## Abstract

A new laser differential reflection-confocal focal-length measurement (DRCFM) method is proposed for the high-accuracy measurement of the lens focal length. DRCFM uses weak light reflected from the lens last surface to determine the vertex position of this surface. Differential confocal technology is then used to identify precisely the lens focus and vertex of the lens last surface, thereby enabling the precise measurement of the lens focal length. Compared with existing measurement methods, DRCFM has high accuracy and strong anti-interference capability. Theoretical analyses and experimental results indicate that the DRCFM relative measurement error is less than 10 ppm.

© 2012 OSA

## 1. Introduction

The focal length is a key parameter that reflects performance of a lens. The focal length is related to the curvature radii of the lens surfaces, refractive index of the lens material, and lens thickness, among others. The actual focal length always deviates from the theoretical value as a result of manufacturing and assembly errors. Therefore, the precise measurement of the lens focal length is a core problem in the field of optical measurement.

Various methods for focal-length measurement have been proposed. Classical methods such as nodal slide and image magnification are based on the principles of geometry optics. Although these methods are easy to setup and manipulate, high accuracy is difficult to achieve because of subjectivity-related measurement errors. To improve the measurement accuracy, some novel methods based on the principle of physical optics have been presented. For example, moiré deflectometry [1] and Talbot interferometry [2–4] use a moiré fringe to determine the lens focal length. Their measurement accuracies (lower than 0.36%) are limited by the manual measurement of the inclination of the moiré fringes [3]. Grating shearing interferometry measures the lens focal length by using the relative lateral shift between the undiffracted zero order and diffracted first order caused by the grating. The measurement accuracy (lower than 0.4%) is limited by the accuracy of charge-coupled device (CCD) measurements [5]. Lau phase interferometry uses the slope of a phase map to determine the lens focal length [6,7]. The measurement accuracy (lower than 0.2%) is influenced by the aberrations of an unwrapped phase map [7]. The circular Dammann grating method uses the diffraction characteristics of a circular Dammann grating at the focal plane of a lens to locate the focus of the test lens. The identification accuracy of the focus (lower than 0.1%) is limited by the definition of a diffraction pattern [8]. Phase-shift interferometry identifies the image point of the test lens using interference fringes, and then calculates the lens focal length by Gaussian or Newtonian equations [9,10]. The measurement accuracy reaches 0.01% [10], which is the highest accuracy among those of existing measurement methods. Other proposed methods include the fiber-optic autocollimation method [11], reference plant method [12] and Hartmann-Shack method [13]. However, their measurement accuracies need to be improved for engineering measurements.

The accuracies of existing focal-length measurement methods are lower than 0.01%, which cannot meet the demands of some sophisticated technologies [14,15]. In the previous work, we have presented a laser differential confocal ultra-long focal length measurement with the relative error of about 0.01% [16], but it is not suitable for the short focal-length measurement. Therefore, a laser differential reflection-confocal focal-length measurement (DRCFM) method is proposed in this paper. DRCFM uses differential confocal intensity detection to identify the lens focus and vertex of the lens last surface with high accuracy. The method also has a strong anti-interference capability [17]. Theoretical analyses and preliminary experiments show that the DRCFM relative measurement error is less than 10 ppm. Thus, DRCFM can enable the high-accuracy measurement of the lens focal length.

## 2. DRCFM principle

Our group has previously established that the null point of the axial intensity response signal corresponds to the focus of the objective in a differential confocal system [17]. On this basis, DRCFM uses null points *Q _{A}* and

*Q*of differential confocal response signals

_{B}*I*and

_{A}*I*to identify precisely the lens focus and vertex of the lens last surface, as shown in Fig. 1 . The distance between these two null positions is then determined to facilitate the high-accuracy measurement of the lens back focal length

_{B}*l*'.

_{F}*z*and

_{A}*z*are the position coordinates of reflector R corresponding to null points

_{B}*Q*and

_{A}*Q*, respectively.

