## Abstract

A new laser differential confocal focal-length measurement method is proposed for the measurement of an ultra-long focal-length. The approach proposed uses the property of an axial intensity curve that the absolute zero precisely corresponds to the focus of the objective in a differential confocal focusing system (DCFS) to measure the variation in position of DCFS focus with and without a measured ultra-long focal-length lens (UFL), uses the distance between the two focuses to obtain the UFL focal-length, and thereby achieving the precise measurement of ultra-long focal-length. The method has a high focusing precision, a strong anti-interference capability and a short measurement light-path. The theoretical analyses and preliminary experimental results indicate that the relative measurement error is about 0.01% when the method is used for the measurement of back-focus-distance (BFD).

©2009 Optical Society of America

## 1.Introduction

It is difficult for the focal-length measurement of an ultra-long focal-length lens (UFL) widely used in large optical systems, such as space optical system, high energy laser system and laser fusion program, because UFL has a small N.A. which makes it difficult to find the exact position of its focus, and has a long measurement light-path so that the measurements is easily affected by the disturbance of environment and cannot get a high precision of length measurement. Now, the methods used for the measurement of an ultra-long focal-length can be classified into two categories.

Some methods are based on the geometric optics. For example, Liana Glatt et al. [1] used moire deflectometry to obtain the focal-length with a theoretical measurement error less than 0.1% by measuring the tilt angle of moire fringes formed by two gratings. V.I. Meshcheryakov et al. obtained the focal-length by two positions of a luminous slit, one with an optical wedge inserted into the light-path and the other without it, and the experiment showed that a focal length of 25m can be measured with an error of 0.1% [2]. The measurement precision of these methods is limited by diffraction to be about 0.1%.

Other methods use the interferometric techniques. For example, Yoshiaki Nakano et al. calculated the focal length by Talbot interferometry, which measured the tilt angle of Ronchi grating Talbot image by moiré technique [3-5]. Based on it, Priti Singh et al. used a Fourier fringe analysis technique to remove the noise caused by the Ronchi grating lines and used a four-step algorithm to obtain the lens’ phase map from the Talbot fringes, and this method evaluated the focal length from the slope of the phase map with a measurement error less than 0.3% [6]. T.G. Parham et al. used Fizeau phase-shifting interferometer (FPSI) to determine the back-focal-distance (BFD) by measuring the radius of wave-front curvature and have obtained a measurement precision of 0.02% [7]. Brian DeBo applied Fresnel-zone hologram to the measurement system used by T. G. instead of the expensive retrosphere to achieve a focal-length measurement with a precision of better than 0.0088% [8]. In comparison with the methods based on geometric optics, the methods using interferometry have a high measurement precision, but the effect of environmental temperature, airflow and vibration on interferogram results in a rigorous requirement for measurement environment and a limitation to further improve the measurement precision.

The key to improve the measurement precision of UFL focal-length is how to improve the focusing precision and shorten the measurement light-path. And therefore, a new laser differential confocal focal-length measurement method is proposed for the precise measurement of UFL focal-length, especially for the BFD measurement which is essential in the assemblage of a large optical system such as the National Ignition Facility [7,8].

In comparison with the existing UFL focal-length measurement methods, the method proposed has a high focusing precision, strong environmental anti-interference capability and a measurement system with compact structure.

## 2. Measurement principle

#### 2.1 Differential confocal focusing principle

Based on the property of a confocal microscopy system (CMS) that the axial offset of a pinhole near the focus only changes its axial light intensity response phase [9], the differential confocal focusing principle as shown in Fig. 1 divides CMS receiving light-path into two paths to make two sets of pinholes and detectors placed behind and before the focus respectively, and improves CMS axial resolution near the focus through differential subtraction of two detection signals with given phases received, thereby achieving the CMS high precise focusing. The focusing process is as below.

When reflector *R* is moved near the focus along the optical axis, intensity response *I*
_{1}(*v*,*u*, + *u _{M}*) received by detector 1 is [9]:

*J*

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

*u*is the axial normalized offset of a pinhole,

_{M}*ρ*is a radial normalized radius,

*p*

_{1}(

*ρ*) is the pupil function of focusing system including RL,

*p*

_{2}(

*ρ*) is the pupil function of collecting system including Lens 2,

*u*and

*v*are axial and lateral normalized coordinates, respectively, and

*z*is the axial displacement of the object,

*r*is the radial coordinate of the objective,

*D*/

*f*is the relative aperture of objective RL used in a differential confocal focusing system (DCFS).

