unit circle, x² +y² ≤ 1 . The binary amplitude transmittance T ′

( )

x,y is generated by the DMD™ as shown in Fig. 4-2.

Figure 4-2 Illustration of the binary amplitude transmittance T ′

( )

x,y for the normalized circular aperture which is generated by the DMD™. T

( )

x,y represents a specifically shaped aperture for a conventional annular apodizer.

We can derive the amplitude transmittance of the shaped aperture ^{T ′}

( )

^x,^y ⁱⁿ

an on-state configuration as follows,

( ) ( ) ( )

)

the amplitude transmittance with a continuous profile at the aperture stop which can extend the focal depth in the imaging system with a conventional annular apodizer [6].

D is the corresponding diameter of the effective aperture stop. c represents the width of each square individual aperture generated by DMD™ in the pupil plane, which is equal to an integer multiple of the value d, with d being the width of each square pixel in the DMD™. ^δ

[

^x^-

(

^2mc/D

) ]

^δ

[

^y^-

(

^2nc/D

) ]

denotes the delta function indicating the location of the individual aperture in the normalized coordinate on the aperture stop.

( )

x,y

[

(

x c/D

) (

-H x- c/D

) ]

[

(

y c/D

) (

-H y -c/D

) ]

E' = + × + is the binary amplitude

transmittance of the individual shaped aperture, which is then scaled and normalized into the pupil coordinate. ^Int

[

(^D/c^-¹)^/²

]

is the interpart of [(^D/c^-¹)^/²]^.^H(x⁺^c/D), H(x- c/D), H(y+ c/D) and H(y- c/D) are the step functions. It is evident that the total aperture function is formed by convolving the individual aperture function with an appropriate array of the delta function, each located at one of the coordinate origins

(

x_m ,y_n

) (

= 2mc/D ,2nc/D

)

, where m, n = …-2, -1, 0, 1, 2, …

The quality and location of the individual aperture on the pupil depends on the scale ratio defined as

( )

c/D

K ≡ (4-4) for the specific diameter of the effective aperture stop on the DMD™ in the imaging system. The value of the scale ratio K determinates how many resolutions, how many gray levels and how fast the DMD™ can dynamically generate the shaped apertures within a specific exposure time.

It is worthwhile to give an example for illustration. If the DMD™ array is 1024×768 with a pixel size of 13.7 µm square, and the active area is 14.03 mm × 10.52 mm = 147.60 mm² [52], then the number of D is ~10.52mm (i.e. equal to the width of the active area of the DMD) if the effective aperture stop is located on the circular area centered at the actual DMD™. In the case of K =0.05, the width of each individual square aperture c is 0.53mm and is equivalent to 38 square pixels with the same amplitude transmittance. There are 10 (i.e.Int[(D/c-1)/2]+1) gray levels for a specific shaped aperture including the full bright mode and full dark mode. The current DMD™- based system can offer 8 bits or 256 gray levels within a time period of 5.6 ms per primary color [52]. Thus, the DMD™ can rapidly generate one shaped aperture with 10 gray levels within the very short exposure time of 0.22 ms (i.e.

5.6×10/256) in the case of K =0.05.

The computer program for evaluating Eq. (4-2)-(4-4) is written in MATHEMATICA software [41]. We assumed D=2 for the simplification and evaluated four cases for the scale ratios K=0, K =0.05, K =0.1 and K =0.3. The binary amplitude transmittances of the shaped apertures T ′

( )

x,y are shown in Fig. 4-3. The scale ratio K=0 stands for the amplitude transmittance with a continuous profile. It is evident that the scale level of the binary amplitude transmission at the aperture stop increases with the reduction of scale ratio K, and the distribution of the binary amplitude transmission gets close to the continuous profile if the scale ratio K decreases.

Figure 4-3 Binary Amplitude transmittance T ′

( )

x,y with T(x, y) =1- (x²+y²) on the normalized pupil in the condition of D=2, and scale ratios at (a) K=0, (b) K=0.05, (c) K=0.1, and (d) K=0.3, which are generated by DMD™.

In order to evaluate the relationship between the image performance and the size of the individual square aperture on the normalized pupil, we modified Eq. (4-2) and (4-3) to the following equations.