_{B}When reflector R moves near position *A* along the optical axis of test lens L_{t}, the measurement beam reflected from R is again reflected by the polarized beam splitter (PBS) onto the beam splitter (BS). The two measurement beams split by the BS are received by the virtual pinholes (VPH1 and VPH2). The normalized intensity signals *I _{A}*

_{1}(

*ν*

_{2},

*u*, +

*u*) and

_{M}*I*

_{A}_{2}(

*ν*

_{2},

*u*,-

*u*) received from CCD1 and CCD2 can be obtained by the Huygens–Fresnel diffraction integral formula.

_{M}*J*

_{0}is a zero-order Bessel function,

*ρ*is the radial normalized radius of a pupil,

*θ*is the angle of variable

*ρ*in the polar coordinate,

*r*

_{1}is the radial coordinate on reflector R,

*v*

_{1}is the normalized coordinate of variable

*r*

_{1},

*z*is the axial displacement between reflector R and position

*A*,

*u*is the normalized coordinate of variable

*z*,

*r*

_{2}is the radial coordinate on the object plane of VPH1 or VPH2,

*v*

_{2}is the normalized coordinate of variable

*r*

_{2},

*M*is the offset of the VPHs from the focus of collimating lens L

_{c},

*u*is the normalized offset of variable

_{M}*M*,

*P*(

_{t}*ρ*) is the pupil function of test lens L

_{t},

*P*(

_{c}*ρ*) is the pupil function of collimating lens L

_{c},

*D*is the effective aperture of L

_{c}and L

_{t}which is equal to the smaller aperture between L

_{c}and L

_{t},

*D*/

*f*is the effective relative aperture of L

_{c}'_{c}, and

*D*/

*f*is the effective relative aperture of L

_{t}'_{t}. To ensure the full aperture measurement of L

_{t}, the clear aperture of L

_{c}should be no smaller than that of L

_{t}.

When *P _{c}*(

*ρ*) = 1 and

*P*(

_{t}*ρ*) = 1, the differential confocal response signal

*I*(

_{A}*u*,

*u*) is obtained through the differential subtraction of

_{M}*I*

_{A}_{1}(0,

*u*, +

*u*) and

_{M}*I*

_{A}_{2}(0,

*u*,-

*u*).

_{M}As shown in Fig. 1, null point *Q _{A}* of differential confocal response signal

*I*(

_{A}*u*,

*u*) precisely corresponds to position

_{M}*A*such that reflector R is exactly at the focus of L

_{t}.

When reflector R moves near position *B* along the optical axis of L_{t}, a part of the measurement beam is reflected by the last surface of L_{t}. The overlap area of the measurement beam and the last surface of L_{t} is so small that the effect of the surface curvature on the differential confocal response signal is negligible. When the distance between reflector R and position *B* is *z*, the distance between the focusing point of the measurement beam and the vertex of the L_{t} last surface is 2*z*. Therefore, the differential confocal response signal *I _{B}*(

*u*,

*u*) obtained through the differential subtraction of

_{M}*I*

_{B}_{1}(0,

*u*, +

*u*) and

_{M}*I*

_{B}_{2}(0,

*u*,-

*u*) can be derived as follows.

_{M}As shown in Fig. 1, null point *Q _{B}* of differential confocal response signal

*I*(

_{B}*u*,

*u*) precisely corresponds to position

_{M}*B*such that the measurement beam is focused on the vertex of the L

_{t}last surface.

When the other parameters of L_{t} are given, the distance between the L_{t} second principle point and the vertex of the L_{t} last surface can be calculated by using ray tracing formulas. Then, the effective focal length of L_{t} can be indirectly measured. Generally, the distance between the L_{t} second principle point and the vertex of the L_{t} last surface is so small that the resulting measurement error is negligible. With a single lens as an example, the effective focal length can be calculated as follows.

*r*

_{1}is the curvature radius of the L

_{t}front surface,

*r*

_{2}is the curvature radius of the L

_{t}back surface,

*n*is the L

_{t}refractive index, and

*b*is the L

_{t}thickness.