Intensity response *I*
_{2}(*v*,*u*,-*u _{M}*) received by detector 2 is [9]:

Signal *I* (*v*,*u*,*u _{M}*) obtained through the differential subtraction of

*I*

_{1}(

*v*,

*u*, +

*u*) and

_{M}*I*

_{2}(

*v*,

*u*,-

*u*) is:

_{M}When RL and Lens 2 have the same parameters and there is no pupil filter in the system, DCFS axial response *I* (0,*u*,*u _{M}*) obtained from Eq. (4) is:

Zero *O* of differential axial response curve *I*(0*,u*,*u _{M}*) precisely corresponds to DCFS focus and the precise focusing of a lens is achieved by exactly determining the position of the reflector corresponding to zero

*O*of

*I*(0,

*u*,

*u*).

_{M}In addition, the laser differential confocal focusing method can reduce the focal-depth and improve the axial focusing sensitivity by using the superresolution pupil filtering [9], and enhance the environmental anti-interference capability by using intensity modulation detection technique.

#### 2.2 Differential confocal ultra-long focal-length focusing measurement principle

The method proposed can be directly used in the precise measurement of BFD, and the classically defined focal length (i.e., distance of focal point from rear principal plane) can be calculated from the measured BFD if the parameters of the lens are known by preliminary measurement.

In a differential confocal ultra-long focal-length focusing measurement system as shown in Fig. 2
, UFL is an ultra-long focal-length lens to be measured and RL is the reference lens with a given focal-length. The calculation of distance *d* between principal points of UFL and RL is simplified when RL is a plano-convex lens with coincided principal point and vertex.

The light is focused on focus *A* by RL without UFL in the light-path, and the differential response curve exactly corresponds to zero *O*
_{1} when reflector *R* passed through point *A* along optical axis. The focus is changed from *A* to *B* when the UFL is combined with RL shown in Fig. 3
, and the differential response curve exactly corresponds to zero *O*
_{2} when reflector *R* passed through point *B* along optical axis. Distance *l = z _{A}-z_{B}* between focuses

*A*and

*B*can be obtained by measuring the movement displacement of reflector

*R*corresponding to two zeroes

*O*

_{1}and

*O*

_{2}of differential response curve.

The diagram of combination lens is shown in Fig. 3. Focal-length of combination lens system *f'*, UFL focal-length *f*
_{1}
*'*, RL focal-length *f*
_{2}
*'* and distance *d* between principal points of UFL and RL satisfy Eq. (6).

And

where*l*is the distance between

*A*and

*B*,

*H*

_{2}

*H*

_{22}is the distance between the right principal points of combination Lens and RL.

As shown in Fig. 3, we have

Hence, UFL focal-length *f*
_{1}
*'*obtained using Eq. (6) ~Eq. (8) can be written as:

*d*=

*d*

_{0}+

*H*

_{12}

*V*

_{12},

*H*

_{12}

*V*

_{12}is the distance between the right principal point and vertex point of UFL,

*d*

_{0}is the distance between the vertex points of UFL and RL.

Then,

When the UFL parameters are known by preliminary measurement, the focal-length of the measured lens *f*
_{1}
*'* can be obtained by:

*b*

_{1}is the UFL thickness,

*n*

_{1}is the refractive index, and

*r*

_{11}and

*r*

_{12}are the radii of curvature of the front and back surface.

## 3. Analyses of key parameters

#### 3.1 Focusing sensitivity

Focusing sensitivity *S*(0,0,*u _{M}*) at absolute zero

*O*can be obtained by differentiating curve

*I*(0,

*u*,

*u*) on

_{M}*u*and is

Focusing sensitivity *S*(0,0,*u _{M}*) is shown in Fig. 4
with different

*u*.

_{M}It can be seen from Fig. 4 that the differential focusing curve has the largest sensitivity *S*
_{max} at zero *O* when *u _{M}* = 5.21.

DCFS focusing error *σ _{z}* and relative aperture

*D*/

*f*′ of the confocal objective satisfy Eq. (14).

It can be seen from Eq. (14) that the focusing error *σ _{z}* decreases as

*D*/

*f '*increases.

In addition to *u _{M}* and

*D*/

*f '*, the pinhole size and the lateral offset of the detector have an effect on the differential measurement sensitivity, but it can be eliminated by readjusting the optical path arrangement [10,11].

#### 3.2 Measurement range of focal-length

As shown in Fig. 5
, if UFL focal-length *f*
_{1}
*'* is short, the combination lens focus *B* will be far from point *A* after UFL is introduced into the light-path. At the same time, the focusing intensity response curve will be out of shape and the focusing sensitivity will decrease obviously. If *f*
_{1}
*'* > 2.5*f*
_{2}
*'* and especially *f*
_{1}
*'*→∞, the focus of the combination lens will be close to point *A* and the focusing sensitivity is the best.

