Conclusions

In conclusion, we derived a radially symmetric phase-only filter that enhances the system tolerance to an SA5 by using the method of stationary phase approximation similar to the approach of Mezouari and Harvey. Two different implementations have been provided. The proposed approaches of extension and its deduced phase filters will be especially useful in the case when the imaging optical system has a large size of aperture. Which one is better is, of course, dependent on the inherent characteristics of optical systems over the leading and different orders of aberration.

Inclusion of Maréchal treatment in the approach of Mezouari and Harveis is an efficient method to deduce a phase filter with superior tolerance in defocus and spherical aberration as well as better system performance of imaging quality.

The critical issue of trade-off is worthwhile to be readdressed; it has been detailed that the use of the proposed phase filter causes a reduction in intensity and the MTF. These reductions form the baselines in practical applications of the use of phase filter. Optimization has to be reconsidered in developing the filters to meet the performance requirements and/or system specifications. It is worth noting that the pupil phase function in a logarithmic form is in fact a particular solution of the developed differential equations. That is to say, this solution form has been obtained by assigning the added constants to some particular values. This is the freedom and the advantage of the approach of Mezouari and Harvey in deducing an optimized solution of phase filter. In this section, in contrast, it is the capability of “superior tolerance” to be addressed. In other words, one of main goals of this section is to

investigate how “superior” can be established in tolerance based on the approach of Mezouari and Harvey (an approach with solving the different equations of aberration coefficients). Finally, as a comment, it is a simple and straightforward matter to determine the extension to a higher order spherical aberration (say, the seventh order spherical aberration, SA7), which could be more critical in the systems with larger aperture.

Chapter 3

Phase pupil filters employed in minimization of variation of Strehl ratio with defocus and spherical

aberration for dual wavelengths

3.1 Basic theory

For the collimated light passing through an objective lens and a phase-shifting apodizer and then converging through onto an optical disk, the normalized amplitude distribution in the image side can be defined as ^[1-3]:

u r rdr

ρ and u are the simplified radial and axial coordinates, respectively, on the image side:

= NA ⋅R u= 2 (NA)²⋅Z numerical aperture of the objective lens. Here, P(r) represents the generalized pupil function, which for a radially symmetric pupil can be represented as

(3.3)

Therefore, the on-axis amplitude distribution function in the focal region can be expressed as:

Now, if we mainly consider about the effect of defocus and third order spherical aberration (SA3) to the on-axis performance of an optical system, the on-axis intensity distribution is then given by:

2

As the frequency of the incident collimated light varies, the resulting magnitude of defocus (or SA3) will change. Here comes an important issue that we have to find out a relationship between those aberration terms and the frequencies. If the amount of variation of the frequency is small, it is reasonable to state that there still remains a linear relation between the aberration terms and the frequency.

Note that the primary aberration terms, up to fourth order in pupil and object or image coordinates, can be expressed as ^[4-8]:

(3.6)

where those five terms refer to spherical aberration, coma, astigmatism, Petzval curvature and distortion. To evaluate the on-axis performance of an optical system, we mainly pay our attention to the first term, SA3. And for a thin lens system, the coefficient ass, is given by:

where p and q are called the position and shape factors, respectively.

p

=(2

f

/

S

)−1=1−2

f

/

S

' (3.8) q=(R₂ +R₁)/(R₂ −R₁) (3.9) S and S’ refer to the object distance and the image distance, and R1 and R2 are the radii of curvature of the two surfaces of the lens.

If the position factor is given, the value of the shape factor which minimizes the spherical aberration is given by the condition:

=0

For the case that an object is at infinity and the image is at the focal plane of the lens, the value of the position factor p is set to be 1. Substituting the above Eq. 3.11 into Eq.3.7, hence the corresponding minimum spherical aberration is obtained:

]

Thus, the wave-front aberrations for two different incident light frequencies can be expressed as:

With a change of variable, ξ= r²-1/2, we get:

By employing the stationary phase approximation, the axial irradiation distribution is given by: Notice that the stationary points for both frequencies are given by:

[

n

₁Φ( )+

a

₂

W

₀₄₀ ²+(

a

₂

W

₄₀+

a

₁

W

₂₀) ] _{= s} =0

If the added phase filter is used to enhance the system tolerance to SA3 for both wavelengths, the following equations must be satisfied simultaneously:

0

ng pupil phase functions that control SA3 when th

4

Then the resulti e optical system is

at the best focal plane will be written as:

θ_λ1(r)= A₂r⁴ + log( )

θ_λ (r)= B₂r⁴ +B₁r⁴log(r) (3.19-2)

Similarly, the independence of the axial irradiation distribution on defocus aberration

