Abstract

In both temporal and spatial carrier phase shifting interferometry, the primary source of phase calculation error results from an error in the relative phase shift between sample points. In spatial carrier phase shifting interferometry, this phase shifting error is caused directly by the wavefront under test and is unavoidable. In order to minimize the phase shifting error, a pixelated spatial carrier phase shifting technique has been developed by 4D technologies. This new technique allows for the grouping of phase shifted pixels together around a single point in two dimensions, minimizing the phase shift change due to the spatial variation in the test wavefront. A formula for the phase calculation error in spatial carrier phase shifting interferometry is derived. The error associated with the use of linear N-point averaging algorithms is presented and compared with those of the pixelated spatial carrier technique.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  2. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  3. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital wave-front measuring interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983).
    [CrossRef] [PubMed]
  4. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase shifting for nonsinusoidal waveforms with phase-shift errors," J. Opt. Soc. Am. A 12, 761-768 (1995).
    [CrossRef]
  5. K. G. Larkin and B. F. Oreb, "Design and assessment of symmetrical phase-shifting algorithms," J. Opt. Soc. Am. A 9, 1740-1748 (1992).
    [CrossRef]
  6. D. W. Phillion, "General methods for generating phase-shifting interferometry algorithms," Appl. Opt. 36, 8098-8115 (1997).
    [CrossRef]
  7. J. Schmit and K. Creath, "Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry," Appl. Opt. 34, 3610-3619 (1995).
    [CrossRef] [PubMed]
  8. M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
    [CrossRef]
  9. Y. Surrel, "Phase stepping: a new self-calibrating algorithm," Appl. Opt. 32, 3598-3600 (1993).
    [CrossRef] [PubMed]
  10. R. Jozwicki, M. Kujawinska, and L. A. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
    [CrossRef]
  11. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  12. D. M. Shough, O. Y. Kwon, and D. F. Leary, "High speed interferometric measurement of aerodynamic phenomena," in Propagation of High-Energy Laser Beams Through the Earth's Atmosphere, P. B. Ulrich and L. E. Wilson, eds., Proc. SPIE 1221, 394-403 (1990).
    [CrossRef]
  13. K. Creath and J. Schmit, "Errors in spatial phase-stepping techniques," in Interferometry '94: New Techniques and Analysis in Optical Measurements, M. Kujawinska and K. Patorski, eds., Proc. SPIE 2340, 170-176 (1994).
    [CrossRef]
  14. J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
    [CrossRef]
  15. P. Hariharan, B. F. Oreb, and T. Eiji, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (1987).
    [CrossRef] [PubMed]
  16. P. de Groot, "Phase-shift calibration errors in interferometers with spherical Fizeau cavities," Appl. Opt. 34, 2856-2863 (1995).
    [CrossRef]
  17. J. D. Tobiason and K. W. Atherton, "Interferometer using integrated imaging array and high-density polarizer array," US Patent 6,850,329 (1 February 2005).
  18. J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

2004 (1)

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

1997 (2)

D. W. Phillion, "General methods for generating phase-shifting interferometry algorithms," Appl. Opt. 36, 8098-8115 (1997).
[CrossRef]

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

1995 (3)

1994 (1)

K. Creath and J. Schmit, "Errors in spatial phase-stepping techniques," in Interferometry '94: New Techniques and Analysis in Optical Measurements, M. Kujawinska and K. Patorski, eds., Proc. SPIE 2340, 170-176 (1994).
[CrossRef]

1993 (1)

1992 (2)

R. Jozwicki, M. Kujawinska, and L. A. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

K. G. Larkin and B. F. Oreb, "Design and assessment of symmetrical phase-shifting algorithms," J. Opt. Soc. Am. A 9, 1740-1748 (1992).
[CrossRef]

1990 (1)

D. M. Shough, O. Y. Kwon, and D. F. Leary, "High speed interferometric measurement of aerodynamic phenomena," in Propagation of High-Energy Laser Beams Through the Earth's Atmosphere, P. B. Ulrich and L. E. Wilson, eds., Proc. SPIE 1221, 394-403 (1990).
[CrossRef]

1987 (1)

1983 (1)

1982 (1)

Atherton, K. W.

J. D. Tobiason and K. W. Atherton, "Interferometer using integrated imaging array and high-density polarizer array," US Patent 6,850,329 (1 February 2005).

Brock, N. J.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

Burow, R.

