4. Pupil filters designed for simultaneously achieving super-resolution for
4.2 Basic Theory for Super-resolution
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 wavelengths. If yes, the goal is accomplished.
4.5.2 Simulation Verification
Step 1:
With the condition a2+b2=1, we adjust the phase factors and the radii of these four zones. For CD surface, t1 and t3 are still assigned both to be 1; and φ1 and φ3 are assigned to be 0 and 0.4π. For DVD surface, T2 and T4 are also assigned both to be 1; and Ψ2 and Ψ4 are assigned to be 0 and 0.35π. The value of radius c is first assumed to be 0.7, while the value of radius a varies from 0.4 to 0.6, and the value of radius b is obtained from the relation b= (1-a2)1/2.
With all the factors being settled down, we obtain the relation among the radius a and the transverse gains and Strehl ratio of such a system for both CD and DVD cases, as shown in Fig. 4.7(a) and 4.7(b), respectively. It can be observed that with increase in radius a, the transverse gain decreases for CD, while one increases first and then decreases for DVD. In the range that a∈[0.4, 0.6], the gains are all greater than 1, which means that superresolution property can be realized simultaneously for both CD and DVD. It can also be seen that, for the case of CD, the Strehl ratio increases with the increment of the radius a, but decreases for the case of DVD.
Fig. 4.7: 1) Relation among the transverse gain and the radius of the first zone.
2) Relation among the Strehl ratio and the radius of the first zone.
(a) for the case of CD (b) for the case of DVD
Figures 4.8(a) and 4.8(b), respectively show the transverse intensity distributions for three particular solutions for CD and DVD, corresponding to three different kinds of set-up of the pupil filter, in comparison to the case of clear pupil. The intensity has been normalized to the clear pupil size.
It should be carefully minded that the simulated results of the intensity distribution, shown in Fig. 4.8, disagree with those derived from Eqs. 4.11 and 4.12.
For instance, when assuming the value of radius a to be 0.6, the predicted value of the Strehl ratio for CD derived from Eq. 4.11 is about 0.9, while that for DVD is approximately equal to 0.5.
Fig. 4.8: The transverse intensity distributions 1) with the clear pupil, 2) with the designed filter, for a=0.5 (solid curve), a=0.57 (dashed curve), a=0.6 (dotted curve) (a) for the case of CD (b) for the case of DVD.
However, as shown in figure 4.8, the central peak value of the normalized intensity distribution for CD is, in fact, less than 0.4, while that for DVD is less than
0.5. Besides, for CD, the set-up of the filter doesn’t achieve the goal of superresolution, but even enlarges the spot size in the focal plane. For DVD, though transverse superresolution is obtained, the transverse side-lobe is tremendously worsened in the focal plane. It is noticed that, for both cases of CD and DVD, the contrast is worsened, which makes it much more difficult in the practical application of reading the data from the disk.
(a) (b)
Fig. 4.9: Relation among the displacement of focus in the axial direction and the radius of the first zone. (a) for the case of CD (b) for the case of DVD
It can be clearly be seen in Fig. 4.9 that the computed value of the axial displacement, uF, is in fact quite apart from zero for both cases of CD and DVD.
The place where maxima intensity occurs has been shifted away from the geometric focus, so that the derived forms of Strehl ratio and transverse gain, based on 2nd-order approximation, become incorrect in describing the focal behavior for the designed cases here.
Step 2:
Now, the intensity distributions along the transverse direction for CD and DVD
at the paraxial focus are obtained directly from the computation of the diffraction theory, shown in Fig. 4.10(a) and (b), respectively. We can see that the results correspond to what we have shown in Fig. 4.6. For CD, the goal of superresolution isn’t achieved, but even enlarges the spot size; for DVD, though transverse superresolution is obtained, the transverse side-lobe is worsened.
Fig. 4.10 (a)
Fig. 4.10(b)
Fig. 4.10: The transverse intensity distributions 1) with the clear pupil, 2) with the designed filter, for a=0.6 (a) for the case of CD (b) for the case of DVD.
