Abstract

The removal of misalignment aberrations is a key problem in the null test of cylindrical surfaces. Although the quadratic polynomial with two variables and the orthogonal Chebyshev polynomials have been used to separate the misalignment aberrations from the extracted phase data, there is no physical meaning corresponding to the polynomials coefficients. Additionally, the Runge phenomenon may occur when the high-order polynomials are employed. In this paper, all the possible aberrations caused by the adjustment errors were analyzed. Based on the first-order approximate principle, the mathematical models, which describe the relationship between the misalignment aberrations and the possible adjustment errors, were deduced. With these mathematical expressions, all the possible adjustment errors can be estimated by using the least-squares fitting algorithm, and then the genuine surface deviations can be obtained by subtracting the misalignment aberrations from the extracted phase data. Computer simulations and experiments have been conducted to demonstrate the validity and feasibility, which show more than 96% misalignment aberrations can be removed. Compared with the existing methods, the proposed model provides a feasible way to estimate adjustment errors with better accuracy.

© 2013 Optical Society of America

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References

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  1. J. M. Geary, “Testing cylindrical lenses,” Opt. Eng. 26, 261219 (1987).
    [CrossRef]
  2. J. M. Geary, “Overview of cylindrical optics testing using a fiber optic reference,” Proc. SPIE 2536, 68–74 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. 42, 4468–4479 (2003).
    [CrossRef]
  6. P. Tam, K. Gross, and J. Bogan, “Interferometric testing of cylinder optics using computer generated hologram (CGH),” Proc. SPIE 3134, 162–166 (1997).
    [CrossRef]
  7. J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10–S16 (2002).
    [CrossRef]
  8. J. Lamprecht, N. Lindlein, and J. Schwider, “Null test measurement of high-numerical-aperture cylindrical microlenses in transmitted light,” Proc. SPIE 5180, 253–260 (2004).
    [CrossRef]
  9. K. Mantel, N. Lindlein, and J. Schwider, “Simultaneous characterization of the quality and orientation of cylindrical lens surfaces,” Appl. Opt. 44, 2970–2977 (2005).
    [CrossRef]
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    [CrossRef]
  11. P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
    [CrossRef]
  12. J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. Eng. Des. 33, 105–125 (1998).
    [CrossRef]
  13. G. Kang, J. Xie, and Y. Liu, “New design techniques and alignment methods for CGH-null testing of aspheric surface,” Proc. SPIE 6624, 66240K (2008).
    [CrossRef]
  14. http://diffraction.com/cylinder.php .
  15. K. Khalsa, Metropro Reference Guide (Zygo Corporation, 2006).
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    [CrossRef]
  17. F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Separating misalignment from misfigure in interferograms on cylindrical optics,” Opt. Express 21, 8856–8864 (2013).
    [CrossRef]
  18. B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt. 52, 2625–2636 (2005).
    [CrossRef]
  19. T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik 112, 304–308 (2001).
    [CrossRef]
  20. D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Misalignment aberrations calibration in testing of high-numerical-aperture spherical surfaces,” Appl. Opt. 50, 2024–2031 (2011).
    [CrossRef]
  21. “Encoding and fabrication report: CGH cylinder null h45f15c,” (Diffraction International, 5810 Baker Road, Suite 225 Minnetonka, Minnesota 55345-5982, USA, 2012).

2013 (1)

2011 (2)

D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Misalignment aberrations calibration in testing of high-numerical-aperture spherical surfaces,” Appl. Opt. 50, 2024–2031 (2011).
[CrossRef]

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

2010 (1)

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

2008 (1)

G. Kang, J. Xie, and Y. Liu, “New design techniques and alignment methods for CGH-null testing of aspheric surface,” Proc. SPIE 6624, 66240K (2008).
[CrossRef]

2006 (1)

2005 (2)

K. Mantel, N. Lindlein, and J. Schwider, “Simultaneous characterization of the quality and orientation of cylindrical lens surfaces,” Appl. Opt. 44, 2970–2977 (2005).
[CrossRef]

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt. 52, 2625–2636 (2005).
[CrossRef]

2004 (1)

J. Lamprecht, N. Lindlein, and J. Schwider, “Null test measurement of high-numerical-aperture cylindrical microlenses in transmitted light,” Proc. SPIE 5180, 253–260 (2004).
[CrossRef]

2003 (1)

2002 (2)

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10–S16 (2002).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158–168 (2002).
[CrossRef]

2001 (1)

T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik 112, 304–308 (2001).
[CrossRef]