(3.20-1)

(3.20-2)

reduce the defocus erro ible yield:

₂ (3.21-2)

.2 Illustration and Simulation Verification

laser ple here. Now, in order to lower down the sensitivity of leads to the following equations:

Then the resulting phase functions that r when SA3 is

neglig

θ_λ1(r)=α(r² −1/2)² (3.21-1) θ_λ (r)=β(r² −1/2)²

3

A DVD/CD optical pick-up head system, containing 635 and 785 nm diodes, is taken as an exam

the on-axis intensity of this optics system to SA3, the circular symmetric logarithmic phase filter, described in Eq.3.19, is employed. We first consider about the system performance for the wavelength which is equal to the average value of the two wavelengths. How the parameters of the pupil function are determined depends on the system requirement of the magnitude of on-axis intensity and the range of

)

tolerance. For instance, if we want to enhance the tolerance to SA3 for at least five times and still keeps the magnitude of the on-axis intensity at least one- tenth of the original value, we may find it a good option to set the parameters α1= 5.6π and α2= 0.401*α1. Then we pay attention to the variation of the intensity and the tolerance to the deviation of the wavelength, as shown in Fig. 3.1. It can be seen that the corresponding changes of those terms to the deviation of the wavelength are still under control.

Fig. 3.1 Relation among 1) the intensity and 2) the enhanced factor of tolerance to the deviation of the wavelength with the use of the designed filter

ystem tolerance to SA3 for DVD is enhanced for about seven times, as shown in Fig.3.1 (a). Meanwhile, the corre

By setting A1 = α1*(1+0.112) and A2=α2*(1+0.112), the s

sponding values of B1 and B2 are also determined, where B1=α1*(1-0.112) and B^B2=α2*(1-0.112), and the tolerance to SA3 for CD is improved by a factor of 6, as shown in Fig.3.2 (b). However, it should be noticed that the application of the phase filter will shift away the position of the central peak in both cases.

Fig. 3.1 (a)

Fig. 3.2 The Strehl ratio as a function of aberration coefficient W040, with zero defocus (W020=0). The solid curve represents an ideal lens, while the dashed one

transfer function (MTF) for different values of SA3 for these two wavelengths is displayed in Fig. 3.3.

Orig

corresponds to the use of the phase filter (where α1= 5.6π and α2= 0.401*α1) (a) for the case: A1 =α1*(1+0.112) and A2=α2*(1+0.112)

(b) for the case: B1=α1*(1-0.112) and B2=α2*(1-0.112)

A comparison between the computed modulation

inally, the MTF is sensitive to the variation of the amount of additional SA3.

When the designed logarithmic phase filter is used, the MTF becomes less sensitive to

SA3 in both cases. However, there is a reduction of the effective cut-off frequency of about 45%, and a reduction in the signal-to-noise ratio.

Fig. 3.3 Computed MTF with initial setting W020=0 with the proposed phase filter (where α1= 5.6π and α2= 0.401*α1), in which the solid curve: W040=0, the dashed curve: W040=0.5λ, the dotted curve: W040=λ, and the dash-dot curve: W040=2λ.) (a) for the case: A1 =α1*(1+0.112) and A2=α2*(1+0.112)

(b) for the case: B1=α1*(1-0.112) and B2=α2*(1-0.112),

3.3 Conclusions

to implement the minimization of variation of Strehl tio with defocus and spherical aberration for dual wavelengths. By using the In this chapter, we present a way

ra

proposed phase pupil filters, we can see that the system tolerance to SA3 is, indeed, enhanced. However, as a trade-off, the use of the phase filters will inevitably drop down the central peak of the intensity and also the effective cut-off frequency, which leads to worsen the image quality. It should be noticed that, we make some simplification and put some constraints in the deduction of the pupil function, which will also limit the use of the designed phase filters.

Chapter 4

esigned for simultaneously achieving super-resolution for two different wavelengths

4.1 Brief History Review

ssed by Toraldo di Francia in 1952 ^[1]. Being ed by diffraction in optical systems, is idea has aroused considerable interest, especially in the fields of optical storage

plitude (variab

Pupil filters d

Superresolution was first being discu

able to overcome the limits of resolution impos th

and optical microscopy ^[2,3]. One can enhance the storage capability of a single compact disk by reducing the size of the focusing laser spot. Thereby, lots of methods have been proposed to designing superresolving pupil filters. At first, these filters were based on am le-transmittance) pupils ^[4-7]. Later on, attention has been shifted to develop pure-phase filters in order to overcome some drawbacks of amplitude filters: for instance, intensity loss issue ^[8-12]. Many phase profiles that achieve transverse superresolution are based on annular designs, such as the diffractive superresolution elements (DSEs) proposed by Sales and Morris ^[2], and

the three-zone binary phase filters reported by Wang ^[13,14]. For the design of continuous superresolving phase-only profiles, the global/local united search algorithm (GLUSA) is generally used ^[15]. However, it requires extremely complex phase masks to achieve the wanted performance. In order to conquer that difficulty, superresolving continuous smoothly varying phase-only filters, obtained by using a series of figures of merit which are properly defined to describe the effect of general complex pupil functions, were proposed ^[16]. The advantages of these kinds of filters are that they don’t produce energy absorption and they are easy to build with a phase-controlling device such as a deformable mirror.

