In this paper, a new calibration method for accurate long focal-length
measurements, based on Talbot
interferometry, is presented. Error analysis is derived in detail by the
numerical method, and an effective way to improve the accuracy is proposed. By
this method, the systematic errors that are the main factors effecting accuracy
are calibrated and reduced. Both simulation and experiments have been carried
out to prove the effectiveness and advantages of the proposed method as compared
to conventional approaches. The experimental results reveal that the relative
error is lower than 0.02%, and the repeatability is better than 0.05%. This
method is especially useful for measuring long focal-length lenses.

B. DeBoo and J. Sasian, “Novel method for precise focal length measurement,” in International Optical Design Conference, 2002 OSA Technical Digest Series (Optical Society of America, 2002), paper IMCS5.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 2002).

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

M. Lurie, “Evaluation of expanded laser beams using an optical flat,” Opt. Eng. 15, 68–69 (1976).

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

M. Lurie, “Evaluation of expanded laser beams using an optical flat,” Opt. Eng. 15, 68–69 (1976).

Bai, J.

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

Bulirsch, R.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 2002).

Chang, C.-W.

C.-W. Chang and D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

B. DeBoo and J. Sasian, “Novel method for precise focal length measurement,” in International Optical Design Conference, 2002 OSA Technical Digest Series (Optical Society of America, 2002), paper IMCS5.

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

B. DeBoo and J. Sasian, “Novel method for precise focal length measurement,” in International Optical Design Conference, 2002 OSA Technical Digest Series (Optical Society of America, 2002), paper IMCS5.

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 2002).

Su, D.-C.

C.-W. Chang and D.-C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

Sun, C.

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

C. Sun, Y. Shen, J. Bai, X. Hou, and G. Yang, “The precison limit analysis of long focal length testing based on Talbot effect of Ronchi grating,” Acta Photon. Sin. 33, 1214–1217 (2004).

B. DeBoo and J. Sasian, “Novel method for precise focal length measurement,” in International Optical Design Conference, 2002 OSA Technical Digest Series (Optical Society of America, 2002), paper IMCS5.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 2002).

Optical configuration for measuring the focal length of a lens using two
Ronchi gratings based on Talbot interferometry. The second grating G2 is
located at the Talbot distance of $d={kp}^{2}/\lambda $. When the grating G2 is rotated
perpendicular to $y\u2013z$ plane, a moiré pattern is fetched
behind G2. The focal length can be computed using the angle of moiré
fringes.

Moiré fringes formed by two gratings: $\theta $ is the inclined angle of two gratings.
$\alpha $ is the angle of the moiré fringes
relative to horizontal direction. The two angles are positive clockwise. G1
is the first grating. G1′ corresponds to the Talbot image of G1 at a
certain Talbot distance. The other identical grating G2 is located closely
to G1′ with a small inclined angle $\theta $ that is relative to G1.

Flow chart of algorithmic processes: $({\alpha}_{i},{\theta}_{i})$ are the tested values;
$(\mathrm{\Delta}\alpha ,\mathrm{\Delta}\theta )$ and $C$ are the previously computed results. By
substituting $({\alpha}_{i},{\theta}_{i})$, $(\mathrm{\Delta}\alpha ,\mathrm{\Delta}\theta )$, and $C$ in Eq. (11), we can get $X={(\delta \mathrm{\Delta}\alpha ,\delta \mathrm{\Delta}\theta ,-{C}^{\prime})}^{T}$. When $|\frac{{C}_{m}-{C}_{m-1}}{{C}_{m}}|<{10}^{-10}$ ($m$ is the number of calculations), iteration
ends and calibration results are obtained as $\mathrm{\Delta}\alpha =\mathrm{\Delta}\alpha +\delta \mathrm{\Delta}\alpha $, $\mathrm{\Delta}\theta =\mathrm{\Delta}\theta +\delta \mathrm{\Delta}\theta $; otherwise, iteration should be continued,
where $\mathrm{\Delta}\alpha $, $\mathrm{\Delta}\theta $, $C$ are replaced by $\mathrm{\Delta}\alpha +\delta \mathrm{\Delta}\alpha $, $\mathrm{\Delta}\theta +\delta \mathrm{\Delta}\theta $, ${C}^{\prime}$, respectively.

