1. Introduction

Optical resolution can be greatly enhanced by use of a solid immersion lens (SIL). A SIL reduces the wavelength of light inside the optically dense medium of the lens and, as a result, reduces the focused spot size, ze, s=λ(nsinθ), where λ is the wavelength in vacuum, n is the index of the SIL, and θ is the marginal ray angle inside the SIL. First investigated by Mansfield and Kino in 1990 [1], SILs have since found use in many applications where increased resolution is desired, such as data storage and lithography [2]. Since a SIL acts as a single refracting surface, the paraxial aberrations introduced by the lens have simple geometric relationships. The following sections investigate the 3rd order aberrations for a single refracting surface and their implications for fabrication and usage of SILs.

2. Geometric study of a single refracting surface

2.1 Spherical parent 3^rd order aberrations

This section outlines and utilizes aberration theory and notation as presented by Roland Shack. [4]

Refraction by a spherical surface aberrates the transmitted wavefront such that rays do not focus to a single point. This wavefront aberration is commonly separated into a polynomial expansion based on pupil and field position. The wavefront function can be expanded in such a way as to define each of the 3^rd order elements in terms of a parent aberration, spherical aberration W ₀₄₀, given in Eq. (1). Thus, coma W ₁₃₁, astigmatism W ₂₂₂, and distortion W ₁₃₁ can be described in relation to the equation for W ₀₄₀, as given in Eqs. (2) through (6). The background derivation to these equations is given in the Appendix.

where

and where the refraction invariant for the marginal and chief rays is given by

and

respectively. The change in u/n is given by

The 3^rd order aberration corresponding to field curvature, W ₂₂₀, is discussed in detail in Section 3.7. The collection of these equations are referred to as spherical parent 3^rd order (SP3O) equations.

It is desirable to find W ₀₄₀ in terms of the known quantities: n, n′, R, l′, NA, ȳ, and u, as defined in Fig. 1. To derive this relationship, A, Ā, y, ū′, u, and u′ are reformulated in terms of the known quantities.

To start, several geometric relationships are defined:

Eq. (14) gives the relationship for y as a function of u′, which is determined in terms of other known quantities in Eq. (29). From Eqs. (8) and (16),

Substituting in Eq. (12) yields an expression for A,

A similar process gives Ā,

To find u′, the refraction equation is used,

where

Solving for ū′,

To findu′, Eq. (17) is reformatted to

Snell’s law states that

and when substituted, along with Eq. (16), into Eq. (25), yields

Utilizing Eqs. (12) and (14), u′ is found:

Δ(u/n) is found by applying the refraction equation to the marginal ray,

The unknown quantities (A, Ā, y, ū′, u, u′) are now known in terms of the known quantities (n, n′, R, l′, NA, ȳ, ū) and the SP3O aberrations can be evaluated.

2.2 Focus position

To find the location of focus in the material given n, n′, R and l, Eq. (14) is reformulated into

Addition of u′ from Eq. (29) gives

Application of Eq. (13) gives the relationship for the focal point,

Solution of l gives the focal point in the absence of the medium as a function of the focal point inside the medium, both measured from the vertex,

where both l and l’ are measured from the vertex.

3. Aberration study

3.1 Hemisphere and Hyperhemisphere

There are two conditions that result in zero spherical aberration in Eq (1): 1) A=0, which corresponds to no refraction across the surface; or 2) Δ(u/n)=0, which is called an aplanatic condition. To find the focus point in the medium under these two circumstances, condition 1) is used along with Eq. (20):

Application of Eq. (14) yields

The result is a concentric surface, or a hemisphere, which agrees with the condition A=0, since no refraction takes place across a surface concentric with focus.

To find l′ for condition 2), the aplanatic condition u′/n′=u/n is assumed and Eq. (31) is solved for u/n :

Application of Eq. (14) gives l′ for the aplanatic condition, which is

Substitution of Eq. (43) into Eq. (37) gives the location of focus in the absence of the refracting surface, which is

A refracting surface that obeys Δ(u/n)=0, with image and virtual object distances given by Eqs. (43) and (44), is called an aplanatic surface and introduces no spherical aberration, coma or astigmatism. A truncated sphere with a thickness given by Eq. (43) is called a hyperhemisphere.

