Abstract

The modulus of the degree of coherence can be derived from interference patterns either by using fringes and next neighbour operations or by using several interferograms produced through phase shifting. Here the latter approach will be followed by using a lateral shearing interferometer exploiting a diffractive grating wedge providing a linearly progressive shear. Phase shifting methods offer pixel-oriented evaluations but suffer from instabilities and drifts which is the reason for the derivation of an error immune algorithm. This algorithm will use five π/2-steps of the reference phase also for the calculation of the modulus of the coherence function.

© 2016 Optical Society of America

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References

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  1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [Crossref] [PubMed]
  2. 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]
  3. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [Crossref] [PubMed]
  4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [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. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [Crossref]
  7. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [Crossref] [PubMed]
  8. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shiftin interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [Crossref] [PubMed]
  9. P. de Groot, “101-frame algorithm for phase shifting interferometry,” Preprint 3098–33 EUROPTO (1997).
  10. K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236–253 (2001).
    [Crossref] [PubMed]
  11. S. K. Debnath and M. P. Kothiyal, “Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer,” Appl. Opt. 46, 5103–5109 (2007).
    [Crossref] [PubMed]
  12. K. Kinnstaetter, A.W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [Crossref] [PubMed]
  13. K. Saastamoinen, J. Tervo, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spatial coherence measurement of polychromatic light with modified Youngs interferometer,” Opt. Express 21, 4061–4071 (2013).
    [Crossref] [PubMed]
  14. S. Divitt, Z. J. Lapin, and L. Novotny, “Measuring coherence functions using non-parallel double slits,” Opt. Express 22, 8277–8290 (2014).
    [Crossref] [PubMed]
  15. J. Schwider, “Continuous lateral shearing interferometer,” Appl. Opt. 23, 4403–4409 (1984).
    [Crossref] [PubMed]
  16. V. Nercissian, I. Harder, K. Mantel, A. Berger, G. Leuchs, N. Lindlein, and J. Schwider, “Diffractive simultaneous bidirectional shearing interferometry using tailored spatially coherent light,” Appl. Opt. 50, 571–578 (2011).
    [Crossref] [PubMed]
  17. J. Schwider, “DOE-based interferometry”, Optik 108, 181–196 (1998).
  18. H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
    [Crossref] [PubMed]
  19. G. Fütterer, “UV-Shearing Interferometrie zur Vermessung lithographischer “Phase Shift” Masken und VUVStrukturierung” Ph.D. thesis University Erlangen-Nuremberg, (2002).
  20. G. Fütterer, M. Lano, N. Lindlein, and J. Schwider, “Lateral shearing interferometer for phase-shift mask measurement at 193 nm,” Proc. SPIE 4691, 541–551 (2002).
    [Crossref]
  21. I. Harder, “Laterales DUV-Shearing Interferometer mit reduzierter zeitlicher und räumlicher Kohärenz” Ph.D. thesis University Erlangen-Nuremberg, (2003).
  22. D. Li, D. P. Kelly, and J. T. Sheridan, “Speckle suppression by doubly scattering systems,” Appl. Opt. 52, 8617–8626 (2013).
    [Crossref]
  23. Photonfocus AG MV1-D1312-80-G2 data sheet.
  24. C. W. McCutchen, “Generalized source and the Van Cittert-Zernike theorem: a study of the spatial coherence required for interferometry,” J. Opt. Soc. Am. A 56, 727–733 (1966).
    [Crossref]
  25. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (1990).
    [Crossref]
  26. J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
    [Crossref]
  27. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
    [Crossref]
  28. M. Born and E. Wolf, Principles of Optics, 7th ed. (PergamonOxford, 1999).
    [Crossref]
  29. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
    [Crossref] [PubMed]

2014 (1)

2013 (2)

2011 (1)

2007 (1)

2002 (1)

G. Fütterer, M. Lano, N. Lindlein, and J. Schwider, “Lateral shearing interferometer for phase-shift mask measurement at 193 nm,” Proc. SPIE 4691, 541–551 (2002).
[Crossref]

2001 (1)

2000 (1)

1998 (1)

