## Abstract

Abstract: In order to achieve the precise measurement of the lenses axial space, a new lenses axial space ray tracing measurement (ASRTM) is proposed based on the geometrical theory of optical image. For an assembled lenses with the given radius of curvature *r _{n}* and refractive index

*n*of every lens, ASRTM uses the annular laser differential confocal chromatography focusing technique (ADCFT) to achieve the precise focusing at the vertex position

_{n}*P*of its inner-and-outer spherical surface

_{n}*S*and obtain the coordinate

_{n}*z*corresponding to the axial movement position of ASRTM objective, and then, uses the ray tracing facet iterative algorithm to precisely determine the vertex position

_{n}*P*of every spherical surface by these coordinates

_{n}*z*, refractive index

_{n}*n*and spherical radius

_{n}*r*and thereby obtaining the lenses inner axial space

_{n},*d*. The preliminary experimental results indicate that ASRTM has a relative measurement error of less than 0.02%.

_{n}© 2010 OSA

## 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*of the lens’ spherical surface

_{n}*S*. But the precision of the vertex position

_{n}*p*depends on the lens’ radii

_{n}*r*and refractive index

_{n}*n*, and the figure error and the axial position

_{n}*z*of the lens L

_{n}_{1}. Therefore, it is difficult to obtain the inner axial space

*d*of the assembled lenses using the direct measurement.

_{n}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*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

_{n}*S*and obtain the precise coordinate

_{n}*z*of the objective movement position corresponding to the vertex position

_{n}*P*of the lenses’ inner-and-outer sphere, where

_{n}*n*=1, 2, 3, 4. And then, ASRTM uses the ray tracing facet iterative algorithm to precisely determine the vertex position

*P*of every spherical surface

_{n}*S*by these coordinates

_{n}*z*, refractive index

_{n}*n*and spherical radius

_{n}*r*and thereby achieving the high-precision measurement of lenses’ inner axial space

_{n},*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*and obviously reduce the adverse effect of the figure error of the front spherical surface

_{n}*S*

_{n-}_{1}on the accuracy of focusing at the back spherical surface

*S*. At present, many methods are used for the high-precision location on the outer surface

_{n}*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*, radius of curvature

_{n}*r*, 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

_{n}*S*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.

_{n}’## 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*of objective L

_{n}_{1}, 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*of ADCFT axial intensity response curve

_{n}*I*(

*z*) at the different focusing plane of the objective L

_{1}to precisely determine the spherical vertex position

*P*of the test lenses and obtain the coordinate

_{n}*z*corresponding to the movement position of objective L

_{n}_{1}where

*n*=1, 2, 3, 4. Finally, ASRTM determines the precise vertex position

*P*of spherical surface

_{n}*S*by the ray tracing, and thereby achieving the precise focusing of the lenses’ inner-and-outer spherical surface.

_{n}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 L_{1} 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 L_{2} 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]

*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 L

_{1},

*p*

_{2}(

*ρ*,

*θ*) is the pupil function of L

_{2},

*p*

_{C}(

*ρ*,

*θ*) is the pupil function of L

_{C},

*u*is the axial normalized optical coordinate, and

*u*=8π

*z*sin

^{2}(

*α*

_{0}/2),

*v*is the lateral normalized optical coordinate and

*v*=(2π

*r*sin

*α*

_{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 L_{1}, L_{2} 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:

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

The numerical analysis on Eq. (3) shows that the differential focusing curve has the best sensitivity *S*max at zero when axial normalized offset *u _{M}* satisfies:

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

*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*with different

_{z}*α*

_{0}and

*ε*is shown in Fig. 3 .

Half-aperture angle *α*
_{0} of objective L_{1} 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 L_{1} 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*of L

_{n}_{1}movement position corresponding to zero point

*O*in turn.

_{n}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.

Where *ρ*
_{pupil} is the radius of objective L_{1} at the exit pupil, *f*
_{1}
*'* is the focal length of objective L_{1}, *r _{n}* is the radius of spherical surface

*S*,

_{n}*n*is the refractive index between the spherical surfaces

_{n}*S*and

_{n}*S*

_{n}_{+1},

*d*is the axial space between

_{n}*S*and

_{n}*S*

_{n}_{+1}, and

*i*and

_{n}*i*´ are the angles of the ray into and outof

_{n}*S*.

_{n}According to the ray tracing formula [9], the variables of spherical surface *S _{n}* satisfy

*l*is the distance from the vertex of

_{n}*S*to the point of intersection of the optical axis with the incidence light on

_{n}*S*,

_{n}*l*´ is the distance from the vertex of

_{n}*S*to the point of intersection of optical axis with the exit light from

_{n}*S*,

_{n}*θ*is the angle between incidence light on

_{n}*S*and optical axis, and

_{n}*θ*´ is the angle between the exit light from

_{n}*S*and the optical axis.

_{n}It is known from Fig. 4 that the relation between *S _{n}*

_{-1}and

*S*satisfies

_{n}*ln=ln-1´-dn-1*and

*θ*=

_{n}*θ*

_{n}_{-1}´. Recursive ray tracing expressions between

*S*

_{n}_{-1}and

*S*obtained by Eq. (6) are

_{n}As shown in Fig. 4, when ASRTM pencil tip is coincident with the vertex of *S _{n}*

_{+1}, then

*θ*

_{0}is the half-aperture angle of the measurement beam from L

_{1}. With Eq. (8) as the initial conditions, the axial space between

*S*and

_{n}*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*} and L

_{n}_{1}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

*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.

#### 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*
^{2}cos^{2}
*θ* have effects on the axial position of diffraction focus, while primary coma *A*
_{031}
*ρ*
^{3}cos*θ* and primary distortion *A*
_{111}
*ρ*cos*θ* have no effect on the axial focusing. Here, only the *A*
_{040}
*ρ*
^{4} and *A*
_{022}
*ρ*
^{2}cos^{2}
*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 −2*A*
_{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

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

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 −2*A*
_{022}
*ρ*
^{2}cos^{2}
*θ* is introduced into the measurement system when only the primary astigmatism of *A*
_{022}
*ρ*
^{2}cos^{2}
*θ* introduced by the test lenses is considered. And ASRTM intensity response curve obtained using Eq. (1) should satisfy

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
.

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 L

_{1}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.

Where the caliber of objective L_{1} 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 L_{1}.

#### 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 L_{1} 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.

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 L_{1} 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.

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.

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