Abstract

A method based on a specific quasi-common-optical-path (QCOP) configuration for two-dimensional displacement measurement is presented. The measurement system consists of a heterodyne light source, two-dimensional holographic grating, specially designed set of half wave plates, and lock-in amplifiers. Two measurement configurations, for single and differential detection, are designed. The sensitivity, resolution and nonlinear phase error of the differential detection type are better than those of the single detection type. The experimental results demonstrate that the QCOP interferometer has the ability to measure two-dimensional displacement while maintaining high system stability.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Montoya, C. H. Chang, R. K. Heilmann, and M. L. Schattenburg, “Doppler writing and linewidth control for scanning beam interference lithography,” J. Vac. Sci. Technol. B 23(6), 232640 (2005).
    [CrossRef]
  2. G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
    [CrossRef]
  3. Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
    [CrossRef]
  4. A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
    [CrossRef] [PubMed]
  5. D. Lin, X. Jiang, F. Xie, W. Zhang, L. Zhang, and I. Bennion, “High stability multiplexed fiber interferometer and its application on absolute displacement measurement and on-line surface metrology,” Opt. Express 12(23), 5729–5734 (2004).
    [CrossRef] [PubMed]
  6. F. Restagno, J. Crassous, E. Charlaix, and M. Monchanin, “A new capacitive sensor for displacement measurement in a surface-force apparatus,” Meas. Sci. Technol. 12(1), 16–22 (2001).
    [CrossRef]
  7. L. Precision, “Lion Precision white paper 2004 User manual and literature,” (2004).
  8. D. Crespo, J. Alonso, and E. Bernabeu, “Reflection optical encoders as three-grating moiré systems,” Appl. Opt. 39(22), 3805–3813 (2000).
    [CrossRef]
  9. J. H. Song, K. C. Kim, and S. H. Kim, “Reducing tilt errors in moire´ linear encoders using phase-modulated grating,” Rev. Sci. Instrum. 71(6), 2296–2300 (2000).
    [CrossRef]
  10. D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
    [CrossRef]
  11. H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
    [CrossRef]
  12. C. M. Wu, “Heterodyne interferometric system with subnanometer accuracy for measurement of straightness,” Appl. Opt. 43(19), 3812–3816 (2004).
    [CrossRef] [PubMed]
  13. J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Act. A 137(1), 185–191 (2007).
    [CrossRef]
  14. H. Packard, “5526A Laser Measurement System User's Guide,” (1980).

2010

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

2007

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Act. A 137(1), 185–191 (2007).
[CrossRef]

2005

J. C. Montoya, C. H. Chang, R. K. Heilmann, and M. L. Schattenburg, “Doppler writing and linewidth control for scanning beam interference lithography,” J. Vac. Sci. Technol. B 23(6), 232640 (2005).
[CrossRef]

2004

2001

F. Restagno, J. Crassous, E. Charlaix, and M. Monchanin, “A new capacitive sensor for displacement measurement in a surface-force apparatus,” Meas. Sci. Technol. 12(1), 16–22 (2001).
[CrossRef]

Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
[CrossRef]

2000

D. Crespo, J. Alonso, and E. Bernabeu, “Reflection optical encoders as three-grating moiré systems,” Appl. Opt. 39(22), 3805–3813 (2000).
[CrossRef]

J. H. Song, K. C. Kim, and S. H. Kim, “Reducing tilt errors in moire´ linear encoders using phase-modulated grating,” Rev. Sci. Instrum. 71(6), 2296–2300 (2000).
[CrossRef]

1996

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[CrossRef]

Alayli, Y.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Alonso, J.

Bennion, I.

Bernabeu, E.

Blaize, S.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Bruyant, A.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Bu, J. U.

Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
[CrossRef]

Chang, C. H.

J. C. Montoya, C. H. Chang, R. K. Heilmann, and M. L. Schattenburg, “Doppler writing and linewidth control for scanning beam interference lithography,” J. Vac. Sci. Technol. B 23(6), 232640 (2005).
[CrossRef]

Charlaix, E.

F. Restagno, J. Crassous, E. Charlaix, and M. Monchanin, “A new capacitive sensor for displacement measurement in a surface-force apparatus,” Meas. Sci. Technol. 12(1), 16–22 (2001).
[CrossRef]

Chassagne, L.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Chen, C. D.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[CrossRef]

Chen, H. Y.

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Act. A 137(1), 185–191 (2007).
[CrossRef]

Chen, J. C.

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

Chiu, M. H.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[CrossRef]

Crassous, J.

