Lenses axial space ray tracing measurement

Weiqian Zhao; Ruoduan Sun; Lirong Qiu; Libo Shi; Dingguo Sha

doi:10.1364/OE.18.003608

1. Introduction

In the assembly of high performance optical systems such as lithography lens, high precision objective lens and satellite camera, and so on, the axial space between lenses must be determined precisely. In the lithography lens, the axial space deviation of each singlet brings about the aberrations such as spherical aberration, astigmatism, coma and distortion, and the measurement accuracy of the axial space between lenses should be achieved at the micrometer level [1]. At present, the inner axial space of lenses are ensured by machining and assembly, and the lens axial space is calculated by the thickness of singlet and the thickness of lenses obtained through the contact altimeter in the assembly [2]. Mirau interferometer can precisely identify the vertex position of the outermost surface in the assembly to control the axial space between lenses, but it cannot precisely identify the vertex position P_n of the inner surface of lenses [3]. Whether the inner axial space of lenses fit the design parameter can be analyzed only by its image property. In fact, the lens axial space is one of the main factors having effect on image property. Therefore, the non-contact and high-precision measurement of the spaces between lenses is needed to reduce the inner axial space error in the assembly, which has not been found so far.

As shown in Fig. 1 , in order to achieve the high-precision measurement of the assembled lenses inner axial space d_n, the non-contact optical chromatography measurement needs to be used for precise obtaining the vertex position p_n of the lens’ spherical surface S_n. But the precision of the vertex position p_n depends on the lens’ radii r_n and refractive index n_n, and the figure error and the axial position z_n of the lens L₁. Therefore, it is difficult to obtain the inner axial space d_n of the assembled lenses using the direct measurement.

Fig. 1 Lenses axial space measurement.

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Here, a new axial space ray tracing measurement (ASRTM) is proposed for the lenses inner axial space measurement based on the geometrical theory of optical image. For an assembled lenses with the given radius of curvature r_n and refractive index n_n of every lens, ASRTM uses the optical focusing technique to achieve the high-precision chromatography focusing on the assembled lenses’ inner-and-outer spherical surface S_n and obtain the precise coordinate z_n of the objective movement position corresponding to the vertex position P_n of the lenses’ inner-and-outer sphere, where n=1, 2, 3, 4. And then, ASRTM uses the ray tracing facet iterative algorithm to precisely determine the vertex position P_n of every spherical surface S_n by these coordinates z_n, refractive index n_n and spherical radius r_n, and thereby achieving the high-precision measurement of lenses’ inner axial space d_n.

The key in ASRTM for the high-precision measurement of lenses’ axial space d_n is how to achieve the precise focusing on the inner spherical surface S_n and obviously reduce the adverse effect of the figure error of the front spherical surface S_n- ₁ on the accuracy of focusing at the back spherical surface S_n. At present, many methods are used for the high-precision location on the outer surface S ₁ [3–5] whereas the precise focusing on the lenses’ inner spherical surface S_n (n=2, 3, 4) is not achieved so far because of the effect of the refractive index n_n, radius of curvature r_n, figure error and so on. Therefore, based on the chromatography property and the property of an axial intensity curve that the absolute zero precisely corresponds to the focus of the objective in our differential confocal system (DCS) [6], a new annular laser differential confocal chromatography focusing technique (ADCFT) is proposed to achieve the precise focusing at the lenses’ inner-and-outer spherical surface and reduce the effect of spherical surface S_n’ figure error on the accuracy of focusing at the inner spherical surface, and thereby achieving the high-precision ray tracing measurement of lenses’ inner axial space.

2. ASRTM principle

2.1 Precise focusing at inner-and-outer spherical surface of the lenses

As shown in Fig. 2 , the zero O_n of ADCFT axial intensity response curve I(z) corresponds to the focus F_n of objective L₁, two virtual pinholes (VPH) with the same offset M are used to receive the intensity signals I _VPH1(0,u,+u_M,ε) and I _VPH2(0,u,-u_M,ε) respectively, and the axial response curve I(0,u,u_M,ε) is obtained through the differential subtraction of I _VPH1(0,u,+u_M,ε) and I _VPH2(0,u,-u_M,ε). And then, ASRTM uses the zero O_n of ADCFT axial intensity response curve I(z) at the different focusing plane of the objective L₁ to precisely determine the spherical vertex position P_n of the test lenses and obtain the coordinate z_n corresponding to the movement position of objective L₁ where n=1, 2, 3, 4. Finally, ASRTM determines the precise vertex position P_n of spherical surface S_n by the ray tracing, and thereby achieving the precise focusing of the lenses’ inner-and-outer spherical surface.