3.2 Spot size reduction

The relationships in the previous section can be used to determine the spot size reduction inside the SIL. The spot size outside and inside the SIL for paraxial angles is

and

respectively. Substitution of u′ from Eq. (29) gives

Conditions on l′ for the hemisphere, Eq. (40), and hyperhemisphere, Eq. (43), are used to find the spot size for both aberration-free cases. The spot size for the hemisphere is given by:

Similarly, the spot size for the hyperhemisphere is:

Given an incident medium of air where n=1, the hemisphere increases NA′ and reduces the spot size by a factor of n′, whereas the hyperhemisphere increases NA′ and reduces the spot size by a factor of n′². Both configurations have the same maximum achievable NA′, since u′ has a maximum physical value of 1. However, since the hyperhemisphere refracts the beam, whereas the hemisphere does not, the NA required to achieve the maximum NA′ is less for the hyperhemisphere. Maximum NA′ conditions for both configurations are shown in Fig. 2.

 

Fig. 1. Geometry of chief and marginal ray bending at a single refracting surface.

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Fig. 2. Maximum NA′ conditions for hemisphere and hyperhemisphere SIL configurations.

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3.3 Spherical Parent 3rd Order (SP3O) equations vs. raytracing

Equations (14), (20), (29), and (32) allow calculation of W ₀₄₀ via Eq. (1) and, as a result of Eqs. (2) through (6), the other SP3O aberration equations. Figure 3 shows a graph of an analytical calculation of W ₀₄₀ superimposed on data numerically generated by ray trace software for n=1, n′=1.5, R=1.5mm, and NA = 0.1. The horizontal axis is shown as l′/R, where the hemisphere point is located at l′/R=1 and the hyperhemisphere at l′/R=1.67.

 

Fig. 3. Graph of W ₀₄₀ generated using raytrace methods and the analytical approach described as a function of SIL thickness for n=1.0, n′=1.5, R=1.5mm, NA=0.1.

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The raytrace and analytical data exhibit excellent agreement for the low NA used in Fig. 3. Since the equations used in Section 2 are defined for the paraxial region, one would expect to see this agreement begin to fail for non-paraxial systems with larger amounts of W ₀₄₀. Figure 4 shows a companion graph to Fig. 3 with an increased input NA of 0.6.

 

Fig. 4. Graph of analytical and raytrace generated W ₀₄₀ at a larger NA than Fig. 3, showing deviation due to paraxial approximations made for the analytical calculations.

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In this scenario, as the curve moves away from the horizontal axis, the analytical method underestimates the amount of aberration compared to raytrace data. However, the analytical curve agrees well with the numerical data near the axis for small aberration, which means the values near the vicinity of the hemisphere and hyperhemisphere points of interest agree well with raytrace values for W ₀₄₀.

The results in Fig. 4 are for an input NA of 0.6, which is close to the maximum NA′ condition of u=-0.667, where u′=-1 inside the SIL at the hyperhemisphere point. However, since the 3^rd order equations are defined for the paraxial region, there is nothing inherent in the equations to show this limitation on u. In fact, non-physical values greater than 1 can be used for input NA or u and the equations will generate values for non-physical conditions.

Superimposing a graph of u′ on the aberration curve serves as an indicator of non-physical results. The condition -1<u′<0, represents the physical limitation on u′ inside the SIL and gives an indicator of where the analytical data are valid.

 

Fig. 5. Graph of W₀₄₀ and u′ for a high input NA that results in a non-physical hyperhemisphere location. Only when -1<u′<0 are the results from the analytical solution valid.