F. Restagno, J. Crassous, E. Charlaix, and M. Monchanin, “A new capacitive sensor for displacement measurement in a surface-force apparatus,” Meas. Sci. Technol. 12(1), 16–22 (2001).
[CrossRef]

Crespo, D.

Dai, G.

G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
[CrossRef]

Danzebrink, H. U.

G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
[CrossRef]

Deturche, R.

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

Hasche, K.

G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
[CrossRef]

Heilmann, R. K.

J. C. Montoya, C. H. Chang, R. K. Heilmann, and M. L. Schattenburg, “Doppler writing and linewidth control for scanning beam interference lithography,” J. Vac. Sci. Technol. B 23(6), 232640 (2005).
[CrossRef]

Hsieh, H. L.

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

Hsu, C. C.

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Act. A 137(1), 185–191 (2007).
[CrossRef]

Jiang, X.

Kim, K. C.

J. H. Song, K. C. Kim, and S. H. Kim, “Reducing tilt errors in moire´ linear encoders using phase-modulated grating,” Rev. Sci. Instrum. 71(6), 2296–2300 (2000).
[CrossRef]

Kim, S. H.

J. H. Song, K. C. Kim, and S. H. Kim, “Reducing tilt errors in moire´ linear encoders using phase-modulated grating,” Rev. Sci. Instrum. 71(6), 2296–2300 (2000).
[CrossRef]

Lee, J. W.

Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
[CrossRef]

Lee, J. Y.

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Act. A 137(1), 185–191 (2007).
[CrossRef]

Lee, S. H.

Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
[CrossRef]

Lerondel, G.

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Lin, D.

Monchanin, M.

F. Restagno, J. Crassous, E. Charlaix, and M. Monchanin, “A new capacitive sensor for displacement measurement in a surface-force apparatus,” Meas. Sci. Technol. 12(1), 16–22 (2001).
[CrossRef]

Montoya, J. C.

J. C. Montoya, C. H. Chang, R. K. Heilmann, and M. L. Schattenburg, “Doppler writing and linewidth control for scanning beam interference lithography,” J. Vac. Sci. Technol. B 23(6), 232640 (2005).
[CrossRef]

Nam, H. J.

Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
[CrossRef]

Pohlenz, F.

G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
[CrossRef]

Restagno, F.

F. Restagno, J. Crassous, E. Charlaix, and M. Monchanin, “A new capacitive sensor for displacement measurement in a surface-force apparatus,” Meas. Sci. Technol. 12(1), 16–22 (2001).
[CrossRef]

Royer, P.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Ruaux, P.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Schattenburg, M. L.

J. C. Montoya, C. H. Chang, R. K. Heilmann, and M. L. Schattenburg, “Doppler writing and linewidth control for scanning beam interference lithography,” J. Vac. Sci. Technol. B 23(6), 232640 (2005).
[CrossRef]

Sinno, A.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Song, J. H.

J. H. Song, K. C. Kim, and S. H. Kim, “Reducing tilt errors in moire´ linear encoders using phase-modulated grating,” Rev. Sci. Instrum. 71(6), 2296–2300 (2000).
[CrossRef]

Su, D. C.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[CrossRef]

Topçu, S.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

Wilkening, G.

G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
[CrossRef]

Wu, C. C.

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Act. A 137(1), 185–191 (2007).
[CrossRef]

Wu, C. M.

Wu, W. T.

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

Xie, F.

Xu, M.

G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
[CrossRef]

Yee, Y.

Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
[CrossRef]

Zhang, L.

Zhang, W.

Appl. Opt.

J. Opt.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[CrossRef]

J. Vac. Sci. Technol. B

J. C. Montoya, C. H. Chang, R. K. Heilmann, and M. L. Schattenburg, “Doppler writing and linewidth control for scanning beam interference lithography,” J. Vac. Sci. Technol. B 23(6), 232640 (2005).
[CrossRef]

Meas. Sci. Technol.

F. Restagno, J. Crassous, E. Charlaix, and M. Monchanin, “A new capacitive sensor for displacement measurement in a surface-force apparatus,” Meas. Sci. Technol. 12(1), 16–22 (2001).
[CrossRef]

H. L. Hsieh, J. Y. Lee, W. T. Wu, J. C. Chen, R. Deturche, and G. Lerondel, “Quasi-common-optical-path heterodyne grating interferometer for displacement measurement,” Meas. Sci. Technol. 21(11), 115304 (2010).
[CrossRef]

Opt. Express

Rev. Sci. Instrum.