Creath, K.

J. Schmit and K. Creath, "Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry," Appl. Opt. 34, 3610-3619 (1995).
[CrossRef] [PubMed]

K. Creath and J. Schmit, "Errors in spatial phase-stepping techniques," in Interferometry '94: New Techniques and Analysis in Optical Measurements, M. Kujawinska and K. Patorski, eds., Proc. SPIE 2340, 170-176 (1994).
[CrossRef]

Cuevas, F. J.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

de Groot, P.

Eiji, T.

Elssner, K. E.

Farrant, D. I.

Grzanna, J.

Hariharan, P.

Hayes, J. B.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

Hibino, K.

Ina, H.

Jozwicki, R.

R. Jozwicki, M. Kujawinska, and L. A. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Kimbrough, B. T.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

Kobayashi, S.

Kujawinska, M.

R. Jozwicki, M. Kujawinska, and L. A. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Kwon, O. Y.

D. M. Shough, O. Y. Kwon, and D. F. Leary, "High speed interferometric measurement of aerodynamic phenomena," in Propagation of High-Energy Laser Beams Through the Earth's Atmosphere, P. B. Ulrich and L. E. Wilson, eds., Proc. SPIE 1221, 394-403 (1990).
[CrossRef]

Larkin, K. G.

Leary, D. F.

D. M. Shough, O. Y. Kwon, and D. F. Leary, "High speed interferometric measurement of aerodynamic phenomena," in Propagation of High-Energy Laser Beams Through the Earth's Atmosphere, P. B. Ulrich and L. E. Wilson, eds., Proc. SPIE 1221, 394-403 (1990).
[CrossRef]

Malacara, D.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Marroquin, J. L.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

Merkel, K.

Millerd, J. E.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

North-Morris, M. B.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

Novak, M.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Oreb, B. F.

Phillion, D. W.

Salbut, L. A.

R. Jozwicki, M. Kujawinska, and L. A. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Schmit, J.

J. Schmit and K. Creath, "Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry," Appl. Opt. 34, 3610-3619 (1995).
[CrossRef] [PubMed]

K. Creath and J. Schmit, "Errors in spatial phase-stepping techniques," in Interferometry '94: New Techniques and Analysis in Optical Measurements, M. Kujawinska and K. Patorski, eds., Proc. SPIE 2340, 170-176 (1994).
[CrossRef]

Schwider, J.

Servin, M.

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Shough, D. M.

D. M. Shough, O. Y. Kwon, and D. F. Leary, "High speed interferometric measurement of aerodynamic phenomena," in Propagation of High-Energy Laser Beams Through the Earth's Atmosphere, P. B. Ulrich and L. E. Wilson, eds., Proc. SPIE 1221, 394-403 (1990).
[CrossRef]

Spolaczyk, R.

Surrel, Y.

Takeda, M.

Tobiason, J. D.

J. D. Tobiason and K. W. Atherton, "Interferometer using integrated imaging array and high-density polarizer array," US Patent 6,850,329 (1 February 2005).

Wyant, J. C.

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

Appl. Opt. (6)

J. Mod. Opt. (1)

M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

R. Jozwicki, M. Kujawinska, and L. A. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Proc. SPIE (3)

D. M. Shough, O. Y. Kwon, and D. F. Leary, "High speed interferometric measurement of aerodynamic phenomena," in Propagation of High-Energy Laser Beams Through the Earth's Atmosphere, P. B. Ulrich and L. E. Wilson, eds., Proc. SPIE 1221, 394-403 (1990).
[CrossRef]

K. Creath and J. Schmit, "Errors in spatial phase-stepping techniques," in Interferometry '94: New Techniques and Analysis in Optical Measurements, M. Kujawinska and K. Patorski, eds., Proc. SPIE 2340, 170-176 (1994).
[CrossRef]

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic interferometer," in Interferometry XII: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Other (4)

J. D. Tobiason and K. W. Atherton, "Interferometer using integrated imaging array and high-density polarizer array," US Patent 6,850,329 (1 February 2005).

J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, B. T. Kimbrough, and J. C. Wyant, "Pixelated phase-mask dynamic interferometers," in Fringe 2005 (Springer, 2005), pp. 640-647.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

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 (8)

Fig. 1
Fig. 1

(Color online) Calculating θ3 using shifted data sets.