In the measurement system proposed, UFL focal-length *f*
_{1}
*'* is limited by RL focal-length *f*
_{2}
*'* and the best measurement range of *f*
_{1}
*'* varies with *f*
_{2}
*'*.

The optimized *u _{M}* satisfies Eq. (15) before UFL is inserted into the measurement light-path and DCFS focusing sensitivity is the best.

After the UFL is inserted into the measurement light-path, the focal length changes from *f*
_{2}
*'* to *f '*, and their ratio is expressed as:

The optimized *u _{M}* changes to

*u*´ when UFL is used in DCFS, and

_{M}*u*´ obtained from Eqs. (15) and (16) is

_{M}According to Eq. (5) and Eq. (17), differential confocal focusing curves are shown in Fig. 6
when *f*
_{1}
*'* is equal to 1.5*f*
_{2}
*'*, 2*f*
_{2}
*'*, 2.5*f*
_{2}
*'*, 3*f*
_{2}
*'*, 5*f*
_{2}
*'*, 10*f*
_{2}
*'*,⋅⋅⋅, or ∞, respectively.

It can be seen from Fig. 6 that when *f*
_{1}
*'* = 1.5*f*
_{2}
*'*, the response curve has more peaks than the other curves and its slope near the point *O* is reverse so that it cannot be used for focusing; when *f*
_{1}
*'*≥2.5*f*
_{2}
*'*, the differential focusing curves all pass through absolute zero *O* although their sensitivity are different, and then, the focusing method using absolute zero *O* on UFL focal-length measurement has a high precision. And therefore, it is that the measurement method proposed is especially fit for measurement of ultra-long focal-length.

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

As shown in Fig. 7
, when two pinholes are placed from the image plane by different amounts, the zero of DCFS axial curve will deviate from the focus of the RL and combination lens, so the measurement results of *l* will be changed.

Let the offset of two pinholes from the focus are *M* + *δ* and *M*, their corresponding normalized offsets are *u _{M}* +

*u*and

_{δ}*u*, the offset of

_{M}*O*from the focus

_{1}*A*and

*O*from the focus

_{2}*B*in the measurement process are Δ

*l*and Δ

_{1}*l*, and their normalized amounts are

_{2}*u*

_{1}and

*u*

_{2}.

Then, according to Eq. (5), the axial response *I* (0,*u*,*u _{M}*) on points

*O*

_{1}and

*O*

_{2}are:

From Eq. (2) and Eq. (20), the offset Δ*l*
_{1} and Δ*l*
_{2} of *O _{1}* and

*O*are respectively:

_{2}*f*

_{2}

*'*,

*f*and

_{Lens2}'*f '*are the focal-length of RL, Lens2 and combination lens respectively.

From Eq. (6), *f '* is

When the axial offset error of pinhole is *δ*, the error of *l* can be obtained from Eq. (21) and Eq. (22):

*δ*is not possible to be eliminate entirely but to be reduced largely in the adjustment of light path. The variation of

*l*is small because offset Δ

*l*and Δ

_{1}*l*is close in amount and the same in direction, so that they are counteracted in part. In a common DCFS, |Δ

_{2}*l|*<0.1

*δ*is obtained from Eq. (23) when

*f*=

_{Lens2}'*f*

_{2}

*'*and UFL focal-length satisfies

*f*

_{1}

*'*>5

*f*

_{2}

*'*and

*f*

_{1}

*'*>>d. Therefore, the effect of

*δ*on

*l*is smaller as the UFL focal-length

*f*

_{1}

*'*is longer.

## 4. Error analyses

From Eq. (10), it can be seen that the error of *l*, *f*
_{2}
*'* and *d*
_{0} has an effect on measurement of UFL *f*
_{BFD}
*'*, and the error propagation coefficient obtained by differentiating Eq. (10) on *d*
_{0}, *l* and *f*
_{2}
*'* is respectively:

The relation between ∂*f*
_{BFD}
*'*/∂*l* and *f*
_{2}
*'*, ∂*f*
_{BFD}
*'*/ *f*
_{2}
*'* and *f*
_{2}
*'* are shown in Fig. 8
respectively when UFL focal-length is 10 m, 20 m or 30 m, respectively.

It can be seen from Fig. 8 that the error propagation coefficient of *l* is the largest in the error sources, and focusing error *σ _{z}* has a direct effect on the measurement precision of focal-length. And

*σ*obtained from Eq. (14) is written as

_{z}*SNR*is the signal noise ratio of the photoelectric detector.