Schematic of the experimental arrangement for measuring the focal length of a
test lens: beam from a He–Ne laser was collimated and expanded. The
collimated beam reflected by the polarization splitter prism passed through
the measured lens and a mirror and then came back to two gratings. At the
diffuse plate behind G2, the moiré pattern was captured by a CCD
camera. The output signal was analyzed with a PC computer. Thus the
parameter $f$ in Eq. (1) could be computed through the angle of the moiré
fringes. Since the light passed through the measured lens twice,
$f$ is the back focal length of two measured
lenses grouped. If a concave mirror with radius $R$ replaces the measured lens and mirror, the
calculated parameter $f$ is half of the radius
$R$.

Photograph of the moiré fringes produced by two Ronchi gratings. The
parameter $\alpha $ in Eq. (1) is the inclined angle of the moiré fringes
relative to the horizontal direction.

Back focal length $f$ calculated by Eq. (4) before and after
calibration, where $l=650\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ and $d=12.3835\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. Before calibration, since systems
errors aren’t removed and $(\mathrm{\Delta}\alpha ,\mathrm{\Delta}\theta )=(0,0)$, the blue curve changes sharply. After
calibration, as systematic errors are calibrated by
$(\mathrm{\Delta}\alpha ,\mathrm{\Delta}\theta )=(2.119,-0.022)$, the red curve of focal length has been
revised and becomes smooth. In practice, we calibrated the systematic
errors several times until the results of focal length varied less than
0.05%.

Experimental data for repeatability of the concave mirror
($R=-6574\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$). The red line corresponds to the
standard value of the mirror radius $R=-6574\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$; the blue curve corresponds to 75 sets
of tested data of radius $R=2f$ computed by Eq. (4), which drift up and down
on the standard value but have little deviation.

Calibrated data computed are close to the assumed ones.
$C$ calibrated from Eq. (11) is also constant. The
calibration method is reasonable and effective to reduce
disturbances.

Table 2.

Experimental Data for Calibration of Measuring the Focal Lengtha

Serial Number

Angle of Moiré Fringes $\alpha (\xb0)$

Inclined Angle between Two Gratings $\theta (\xb0)$

Serial Number

Angle of Moiré Fringes $\alpha (\xb0)$

Inclined Angle between Two Gratings $\theta (\xb0)$

1

16.2975

0.4050

14

24.2451

0.2775

2

16.7421

0.3950

15

24.2508

0.2775

3

17.1214

0.3855

16

25.3391

0.2655

4

17.6679

0.3750

17

26.2597

0.2565

5

18.2286

0.3655

18

27.0789

0.2485

6

18.6087

0.3565

19

28.0469

0.2390

7

19.3627

0.3435

20

29.1989

0.2305

8

20.1763

0.3340

21

30.5557

0.2190

9

20.5302

0.3250

22

31.7963

0.2100

10

21.2665

0.3150

23

32.9906

0.2000

11

22.1504

0.3045

24

34.8245

0.1900

12

22.7473

0.2950

25

36.3641

0.1805

13

23.7040

0.2860

26

38.6562

0.1690

Systematic error $\mathrm{\Delta}\theta (\xb0)$

$-0.022$

Systematic error $\mathrm{\Delta}\alpha (\xb0)$

2.119

Twenty-six sets of data were caught in about 15 min. Systematic
errors $\mathrm{\Delta}\theta $ and $\mathrm{\Delta}\alpha $ were calculated by the calibration
method discussed in the text.

Table 3.