1998 (1)

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. Eng. Des. 33, 105–125 (1998).
[CrossRef]

1997 (1)

P. Tam, K. Gross, and J. Bogan, “Interferometric testing of cylinder optics using computer generated hologram (CGH),” Proc. SPIE 3134, 162–166 (1997).
[CrossRef]

1995 (1)

J. M. Geary, “Overview of cylindrical optics testing using a fiber optic reference,” Proc. SPIE 2536, 68–74 (1995).
[CrossRef]

1987 (1)

J. M. Geary, “Testing cylindrical lenses,” Opt. Eng. 26, 261219 (1987).
[CrossRef]

1972 (1)

Bennett, V.

Bogan, J.

P. Tam, K. Gross, and J. Bogan, “Interferometric testing of cylinder optics using computer generated hologram (CGH),” Proc. SPIE 3134, 162–166 (1997).
[CrossRef]

Chen, C.

Dresel, T.

T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik 112, 304–308 (2001).
[CrossRef]

Geary, J.

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

Geary, J. M.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Separating misalignment from misfigure in interferograms on cylindrical optics,” Opt. Express 21, 8856–8864 (2013).
[CrossRef]

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

J. M. Geary, “Overview of cylindrical optics testing using a fiber optic reference,” Proc. SPIE 2536, 68–74 (1995).
[CrossRef]

J. M. Geary, “Testing cylindrical lenses,” Opt. Eng. 26, 261219 (1987).
[CrossRef]

Gross, K.

P. Tam, K. Gross, and J. Bogan, “Interferometric testing of cylinder optics using computer generated hologram (CGH),” Proc. SPIE 3134, 162–166 (1997).
[CrossRef]

Huntley, J. M.

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. Eng. Des. 33, 105–125 (1998).
[CrossRef]

Kang, G.

G. Kang, J. Xie, and Y. Liu, “New design techniques and alignment methods for CGH-null testing of aspheric surface,” Proc. SPIE 6624, 66240K (2008).
[CrossRef]

Khalsa, K.

K. Khalsa, Metropro Reference Guide (Zygo Corporation, 2006).

Lamprecht, J.

K. Mantel, J. Lamprecht, N. Lindlein, and J. Schwider, “Absolute calibration in grazing incidence interferometry via rotational averaging,” Appl. Opt. 45, 3740–3745 (2006).
[CrossRef]

J. Lamprecht, N. Lindlein, and J. Schwider, “Null test measurement of high-numerical-aperture cylindrical microlenses in transmitted light,” Proc. SPIE 5180, 253–260 (2004).
[CrossRef]

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10–S16 (2002).
[CrossRef]

Lindlein, N.

K. Mantel, J. Lamprecht, N. Lindlein, and J. Schwider, “Absolute calibration in grazing incidence interferometry via rotational averaging,” Appl. Opt. 45, 3740–3745 (2006).
[CrossRef]

K. Mantel, N. Lindlein, and J. Schwider, “Simultaneous characterization of the quality and orientation of cylindrical lens surfaces,” Appl. Opt. 44, 2970–2977 (2005).
[CrossRef]

J. Lamprecht, N. Lindlein, and J. Schwider, “Null test measurement of high-numerical-aperture cylindrical microlenses in transmitted light,” Proc. SPIE 5180, 253–260 (2004).
[CrossRef]

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10–S16 (2002).
[CrossRef]

T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik 112, 304–308 (2001).
[CrossRef]

Liu, F.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Separating misalignment from misfigure in interferograms on cylindrical optics,” Opt. Express 21, 8856–8864 (2013).
[CrossRef]

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

Liu, Y.

G. Kang, J. Xie, and Y. Liu, “New design techniques and alignment methods for CGH-null testing of aspheric surface,” Proc. SPIE 6624, 66240K (2008).
[CrossRef]

Mantel, K.

Pruss, C.

Pruß, C.

S. Reichelt, C. Pruß, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158–168 (2002).
[CrossRef]

Reardon, P.

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

Reardon, P. J.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Separating misalignment from misfigure in interferograms on cylindrical optics,” Opt. Express 21, 8856–8864 (2013).
[CrossRef]

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt. 52, 2625–2636 (2005).
[CrossRef]

Reichelt, S.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. 42, 4468–4479 (2003).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158–168 (2002).
[CrossRef]

Robinson, B. M.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Separating misalignment from misfigure in interferograms on cylindrical optics,” Opt. Express 21, 8856–8864 (2013).
[CrossRef]

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt. 52, 2625–2636 (2005).
[CrossRef]

Schreiner, R.