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Figure 5 shows W ₀₄₀ for a high NA geometry using a high index material (Gallium Phosphide, or GaP) for the SIL medium. For a GaP hyperhemisphere SIL, the maximum input NA (assuming air incidence) is 1/3.3=0.303. Since the input NA used in Fig. 5 is 0.5, the hyperhemisphere point is invalid. The graph of the marginal ray inside the SIL verifies this condition. The u′ curve goes below -1 before the hyperhemisphere point, so all data beyond, as shown in the grayed box, is non-physical and invalid. From the graph, u′=-NA at the hemisphere point, l′/R=1, verifying that u=u′ and that no refraction takes place.

3.4 Agreement with previous work

In the aberration study by Baba, [5] which agreed with the previous aberration treatment by Mansfield, [6] the form of the induced spherical aberration based on the SIL thickness is given by,

for spherical aberration present in a near-hemispherical SIL in air. This expression is in agreement with the SP3O equations very close to l′=R, as shown in Fig. 6, but the implicit assumption that the SIL is a hemisphere leads to a symmetric aberration function, which is not strictly the case.

 

Fig. 6. Comparison of spherical aberration curves for the current 3rd order treatment vs. the previous treatments by Baba and Mansfield.

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The overall thickness tolerance near the hemisphere point is in close agreement for both treatments, but the equation put forth by Baba and Mansfield slightly overestimates W ₀₄₀ when l′<R and underestimates for l′>R. The SP3O treatment results in asymmetrical limits to the thickness tolerance, explained in the next section, whereas the Baba and Mansfield treatments assume the thickness tolerance is symmetrical about the hemisphere point. The previous treatments, being hemisphere approximations, also do not correctly model the aberration behavior further away from the l′=R point.

In another aberration study in the literature, the equations used by Jo [7] to arrive at geometric aberrations, based on techniques put forth by Welford, [8] match the SP3O equations exactly. The most notable difference is, of course, that all the previous aberration treatments assume air for the incident medium. The SP3O equations have no such limitation and can be used to generate the geometric aberrations used as the input to the vector diffraction tolerance study done by Jo, including diffraction and polarization effects.

3.5 Tolerancing

The third order aberration curve also gives information about the tolerance on SIL thickness. The majority of SILs are fabricated by lapping and polishing spheres to a desired thickness. Imperfections and errors in the fabrication process result in an error in SIL thickness, which induces aberration.

 

Fig. 7. Graph of W ₀₄₀ for fixed u, showing λ/20 thickness tolerance windows.

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If a tolerance window is added to the graph of W ₀₄₀, the thickness values at the intersection of the aberration curve with the tolerance window gives the thickness tolerance. Figure 7 shows the aberration curve for a 500µm radius GaP SIL with NA=0.25 and ±λ/20 tolerance windows. Due to the high slope of the aberration curve at the hyperhemisphere point, the thickness tolerance is very tight with Δt=317nm, or just 0.05% of the SIL thickness at that location. Table 1 shows a compilation of tolerance values for a 500µm radius SIL and differing SIL and incident indices. It displays the thickness tolerance calculated through the 3^rd order equations as well as tolerances achieved through an inverse limit tolerance analysis in an optical design program. Zemax [9] is used to generate the raytrace tolerance data using a user script for the tolerancing routine optimized on a paraxial marginal ray height of zero at the image plane and the full pupil marginal ray optical path difference of λ/20 as the tolerance criteria. The illumination NA, which is represented by |u|, is chosen to achieve a |u′| of 0.65 at the hyperhemisphere point.

Table 1. Third order and raytrace derived tolerance data for several R=500µm radius SIL configurations and fixed u.

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The thickness tolerance windows listed in Table 1 are not necessarily symmetrical about the nominal thickness value since the curve is not symmetrical, as discussed in the previous section.

Since the angles chosen for comparison are far away from paraxial, the underestimation of the aberrations by the 3^rd order equations is apparent in Table 1. Though raytrace and analytical approaches exhibit good agreement near hemisphere and hyperhemisphere points, the slight underestimation of the 3^rd order equations reduces slope of the aberration curve away from the axis, resulting in larger tolerances. This underestimation is more apparent for the hyperhemisphere, as the very small tolerances due to the very steep angles involved are affected greatly by small changes in slope. Regardless of the differences in 3^rd order and raytrace tolerances, Table 1 shows the interesting result that the hyperhemisphere tolerances are more than two orders of magnitude tighter than the hemisphere configuration.

