Abstract

A method for measuring long focal lengths using a double-grating interferometer is proposed. The intensity distribution of a double-grating interferometer illuminated by a spherical beam is derived with diffraction theory. A tiny rotation angle is set between the two gratings, and a high-contrast interference pattern is produced by the adjacent diffracted orders. The angular change in the fringes after insertion of a test lens is a measure of the focal length. The uncertainty due to aberration of the collimated beam was analyzed by measuring a series of lenses. The relative deviations are less than 0.1%.

© 2013 Optical Society of America

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References

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  1. M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
    [CrossRef]
  2. H. E. Bennett and J. J. Shaffer, “Test facility for long-focal-length mirrors,” Proc. SPIE 1848, 117–124 (2005).
    [CrossRef]
  3. H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
    [CrossRef]
  4. B. De Boo and J. Sasián, “Precision focal-length measurement technique with a relative Fresnel-zone hologram,” Appl. Opt. 42, 3903–3909 (2003).
    [CrossRef]
  5. B. De Boo and J. Sasián, “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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    [CrossRef]
  10. P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, “Measurement of focal length with phase-shifting Talbot interferometry,” Appl. Opt. 44, 1572–1576 (2005).
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    [CrossRef]
  12. Y. Song, Y. Y. Chen, A. He, and Z. Zhao, “Spatial phase-shifting characteristic of double grating interferometer,” Opt. Express 17, 20415–20429 (2009).
    [CrossRef]
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    [CrossRef]
  14. Y. Xu, O. Sasaki, and T. Suzuki, “Double-grating interferometer for measurement of cylinder diameters,” Appl. Opt. 43, 537–541 (2004).
    [CrossRef]
  15. J. Dhanotia, S. Prakash, S. Rana, and O. Sasaki, “Slope measurement of bent plates using double-grating shearing interferometry,” Appl. Opt. 50, 2958–2963 (2011).
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  16. J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express 19, 584–596 (2011).
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2011 (2)

2010 (1)

2009 (1)

2005 (2)

2004 (1)

2003 (2)

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

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

1998 (1)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Analysis of moiré fringes for measuring the focal length of lenses,” Opt. Lasers Eng. 30, 279–286 (1998).
[CrossRef]

1997 (1)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

1991 (1)

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

1985 (1)

1984 (1)

1974 (1)

P. Hariharan, W. H. Steel, and J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Allen, R. G.

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Álvarez-Herrero, A.

Belenguer, T.

Bennett, H. E.

H. E. Bennett and J. J. Shaffer, “Test facility for long-focal-length mirrors,” Proc. SPIE 1848, 117–124 (2005).
[CrossRef]

Burge, J. H.

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Chen, Y. Y.

de Angelis, M.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Analysis of moiré fringes for measuring the focal length of lenses,” Opt. Lasers Eng. 30, 279–286 (1998).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

De Boo, B.

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

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

De Nicola, S.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Analysis of moiré fringes for measuring the focal length of lenses,” Opt. Lasers Eng. 30, 279–286 (1998).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Dettmann, L. R.

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Dhanotia, J.

Faridi, M. S.

Ferraro, P.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Analysis of moiré fringes for measuring the focal length of lenses,” Opt. Lasers Eng. 30, 279–286 (1998).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Finizio, A.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Analysis of moiré fringes for measuring the focal length of lenses,” Opt. Lasers Eng. 30, 279–286 (1998).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Hariharan, P.

P. Hariharan, W. H. Steel, and J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

He, A.

Ketelsen, D. A.

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Krishnaswamy, S.

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

Martin, H. M.

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Miller, S. M.

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Murata, K.

Nakano, Y.

Pierattini, G.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Analysis of moiré fringes for measuring the focal length of lenses,” Opt. Lasers Eng. 30, 279–286 (1998).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Prakash, S.

Qiu, L.

Quiroga, J. A.

Rana, S.

Rosakis, A. J.

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

Sasaki, O.

Sasián, J.

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

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

Sasián, J. M.

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Sha, D.

Shaffer, J. J.

H. E. Bennett and J. J. Shaffer, “Test facility for long-focal-length mirrors,” Proc. SPIE 1848, 117–124 (2005).
[CrossRef]

Shakher, C.

Singh, P.

Sirohi, R. S.

Song, Y.

Steel, W. H.

P. Hariharan, W. H. Steel, and J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Sun, R.

Suzuki, T.

Tippur, H. V.

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

Vargas, J.

Wyant, J. C.

P. Hariharan, W. H. Steel, and J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Xu, Y.

Zhao, W.

Zhao, Z.