( ) ( ) ( )

⁾ amplitude transmittance with a clear aperture. D is the corresponding diameter of the effective aperture stop. c represents the width of each square individual aperture generated by DMD™ in the pupil plane. The parameter a represents the distance between each square individual aperture as shown in Fig.

4-2.^δ

[

^x^-

(

^2ma/D

) ]

^δ

[

^y^-

(

^2na/D

) ]

denotes the delta function indicating the location of the individual aperture in the normalized coordinate on the aperture stop.

( )

x,y

[

(

x c/D

) (

-H x- c/D

) ]

[

(

y c/D

) (

-H y -c/D

) ]

E' = + × + is the binary amplitude

transmittance of the individual shaped aperture, which is then scaled and normalized into the pupil coordinate. ^Int

[ (

^D/a^-¹

)

^/²

]

is the interpart of

[ (

D/a-1

)

]

. H(x⁺c/D), H(x- c/D), H(y+ c/D) and H(y- c/D) are the step functions. It is evident that the total aperture function is formed by convolving the individual aperture function with an appropriate array of the delta function, each located at one of the coordinate origins

(

x_m ,y_n

) (

= 2ma/D ,2na/D

)

, where m, n = …-2, -1, 0, 1, 2, … We assumed D=2 and a=0.25 in Eq. (4-5) and (4-6). Four cases of the binary amplitude transmittances

( )

^x,^y

T ′ for c=0.05, c =0.1, c =0.15 and c =0.2 were computed and shown in Fig. 4-4.

There are nine individual apertures along x-axis and y-axis within the pupil, respectively. The results show that the individual aperture size on the normalized pupil is shrunk with the width of each square individual aperture (i.e. the value c) generated by DMD™. That is equivalent to the term E'

( )

x,y varied with c in Eq.

(4-5).

Figure 4-4 Binary Amplitude transmittance T ′

( )

x,y with T(x, y) =1 on the normalized pupil in the condition of D=2 in different conditions of (a) c=0.05, (b) c=0.1, (c) c=0.15 and (d) c=0.2, which are generated by DMD™.

4.5 Optical transfer function and image performance evaluation

The OTF is derived from the autocorrelation of the pupil function by using the Hopkins canonical coordinate [37, 48-49] and is given by

( ) ( )

the f-number of the imaging lens system, λ is the wavelength and N is the number of cycles per unit length in the image plane. The denominator of Eq. (4-7) is the

normalizing factor for making τ0 (0) = 1. The g (s, 0) and g(0, 0) in the OTF for the

Eqs. (2-9) and (2-10) can be further modified as

( )

[

( )

]

and (2-10) with the y-axis for the summation in Eqs. (2-11) and (2-12), an initial setting of p=100 is made for the number of intervals used to find the value of

[

¹ ^-

( )

^s/2

]

^/ ^p'

∆ = ² ^1/2 and ∆y =1/p for g(s, 0) and g (0, 0), respectively. Different

numbers of y, from -

[

(

s/2

)

]

¹^/² to

[

(

s/2

)

]

¹^/² , are then used to calculate the OTF.

The OTF of the different pupil functions are numerically computed using MATHEMATICA software [41] based on Eqs. (4-2)-(4-4), (4-11) and (4-12) for binary amplitude transmittances of the aperture functions for K=0.05, K=0.1, and K=0.3 which are shown in Fig. 4-3. The OTF of the aberration-free and defocused system with a clear aperture T(x, y) = 1 (i.e. a uniform-shaped aperture) and one specific shaped aperture T(x, y) =1- (x²+y²), as x²+y²≦ 1, and T(x,y)=0 as x²+y²> 1, over the normalized pupil coordinate which is scaled to make the outer periphery as one unit circle, i.e., x² + y² ≤ 1, respectively, already investigated theoretically and experimentally in the literature [6], are calculated again here for comparison.