Fig. 2 ASRTM principle. L_C is collimating lens, B is an annular pupil, PBS is polarized beam splitter, P is λ/4, L₁ is objective, L₂ is collecting lens, BS is beam splitter, CCD1 and CCD2 are detectors, MO1 and MO2 are microscope objective, M is the offset of pinhole from the focus of L₂.

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VPH consists of an objective and a CCD, and achieves the intensity point detection in a confocal microscopy system by obtaining the total intensity within the designated areas near the energy centre of the spot.

When the convergence point of beam passing though objective L₁ is moved near each spherical surface along the optical axis of the test lenses, the measurement beam is partly reflected back and then reflected by the polarized beam splitter (PBS) to the collecting lens L₂ and beam splitter (BS), and the two measurement beam split by BS are received by VPHs.

The overlap area between the measurement spot and the test spherical surface is so small that the effect of the surface’s figure and curvature on the ADCFT intensity response is negligible. Then the signals obtained from detectors CCD1 and CCD2 are normalized as I _VPH1 (0,u,+u_M,ε) and I _VPH2(0,u,-u_M,ε), and the signal obtained through their differential subtraction is [6]

\begin{array}{l} I (v, u, u_{M}, ε) = I_{V P H 1} (v, u, + u_{M}, ε) - I_{V P H 2} (v, u, - u_{M}, ε) \\ = {| \frac{2}{(1 - ε^{2})} \int_{ε}^{1} p_{C} (ρ) \cdot p_{1}^{2} (ρ) \cdot p_{2} (ρ) \cdot e^{j ρ^{2} (2 u + u_{M}) / 2} \cdot J_{0} (ρ v) ρ d ρ |}^{2} \\ - {| \frac{2}{(1 - ε^{2})} \int_{ε}^{1} p_{C} (ρ) \cdot p_{1}^{2} (ρ) \cdot p_{2} (ρ) \cdot e^{j ρ^{2} (2 u - u_{M}) / 2} \cdot J_{0} (ρ v) ρ d ρ |}^{2} . \end{array}

where ε is the normalized radius of the annular pupil, J ₀ is a zero-order Bessel function, ρ is the radial normalized radius of a pupil, p ₁(ρ,θ) is the pupil function of L₁, p ₂(ρ,θ) is the pupil function of L₂, p _C(ρ,θ) is the pupil function of L_C, u is the axial normalized optical coordinate, and u=8πzsin²(α ₀/2), v is the lateral normalized optical coordinate and v=(2πrsinα ₀)/λ, z is the axial displacement of the objective, r is the radial coordinate of the objective and α ₀ is the half-aperture angle.

When lenses L₁, L₂ and L_C have the same calibers, p _C(ρ)=1, p ₁(ρ)=1 and p ₂(ρ)=1, I(0,u,u_M,ε) obtained from Eq. (1) is:

I (0, u, u_{M}, ε) = {[\sin c (\frac{2 u + u_{M}}{4 π} \cdot (1 - ε^{2}))]}^{2} - {[\sin c (\frac{2 u - u_{M}}{4 π} \cdot (1 - ε^{2}))]}^{2} .

As shown in Fig. 2, the zeroes of curves I(0,u,u_M,ε) precisely correspond to the vertex positions P ₁, P ₂, P ₃, and P ₄ of lens surfaces, and the sensitivity S(0,0,u_M,ε) at the zero can be obtained by differentiating Eq. (2) on u and is

S (0, 0, u_{M}, ε) = {\frac{\partial I (0, u, u_{M}, ε)}{\partial u} |}_{u = 0} = 2 \cdot \sin c (\frac{u_{M}}{4 π} \cdot (1 - ε^{2})) \cdot \frac{\cos (\frac{u_{M}}{4} \cdot (1 - ε^{2})) \frac{u_{M}}{4} \cdot (1 - ε^{2}) - \sin (\frac{u_{M}}{4} \cdot (1 - ε^{2}))}{{(\frac{u_{M}}{4})}^{2} \cdot (1 - ε^{2})} .

The numerical analysis on Eq. (3) shows that the differential focusing curve has the best sensitivity Smax at zero when axial normalized offset u_M satisfies:

u_{M} \approx 5.21 \times \frac{1}{1 - ε^{2}} .