We are now interested in where the best image plane (BIP) appears after we add the designed filter into the system. Radial intensity scans at various axial coordinates are shown in Fig. 4.11(a) and(b), for CD and DVD, respectively. The range of axial coordinates, u, explored here is -5 to 5, which would certainly seem to cover the transition region we are interested in. From the figure, it seems reasonable to state that the BIP, where best image performance is gotten for both CD and DVD, occurs around u= 3.
For CD, we can see that the peak value of the main lobe is 0.5 and the side-lobe is extremely small when the radial distance, v, ranges between -10 to 10. The size of blur circle remains the same as the one without adding the filter. For DVD,
transverse superresolution is obtained, but the increased transverse side-lobe will still be a big concern. The contrast of the system performance will be lowered down.
Fig. 4.11: Radial intensity scans at various planes (the range of axial coordinates, u, explored here is -6 to 6) (a) for the case of CD (b) for the case of DVD
Step 3:
When dealing with the case in which the best image plane is not near the paraxial focus, it’ll be more proper to use the modified 2nd order approximation method. For CD, we find that the BIP has been shifted to u=3.527, while that occurs at u=3.1 for DVD. Once the location of the BIP is found, we then fine tune the transmittances of the zones. Here, in this case, we set the transmittance of first zone for CD surface to be 0.7. The computed results of the superresolution factors are shown in Fig. 4.12.
(a) (b)
Fig. 4.12: 1) Relation among the displacement and the radius of the first zone.
2) Relation among the transverse gain and the radius of the first zone.
3) Relation among the Strehl ratio and the radius of the first zone.
(a) for the case of CD (b) for the case of DVD
It can be seen that the computed value of the axial displacement is very close to zero when the radius of the first zone is set to be 0.6. The values of transverse gains are all greater than 1 for CD and DVD, which means that superresolution property can be achieved in this case. However, as a drawback, the Strehl ratios for both CD and DVD have dropped greatly, lower than half value of that without a filter.
Step 4:
Similar to what we have done in step 2, now we try to verify the accuracy of the computed results gotten in step 3. The intensity distributions along the transverse direction for CD and DVD at the shifted focus are shown in Fig. 4.13(a) and (b), respectively.
Fig. 4.13 The transverse intensity distributions 1) with the clear pupil, 2) with the designed filter, for radius a=0.5 ; transmittance t1= 0.7 (a) for the case of CD (b) for the case of DVD.
For CD, we find that the transverse gain, in fact, is equal to 1.038, being lower than the expected value; while for DVD, that is equal to 1.259. Besides, the size of the main-lobe of the diffracted pattern has been narrowed down, achieving superresolution in both cases.
Within the range , we can see that, for CD, the transverse side-lobes become relatively smaller than the central peak value (being approximately 1% of the central peak value). But, for DVD, the transverse side-lobe isn’t alleviated but enhanced, which leads to worsen the contrast of the final image.
] 10 , 10 [ '∈ −
u
4.6 Conclusions
In this section, we have shown that, with the use of a rotationally symmetric four-zone pupil filter, transverse superresolution can be both realized for two different wavelengths. Notice that the expressions of the factors, like gains, Strehl ratio, and axial displacement, have to be modified for the case in which the best image plane is not near the paraxial focus. It should also be mentioned that, the transverse sidelobe is somehow troublesome, especially for the case of DVD. In the future work, the main aim in optical superresolution is to reduce the main-lobe size of the point-spread function while increasing the central intensity and suppressing the sidelobes.
Chapter 5
5.1 Conclusions
In the first part of this thesis, we have first presented a radially symmetric phase-only filter to help alleviate the effects caused by the fluctuation of third- and fifth-order spherical aberrations simultaneously. It can be clearly seen that the system tolerance to SA5 is improved by several times. Meanwhile, the trade-off issue, i.e., a reduction in intensity and the MTF, has also been discussed in details. These reductions form the baselines in practical applications of the use of phase filter.
Following the ideas mentioned above, we get interested in seeking a way to enhance the tolerance of the system to defocus or SA3 for both wavelengths. Phase pupil filters used to minimize the variation of Strehl ratio with defocus and third-order spherical aberration (SA3) for dual wavelengths are being discussed. Notice that if the amount of variation of the frequency is small, we may find a form of pupil phase function to achieve that goal.