The results for the aberration-free system i.e., ω20= 0 is shown in Fig. 4-5(a), and the defocused systems with ω20= λ/π, 3λ/π, 5λ/π, 10λ/π and 15λ/π are shown in Figs. 4-5 (b)-(f), respectively. We compared the OTF of the different scale ratios K to the OTF of a clear aperture T(x, y) = 1. For the large values of ω20 , e.g., 5λ/π, 10λ/π and 15λ/π as shown in Fig. 4-5(d)-(f), the spatial frequency corresponding to the first zero becomes smaller. Since the spatial frequency of the first zero generally represents the resolution limit of a defocused imaging system, we can take the first zero as defining the degree of focus. The larger degree of focus in the larger value of ω20 commonly represents the longer depth of focus in a defocused system. For the large values of ω20 , the degree of focus for the aperture with scale ratio K less than 0.3 is larger than for the aperture T(x, y) = 1. It is evident that the specific shaped aperture, which is generated by the DMD™ with scale ratio K=0.3 or less, can then extend the depth of focus compared to a clear aperture in the conventional imaging system. We also compared the OTF of the different scale ratios K to the OTF of one specific shaped aperture T(x, y) =1- (x²+y²). The OTF value of the former increased and got close to the OTF value of the latter when the scale ratio K decreased gradually, especially in the low spatial frequency region. This

shows that the OTFs of the specifically shaped aperture, which are generated by the DMD™, with scale ratio K=0.05 or less can coincide with the OTF of the conventional annular apodizer with continuously-shaped aperture.

Figure 4-5 Optical transfer functions in an aberration-free system and a defocused system with different defocused coefficients (a) ω₂₀= 0, (b) ω₂₀= λ/π, (c) ω₂₀= 3λ/π, (d) ω₂₀= 5λ/π, (e) ω₂₀= 10λ/π and (f) ω₂₀= 15λ/π, for binary amplitude transmittances of the aperture functions for K=0.05, K=0.1, and K=0.3, which are generated by the DMD™, and for a clear aperture T(x, y) = 1 and one specific shaped aperture T(x, y) =1- (x²+y²) of a conventional annular apodizer, respectively.

To highlight the capability of our approach, we take a spoke pattern to explore imaging performance. Referring to Fig. 4-6, in column A, one could see the images for the clear aperture, while in column B, the images for the specific shaped aperture with scale ratio K=0.05 are shown. Furthermore, the lines (a)-(d) correspond to the images obtained with the defocused coefficients of ω20= 0, ω20= 5λ/π, ω20= 10λ/π, and ω20= 15λ/π, respectively. Comparing with the images for the specific shaped aperture, the images for the clear aperture show a more critical loss of contrast at high spatial frequencies with larger ω20. Hence, we can conclude that the image quality will be enhanced as the specific shaped aperture is used, especially for the large defocus coefficients in an imaging system. In other words, as for a real implementation of the DMD™, the specific shaped aperture can extend the depth of focus compared to a clear aperture in the conventional imaging system.

Figure 4-6 The computer-simulated images of spoke patterns for A, a clear aperture, and B, a specific shaped aperture with the scale ratio K=0.05 obtained with different defocused coefficients: (a) ω₂₀= 0, (b) ω₂₀= 5λ/π, (c) ω₂₀= 10λ/π and (d) ω₂₀= 15λ/π.

In order to evaluate the relationship between the image performance and the size of the individual square aperture on the normalized pupil, we computed the OTF of the other types of pupil functions based on Eqs. (4-5)-(4-6), (4-11) and (4-12) for

the binary amplitude transmittances ^{T ′}

( )

^x,^y for c=0.05, c =0.1, c =0.15 and c =0.2 in the conditions of T( )x,y =1, D=2 and a=0.25 which is shown in Fig. 4-4. The OTF of the aberration-free system with a clear aperture T(x, y) = 1 was also calculated here for comparison. The OTF’s calculation results for the aberration-free system are shown in Fig. 4-7. The general tendency of the OTF curve versus the number of the individual aperture within the pupil is readily evident. There are nine individual apertures along x-axis and y-axis within the pupil, respectively. Meanwhile, there are nine peaks on the OTF curve at the near-periodic 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 individual apertures (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 decrease in the size of the individual aperture. Therefore, if the size of an individual aperture is much less than that of pupil, then OTF values of the even-peak frequencies

the binary amplitude transmittances ^{T ′}

( )

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