Fig. 2
Fig. 2

The 5-point algorithm theoretical phase calculation error for a wavefront with tilt. t = π∕64 rad∕pix. Values calculated with Eq. (20).

Fig. 3
Fig. 3

The 5-point algorithm phase calculation error for a wavefront with tilt. t = π∕64 rad∕pix. Values obtained through computer simulation.

Fig. 4
Fig. 4

(Color online) Peak-to-valley phase calculation error as a function of wavefront tilt for the 4-, 5-, 6-, and 7-point algorithms. The solid curves represent the theoretical values determined with Eq. (33). The points indicate values obtained via computer simulation.

Fig. 5
Fig. 5

(Color online) Linear versus pixelated spatial carrier. The arrows represent the phase shift error associated with the wavefront being measured. The pixelated spatial carrier groups four phase shifted pixels about a center point in order to minimize accumulative phase shift error.

Fig. 6
Fig. 6

Stacked and circular pixelated mask orientations.

Fig. 7
Fig. 7

(Color online) A 4 × 4 section of a circular pixelated mask carrier. The phase at point-5 can be calculated using the intensity values obtained at pixels 1–9 in the circular-9 algorithm.

Fig. 8
Fig. 8

(Color online) Peak-to-valley phase calculation error as a function of wavefront tilt for the circular-4, stacked-4, circular-9, and stacked-9 algorithms. The solid curves represent theoretical values. Points indicate values obtained via computer simulation. Tilt orientation is at 0° and 45° for the stacked and circular algorithms, respectively.

Tables (1)

Tables Icon

Table 1 Spatial Carrier Phase Shifting Calculation Error for a Wavefront with Tilt

Equations (54)