The best sensitivity obtained from Eqs. (3) and (4) meets S _max=−0.54×(1-ε ²), and the annular laser differential confocal focusing resolution σ_z satisfy

σ_{z} = \frac{δ I (0, u, u_{M}, ε)}{| 0.54 \cdot (1- ε^{2}) |} \cdot \frac{λ}{8 π \cdot \sin^{2} (α_{0} / 2)} = \frac{0.0737 λ}{(1- ε^{2}) \cdot S N R \cdot \sin^{2} (α_{0} / 2)}

where δI(0,u,u_M,ε) is the grey resolution of CCD in VPH, SNR is the signal noise ratio of VPH and SNR=1/δI(0,u,u_M,ε).

It can be seen from Eq. (5) that focusing resolution σ_z depends on half-aperture angle α ₀ and normalized radius ε, and focusing resolution σ_z with different α ₀ and ε is shown in Fig. 3 .

Fig. 3 Differential confocal focusing resolution σ_z with different α ₀ and ε.

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Half-aperture angle α ₀ of objective L₁ should be reduced to increase the ADCFT working distance while the normalized radius ε should be increased to reduce the aberration induced by the test lens surface.

It can be seen from Fig. 3 that ADCFT focusing resolution σ_z is better than 200 nm when α ₀>0.114 rad and ε<0.8, which can meet the requirement of assembly for the high-precision lenses. Therefore, there should be ε<0.8 in ADCFT.

And furthermore, ADCFT can be combined with a phase pupil filter to shorten the focal-depth so as to further improve the axial sensitivity at the surface vertex [7,8].

2.2 Calculation of axial space

When the test lens L₁ is moved along the optical axis shown in Fig. 2, ASRTM uses the absolute zero of ADCFT response curve to determine the overlapping position of pencil tip and the vertex P_n of the test lens spherical surface, and records the coordinate z_n of L₁ movement position corresponding to zero point O_n in turn.

As shown in Fig. 4 , supposing the lenses are spherical lens and the total surface number is n+1, the measurement ray has a deflection after passing through each spherical surface and the half-aperture angle θ_n changes due to the refractive index when the measurement beam with half-aperture angle θ ₀ is converged on S_n ₊₁ of the test lenses.

Fig. 4 Axial space calculation principle using ray tracing.

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Where ρ _pupil is the radius of objective L₁ at the exit pupil, f ₁ ' is the focal length of objective L₁, r_n is the radius of spherical surface S_n, n_n is the refractive index between the spherical surfaces S_n and S_n ₊₁, d_n is the axial space between S_n and S_n ₊₁, and i_n and i_n´ are the angles of the ray into and outof S_n.

According to the ray tracing formula [9], the variables of spherical surface S_n satisfy

{\begin{cases} i_{n} = arc \sin (\frac{l_{n} - r_{n}}{r_{n}} \cdot \sin θ_{n}) \\ i_{n}^{'} = arc \sin (\frac{n_{n - 1}}{n_{n}} \cdot \sin i_{n}) \\ θ_{n}^{'} = θ_{n} + i_{n} - i_{n}^{'} \\ l_{n}^{'} = r_{n} + \frac{\sin i_{n}^{'}}{\sin θ_{n}} \cdot r_{n} \end{cases}

where l_n is the distance from the vertex of S_n to the point of intersection of the optical axis with the incidence light on S_n, l_n´ is the distance from the vertex of S_n to the point of intersection of optical axis with the exit light from S_n, θ_n is the angle between incidence light on S_n and optical axis, and θ_n´ is the angle between the exit light from S_n and the optical axis.

It is known from Fig. 4 that the relation between S_n _-1 and S_n satisfies ln=ln-1´-dn-1 and θ_n=θ_n _-1´. Recursive ray tracing expressions between S_n _-1 and S_n obtained by Eq. (6) are

{\begin{cases} θ_{n}^{'} = θ_{n - 1}^{'} + arc \sin (\frac{l_{n - 1}^{'} - d_{n - 1} - r_{n}}{r_{n}} \cdot \sin θ_{n - 1}^{'}) - arc \sin (\frac{n_{n - 1}}{n_{n}} \cdot \frac{l_{n - 1}^{'} - d_{n - 1} - r_{n}}{r_{n}} \cdot \sin θ_{n - 1}^{'}) \\ l_{n}^{'} = r_{n} + \frac{n_{n - 1}}{n_{n}} \cdot \frac{\sin θ_{n - 1}^{'}}{\sin θ_{n}^{'}} \cdot (l_{n - 1}^{'} - d_{n - 1} - r_{n}) \end{cases} .