Another item of note is that tolerances for the last two hyperhemisphere conditions are different, even though NA′ remains constant. This result is due to the fact that Δn across the surface is less for case #3 than case #2. In reality, the higher index incident medium would most likely be due to some attached conformal material, such as epoxy, separated in air from the final element of the illumination system. This interface would have its own spherical aberration component and would be added to the contribution of the SIL.

In the design stage of a system incorporating a SIL, if the NA of the illumination system is fixed and unchangeable, Table 1 presents the fabrication tolerance and possible NA′ for several SIL materials and illumination configurations. However, for designs where parameters such as illumination NA are variable, it is desirable to compare the fabrication tolerances for hemisphere and hyperhemisphere with the same NA′, since this value is proportional to the actual spot size reduction.

To accomplish this analysis, W ₀₄₀ is required as a function of l’ for fixed u, instead of fixed u′. This condition is accomplished by setting u′ as fixed and solving for u in Eq. (29),

Figure 8 shows W ₀₄₀(l′)u′ for R=0.5mm SIL with index n′=1.5.

 

Fig. 8. Graph of analytical and raytrace generated spherical aberration and input marginal ray angle u for fixed u′.

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Figure 8 can be best interpreted and thought of in the reverse optical direction compared to Fig. 3. Figure 8 represents the amount of spherical aberration in a beam originating from a source point l′ away from the vertex of a spherical refracting surface with a marginal ray angle of u′. As the point source is moved closer to the vertex, the marginal ray angle of the beam exiting the surface, given by u, increases, as is also shown in Fig. 8.

Since a typical function of a lens with a point source is collimation, an interesting side note is to look at the condition when u=0. Setting Eq. (54) to zero and solving for l′ gives the condition for collimation,

Setting u=0 in Eq. (32), and the non-primed version of Eq. (20) and applying Eq. (55) yields

To minimize Eq. (56), one must minimize u′, R, and maximize Δn. To maximize Δn, an obvious choice is to use a high index lens material and air as the exit medium. Since laser diodes are a common point-like source that require collimation, a maximum u′ can be approximated. A typical large fast axis FWHM divergence is 40°, which, immersed in a high index like GaP, reduces to approximately 12°, which gives an approximate maximum |u′| of 0.2. With this information and a λ/10 aberration tolerance, a maximum radius value is calculated, R ≤ 560λ, which for λ=500nm, R≤280µm. To a geometrical and paraxial approximation, this is the maximum radius for a GaP lens required to achieve less than λ/10 spherical aberration from a purely spherical collimating lens.

The aberration curve given in Fig. 8 can be used to compare the thickness tolerance for a hemisphere and hyperhemisphere for identical NA′. Using u′ equal to the hyperhemisphere point and the system parameters for Fig. 7, Fig. 9 shows the tolerance windows for constant NA′.

 

Fig. 9. Graph of W ₀₄₀ for fixed u′ showing λ/20 thickness tolerance windows.

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The tolerance for the hyperhemisphere point is the same in Fig. 9 as in Fig. 7. However, the tolerance for the hemisphere case is dramatically reduced due to the larger input marginal ray angle, shown by the line of u crossing -0.825 at the hemisphere point. Even with identical NA′, the hyperhemisphere has a thickness tolerance 30 times smaller than the hemisphere. Table 2 shows the same data as Table 1 with fixed u′ instead of u. The result is similar to the comparison between Fig. 7 and Fig. 9. The hyperhemisphere tolerances stay the same, while the hemisphere tolerances are reduced. Because u′ is fixed, both hemisphere and hyperhemisphere configurations are compared at the same final system NA′.

Table 2. 3rd order and raytrace derived tolerance data for several R=1.5mm SIL configurations and fixed u.

