## Lenses axial space ray tracing measurement

Optics Express, Vol. 18, Issue 4, pp. 3608-3617 (2010)

http://dx.doi.org/10.1364/OE.18.003608

Acrobat PDF (402 KB)

### 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

*P*of the inner surface of lenses [3

_{n}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]

*d*, the non-contact optical chromatography measurement needs to be used for precise obtaining the vertex position

_{n}*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}*r*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

_{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}*d*is how to achieve the precise focusing on the inner spherical surface

_{n}*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

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]

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]

*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

_{n}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]

*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

*O*of ADCFT axial intensity response curve

_{n}*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}_{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.

*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

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]

*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.

_{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:

*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*max at zero when axial normalized offset

*u*satisfies:

_{M}*S*

_{max}=−0.54×(1-

*ε*

^{2}), and the annular laser differential confocal focusing resolution

*σ*satisfywhere

_{z}*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*).

*σ*depends on half-aperture angle

_{z}*α*

_{0}and normalized radius

*ε*, and focusing resolution

*σ*with different

_{z}*α*

_{0}and

*ε*is shown in Fig. 3 .

*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.

*σ*is better than 200 nm when

_{z}*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.

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]

## 2.2 Calculation of axial space

_{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*of the test lens spherical surface, and records the coordinate

_{n}*z*of L

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

*O*in turn.

_{n}*n*+1, the measurement ray has a deflection after passing through each spherical surface and the half-aperture angle

*θ*changes due to the refractive index when the measurement beam with half-aperture angle

_{n}*θ*

_{0}is converged on

*S*

_{n}_{+1}of the test lenses.

*ρ*

_{pupil}is the radius of objective L

_{1}at the exit pupil,

*f*

_{1}

*'*is the focal length of objective L

_{1},

*r*is the radius of spherical surface

_{n}*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}*S*satisfywhere

_{n}*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}*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}*S*

_{n}_{+1}, thenwhere

*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}*d*is obtained through Eq. (7) when the parameters of each singlet

_{n}

*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}).

*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}satisfieswhere 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

*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.

*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

*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

*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

*A*

_{040}

*=*1.5λ, the axial intensity response curves with different

*ε*are shown in Fig. 6 .

*ε*increases, whereas the sensitivity at the zero decreases excessively when

*ε>*0.8.

#### 3.2.2 Suppression on primary astigmatism

*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

*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 .

*ε*increases.

## 4. Experiments

### 4.1 Experimental setup

_{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

*n*

_{1}=1.5143, the radius of its front surface is

*r*

_{1}=90.7908 mm, and its thickness is 4.0060 mm.

_{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.

*d*

_{1}=4.0068 mm.

### 4.3 Axial space measurement of lenses

*S*=0.323 mm in the assembly of lens.

_{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*of the four spherical surfaces.

_{n}*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.

*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

- 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.

## Acknowledgment

## References and links

1. | D. M. Williamson, “Compensator selection in the tolerancing of a microlithographic lens,” Proc. SPIE |

2. | K. K. Westort, “Design and fabrication of high performance relay lenses,” Proc. SPIE |

3. | T. Sure and J. Heil, “Microscope objective production: On the way from the micrometer scale to the nanometer scale,” Proc. SPIE |

4. | L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. |

5. | W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal ultra-long focal length measurement,” Opt. Express |

6. | W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express |

7. | W. Zhao, L. Qiu, and Z. Feng, “Effect of fabrication errors on superresolution property of a pupil filter,” Opt. Express |

8. | L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. |

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

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(180.1790) Microscopy : Confocal microscopy

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: December 14, 2009

Revised Manuscript: January 18, 2010

Manuscript Accepted: January 19, 2010

Published: February 4, 2010

**Citation**

Weiqian Zhao, Ruoduan Sun, Lirong Qiu, Libo Shi, and Dingguo Sha, "Lenses axial space ray tracing measurement," Opt. Express **18**, 3608-3617 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3608

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### References

- D. M. Williamson, “Compensator selection in the tolerancing of a microlithographic lens,” Proc. SPIE 1049, 178–186 (1989).
- K. K. Westort, “Design and fabrication of high performance relay lenses,” Proc. SPIE 548, 40–47 (1984).
- 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]
- L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31(9), 1961–1966 (1992). [CrossRef]
- 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]
- 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]
- 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]
- 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).
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), Chap. 4, Chap. 9.

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