In this chapter, we present a way of how a rotationally symmetric four-zone pupil is being designed, to both achieve superresolution property for two different

.2 Basic theory for super-resolution

T on for small

displacements of the focus position from geometrical focus

The diff the

tationally symmetric amplitude pupil functions, can be analyzed as follows.^[17]

filter

wavelengths. The problems that we face in the design will be carefully discussed in the following pages.

4

4.2.1 he 2

^nd

order expansion of the intensity distributi

racted intensity distribution near the geometrical focus, while applying ro

The general complex pupil function can be written as:

P(ρ)=T(ρ)⋅exp[iϕ(ρ)] (4.1)

where T(ρ) is the transmittance function, and φ(ρ) is the phase function. Then the itude U in the focal reg

normalized complex field ampl ion can be written as:

∫

⁻

Here υ and u are radial and axial optical coordinates, respectively ); focal plane, the diffracted field distr

U

(

υ

,0) 2

P

(

ρ

)

J

(

υρ

)

ρ d ρ

(4.4)

Notice that the above equation is the Hankel transform of the pupil function. Along the axis, we will get:

Where sin α is the numerical aperture of the system, and r and z denote the rad

axial distances. In the ibution will be:

Then the field distribution along the axis can be written as:

⁼

∫

^{1 −}

Therefore, it’s clear to see that the above equation is the Fourier transform of the equivalent pupil function Q(t). The pupil function is assumed as P(ρ) =1 for the case of achieving the diffraction limit . Somehow, if we carefully modify the pupil function P(ρ), super-resolution can be realized.

According to the theories of Sheppard and Hegedus and De Juan et al. ^[7], within the 2nd order approximation, the transverse and axial intensity distributions can be expressed as:

where * denotes the complex conjugate and In is the nth moment of the pupil function

=

It can be seen that the transverse intensity is symmetrical with respect to the geometrical focus (v = 0, u = 0) [see Appendix B.1]. However, for the axial intensity this

given by:

∫

^{1 2n+1}

is not true in general. The displacement of focus in the axial direction and the Strehl ratio are given by:

2 The transverse and axial gains, which are defined as the ratio between the

approximation of th

squared width of the parabolic e intensity PSF without the filter ith the filter, are given by:

GT and GA are greater than unity for transverse or axial superresolution, respectively.

It should be noted that Eq. 4.10 is valid only for small displacements of the focus position from the geometrical focus, where the second-order expansion of the

.2.2 The 2

^nd

order expansion of the intensity distribution to the case in which the best image plane is not near the paraxial focus

J.Camp xial

nd the transverse gain to the case in which the best image plane is not near the

intensity distribution is a good approximation to describe the focal behavior. The position of the maximum intensity is given by the coordinates (0, uF). Analogously to the development by Sheppard and Hegedus, expressions for the transverse and axial gains corresponding to complex pupil functions are obtained from the second-order expansion of the intensity with respect to this point.

4

os, J. C. Escalera, and M. J. Yzuel have extended the expressions for the a a

paraxial focus [see Appendix B.2]. They first search for the maximum of the on-axis intensities, and then they develop up to the second order superresolution factors around that point, say umax. The generalized expressions for those factors are expressed as below:

Note that u0 is measured from the BIP centered at u to zero for most functions of an optical system.

sverse intensity distributions directly the basic diffraction theory. The transverse intensity distribution of the image

p

.3 Structure of Hybrid Dual Focus Lens

CD for D

max, so its values will be close

4.2.3 Fourier Optical Transformations

It is necessary for us to give a check to the tran from

intensity can be obtained by directly convoluting the object intensity distribution function with the point spread function of the optical system. However, it could be a terrible job, implementing the convolution operation [19]. As an alternative choice, we take the Fourier transform of the object intensity distribution function first, and then multiply it with the optical transfer function of the o tical system. Next, by taking an inverse Fourier transform of it, the transverse intensity distribution is obtained. Avoiding taking the convolution operation but alternatively implementing the FFT (inverse FFT), it’ll help save lots of computing time.