A. Sinno, P. Ruaux, L. Chassagne, S. Topçu, Y. Alayli, G. Lerondel, S. Blaize, A. Bruyant, and P. Royer, “Enlarged atomic force microscopy scanning scope: novel sample-holder device with millimeter range,” Rev. Sci. Instrum. 78(9), 095107 (2007).
[CrossRef] [PubMed]

G. Dai, F. Pohlenz, H. U. Danzebrink, M. Xu, K. Hasche, and G. Wilkening, “Metrological large range scanning probe microscope,” Rev. Sci. Instrum. 75(4), 962–969 (2004).
[CrossRef]

J. H. Song, K. C. Kim, and S. H. Kim, “Reducing tilt errors in moire´ linear encoders using phase-modulated grating,” Rev. Sci. Instrum. 71(6), 2296–2300 (2000).
[CrossRef]

Sens. Act. A

Y. Yee, H. J. Nam, S. H. Lee, J. U. Bu, and J. W. Lee, “PZT actuated micromirror for fine-tracking mechanism of high-density optical data storage,” Sens. Act. A 89(1-2), 166–173 (2001).
[CrossRef]

J. Y. Lee, H. Y. Chen, C. C. Hsu, and C. C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Act. A 137(1), 185–191 (2007).
[CrossRef]

Other

H. Packard, “5526A Laser Measurement System User's Guide,” (1980).

L. Precision, “Lion Precision white paper 2004 User manual and literature,” (2004).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Schematic view of the heterodyne four beams light source comprising an He-Ne Laser, an Electro-Optic Modulator (EOM), a Beam Expander, and two Half Wave Plates (HWP1, HWP2). The different temporal frequencies after passing through the HWPs are given for illustration.

Fig. 2
Fig. 2

Schematic representation of the single type QCOP heterodyne grating straightness interferometer comprising in addition to the four beams heterodyne light source (see Fig. 1), a mirror, one focusing Lens (L1), a 2D grating, three polarizers (P1, P2, P3), and three detectors (D1, D2, D3).

Fig. 3
Fig. 3

Schemes of the single and differential type QCOP methods.

Fig. 4
Fig. 4

Experimental setup of the single type 2D QCOP heterodyne grating interferometer.

Fig. 5
Fig. 5

(a). Experimental results for 1D W motion; (b). The deviation between the QCOP method and HP 5529A.

Fig. 6
Fig. 6

Experimental results for quadrangular motion.

Fig. 7
Fig. 7

Top view of experimental results for quadrangular motion.

Fig. 8
Fig. 8

(a). The deviation (straightness error) of the Path 1 in the Y direction when the XY stepper moves along the X direction; (b). The deviation (straightness error) of the Path 2 in the X direction when the XY stepper moves along the Y direction.

Fig. 9
Fig. 9

(a). Top view of experimental results for octagonal motion; (b). The deviation (straightness error) of the Path 1 in the Y direction when the XY stepper moves along the X direction.

Fig. 10
Fig. 10

The simulation results of the non-linear periodic error of the differential and single types.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

E H = ( e i ω t / 2 e i ω t / 2 ) .
E B = E C = J ( 180 ) E H = ( e i ω t / 2 e i ω t / 2 ) , E A = E D = J ( 0 ) E H = ( e i ω t / 2 e i ω t / 2 ) ,
ϕ q m = 2 m π l q / p ,
I 1 = 1 + cos [ ω t k ( Δ d ) ( ϕ x 1 ϕ x 0 ) ] .         = 1 + cos ( ω t ϕ x 1 )
I 2 = 1 + cos ( ω t ϕ y 1 ) .
I 3 = 1 + cos ( ω t ) .
Φ q = 2 π l q / p ,
l q = p × Φ q / 2 π .
I 4 = 1 + cos ( ω t ϕ x 1 ) ,   and I 5 = 1 + cos ( ω t + ϕ x 1 ) .
I 6 = 1 + cos ( ω t ϕ y 1 ) ,   and   I 7 = 1 + cos ( ω t + ϕ y 1 ) .
Φ q = 4 π l q / p .
E = P( α , θ p ) [ HWP(180+ δ , 45 o + ε ) ( e i ω t/2 e i ω t/2 ) + ( e i ω t/2 e i ω t ) e i ϕ q 1 ] ,
I = | E | 2 A C cos ( ω t + Φ ' ) ,
Δ Φ = Φ ' ( α , θ p , δ , ε ) ϕ q 1 ,
Δ Φ = Φ ' ( α , θ 0 , δ , ε ) Φ ' ( α , θ 90 , δ , ε ) 2 ϕ q 1.

Metrics