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10–S16 (2002).
[CrossRef]

Schwider, J.

K. Mantel, J. Lamprecht, N. Lindlein, and J. Schwider, “Absolute calibration in grazing incidence interferometry via rotational averaging,” Appl. Opt. 45, 3740–3745 (2006).
[CrossRef]

K. Mantel, N. Lindlein, and J. Schwider, “Simultaneous characterization of the quality and orientation of cylindrical lens surfaces,” Appl. Opt. 44, 2970–2977 (2005).
[CrossRef]

J. Lamprecht, N. Lindlein, and J. Schwider, “Null test measurement of high-numerical-aperture cylindrical microlenses in transmitted light,” Proc. SPIE 5180, 253–260 (2004).
[CrossRef]

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10–S16 (2002).
[CrossRef]

T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik 112, 304–308 (2001).
[CrossRef]

Tam, P.

P. Tam, K. Gross, and J. Bogan, “Interferometric testing of cylinder optics using computer generated hologram (CGH),” Proc. SPIE 3134, 162–166 (1997).
[CrossRef]

Tiziani, H. J.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. 42, 4468–4479 (2003).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158–168 (2002).
[CrossRef]

Wang, D.

Wyant, J.

Xie, J.

G. Kang, J. Xie, and Y. Liu, “New design techniques and alignment methods for CGH-null testing of aspheric surface,” Proc. SPIE 6624, 66240K (2008).
[CrossRef]

Yang, Y.

Zhuo, Y.

Appl. Opt. (5)

J. Mod. Opt. (1)

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt. 52, 2625–2636 (2005).
[CrossRef]

J. Opt. A (1)

J. Schwider, N. Lindlein, R. Schreiner, and J. Lamprecht, “Grazing-incidence test for cylindrical microlenses with high numerical aperture,” J. Opt. A 4, S10–S16 (2002).
[CrossRef]

J. Strain Anal. Eng. Des. (1)

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. Eng. Des. 33, 105–125 (1998).
[CrossRef]

Opt. Eng. (3)

J. M. Geary, “Testing cylindrical lenses,” Opt. Eng. 26, 261219 (1987).
[CrossRef]

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng. 49, 053002 (2010).
[CrossRef]

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50, 043609 (2011).
[CrossRef]

Opt. Express (1)

Optik (1)

T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik 112, 304–308 (2001).
[CrossRef]

Proc. SPIE (5)

J. Lamprecht, N. Lindlein, and J. Schwider, “Null test measurement of high-numerical-aperture cylindrical microlenses in transmitted light,” Proc. SPIE 5180, 253–260 (2004).
[CrossRef]

J. M. Geary, “Overview of cylindrical optics testing using a fiber optic reference,” Proc. SPIE 2536, 68–74 (1995).
[CrossRef]

S. Reichelt, C. Pruß, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158–168 (2002).
[CrossRef]

P. Tam, K. Gross, and J. Bogan, “Interferometric testing of cylinder optics using computer generated hologram (CGH),” Proc. SPIE 3134, 162–166 (1997).
[CrossRef]

G. Kang, J. Xie, and Y. Liu, “New design techniques and alignment methods for CGH-null testing of aspheric surface,” Proc. SPIE 6624, 66240K (2008).
[CrossRef]

Other (3)

http://diffraction.com/cylinder.php .

K. Khalsa, Metropro Reference Guide (Zygo Corporation, 2006).

“Encoding and fabrication report: CGH cylinder null h45f15c,” (Diffraction International, 5810 Baker Road, Suite 225 Minnetonka, Minnesota 55345-5982, USA, 2012).

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Figures (13)

Fig. 1.
Fig. 1.

Schematic diagram for null testing of the cylindrical surface using a CGH: (a) layout of the measurement system and (b) layout of the CGH designed by the Diffraction Corporation.

Fig. 2.
Fig. 2.

Schematic diagram of all relevant degrees of freedom. Only translation along as well rotation around the x axis creates no aberrations.

Fig. 3.
Fig. 3.

Geometry for misalignment aberrations analysis: (a) translation along the y axis, (b) translation along the z axis, (c) rotation around the y axis, and (d) rotation around the z axis.

Fig. 4.
Fig. 4.

Layout of the simulated system for null testing of the cylindrical surface.

Fig. 5.
Fig. 5.