As shown in Fig. 4, when ASRTM pencil tip is coincident with the vertex of S_n ₊₁, then

{\begin{cases} l_{0}^{'} = l_{0} = z_{n + 1} - z_{1} \\ θ_{0}^{'} = θ_{0} \end{cases}

where θ ₀ is the half-aperture angle of the measurement beam from L₁. With Eq. (8) as the initial conditions, the axial space between S_n and S_n ₊₁ obtained through Eq. (7) is d_n=l_n´.

Therefore, each axial space of the test lenses d_n is obtained through Eq. (7) when the parameters of each singlet r ={r ₁,r ₂,⋅⋅⋅⋅⋅⋅,r_n}, n={n ₁,n ₂,⋅⋅⋅⋅⋅⋅,n_n} and L₁ position coordinates z={z ₁, z ₂,⋅⋅⋅⋅⋅⋅, z_n, z_n ₊₁ } are known. The ray tracing process can be expressed as a function T and d_n=T(r, n, z, θ ₀).

Function T only corresponds to the measurement result of a ray at half-aperture angle θ ₀, and the axial space should be calculated by integrating on the measurement rays within the whole pupil plane. Assuming that the rays are uniformly distributed in the pupil plane, the axial space of lenses d_n _AXIAL satisfies

d_{n AXIAL} = \frac{\int_{ε ρ_{pupil}}^{ρ_{pupil}} T (r, n, z, arc \tan (\frac{ρ}{{f^{'}}_{1}})) \cdot 2 π ρ \cdot d ρ}{π \cdot ρ_{pupil}^{2} \cdot (1 - ε^{2})}

where the half-aperture angle of the measurement beam meets θ ₀=arctan(ρ/f ₁ ').

3. Effect of annular pupil on ASRTM sensitivity

3.1 Reducing the wave aberration

ASRTM is simulated using Zemax. The laser wavelength is 632.8 nm and the test lenses consists of two lenses, where caliber D ₁ of lens L ₁ is 9.6 mm, f ₁' is 35 mm, the radii of four test spherical surfaces consisting of two lenses are 195.421 mm, −140.234 mm, −140.234 mm and −400.731 mm in turn, the spaces between the surfaces of the two lenses are 12 mm, 0.323 mm and 10 mm in turn, and the refractive indices of the two lenses are 1.5143 and 1.6686.

When the annular pupil B is not used and the measurement beam is focused on the fourth surface of the test lenses, ASRTM wave aberration shown in Fig. 5(a) is PV=0.1243λ. And when the annular pupil B with ε=0.7 is used, ADCFT wave aberration shown in Fig. 5(b) is PV=0.0321λ which is about four times smaller than that of ASRTM. It is obvious that the annular illumination can improve the ASRTM measurement accuracy.

Fig. 5 Effect of annular pupil on wave aberration.a) Without annular pupil B, b) With annular pupil B.

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3.2 Effect of annular pupil on axial sensitivity

Simulations using Zemax indicate that the spherical aberration is the major aberration for ASRTM, and there are some other aberrations, e.g. astigmatism, coma and distortion, accompanying the misalignment of optical setup and the defects of the lens components in the measurement.

It can be seen from displacement theorem [9] that the primary spherical aberration A ₀₄₀ ρ ⁴ and primary astigmatism A ₀₂₂ ρ ²cos² θ have effects on the axial position of diffraction focus, while primary coma A ₀₃₁ ρ ³cosθ and primary distortion A ₁₁₁ ρcosθ have no effect on the axial focusing. Here, only the A ₀₄₀ ρ ⁴ and A ₀₂₂ ρ ²cos² θ obviously affecting the axial space measurement are considered.

3.2.1 Suppression on primary spherical aberration

The annular beam can reduce the ASRTM wave aberration and suppress the side lobe of axial intensity response curve whereas it reduces the ASRTM focusing sensitivity. The wave aberration of −2A ₀₄₀ ρ ⁴ is introduced into the measurement system when only the primary spherical aberration of A ₀₄₀ ρ ⁴ introduced by the test lenses is considered. And ASRTM intensity response curve obtained using Eq. (1) should satisfy

I (0, u, u_{M}, ε) = {| \frac{2}{1 - ε^{2}} \int_{ε}^{1} e^{- 2 j A_{040} ρ^{4}} \cdot e^{j ρ^{2} (2 u + u_{M}) / 2} ρ d ρ |}^{2} - {| \frac{2}{1 - ε^{2}} \int_{ε}^{1} e^{- 2 j A_{040} ρ^{4}} \cdot e^{j ρ^{2} (2 u - u_{M}) / 2} ρ d ρ |}^{2} .