In the last part of this thesis, we have presented a four-zone filter to help achieve transverse superresolution for two different wavelengths, simultaneously. There still
remains some room for improvement. While reducing the main-lobe size of the point-spread function, we need to search for a way to increase the central intensity and suppress the sidelobes.
5.2 Future Work
For the work in Chapter 2:
It’s interesting that if we can derive the form of the pupil phase function that extended to even higher order spherical aberration (say, the seventh order spherical aberration, SA7) by using the method we mentioned here. Besides, an investigation similar to that above may also be carried out for the odd-aberration case, i.e., the focal shift, primary and secondary circular coma.
For the work in Chapter 3:
It may be worthwhile of considering the use of hybrid surface lens, an objective lens of combining aspheric surfaces of DVD and CD. In that way, for those two different frequencies, the tolerance of the axial intensity to defocus or SA3 may be simultaneously enhanced.
For the work in Chapter 4:
A global optimization process is needed in the design of the superresolving pupil plate. Once we develop up a reliable optimization method, it’ll be much easier for us to find out the best set-up configuration to meet the requirement.
Appendix A: Derivation of The Pupil Phase Functions
A.1 Derivation of the phase filter that has been developed for W040=W020=0 By substituting of Eq.(2.9) into Eq.(2.6), it leads to the expression:
0
Note that Eq. (A 1.1) is equivalent to the following equation:
0 The solution of the above equation will be:
]}
By choosing α=-5, C = -3/8, and C1=0, a simple solution form will be obtained, which is expressed as:
Note that Eq.(A 1.7) is a particular solution of Eq.(A 1.6), being one of the possible pupil phase functions. With a change of variables, we have the phase pupil function as:
respectively in Eq.(B 1.7)
A.2 Derivation of the phase filter with Maréchal treatment By substituting Eq.(2.14) into Eq.(2.6) leads to the expression:
) ] 6 } 0
Note that Eq. (A 2.1) is equivalent to the following equation:
0 The solution of the above equation will be:
) ] }
Notice that . With a change of variables, the pupil phase function is expressed as the following form:
2
Appendix B: More Information for 2nd order Approximation Theory B.1 Further Discussion to The 2nd-order Approximation Theory
We now reexamine the derivation of Eqs. 4.10 to 4.13. As a reminder, in the focal plane, the field distribution can be expressed as [7]:
=
∫
1 (B 1.1)Notice that the Bessel function of the first kind is expressed as:
∑
∞ For small distances from the focus, we can expand the expressions for the focal-plane and axial amplitudes as a power series:Omitting the higher order terms, we get:
0 2 1 4 2
So the transverse (focal) plane intensity will be:
0 2 Re( 0 1 ) 2 Similarly, the axial intensity will be:
0 2 0 1 [Re( 2 0 | 1|2)] 2
B.2 Modification to The Expressions of The Axial and The Transverse Gain
In the recent study, Silvia Ledesma and Juan Campos have extended the expressions for the axial and the transverse gain to the case in which the best image plane is not near the paraxial focus [20]. The content of the study is shown below.
For the case in which the best image plane is not near the paraxial focus, the expressions for the axial gain, transverse gain, and the Strehl ratio need some modifications. Recall that the field along the axis is:
=
∫
1 (B 2.1)Where u is the axial coordinate centered at the focal plane without the filter. By evaluating |U(0,u)2| numerically from Eq. B 2.1, we find the position umax where the axial intensity is maximum. Then we calculate those factors of superresolution by use of the expansions around this point.
The second-order expansion for the axial response around umax will be:
≅
∫
1 + − − − (B 2.2)The nth moments of the pupil around umax is defined as:
=
∫
1 (B 2.3)Now the terms taken into account is just up to second order in u’= u- umax, Then the axial intensity is approximated as:
[| |' Re( ' '*) ' ]
For transverse response, we expand the field to second order corresponding to umax:
=
∫
1 −Then the transverse intensity can be expressed as
Re( '* ') ]
Note that u0 corresponds to the center of the parabola defined in Eq. 4.41:
Note that u0 corresponds to the center of the parabola defined in Eq. 4.41: