Abstract

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.

© 2012 Optical Society of America

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    [CrossRef]
  4. B. DeBoo and J. Sasian, “Precision focal-length measurement technique with a relative Fresnel-zone hologram,” Appl. Opt. 42, 3903–3909 (2003).
    [CrossRef]
  5. 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.
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  13. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 2002).
  14. 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).
  15. M. Lurie, “Evaluation of expanded laser beams using an optical flat,” Opt. Eng. 15, 68–69 (1976).

2010 (1)

2009 (1)

2005 (1)

2004 (1)

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).

2003 (1)

1999 (1)

1992 (1)

1989 (1)

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]

1987 (1)

1986 (1)

1985 (1)

1984 (1)

1976 (1)

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]

DeBoo, B.

B. DeBoo and J. Sasian, “Precision focal-length measurement technique with a relative Fresnel-zone hologram,” Appl. Opt. 42, 3903–3909 (2003).
[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.

Faridi, M. S.

Filippov, O. K.

Hou, X.

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).

Kothiyal, M. P.

Lurie, M.

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

Meshcheryakov, V. I.

Murata, K.

Murate, K.

Nakano, Y.

Qiu, L.

Sasian, J.

B. DeBoo and J. Sasian, “Precision focal-length measurement technique with a relative Fresnel-zone hologram,” Appl. Opt. 42, 3903–3909 (2003).
[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.

Sha, D.

Shakher, C.

Shen, Y.

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).

Sinel’nikov, M. I.

Singh, P.

Sirohi, R. S.

Sriram, K. V.

Stoer, J.

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).

Sun, R.

Yang, G.

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).

Zhao, W.

Acta Photon. Sin. (1)

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).

Appl. Opt. (7)

J. Opt. Technol. (1)

Opt. Commun. (1)

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]

Opt. Eng. (1)

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

Opt. Express (2)

Other (2)

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).

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Figures (7)

Fig. 1.
Fig. 1.

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 = k p 2 / λ . When the grating G2 is rotated perpendicular to y z plane, a moiré pattern is fetched behind G2. The focal length can be computed using the angle of moiré fringes.

Fig. 2.
Fig. 2.

Moiré fringes formed by two gratings: θ is the inclined angle of two gratings. α 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 θ that is relative to G1.

Fig. 3.
Fig. 3.

Flow chart of algorithmic processes: ( α i , θ i ) are the tested values; ( Δ α , Δ θ ) and C are the previously computed results. By substituting ( α i , θ i ) , ( Δ α , Δ θ ) , and C in Eq. (11), we can get X = ( δ Δ α , δ Δ θ , C ) T . When | C m C m 1 C m | < 10 10 ( m is the number of calculations), iteration ends and calibration results are obtained as Δ α = Δ α + δ Δ α , Δ θ = Δ θ + δ Δ θ ; otherwise, iteration should be continued, where Δ α , Δ θ , C are replaced by Δ α + δ Δ α , Δ θ + δ Δ θ , C , respectively.

Fig. 4.
Fig. 4.

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 .

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

Back focal length f calculated by Eq. (4) before and after calibration, where l = 650 mm and d = 12.3835 mm . Before calibration, since systems errors aren’t removed and ( Δ α , Δ θ ) = ( 0 , 0 ) , the blue curve changes sharply. After calibration, as systematic errors are calibrated by ( Δ α , Δ θ ) = ( 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%.

Fig. 7.
Fig. 7.

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

Tables (4)

Tables Icon

Table 1. Simulation Data to Verify Calibration Methoda

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Table 2. Experimental Data for Calibration of Measuring the Focal Lengtha

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Table 3. Experimental Data of the Concave Mirror ( R = 6574 mm )a

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Table 4. Experimental Results for the Repeatability of the Concave Mirror ( R = 6574 mm )a

Equations (17)

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f = l 1 + d sin θ tan α + cos θ 1 ,
c = | φ ( x ) φ ( x ˜ ) φ ( x ) | / | Δ x x | | x φ ( x ) φ ( x ) x | .
( f l 1 = 1 f d = 1 sin θ tan α + cos θ 1 f θ = d ( tan α cos θ sin θ ) ( sin θ tan α + cos θ 1 ) 2 f α = d sin θ sec 2 α ( sin θ tan α + cos θ 1 ) 2 ) .
f = l 1 + d sin ( θ + Δ θ ) tan ( α + Δ α ) + cos ( θ + Δ θ ) 1 .
sin ( θ + Δ θ ) tan ( α + Δ α ) + cos ( θ + Δ θ ) = C ,
( sin ( θ + Δ θ ) tan ( α + Δ α ) + cos ( θ + Δ θ ) + sin ( θ + Δ θ ) sec 2 ( α + Δ α ) δ Δ α + [ cos ( θ + Δ θ ) tan ( α + Δ α ) sin ( θ + Δ θ ) ] δ Δ θ ) = C ,
k δ Δ α + l δ Δ θ C = b ,
k = sin ( θ + Δ θ ) sec 2 ( α + Δ α ) ,
l = cos ( θ + Δ θ ) tan ( α + Δ α ) sin ( θ + Δ θ ) ,
b = sin ( θ + Δ θ ) tan ( α + Δ α ) cos ( θ + Δ θ ) .
A X = B ,
A = ( k 1 k 2 l 1 l 2 1 1 k n 1 k n l n 1 l n 1 1 ) ,
X = ( δ Δ α , δ Δ θ , C ) T ,
B = ( b 1 , b 2 b n 1 , b n ) T .
Δ α = δ Δ α 1 + δ Δ α 2 + + δ Δ α m ,
Δ θ = δ Δ θ 1 + δ Δ θ 2 + + δ Δ θ m .
1 f 1 F 2 D = 1 F ,

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