J. Schwider, “DOE-based interferometry”, Optik 108, 181–196 (1998).

1997 (1)

1995 (2)

1993 (2)

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[Crossref] [PubMed]

1992 (1)

1990 (2)

1989 (1)

1988 (1)

1987 (1)

1984 (1)

1983 (1)

1974 (1)

1966 (1)

C. W. McCutchen, “Generalized source and the Van Cittert-Zernike theorem: a study of the spatial coherence required for interferometry,” J. Opt. Soc. Am. A 56, 727–733 (1966).
[Crossref]

Berger, A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (PergamonOxford, 1999).
[Crossref]

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Creath, K.

de Groot, P.

P. de Groot, “101-frame algorithm for phase shifting interferometry,” Preprint 3098–33 EUROPTO (1997).

Debnath, S. K.

Divitt, S.

Eiju, T.

Elssner, K.-E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Freischlad, K.

Friberg, A. T.

Fütterer, G.

G. Fütterer, M. Lano, N. Lindlein, and J. Schwider, “Lateral shearing interferometer for phase-shift mask measurement at 193 nm,” Proc. SPIE 4691, 541–551 (2002).
[Crossref]

G. Fütterer, “UV-Shearing Interferometrie zur Vermessung lithographischer “Phase Shift” Masken und VUVStrukturierung” Ph.D. thesis University Erlangen-Nuremberg, (2002).

Gallagher, J. E.

Grzanna, J.

Harder, I.

Hariharan, P.

Herriott, D. R.

Kelly, D. P.

Kinnstaetter, K.

Kirchner, M.

Koliopoulos, C. L.

Kothiyal, M. P.

Lano, M.

G. Fütterer, M. Lano, N. Lindlein, and J. Schwider, “Lateral shearing interferometer for phase-shift mask measurement at 193 nm,” Proc. SPIE 4691, 541–551 (2002).
[Crossref]

Lapin, Z. J.

Larkin, K. G.

Leuchs, G.

Leushacke, L.

Li, D.

Lindlein, N.

Lohmann, A.W.

Mantel, K.

McCutchen, C. W.

C. W. McCutchen, “Generalized source and the Van Cittert-Zernike theorem: a study of the spatial coherence required for interferometry,” J. Opt. Soc. Am. A 56, 727–733 (1966).
[Crossref]

Merkel, K.

Nercissian, V.

Novotny, L.

Oreb, B. F.

Rosen, J.

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[Crossref]

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[Crossref]

Rosenfeld, D. P.

Saastamoinen, K.

Schmit, J.

Schreiber, H.

H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
[Crossref] [PubMed]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Schwider, J.

V. Nercissian, I. Harder, K. Mantel, A. Berger, G. Leuchs, N. Lindlein, and J. Schwider, “Diffractive simultaneous bidirectional shearing interferometry using tailored spatially coherent light,” Appl. Opt. 50, 571–578 (2011).
[Crossref] [PubMed]

G. Fütterer, M. Lano, N. Lindlein, and J. Schwider, “Lateral shearing interferometer for phase-shift mask measurement at 193 nm,” Proc. SPIE 4691, 541–551 (2002).
[Crossref]

J. Schwider, “DOE-based interferometry”, Optik 108, 181–196 (1998).

H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
[Crossref] [PubMed]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[Crossref] [PubMed]

K. Kinnstaetter, A.W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[Crossref] [PubMed]

J. Schwider, “Continuous lateral shearing interferometer,” Appl. Opt. 23, 4403–4409 (1984).
[Crossref] [PubMed]

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]

Sheridan, J. T.

Spolaczyk, R.

Streibl, N.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

K. Kinnstaetter, A.W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[Crossref] [PubMed]

Surrel, Y.

Takeda, M.

Tervo, J.

Turunen, J.

Vahimaa, P.

White, A. D.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (PergamonOxford, 1999).
[Crossref]

Yariv, A.