Experimental Data of the Concave Mirror ($R=-6574\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$)a

Serial Number

Angle of Moiré Fringes
$\alpha +\mathrm{\Delta}\alpha (\xb0)$

Inclined Angle between
Two Gratings
$\theta +\mathrm{\Delta}\theta (\xb0)$

Radius of Tested Concave
Mirror
$R=2f$ (mm)

Mean
$R(\mathrm{mm})$

Error

1

35.6176

0.3489

6573.4472

6574.0047

0.017%

2

35.6118

6574.7154

3

35.6133

6574.3873

4

35.6175

6573.4690

The systematic errors had been calibrated. Substituting these data
into Eq. (4), the
focal length was calculated, where $d=12.8235\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, ${l}_{1}=668.44\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. Relative accuracy is about 0.017%,
better than 0.02%. Therefore, the calibration method is effective,
and the measurement of the long focal length has high precision.

Table 4.

Experimental Results for the Repeatability of the Concave Mirror
($R=-6574\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$)a

Test Times

Radius Mean

Standard Radius

Relative Error

6.25 hours

6573.8921 mm

6574 mm

0.049%

Seventy-five sets of data were fetched in about 6.25 h. The
mirror radius $R=2f$ was calculated by Eq. (4). Relative error is
about 0.049%, lower than 0.05%. This method is insensitive to the
testing environment.

Calibrated data computed are close to the assumed ones.
$C$ calibrated from Eq. (11) is also constant. The
calibration method is reasonable and effective to reduce
disturbances.

Table 2.

Experimental Data for Calibration of Measuring the Focal Lengtha

Serial Number

Angle of Moiré Fringes $\alpha (\xb0)$

Inclined Angle between Two Gratings $\theta (\xb0)$

Serial Number

Angle of Moiré Fringes $\alpha (\xb0)$

Inclined Angle between Two Gratings $\theta (\xb0)$

1

16.2975

0.4050

14

24.2451

0.2775

2

16.7421

0.3950

15

24.2508

0.2775

3

17.1214

0.3855

16

25.3391

0.2655

4

17.6679

0.3750

17

26.2597

0.2565

5

18.2286

0.3655

18

27.0789

0.2485

6

18.6087

0.3565

19

28.0469

0.2390

7

19.3627

0.3435

20

29.1989

0.2305

8

20.1763

0.3340

21

30.5557

0.2190

9

20.5302

0.3250

22

31.7963

0.2100

10

21.2665

0.3150

23

32.9906

0.2000

11

22.1504

0.3045

24

34.8245

0.1900

12

22.7473

0.2950

25

36.3641

0.1805

13

23.7040

0.2860

26

38.6562

0.1690

Systematic error $\mathrm{\Delta}\theta (\xb0)$

$-0.022$

Systematic error $\mathrm{\Delta}\alpha (\xb0)$

2.119

Twenty-six sets of data were caught in about 15 min. Systematic
errors $\mathrm{\Delta}\theta $ and $\mathrm{\Delta}\alpha $ were calculated by the calibration
method discussed in the text.

Table 3.

Experimental Data of the Concave Mirror ($R=-6574\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$)a

Serial Number

Angle of Moiré Fringes
$\alpha +\mathrm{\Delta}\alpha (\xb0)$

Inclined Angle between
Two Gratings
$\theta +\mathrm{\Delta}\theta (\xb0)$

Radius of Tested Concave
Mirror
$R=2f$ (mm)

Mean
$R(\mathrm{mm})$

Error

1

35.6176

0.3489

6573.4472

6574.0047

0.017%

2

35.6118

6574.7154

3

35.6133

6574.3873

4

35.6175

6573.4690

The systematic errors had been calibrated. Substituting these data
into Eq. (4), the
focal length was calculated, where $d=12.8235\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, ${l}_{1}=668.44\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. Relative accuracy is about 0.017%,
better than 0.02%. Therefore, the calibration method is effective,
and the measurement of the long focal length has high precision.

Table 4.

Experimental Results for the Repeatability of the Concave Mirror
($R=-6574\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$)a

Test Times

Radius Mean

Standard Radius

Relative Error

6.25 hours

6573.8921 mm

6574 mm

0.049%

Seventy-five sets of data were fetched in about 6.25 h. The
mirror radius $R=2f$ was calculated by Eq. (4). Relative error is
about 0.049%, lower than 0.05%. This method is insensitive to the
testing environment.