When the primary spherical aberration coefficient is A ₀₄₀ =1.5λ, the axial intensity response curves with different ε are shown in Fig. 6 .

Fig. 6 Intensity curves with different ε and A ₀₄₀ =1.5λ .

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It can be seen from Fig. 6 that the side lobe of axial intensity curve is suppressed well as ε increases, whereas the sensitivity at the zero decreases excessively when ε>0.8.

3.2.2 Suppression on primary astigmatism

The wave aberration of −2A ₀₂₂ ρ ²cos² θ is introduced into the measurement system when only the primary astigmatism of A ₀₂₂ ρ ²cos² θ introduced by the test lenses is considered. And ASRTM intensity response curve obtained using Eq. (1) should satisfy

\begin{array}{l} I (0, u, u_{M}, ε) = {| \frac{1}{π (1 - ε^{2})} \int_{0}^{2 π} \int_{ε}^{1} e^{j ρ^{2} (2 u + u_{M}) / 2} \cdot e^{- j 2 k A_{022} ρ^{2} \cos θ^{2}} ρ d ρ d θ |}^{2} - \\ {| \frac{1}{π (1 - ε^{2})} \int_{0}^{2 π} \int_{ε}^{1} e^{j ρ^{2} (2 u - u_{M}) / 2} \cdot e^{- j 2 k A_{022} ρ^{2} \cos θ^{2}} ρ d ρ d θ |}^{2} . \end{array}

It is known through simulation using Zemax that the astigmatism coefficient A ₀₂₂ usually is an order of magnitude smaller than spherical aberration coefficient in the aberrations produced by the deviation of light-path adjustment. Substituting A ₀₂₂=0.2λ for Eq. (11), the axial intensity response curves with different ε are shown in Fig. 7 .

Fig. 7 Intensity curves with different ε and A ₀₂₂ =0.2λ.

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It can be seen from Fig. 7 that the sensitivity at the zero decreases whereas the zero position of differential confocal curves has no change as ε increases.

From Figs. 2, 6 and 7, the optimized ε≈0.7 is obtained under the consideration of focusing resolution σ_z, half-aperture angle α ₀ of objective L₁ and the primary spherical aberration and astigmatism.

4. Experiments

4.1 Experimental setup

The experimental setup shown in Fig. 8 is established based on Fig. 2 to verify the ASRTM validity.

Fig. 8 Experimental setup. 1. X80 interferometer produced by RENISHAW. 2. Interferometer measurement prisms. 3. Air bearing slider. 4. Test lens. 5. Differential confocal objective. 6. Annular pupil. 7. Collimator. 8. Laser. 9. Single-mode fiber. 10. Monitor. 11. Image capture and process software. 12. VPH 1. 13. VPH 2. 14. Material temperature sensor. 15. Air sensor.

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Where the caliber of objective L₁ is D=9.6 mm, the focal-length is f ₁ '=35 mm, λ=632.8 nm, the normalized radius of the annular pupil is ε=0.71 and X80 interferometer is used for the axial displacement measurement of objective L₁.

4.2 Thickness measurement of singlet

A plane-convex lens is used as the test lens, its refractive index is n ₁=1.5143, the radius of its front surface is r ₁=90.7908 mm, and its thickness is 4.0060 mm.

When objective L₁ is moved along the optical axis, ADCFT obtains the response curves I ₁(z) and I ₂(z) near the front and back surfaces of the test lens and uses their zeroes O ₁ and O ₂ to precisely identify the positions of the surface vertex. The focusing curves are shown in Fig. 9 , where the axial coordinate of point O ₁ is z ₁=−0.0018 mm and the axial coordinate of point O ₂ is z ₂=2.6745 mm.

Fig. 9 Thickness Measurement of singlet.

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When 1000 rays uniformly distributed in the annular pupil plane are used for the ray tracing, the result calculated using Eq. (9) is d ₁=4.0068 mm.

It is obvious that the thickness error obtained by ASRTM is δ =(4.0060-4.0068) mm= −0.0008 mm, i.e. it nearly coincides with the nominal thickness, and the relative error is |(4.0060-4.0068)|/4.0060×100%≈0.02%. And it is therefore that the Eq. (7) and Eq. (9) is right.