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Compared with the hemisphere tolerances at the same u′, the hyperhemisphere tolerances are more than 40 times smaller. On a conceptual level, the tighter tolerances are due to the fact that, to create the equivalent angle u′ inside the SIL as the hemisphere, the hyperhemisphere requires strong bending of the rays. For a spherical surface, the strong bending results in larger values of spherical aberration for an equivalent change in l′.

3.6 Chromatic aberrations

In addition to wavefront aberrations, dispersive chromatic effects are also modeled using the spherical parent 3^rd order equations. Specifically, chromatic aberration is calculated as the spread of focus in the longitudinal direction along the axis and in the transverse direction. The longitudinal and transverse aberrations, respectively, are defined as follows:

and

where

and

where ν is a generalized dispersion coefficient and n_S, n_M, and n_L are indices at three arbitrary wavelengths: short, medium and long.

There are several conditions that result in zero chromatic aberration. Both longitudinal and transverse aberration are zero if Δ(δn/n)=0. Setting both parameters equal,

Cross multiplying and solving for n yields an expression for n(ν) for a given n′ and ν′,

Using Eq. (62), a material can be found that, if used as the incident medium, will cancel chromatic aberration. Unfortunately, the slope of the Δ(δn/n)=0 curve is opposite that of the traditional glass line which greatly limits the glass types available for dispersion matching. The achromatization process is similar to other methods that require dispersion matching and can be considered a special case of an achromatic doublet with matching incident and exit medium indices of refraction.

There are two more conditions that result in zero δ_λW ₀₂₀ and δ_λW ₁₁₁. Longitudinal chromatic aberration is zero when A=0. This condition is the same as condition 1) for W ₀₄₀=0 and is satisfied when the SIL is a hemisphere, l′=R. Transverse chromatic aberration, in contrast, is also zero when A=0. This condition means the chief ray does not undergo refraction at the SIL surface. In other words, the chief ray is always normal to the SIL surface, and as a result, passes through the center of curvature of the lens. Setting Eq. (21) equal to zero and solving for u gives the condition to satisfy Ā=0 for zero transverse chromatic aberration, which is

This condition is equivalent to saying that ȳ=0 at the image plane, or that the system aperture stop is located at the bottom of the SIL. Since the stop cannot be coincident with the image plane, Eq. (63) is only possible for a system with an infinite conjugate source, i.e. a plane wave coming to focus at the center of curvature.

 

Fig. 10. A graph of W ₀₄₀, δ_λW ₀₂₀ and δ_λW ₁₁₁ as a function of SIL thickness for a SIL with R=0.5mm, NA=0.2, and ȳ=10µm.

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Figure 10 shows the two chromatic aberrations superimposed on the curve of W ₀₄₀ as a function of SIL thickness for a 0.5mm radius SIL with n′=1.5 and ν′=62. Since transverse chromatic aberration is zero unless the chief ray height or angle is non-zero at the SIL surface, a marginal ray height of 10µm is used, which emulates a telecentric illumination system.

Since the system is telecentric and Δ(δn/n)≠0, δ_λW ₁₁₁ is nonzero everywhere except at the trivial point where the SIL has zero thickness, l′=0. The δ_λW ₀₂₀ curve crosses the axis at the hemisphere point where A=0. However, the slope of the δ_λW ₀₂₀ curve increases rapidly, resulting in very large longitudinal chromatic aberration at the hyperhemisphere point. In addition to the tighter fabrication tolerances for the hyperhemisphere point discussed in Section 3.5, the amount of longitudinal chromatic aberration makes the hyperhemisphere impractical as a SIL for some applications.

3.7 Field of view considerations

Section 3.5 shows how Eqs. (1) through (6) help determine fabrication tolerances on thickness. They can also be used to determine the field of view inside the SIL for a particular aberration tolerance. Figure 11 shows two simple methods of obtaining non-zero image height. One method is to vary ȳ, which corresponds to a system that is telecentric, and the other method is to vary ū, which corresponds to a system with the stop at the vertex of the SIL. A real optical system that is neither telecentric nor has the stop at the SIL vertex employs a combination of both ȳ and u to achieve field angle, but for the sake of discussion and presentation, the chief ray parameters are discussed in a decoupled manner.