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[Crossref]

Zoeller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Appl. Opt. (13)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[Crossref] [PubMed]

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]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[Crossref] [PubMed]

S. K. Debnath and M. P. Kothiyal, “Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer,” Appl. Opt. 46, 5103–5109 (2007).
[Crossref] [PubMed]

K. Kinnstaetter, A.W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[Crossref] [PubMed]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[Crossref] [PubMed]

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shiftin interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[Crossref] [PubMed]

J. Schwider, “Continuous lateral shearing interferometer,” Appl. Opt. 23, 4403–4409 (1984).
[Crossref] [PubMed]

V. Nercissian, I. Harder, K. Mantel, A. Berger, G. Leuchs, N. Lindlein, and J. Schwider, “Diffractive simultaneous bidirectional shearing interferometry using tailored spatially coherent light,” Appl. Opt. 50, 571–578 (2011).
[Crossref] [PubMed]

H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
[Crossref] [PubMed]

D. Li, D. P. Kelly, and J. T. Sheridan, “Speckle suppression by doubly scattering systems,” Appl. Opt. 52, 8617–8626 (2013).
[Crossref]

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[Crossref]

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

J. Rosen and A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[Crossref]

Opt. Eng. (1)

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoeller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[Crossref]

Opt. Express (3)

Optik (1)

J. Schwider, “DOE-based interferometry”, Optik 108, 181–196 (1998).

Proc. SPIE (1)

G. Fütterer, M. Lano, N. Lindlein, and J. Schwider, “Lateral shearing interferometer for phase-shift mask measurement at 193 nm,” Proc. SPIE 4691, 541–551 (2002).
[Crossref]

Other (5)

I. Harder, “Laterales DUV-Shearing Interferometer mit reduzierter zeitlicher und räumlicher Kohärenz” Ph.D. thesis University Erlangen-Nuremberg, (2003).

G. Fütterer, “UV-Shearing Interferometrie zur Vermessung lithographischer “Phase Shift” Masken und VUVStrukturierung” Ph.D. thesis University Erlangen-Nuremberg, (2002).

P. de Groot, “101-frame algorithm for phase shifting interferometry,” Preprint 3098–33 EUROPTO (1997).

M. Born and E. Wolf, Principles of Optics, 7th ed. (PergamonOxford, 1999).
[Crossref]

Photonfocus AG MV1-D1312-80-G2 data sheet.

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Figures (14)

Fig. 1
Fig. 1 Scheme of the setup for preparing and measuring the degree of spatial coherence using a continuous lateral shearing interferometer based on a grating wedge.
Fig. 2
Fig. 2 Core of the grating wedge interferometer providing continuous lateral shear increasing linearly with the distance of the impinging ray from the edge of the grating wedge. The wedge angle α between the identical gratings G1 and G2 is made equal to the angle of the first diffraction order for the case of perpendicular incidence.
Fig. 3
Fig. 3 Left: Schematic representation of a binary encoded sine grating with a period d = 500λ and integration of the transmissive area perpendicular to the grating vector as blue curve. Right: Experimental realisation showing an enlarged segment of the transparency mask.
Fig. 4
Fig. 4 Left: Raster of points allowing the measurement of the shear. Right: Demonstration of the periodic reoccurrence of the contrast in the exit plane of the progressive shearing interferometer produced by a periodic light source.
Fig. 5
Fig. 5 Left: Visibility-curves for a small shear value parallel to the edge of the grating wedge are shown for the 4-step and the 5-step algorithm. The error reduction by a factor of 5–6 for the 5-step algorithm is clearly visible. Right: Scans parallel to the edge of the grating wedge of the intensity sums I1 + I3 versus I3 + I5 are shown indicating the phase opposition due to the slope error of the piezo transducer.
Fig. 6
Fig. 6 Left: Periodic artefacts in the V4-values for a phase shifter mis-calibration of about 10% and V4 after a lsq-correction with a fitted first order functional (Eq. 14). A reduction of the artefacts by nearly one order of magnitude is clearly indicated and especially values greater V4 = 1 (physically forbidden) are absent in the corrected V4-values. Right: Periodic artefacts in the V5-values for a phase shifter mis-calibration of about 10% and V5 after a lsq-correction with a fitted second order functional (Eq. 19). A reduction of the artefacts by a factor of 5 is clearly indicated resulting in a remaining 1% variation of the modulus data.
Fig. 7
Fig. 7 3-D-graph of the modulus of the degree of coherence for an incoherently radiating double slit with a slit width of 1000λ and a slit distance of 1000λ. The position of zero-shear is located at the left rim where this region is not accessible due to the real edge character of the shearing interferometer.
Fig. 8
Fig. 8 Single slit with width 100λ. Left: Interferogram (top) and phase (mod2π) (bottom) where the zero-shear is on the left hand side (in fact the exact zero-shear position is not accessible because there has to be a small distance at the edge position of the grating wedge). Right: Measured modulus of the coherence function (black curve) and calculated modulus (red curve).
Fig. 9
Fig. 9 Single slit with width of 250λ. Same order of displayed data as in Fig. 8. Note the re-occurrence of regions with non-vanishing modulus.
Fig. 10
Fig. 10 Double slit with width 100λ and center-center distance 1000λ. Same order of displayed data as in Fig. 8. Note that the modulus spans the whole scale from zero to one.
Fig. 11
Fig. 11 Two slits with widths 100λ and 250λ being positioned with a center-center distance of 1000λ. Same order of displayed data as in Fig. 8.
Fig. 12
Fig. 12 Left: Periodic interference fringes produced by grid source having a slit width of 20λ and a period of 500λ where the zero-shear is on the left hand side. Right: Measured modulus of the coherence function (black) and calculated modulus (red).
Fig. 13
Fig. 13 Grid source with slit width 100λ and period 500λ. Same order of displayed data as in Fig. 12
Fig. 14
Fig. 14 Sinusoidal grid source with period of 500λ. Same order of displayed data as in Fig. 12. Please note that the zero-shear position is not fully recorded for the discussed reason.