4.3 Axial space measurement of lenses

An air-spaced achromatic doublets is used as the test lens, the radii of its spherical surfaces are 195.426 mm, −140.270 mm, −140.258 mm and −400.906 mm in turn, its refractive indices are 1.5143 and 1.6686, and its inner space obtained indirectly by conventional contact measurement is S=0.323 mm in the assembly of lens.

When objective L₁ is moved along the optical axis, ASRTM obtains the response curves I ₁(z), I ₂(z), I ₃(z) and I ₄(z) near the front and back surfaces of the test lenses and uses their zeroes O ₁, O ₂, O ₃ and O ₄ to precisely identify the vertex positions P_n of the four spherical surfaces.

The focusing curves are shown in Fig. 10 , where the axial coordinate of point O ₁ is z ₁=−0.1622 mm, the axial coordinate of point O ₂ is z ₂=7.8946 mm, the axial coordinate of point O ₃ is z ₃=8.2271 mm and the axial coordinate of point O ₄ is z ₄=14.5258 mm.

Fig. 10 Space measurement of lenses.

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When 1000 rays uniformly distributed in the annular pupil plane are used for the ray tracing, the result calculated using Eq. (7) and Eq. (9) is d ₁=11.9892 mm, d ₂=0.3178 mm and d ₃=9.9751 mm. The axial space d ₂=0.3178 mm is close to S=0.323 mm, and the main reason is that the result S=0.323 mm itself has an extent error. Consequently, ASRTM is valid.

5. Conclusions

A new lenses inner axial space ray tracing measurement is proposed based on the geometrical theory of optical image, which uses the zero points of ADCFT axial intensity response curves to precisely identify the vertex positions of the test lenses surfaces, and then, calculates the axial space between the surfaces of lenses using the ray tracing. Preliminary experiments indicate that ASRTM has a relative error of less than 0.02%. ASRTM has some advantages as below.

1) The axial chromatography capability enables it to achieve the high-precision focusing at the vertex positions of the lenses inner-and-outer spherical surfaces.
2) Annular laser differential confocal intensity curve has a better linearity and the best sensitivity at the zero point, so it has high focusing accuracy at the vertex of the test lenses inner-and-outer spherical surface.
3) The annular pupil reduces the wave aberration of measurement system and suppresses the side lobe.

The ASRTM provides an effective approach for the high-precision measurement of the lenses axial space which has not been achieved so far.

Acknowledgment

Thanks to National Science Foundation of China (No.60708015, 60927012), Beijing Science Foundation of China (No.3082016), and Excellent Young Scholars Research Fund of Beijing Institute of Technology for the support.

References and links

1. D. M. Williamson, “Compensator selection in the tolerancing of a microlithographic lens,” Proc. SPIE 1049, 178–186 (1989).

2. K. K. Westort, “Design and fabrication of high performance relay lenses,” Proc. SPIE 548, 40–47 (1984).

3. T. Sure and J. Heil, “Microscope objective production: On the way from the micrometer scale to the nanometer scale,” Proc. SPIE 5180, 283–292 (2003). [CrossRef]

4. L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31(9), 1961–1966 (1992). [CrossRef]

5. W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal ultra-long focal length measurement,” Opt. Express 17(22), 20051–20062 (2009). [CrossRef] [PubMed]

6. W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004). [CrossRef] [PubMed]

7. W. Zhao, L. Qiu, and Z. Feng, “Effect of fabrication errors on superresolution property of a pupil filter,” Opt. Express 14(16), 7024–7036 (2006). [CrossRef] [PubMed]

8. L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001).

9. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 1999), Chap. 4, Chap. 9.

Lenses axial space ray tracing measurement

Abstract

1. Introduction

2. ASRTM principle

2.1 Precise focusing at inner-and-outer spherical surface of the lenses

2.2 Calculation of axial space

3. Effect of annular pupil on ASRTM sensitivity

3.1 Reducing the wave aberration

3.2 Effect of annular pupil on axial sensitivity

3.2.1 Suppression on primary spherical aberration

3.2.2 Suppression on primary astigmatism

4. Experiments

4.1 Experimental setup

4.2 Thickness measurement of singlet

4.3 Axial space measurement of lenses

5. Conclusions

Acknowledgment

References and links

Cited By

Figures (10)

Equations (11)

Optics Express