4

A dual focus objective lens of combining aspheric surfaces of DVD and DVD/C pick-up head has been proposed in 1996 ^[18].

Fig. 4.1 Schematics of (a) the single-ring and (b) the double-ring dual focus lens.

As shown in Fig. 4.1 (a), for a lens consisting of two zones, the CD aspheric surface is located in the central region, while the DVD aspheric surface is located in the outward region. In this way, the objective lens for both CD and DVD can be made into a single lens. However, as a drawback, the lens used in DVD will suffer an obvious side-lobe of the focusing spot. In order to reduce the side-lobe effect of the DVD spot, a four-zone scheme is used, which is shown in Fig. 4.1 (b). The CD aspheric surface portion is composed of the central circle area and the middle zone, while the inner and outward zones form an aspheric surface for DVD lens.

The quality of focus spot of the designed objective lens is a function of the width and position of these zones, which can be numerically calculated based on the scalar diffraction model. Theoretically, the focus spot of DVD and CD can be calculated independently when the focus lengths of DVD and CD are different. Therefore, to calculate the focus spot of DVD, the region of CD is viewed as a mask, and vice versa.

In fig. 4.2, we show the resulting schematic of ray tracing, for which light propagates through a singlet lens with hybrid structure designed for two wavelengths.

The simulated result of the transverse intensity distribution at the focal plane is shown in Fig. 4.3. It tells that the intensity near the optical axis is so strong so that that the effect introduced by the defocusing light can entirely be neglected. This result confirms the validity of making the previous assumptions in the last paragraph.

Fig. 4.2 Schematic of the ray trace of a hybrid lens system

Fig. 4.3 Raial intensity distribution of a hybrid lens system at the focal plane

4.4 Set-up of The Four-zone Pupil Filter

In order to both obtain the super-resolution property for the two DVD/CD wavelengths, the objective lens system is modified by adding a complex pupil filters.

The structure of the complex pupil filter is shown in Fig. 4.4.

Fig. 4.4 Structure of the four-zone pupil filter

For the portion of CD surface, as shown in Fig. 4.5, since the second zone corresponds to the region of DVD surface, the transmittance of the second zone is assumed to be zero. The transmittances of the first zone and the third zone are assumed to be t1 and t3; and the corresponding phases of the first zone and the third zone are ψ1 and ψ3. The radii are a, b, and 1. The pupil function for CD can be

Fig. 4.5 Schematics of the portions of the filter for CD

Then the following moments of the pupil function for CD are obtained:

For the portion of DVD surface, as shown in Fig. 4.6, since the first and third zone correspond to the region of CD surface, so the transmittance of these two zones are assumed to be zero. The transmittances of the second zone and the fourth zone are T2 and T4; and the corresponding phases of the second zone and the fourth zone are Ψ2 and Ψ4. The radii are c∙a, c∙b, c, and 1. The pupil function for CD can be

Fig. 4.6 Schematics of the portions of the filter for DVD

Then the following moments of the pupil function for DVD are obtained:

Substituting those moments into Eqs. 4.11 and 4.12, the Strehl ratio, the transverse and the axial gains will be expressed as:

On this condition, substituting Eqs. 4.14 and 4.18 into Eqs. 4.4 and 4.5, the expression of the transverse amplitudes for CD and DVD are obtained:

For CD: Thus, we can obtain the intensity distributions along the transverse direction.

4.5 Illustration and Simulation Verification

4.5.1 Design procedure

The design procedure that we propose here has the following steps:

Step 1:

After carefully setting up all the parameters, like the radius and transmittance of each zone, we use the second order approximation theory, mentioned in section 4.2.1, to calculate the superresolution factors, like the transverse gain, the Strehl ratio, and the displacement of axial focus. Then we can narrow down the range of the parameter in which the wanted superresolution property may be possibly achieved.

Step 2:

To check the accuracy of the computed result gotten in step 1, the transverse intensity distributions are computed directly from the basic diffraction theory without any approximation. We make radial intensity scans at various axial coordinates to find out in where the best image plane (BIP) appears with the applicant the designed filter.

Step 3:

Once the location of the BIP is found, we calculate the gain parameters for the filter in the surrounding of the shifted focus, as what is mentioned in section 4.2.2.

Then we fine tune the parameters, like the transmittances of the zones, and observe

the corresponding changes of the superresolution factors.

Step 4:

Similar to the action we list in step two, we give a further check to see if the transverse intensity distributions really achieve transverse superresolution for both

Similar to the action we list in step two, we give a further check to see if the transverse intensity distributions really achieve transverse superresolution for both

在文檔中 針對光學成像系統不同應用之相位片之分析與設計 (頁 41-0)