 

Fig. 11. Illustration of two methods to achieve non-zero image height in a SIL: a) lateral chief ray displacement in a telecentric system; and b) changing chief ray angle for a system with the stop at the SIL vertex.

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For a hemisphere, A=0, which means that off axis, all 3^rd order aberrations are zero, except for W ₂₂₂. Figure 12 shows the W ₂₂₂ curve for a glass hemisphere with R=1.5mm and NA=0.36.

 

Fig. 12. Third order aberrations vs. image coordinate for a hemisphere. W ₀₄₀ and W ₁₃₁ are both zero, and the W ₂₂₂ curves for the telecentric and stop at vertex conditions are degenerate.

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W ₀₄₀ and W ₁₃₁ lie along the axis since both are zero for A=0. W ₂₂₂ determines the field of view according to the aberration tolerance. Since l′=R, the curves for the telecentric and vertex stop conditions are degenerate.

The aberration behavior grows more interesting, however, when l′ is perturbed slightly to either side of the hemisphere condition. Figure 13 shows a SIL with a thickness thinner than a hemisphere at l′=0.9R. W ₀₄₀ is non-zero, represented by the dashed horizontal line below the axis, but is constant with field since W ₀₄₀ has no field dependence.

 

Fig. 13. Third order aberrations vs. image coordinate for l′=0.9R.

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Since both A and Ā are non-zero, coma is visible now as the dash-dot line. The sign of the aberration changes across the field, since it depends on an odd power of AĀ, which changes sign itself at the optical axis. Also visible is the fact that the telecentric and vertex stop conditions do not create the same amounts of aberration for the same image coordinate, which is evident by the splitting of the W ₁₃₁ and W ₂₂₂ lines. With l′<R, the telecentric condition creates less W ₂₂₂ than at l′=R. However, using a truncated hemisphere with a telecentric system does not equate to larger field of view, since the added coma increases the overall aberration. Figure 14 shows the other case of an augmented hemisphere where l′=1.1R.

 

Fig. 14. Third order aberrations vs. image coordinate for l′=1.1R.

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A given chief ray angle ū′ inside the SIL that creates an image point on one side of the optical axis for a truncated hemisphere creates an image point on the opposite side of the axis for an augmented hemisphere. As a result, any odd 3^rd order aberrations change sign on either side of the hemisphere condition. The sign change is shown by the change in slope of the W ₁₃₁ curve in Fig. 14 from Fig. 13. This behavior also results in identical splitting of the aberration curves for the telecentric and vertex stop conditions, except the telecentric method now creates more W ₂₂₂ for the augmented hemisphere than the vertex stop method for a particular image coordinate. Conceptually, the lower aberration for the vertex stop method is because of Eq. (15), since a longer l′ requires a smaller ū′ to achieve the same y′, which reduces W ₂₂₂.

Since a hyperhemisphere obeys the Δ(u/n)=0 condition, all three main 3^rd order aberrations are zero when Eq. (43) is met. In this case, W ₀₄₀, W ₁₃₁, and W ₂₂₂ do not limit the field of view, rather the tolerance on the field curvature, W ₂₂₀, specifies the acceptable field of view. The total field curvature for sagittal (W _220S), medial (W _220M) and tangential (W _220T) foci are given by [4]:

and

The Petzval curvature is given by

where א is the Lagrange invariant

and

For a hemisphere, W _220P=-½ W₂₂₂, resulting in a flat sagittal field (W_220S=0). As previously discussed (Fig. 12), it is the astigmatism present that limits the field of view for the hemisphere configuration. The sole presence of W ₂₂₂ off axis for the hemisphere brings up a point worth noting for the particular application of solid immersion lithography. Since the sagittal foci are coradial with the optical axis, if the pattern area is limited to an area immediately surrounding a meridional plane, features that are parallel to the plane (such as lines patterned for memory) can be patterned using the higher resolution of the SIL with no loss of field of view in the direction of a meridional plane.