Appl. Opt. (6)

Int. J. Fract. (1)

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

Opt. Commun. (2)

P. Hariharan, W. H. Steel, and J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal length of lenses using a digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Opt. Express (3)

Opt. Lasers Eng. (1)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Analysis of moiré fringes for measuring the focal length of lenses,” Opt. Lasers Eng. 30, 279–286 (1998).
[CrossRef]

Proc. SPIE (2)

H. E. Bennett and J. J. Shaffer, “Test facility for long-focal-length mirrors,” Proc. SPIE 1848, 117–124 (2005).
[CrossRef]

H. M. Martin, R. G. Allen, J. H. Burge, L. R. Dettmann, D. A. Ketelsen, S. M. Miller, and J. M. Sasián, “Fabrication of mirrors for the Magellan Telescopes and the Large Binocular Telescope,” Proc. SPIE 4837, 609–618 (2003).
[CrossRef]

Other (1)

B. De Boo and J. Sasián, “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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Figures (11)

Fig. 1.
Fig. 1.

Optical configuration of double-grating interferometer.

Fig. 2.
Fig. 2.

Pattern of the two gratings.

Fig. 3.
Fig. 3.

Distribution of diffraction orders.

Fig. 4.
Fig. 4.

Interference patterns.

Fig. 5.
Fig. 5.

Principle of double-grating interferometer with a spherical beam.

Fig. 6.
Fig. 6.

Interference patterns for different values of d1.

Fig. 7.
Fig. 7.

Measurement uncertainty affected by collimation beam aberration, where d1=36mm, θ=1°, f=10,000mm.

Fig. 8.
Fig. 8.

Experimental setup.

Fig. 9.
Fig. 9.

Interference patterns for different values of θ.

Fig. 10.
Fig. 10.

Interference patterns (a) without and (b) with the test lens.

Fig. 11.
Fig. 11.

Measuring result distribution for every 30 min.

Tables (1)

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Table 1. Results of Focal-Length Measurementsa

Equations (27)

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g1(x,y)=n=+bnexp(i2πpnx),
g2(x,y)=m=cmexp[i2πpm(xcosθysinθ)].
U(x,y,d2)=Cexp[i2πλd1]g1(x,y)g2(x,y)exp[i2πλd2]=Cexp[i2πλ(d1+d2)]×m=n=bncmexp[i2πp(nx+m(xcosθysinθ))].
I1=2C2b02b12{1+cos[2πp(x(1cosθ)ysinθ)]},
I2=C2+2C2b02b12{1+2cos[2πp(x(1cosθ)ysinθ)]}+2C2b02b12cos[4πp(x(1cosθ)ysinθ)],
I3=2C2b02b12{1+cos[2πp(x(1cosθ)ysinθ)]}.
U(x,y,0)=a0fexp(i2πλx2+y22f).
U(x,y,0+)=U(x,y,0)g1(x,y)=n=bna0fexp(in2πλfp2)exp[i2πλ(x+nλfp)2+y22f].
U(x,y,d1+)=U(x,y,d1)g2(x,y)=n=m=bncma0fd1exp(in2πλfp2)exp{i2πλ[(x+nλfp)2+y22(fd1)+mλ(xcosθysinθ)p]}.
U(x,y,d2)=n=m=bncma0fd1exp{i2πλ[n2λ2f+m2λ2(fd1)+2mnfλ2cosθ2p2]}×exp{i2πλ[(x+nλf+mλcosθ(fd1)p)2+(ymλsinθ(fd1)p)22(fd2)]}.
I1=2a02b02b12(fd2)2{1+cos[2π((fd1)(xcosθysinθ)fxp(fd2)+λfd2λ(fd1)(d2d1)2p2(fd2))]},
I2=4a02b02b12(fd2)2cos[πλfd2+λ(fd1)(d2d1)2λcosθf(d2d1)p2(fd2)]{1+cos[2π(fd1)(xcosθysinθ)fxp(fd2)]},
I3=2a02b02b12(fd2)2{1+cos[2π((fd1)(xcosθysinθ)fxp(fd2)λfd2λ(fd1)(d2d1)2p2(fd2))]}.
λfd2+λ(fd1)(d2d1)2λcosθf(d2d1)p2(fd2)=k,
d1=kp2/λ.
x(1cosθ)ysinθ=2kp,
φ1=θ/2.
(fd1)(xcosθysinθ)fxp(fd2)λfd2λ(fd1)(d2d1)2p2(fd2)=k,
[(fd1)cosθf]x(fd1)(sinθ)y=kp(fd2)+λfd2λ(fd1)(d2d1)2p.
tanφ2=(fd1)cosθf(fd1)sinθ.
f=s+d1+d1sinθtan(φ+θ/2)+cosθ1
fe=ϕ28Δh.
fcffef+fe.
tanφ1=(fed1)cosθfe(fed1)sinθ,
tanφ2=(fcd1)cosθfc(fcd1)sinθ.
ftest=s+d1+d1sinθtan(φ1φ2+θ/2)+cosθ1.
Δf/f=(fftest)/f.

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