Supposing the measurement error of the length between focuses *A* and *B* is *σ _{L}*, the measurement error of

*l*obtained through two focusing is:

RL focal-length *f*
_{2}
*'* can be obtained through measuring the BFD of RL by using the measurement principle as shown in Fig. 1, and the measurement error of *f*
_{2}
*'* is:

Supposing the measurement error of distance *d*
_{0} is *σ _{d0}*, then UFL focal-length measurement error

*σ*

_{f}_{BFD}

*can be obtained by considering the three error sources above, which are described in Eqs. (24)~(29):*

_{'}Supposing the emitting light from collimating system is ideal parallel light, the dual-frequency laser interferometer with measurement precision of 1 ppm is used for the length measurement of *l* and the effect of assembling error and vibration is neglected, the error is *σ _{L}*≈0.2 μm,

*σ*≈1.5 μm,

_{RL}*σ*≈0.5 μm and

_{d0}*σ*≈0.2 μm when

_{z}*f*

_{BFD}

*'*≈10 m,

*D*= 200 mm,

*f*

_{2}

*'*= 1.5 m,

*d*

_{0}≈500 mm and

*SNR*= 200:1, and therefore, the relative measurement error obtained using Eq. (30) is less than 0.001%.

When the parallax of collimating beam and other environmental factors are considered, the each measurement error increases, and the variation of measurement error *σ _{f}*

_{2'}is the largest. And therefore, when the dual-frequency laser interferometer is used for calibrating the RL focal-length, the error is

*σ*≈0.5 μm,

_{l}*σ*

_{f}_{2´}≈50 μm and

*σ*≈5 μm, and UFL focal-length relative measurement error is

_{d0}*σ*

_{f}_{1´}= 0.007%.

## 5. Experiments

UFL often has a focal-length of from 5 to 30 meter and a small relative aperture, so the main factor of measurement error is long focal-depth and large focusing error. And therefore, the set-up with a 1/50 relative aperture and a focal-length reduced at a 1:10 ratio is formed based on Fig. 2 to verify the validity of the method proposed.

In the experiment, He-Ne laser with a wavelength of 632.8nm is used. The UFL is a plano-convex lens, the parameters are *Φ* = 20 mm, *f*
_{BFD}
*'* = 988.73mm, *f*
_{1}
*´* = 995.33 mm, *b*
_{1} = 10mm, *n*
_{1} = 1.5164, *r*
_{11} = 513.99mm and *r*
_{12}→∞, respectively. Both RL and colleting lens are *Φ* = 20 mm and *f _{2}´* = 164.40 mm, and RL focal-length error σ

*= 0.008 mm. And the distance between their vertexes is*

_{f2´}*d*

_{0}= 26.40 mm and σ

*= 0.005 mm.*

_{d0}As shown in Fig. 1, when the reflector *R* is moved along the RL optical axis, the focus *A* of RL is precisely determined by the zero of differential confocal response signal in DCFS. The differential confocal focusing curve *I*(*z*) is shown in Fig. 9
, where *I*
_{1}(*z*) and *I*
_{2}(*z*) are the intensity signals received by two detectors, *I _{A}*(

*z*) is the differential confocal focusing curve and the coordinate corresponding to the focus

*A*is 25.9998 mm.

As shown in Fig. 2, UFL is inserted into the measurement light-path and its optical axis is the same as RL. When the reflector *R* is moved along the optical axis, the focus *B* is precisely determined by the zero of differential confocal focusing curve *I _{B}*(

*z*), and the coordinate corresponding to the focus

*B*is 2.0117 mm.

Hence we got the distance between two focuses *l* = 23.9881 mm, and the repetition measurement error of *l* is *σ _{l}* = 0.6 μm. BFD of the measured lens

*f*

_{BFD}

*´*obtained from Eq. (10) is:

The relative measurement error is:

Then, according to the given parameters including *b*
_{1}, *n*
_{1}, *r*
_{11} and *r*
_{12}, the focal-length of the measured lens *f*
_{1}
*´* can be obtained from Eq. (11):

## 6. Conclusion

Based on the property that the zero of DCFS axial response precisely corresponds to the focus of the objective, a new method of laser differential confocal focal-length measurement is proposed to precisely determine the focus position, and achieve the precise measurement of UFL focal-length by combination lens. Preliminary experiments indicate that the relative measurement error of the method is about 0.01% when it is used for the BFD measurement, and it

- 1) has a measurement light-path shorter than the measured focal-length so that the effect of environment on the measurement precision of the focal-length is small.
- 2) can reduce the focal-depth and improve the focusing sensitivity by using the pupil filtering technique and differential confocal technique.
- 3) can improve the environmental anti-interference capability by using the intensity modulation technique.

The method provides a new and highly precise approach for measurement of ultra-long focal-length.

## Acknowledgment

Thanks to National Science Foundation of China (No.60708015, 60927012) and the Beijing Science Foundation of China (No.3082016) for the support.

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