Since W ₂₂₂ is zero for the hyperhemisphere, W _220S, W _220M, and W _220T are degenerate and the contribution to field curvature due to the any of the astigmatic fields do not balance out the Petzval field curvature. Thus, W ₂₂₀ is the limiting aberration determining the field of view for a hyperhemisphere. Figure 15 shows the acceptable field of view due to field curvature for a hyperhemisphere with the same system parameters and u′ as the hemisphere in Fig. 12.

 

Fig. 15. Third order aberrations vs. image coordinate for a hyperhemisphere.

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Because the Lagrange invariant and the P parameter in Eq. (67) do not change with l′, W _220P is constant with l′. Since W _220P=-½ W ₂₂₂ at l′=R, there is half the amount of field curvature at the hyperhemisphere point as there is astigmatism at the hemisphere point. Depending on actual SIL usage, the lower amount of aberration means the hyperhemisphere has a larger field of view than a hemisphere, due solely to the tolerance on monochromatic 3^rd order aberrations.

This section ends with Fig. 16, showing an overview of the four main 3^rd order aberrations as a function of l′, with a telecentric offset of y=80µm.

 

Fig. 16. Third order aberrations as a function of SIL thickness with n′=1.5, R=1.5mm, NA=0.36, and ȳ=80µm.

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4. Conclusions

The SIL aberration study in this paper arrives at expressions for all 3^rd order aberrations, conditions on radii and thickness for aberration free use, spot size change, and tolerances on SIL thickness and field of view. Unlike previous treatments, this study is completely general and, to a paraxial approximation, valid for any incident and image space media and any chief ray specification. The results show excellent agreement with paraxial numerical ray-trace values and the hemisphere-only expressions for aberration given in the literature. In addition, the current treatment correctly models the asymmetric aberration behavior surrounding the hemisphere point. This behavior can be important for micro SILs when trying to accurately model the thickness tolerance. Results show the thickness tolerance aberration asymmetry and field of view tolerances in a telecentric configuration all favor a tolerance window skewed toward SIL thicknesses less than the radius.

The treatment also agrees with previous findings that aplanatic hyperhemisphere SILs are less practical for some applications due to severe chromatic aberration and a thickness tolerance much tighter than a hemisphere SIL.

For the hemisphere, it is shown that the field of view is determined by astigmatism. The saggital field is flat, however, so for the particular case of imaging lines in the direction of meridional planes, the only limit to the field of view is the size of the lens, within a paraxial geometric approximation. For the hyperhemisphere, field curvature limits the field of view, but there is half the amount of field curvature as there is astigmatism for the hemisphere, resulting in a larger field of view for the hyperhemisphere.

Appendix-Wavefront aberration theory

This appendix follows the aberration treatment given in Roland Shack’s aberration theory notes [4] that results in the spherical aberration parent relationships listed in Eqs. (2) through (6).

The wavefront aberration function can be separated into a polynomial expansion based on powers of the field vector $\stackrel{\rightharpoonup }{H}$, pupil vector $\stackrel{\rightharpoonup }{\rho }$, and the cosine of the angular separation of the field and pupil vectors cosφ. Each term is denoted by the dependence on the powers of each of the expansion variables,

(70)$$W=\sum _{j,m,n}{W}_{\mathrm{klm}}{H}^k{\rho }^l{\cos }^m\phi ,$$

where

(71)$$k=2j+m,$$

and

(72)$$l=2n+m.$$

To a 3^rd order approximation, the aberration due to a spherical surface is equivalent to the aberration introduced by an aspheric plate located on-axis at the center of curvature with the spherical surface contributing only 1^st order properties to the system, such as image location and magnification. In this context, the wavefront departure in the projected pupil footprint on the aspheric plate, shown in Fig. 17 (copied from [2] for convenience), will generate the same 3^rd order aberrations as the original spherical surface.

 

Fig. 17. Schematic of aspheric plate at the center of curvature with field and pupil footprints showing the relationship between pupil and field vectors and the wavefront aberration expansion variable, Ḡ.