## 3. Effect analyses of key parameters

#### 3.1 Focusing sensitivities

Sensitivities *S _{A}*(0,

*u*) at null point

_{M}*Q*and

_{A}*S*(0,

_{B}*u*) at null point

_{M}*Q*can be obtained by differentiating Eqs. (5) and (6) on

_{B}*u*, respectively.

Figure 2(a)
shows that differential confocal response signals *I _{A}*(

*u*,

*u*) and

_{M}*I*(

_{B}*u*,

*u*) have the largest sensitivities

_{M}*S*

_{A}_{max}and

*S*

_{B}_{max}at null points

*Q*and

_{A}*Q*when

_{B}*u*= 5.21.

_{M}Figure 2(b) shows simulation curves *I _{A}*(

*u*,

*u*) and

_{M}*I*(

_{B}*u*,

*u*) with

_{M}*u*= 5.21. The identification at the vertex of the L

_{M}_{t}last surface is clearly much more sensitive than that at the L

_{t}focus.

When *u _{M}* = 5.21, DRCFM focusing errors

*σz*and

_{A}*σz*at the L

_{B}_{t}focus and the vertex of the L

_{t}last surface can be described as follows.

*δI*(

_{A}*u*,

*u*) and

_{M}*δI*(

_{B}*u*,

*u*) are the random noises of the differential confocal response signals caused by the CCDs, and

_{M}*SNR*is the signal-to-noise ratio of the VPHs. Equations (11) and (12) show that the identification precision increases as relative aperture

*D*/

*f*increases.

_{t}'#### 3.2 Effect of axial alignment errors

As shown in Fig. 3
, axis *m* of the measurement beam, optical axis *s* of L_{t}, measurement axis *p* of the distance measurement interferometer (DMI), and optical axis *q* of reflector R should be coincident, whereas the deviation angles between them always exist in practice. Using the autocollimation method without L_{t} in the measurement path, the angle between axes *m* and *q* can be aligned such that the effect of this angle on the measurement can be negligible. If the angle between axes *m* and *s* is defined as *α*, and the angle between axes *m* and *p* is defined as *β*, the back focal length measurement error can be calculated using the following geometric relationship.

Angle *β* is diminished through the careful alignment of axes *p* and *m*, and it should be adjusted only while establishing the system. When the focal length of L_{c} in the DRCFM system is 1000 mm, angle *β* can be reduced to less than 3 second. Angle *α* between axes *m* and *s* can be easily controlled within 2 minute by careful alignment.

#### 3.3 Effect of two VPHs with different offsets

As shown in Fig. 4
, when the offsets of the two VPHs from the L_{c} focus are different, null points *Q _{A}* and

*Q*of the differential confocal response signals deviate from positions

_{B}*A*and

*B*, respectively. Thus, the

*l*' measurements change.

_{F}Let the offsets of two VPHs from the L_{c} focus be *M* and -*M* + *δ _{M}*, their corresponding normalized offsets be

*u*and -

_{M}*u*+

_{M}*u*, the offsets of

_{δM}*Q*from position

_{A}*A*and of

*Q*from position

_{B}*B*in the measurement process be Δ

*l*

_{1}and Δ

*l*

_{2}, and their normalized values be Δ

*u*

_{1}and Δ

*u*

_{2}.

Then, the differential confocal response signals *I _{A}*(

*u*,

*u*) and

_{M}*I*(

_{B}*u*,

*u*) are obtained according to Eqs. (5) and (6).

_{M}From Eq. (4), the respective actual offsets of the corresponding positions *Q _{A}* and

*Q*are

_{B}Therefore, when the deviation of VPHs axial offset is *δ _{M}*, back focal length measurement error

*σ*caused by

_{offset}*δ*can be obtained through Eq. (1).

_{M}Using a grating scale as an aided adjustment of the VPH offsets, offset error *δ _{M}* can be easily controlled within 10 μm, which is limited by both the position accuracy of the grating scale and the focusing accuracy of the L

_{c}focus.

#### 3.4 Distance measurement error

As shown in Fig. 1, the distance between positions *A* and *B* is measured by a single-frequency laser interferometer, and its measurement error is

*L*is the measurement distance.