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

tan ( θ ) = I 2 I 4 I 3 I 1 ,
tan ( θ 3 + ε ) = ( I 2 I 4 ) / ( I 3 I 1 ) ,
tan ( θ 3 ε ) = ( I 2 I 4 ) / ( I 3 I 5 ) ,
tan ( θ ) = 2 I 2 2 I 4 2 I 3 I 1 I 5 .
tan ( a + b 2 ) = sin ( a ) + sin ( b ) cos ( a ) + cos ( b ) .
tan ( θ ) = 3 I 2 4 I 4 I 6 4 I 3 I 1 3 I 5 ,
tan ( θ ) = 4 I 2 8 I 4 + 4 I 6 I 1 7 I 3 + 7 I 5 I 7 ,
tan ( θ m + ε ) = n = 1 N I n W n sin ( φ n , m ) n = 1 N I n W n cos ( φ n , m ) ,
φ n , m = φ n φ m = ( n m ) π 2 .
tan ( ε ) = num ( 8 ) cos ( θ m ) den ( 8 ) sin ( θ m ) den ( 8 ) cos ( θ m ) + num ( 8 ) sin ( θ m ) ,
num ( 8 ) = n = 1 N I n W n sin ( φ n , m ) ,
den ( 8 ) = n = 1 N I n W n cos ( φ n , m ) .
I n = I avg [ 1 + γ cos ( θ n + φ n ) ] ,
θ n = θ m + Δ n , m ,
I n = 1 + cos ( θ m + φ m + Δ n , m + φ n , m ) .
num ( 8 ) = n = 1 N W n 1 2 cos ( θ m + φ m + Δ n , m ) sin ( 2 φ n , m ) + n = 1 N W n sin ( θ m + φ m + Δ n , m ) sin 2 ( φ n , m ) n = 1 N W n sin ( φ n , m ) .
sin ( 2 φ n , m ) = 0.
j = 1 4 ( 1 ) j W 2 j 1 = 1 7 + 7 1 = 0 .
j = 1 3 ( 1 ) j W 2 j = 4 + 8 4 = 0 .
num ( 8 ) = n = 1 N W n sin ( θ m + φ m + Δ n , m ) sin 2 ( φ n , m ) .
den ( 8 ) = n = 1 N W n cos ( θ m + φ m + Δ n , m ) cos 2 ( φ n , m ) .
tan ( ε m ) =
1 - [ 2 cos ( 2 θ m ) + 3 sin ( 2 θ m ) ] cos ( 2 φ m ) 4 + [ 3 cos ( 2 θ m ) 2 sin ( 2 θ m ) ] cos ( 2 φ m ) ,
1 = n = 1 N W n sin ( Δ n , m ) ,
2 = n = 1 N W n cos ( 2 φ n , m ) sin ( Δ n , m ) ,
3 = n = 1 N W n cos ( 2 φ n , m ) cos ( Δ n , m ) ,
4 = n = 1 N W n cos ( Δ n , m ) .
1 = n = 1 N W n sin ( Δ n , m ) = 0 ,
2 = n = 1 N W n cos ( 2 φ n , m ) sin ( Δ n , m ) = 0 ,
3 = n = 1 N W n cos ( 2 φ n , m ) cos ( Δ n , m ) = 8 cos ( t ) sin 2 ( t / 2 ) ,
4 = n = 1 N W n cos ( Δ n , m ) = 8 cos ( t ) cos 2 ( t / 2 ) .
tan ( ε m ) = sin 2 ( t / 2 ) sin ( 2 θ m ) cos ( 2 φ m ) cos 2 ( t / 2 ) sin 2 ( t / 2 ) cos ( 2 θ m ) cos ( 2 φ m ) .
tan ( ε ) max = sin 2 ( t / 2 ) cos ( t ) .
ε Pk-v = 1 π arctan ( sin 2 ( t / 2 ) cos ( t ) ) .
ε Pk–v 1 π ( t 2 ) 2 .
N   even tan ( ε ) = sin ( t / 2 ) N 3 cos ( 2 θ ) cos ( 2 φ ) cos ( t / 2 ) N 3 + sin ( t / 2 ) N 3 sin ( 2 θ ) cos ( 2 φ ) ,
N odd tan ( ε ) = sin ( t / 2 ) N 3 sin ( 2 θ ) cos ( 2 φ ) cos ( t / 2 ) N 3 + sin ( t / 2 ) N 3 cos ( 2 θ ) cos ( 2 φ ) ,
tan ( ε ) max = sin ( t / 2 ) N 3 cos ( t / 2 ) 2 ( N 3 ) sin ( t / 2 ) 2 ( N 3 ) .
ε Pk–v = 1 π arctan ( sin ( t / 2 ) N 3 cos ( t / 2 ) 2 ( N 3 ) sin ( t / 2 ) 2 ( N 3 ) ) .
ε Pk–v = 1 π arctan [ tan ( t / 2 ) N 3 ] .
tan [ θ 5 ] = 2 ( I 2 + I 8 I 4 I 6 ) I 1 I 3 + 4 I 5 I 7 I 9 .
tan [ θ 5 ] = I 1 + I 3 2 I 4 2 I 6 + I 7 + I 9 2 I 2 + 4 I 5 2 I 8 .
tan ( ε ) = sin ( t x / 2 ) cos ( 2 θ ) cos ( 2 φ ) cos ( t x / 2 ) + sin ( t x / 2 ) sin ( 2 θ ) cos ( 2 φ ) ,
tan ( ε ) max = sin ( t x / 2 ) cos ( t x ) .
tan ( ε ) =
sin ( t x / 2 ) sin ( t y / 2 ) sin ( 2 θ ) cos ( 2 φ ) cos ( t x / 2 ) cos ( t y / 2 ) sin ( t x / 2 ) sin ( t y / 2 ) cos ( 2 θ ) cos ( 2 φ ) .
tan ( ε ) max = 2 sin ( t x / 2 ) sin ( t y / 2 ) cos ( t x ) + cos ( t y )
tan ( ε ) max = sin 2 ( t x / 2 ) cos ( t x ) ,
tan ( ε ) =
sin 2 ( t x / 2 ) cos ( 2 θ ) cos ( 2 φ ) cos 2 ( t x / 2 ) + sin 2 ( t x / 2 ) sin ( 2 θ ) cos ( 2 φ ) .
tan ( ε ) = sin 2 ( t x / 2 ) sin 2 ( t y / 2 ) sin ( 2 θ ) cos ( 2 φ ) cos 2 ( t x / 2 ) cos 2 ( t y / 2 ) + sin 2 ( t x / 2 ) sin 2 ( t y / 2 ) cos ( 2 θ ) cos ( 2 φ ) .
tan ( ε ) max =
sin 2 ( t x / 2 ) sin 2 ( t y / 2 ) cos 4 ( t x / 2 ) cos 4 ( t y / 2 ) sin 4 ( t x / 2 ) sin 4 ( t y / 2 ) .
tan ( ε ) max = sin 4 ( t x / 2 ) cos 8 ( t x / 2 ) sin 8 ( t x / 2 ) .

Metrics