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Since the aspheric plate is on-axis, the wavefront aberration function at the plate is in the form of spherical aberration based on a new variable Ḡ, which is a combination of the pupil vector and a scaled field vector,

(73)$$\stackrel{\rightharpoonup }{G}=\stackrel{\rightharpoonup }{\rho }+\gamma \stackrel{\rightharpoonup }{H},$$

where γ is given by Eq. (7).

The wavefront aberration function is written in terms of the new aspheric plate variable,

(74)$$W=W_{040}{G}^4$$
$$=W_{040}{\left(\stackrel{\rightharpoonup }{G}·\stackrel{\rightharpoonup }{G}\right)}^2.$$

Expanding the (Ḡ·Ḡ) term in Eq. (74) using Eq. (73) gives the wavefront function in terms of the original pupil and field vectors,

(75)$$\stackrel{\rightharpoonup }{G}·\stackrel{\rightharpoonup }{G}=\left(\stackrel{\rightharpoonup }{\rho }+\gamma \stackrel{\rightharpoonup }{H}\right)·\left(\stackrel{\rightharpoonup }{\rho }+\gamma \stackrel{\rightharpoonup }{H}\right)$$
$$=\stackrel{\rightharpoonup }{\rho }·\stackrel{\rightharpoonup }{\rho }+2\gamma \stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{\rho }+{\gamma }^2\stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{H}.$$

Squaring Eq. (75) shows how the expansion based on Ḡ is equivalent to the traditional forms of the 3^rd order aberrations,

(76)$${\left(\stackrel{\rightharpoonup }{G}·\stackrel{\rightharpoonup }{G}\right)}^2=\begin{array}{cc}{\left(\stackrel{\rightharpoonup }{\rho }·\stackrel{\rightharpoonup }{\rho }\right)}^2& (W_{040},\mathrm{Spherical})\end{array}$$
$$\begin{array}{cc}+4\gamma \left(\stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{\rho }\right)\left(\stackrel{\rightharpoonup }{\rho }·\stackrel{\rightharpoonup }{\rho }\right)& (W_{131},\mathrm{Coma})\end{array}$$
$$\begin{array}{cc}+4{\gamma }^2{\left(\stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{\rho }\right)}^2& (W_{222},\mathrm{Astigmatism})\end{array}$$
$$\begin{array}{cc}+2{\gamma }^2\left(\stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{H}\right)\left(\stackrel{\rightharpoonup }{\rho }·\stackrel{\rightharpoonup }{\rho }\right)& (W_{220},\mathrm{Field}\mathrm{Curvature})\end{array}$$
$$\begin{array}{cc}+4{\gamma }^2\left(\stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{H}\right)\left(\stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{\rho }\right)& (W_{311},\mathrm{Distortion})\end{array}$$
$$\begin{array}{cc}+{\gamma }^2{\left(\stackrel{\rightharpoonup }{H}·\stackrel{\rightharpoonup }{H}\right)}^2.& (W_{400},\mathrm{Piston})\end{array}$$

Placing this expansion into the (Ḡ·Ḡ)² term in Eq. (74) and solving for each of the third order aberration coefficients in terms of W ₀₄₀ yields the spherical parent aberration relationships given in Eqs. (2) through (6).

Acknowledgments

Dr. Roland Shack who did the fundamental ground work upon which this extension of his theories is based.

References and links

1. S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990). [CrossRef]  

2. B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994). [CrossRef]  

4. R. Shack, Introduction to Abberations, personal communication.

5. M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, “Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy,” Appl. Phys. 85, 6923–6925 (1999).

6. S. M. Mansfield, Solid Immersion Microscopy, PhD Dissertation, Standford University (1992)

7. J. Jo, The Vector Behavior of Aberrations in High Numerical Aperture (0.9<NA<3.1) Laser Focusing Systems, PhD Dissertation (2001)

8. W. T. Welford, Aberrations of Optical Systems, (Adam Hilger, 1986) pp. 130–161.

9. Zemax: commercially available software from the Zemax development corporation