#### 3.5 Synthesis error

Considering the effect of the aforementioned errors on the measurement results of *l _{F}*', the DRCFM total measurement error

*σl*' is

_{F}According to Eq. (1), the respective error transfer coefficients of *σ _{L}*,

*σz*, and

_{A}*σz*are

_{B}Using Eqs. (20) and (21), the DRCFM synthesis measurement error is

## 4. Experiments

#### 4.1 Experimental setup

The experimental setup shown in Fig. 5 is established based on Fig. 1 to verify the validity of DRCFM.

The environmental conditions in the measurement laboratory are pressure = (102540 ± 60) Pa, temperature = (20 ± 0.2) °C, and relative humidity = (40 ± 4) %. In the experiment, a He-Ne laser with a wavelength of 632.8 nm is used as the light source. An achromatic lens (produced by LINOS Photonics GmbH & Co.) with a focal length of 1000 mm and a diameter of 100 mm is used as collimating lens L_{c}. An XL-80 laser interferometer (produced by Renishaw) is used as the distance measurement interferometer. A high-accuracy air bearing slider with a range of 1300 mm and a straightness of 0.1 μm is used as the motion rail. The CCDs used are OK-AM1100 with a pixel size of 8 μm. The *SNR* of the VPHs is 100:1, and the magnification of the VPH microscope objectives is 10 × . Test lens L_{t} is a cemented doublet lens with a diameter of 20 mm and a focal length of 200 mm, and its nominal back focal length *l _{F}*' is 197.3 mm ( ± 2%).

#### 4.2 Experiment results

As shown in Fig. 6
, when reflector R moves across position *A* along the optical axis during measurement, DRCFM uses null point *Q _{A}* of differential confocal response signal

*I*(

_{A}*z*) to determine precisely the focus of L

_{t}. The position coordinate of reflector R corresponding to point

*Q*is

_{A}*z*= −99.17363 mm.

_{A}Similarly, when reflector R moves across position *B*, DRCFM uses null point *Q _{B}* of differential confocal response signal

*I*(

_{B}*z*) to determine precisely the vertex of the L

_{t}last surface. The position coordinate of reflector R corresponding to point

*Q*is

_{B}*z*= 0.17223 mm.

_{B}Therefore, the back focal length *l _{F}*' of L

_{t}is 2|

*z*-

_{A}*z*| = 198.6917 mm, and the repeatability achieved from ten measurements is

_{B}*σ*= 1.6 μm.

_{test}Using Eqs. (11)–(13), (18), and (19), the errors existing in DRCFM are *σ _{L}* = 0.1 μm,

*σz*= 0.75 μm,

_{A}*σz*= 0.37 μm,

_{B}*σ*= 0.03 μm, and

_{axial}*σ*= 0.1 μm. The system error obtained using Eq. (22) is

_{offset}The relative error is

Considering the environmental effects and some negligible errors, the relative error of DRCFM can be less than 10 ppm.

It can be concluded from the above error analyses that the focusing errors *σz _{A}* and

*σz*have more significant effect on the measurement accuracy. So, the measurement accuracy increases as the relative aperture of test lens increases.

_{B}## 5. Conclusions

The proposed DRCFM uses the null points of differential confocal response signals to identify precisely the lens focus and vertex of the lens last surface. Consequently, the lens focal length can be measured with high accuracy. Theoretical analyses and experimental results show that the proposed approach has a relative error of less than 10 ppm and has the following advantages.

- 1) It improves significantly the identification precision at the lens focus and vertex of the lens last surface because it has the best linearity and sensitivity at the null point of a differential confocal response signal.
- 2) It can measure the lens spherical aberration in combination with annular pupil filtering technology.
- 3) It has higher measurement accuracy and stronger anti-interference capability than existing approaches.

Therefore, DRCFM is a feasible method for the high-accuracy measurement of the lens focal length.

## Acknowledgments

The authors gratefully acknowledge the support of the Major National Scientific Instrument and Equipment Development Project of China (No. 2011YQ040136), the National Science Foundation of China (No. 91123014, 60927012), and the Nature Science Association Foundation (No. 11076004).

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