Misalignment aberrations removing with different methods. The first column: the recorded interferograms caused by the misalignments in ty=0.001mm, tz=0.01mm, θy=0.001°, θz=0.01°; the second column: the corresponding phase map extracted based on the interferograms of first column; the third column: residual errors with the proposed method; the fourth column: residual errors with the Zygo method.

Fig. 6.
Fig. 6.

Residual errors comparison between different misalignment removal method: (a) misalignment by translating along the y axis, (b) misalignment by translating along the z axis, (c) misalignment by rotating around the y axis, and (d) misalignment by rotating around the z axis.

Fig. 7.
Fig. 7.

Null test of the cylindrical lens: (a) the recorded interferogram, (b) the corresponding phase map, (c) the remaining phase map with the proposed method, (d) the remaining phase map with the Zygo method, (e) the remaining phase map with the OCPs method, and (f) the remaining phase map with the quadratic polynomials without the x2 term.

Fig. 8.
Fig. 8.

Comparison of the PV and RMS values of the remaining phase data by using the proposed method, the Zygo method, and the OCPs method: (a) comparison of the PV values and (b) comparison of the RMS values.

Fig. 9.
Fig. 9.

Misalignment aberrations removal under introducing certain adjustment errors: (a) the recorded interferogram, (b) the corresponding phase map, and (c) the remaining phase map after removing the misalignment aberrations with the proposed method.

Fig. 10.
Fig. 10.

Reproducibility evaluation with different methods: (a) PV values comparison of the extracted phase data and the remaining phase data with the proposed method and the existing methods and (b) RMS values comparison of the extracted phase data and the remaining phase data with the proposed method and the existing method.

Fig. 11.
Fig. 11.

Schematic of the arrangement of the three subaperture measurements, 24 deg apart.

Fig. 12.
Fig. 12.

Experimental results of a cylindrical lens with rectangular pupil: (a)–(c) interferograms of three apertures; (d), (e) measured phase distributions of three apertures; (g)–(i) remaining phase distributions of three apertures after removing misalignment aberrations with the proposed method.

Fig. 13.
Fig. 13.

Full aperture map of the tested cylindrical lens.

Tables (6)

Tables Icon

Table 1. Comparison of the PV and RMS Values by Using Different Methods

Tables Icon

Table 2. Comparison of the Adjustment Errors between the Nominal Values and the Calculated Results

Tables Icon

Table 3. Comparison of the PV and RMS Values with Different Methods at the Null Position

Tables Icon

Table 4. Comparison of the Repeatability Validation of the PV and RMS Values with Different Methods at the Null Position

Tables Icon

Table 5. Comparison of the PV and RMS Values with Different Methods when Certain Misalignments Induced

Tables Icon

Table 6. Comparison of the PV and RMS Values of the Reproducibility Validation with Different Methods

Equations (13)

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

[x,y,z]T=A[x,y,z]T+V,
OPD=2(|OiP||OiPi|+|OiOi|z),
OPDty=2((y+ty)2+z2R)=2(R1+2ytyR2+(tyR)2R)2(R(1+ytyR2)R)=2yRty=2tysinθ,
OPDtz=2(y2+(ztz)2(Rtz)+tz)=2(R12ztzR2+(tzR)2R+2tz)2(R(1ztzR2)R+2tz)=2((2zR)tz)=2tz(2cosθ),
[xyz]=[cosθy0sinθy010sinθy0cosθy][xyz].
[xyz]=[10θy010θy01][xyz]=[x+zθyyxθy+z].
OPDθy=2(y2+(zxθy)2(RΔzi)+Δzi)=2(y2+z22xzθy+(xθy)2R+2Δzi)=2(R12xzθyR2+(xθyR)2R+2Δzi)2(R(1xzθyR2)R+2Δzi)=2(RxzθyRR+2Δzi)=2(xθycosθy+2xθy)=2xθy(2cosθ),
[xyz]=[cosθzsinθz0sinθzcosθz0001][xyz].
[xyz]=[1θz0θz10001][xyz]=[xyθzxθz+yz].
OPDθz=2((xθz+y)2+z2R)=2(y2+z2+2xyθz+(xθz)2R)=2(R1+2xyθzR2+(xθzR)2R)2(R(1+xyθzR2)R)=2xθzsinθ,
Δϕ=2(P0+tysinθ+tz(2cosθ)+xθy(2cosθ)+xθzsinθ),
S=[ϕmeasP0tysinθtz(2cosθ)xθy(2cosθ)xθzsinθ]2,
ψ=ϕmeasΔϕ.

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