Equations (22)

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I r = I 0 [ 1 + V cos ( Φ ψ r ) ] ,
I 0 = i 1 + i 2 ; V c = 2 i 1 i 2 i 1 + i 2
I r = I 0 + I 0 V c γ cos ( Φ ψ r ) + I 0 V c γ sin ( Φ ψ r ) .
Φ = arctan ( N D ) ; V = N 2 + D 2 I 0 = V c γ .
I 0 = 1 R r = 1 R I r .
ε r = ε ¯ ( r 1 ) ; r = 1 , , R ; ergo : ε 1 = 0 , ε 2 = ε ¯ , ε 3 = 2 ε ¯ ,
I 1 = I 0 + I 0 V c γ cos ( Φ ) , I 2 = I 0 + I 0 V c γ sin ( Φ ) cos ( ε ¯ ) I 0 V c γ cos ( Φ ) sin ( ε ¯ ) , I 3 = I 0 I 0 V c γ cos ( Φ ) cos ( 2 ε ¯ ) I 0 V c γ sin ( Φ ) sin ( 2 ε ¯ ) , I 4 = I 0 I 0 V c γ sin ( Φ ) cos ( 3 ε ¯ ) + I 0 V c γ cos ( Φ ) sin ( 3 ε ¯ ) , I 5 = I 0 + I 0 V c γ cos ( Φ ) cos ( 4 ε ¯ ) + I 0 V c γ sin ( Φ ) sin ( 4 ε ¯ ) .
Δ Φ = arctan ( N D ) arctan ( tan ( Φ ) ) = arctan ( N cos ( Φ ) D sin ( Φ ) D cos ( Φ ) + N sin ( Φ ) ) .
cos ( ε ¯ ) 1 ε ¯ 2 2 and sin ( ε ¯ ) ε ¯ .
Φ 4 = arctan ( I 2 I 4 I 1 I 3 ) = arctan ( N 4 D 4 ) ; V 4 = ( I 2 I 4 ) 2 + ( I 1 I 3 ) 2 I 1 + I 3 = N 4 2 D 4 2 I 04 .
N 4 = I 2 I 4 = I 0 V c γ sin ( Φ ) [ cos ( ε ¯ ) + cos ( 3 ε ¯ ) ] I 0 V c γ cos ( Φ ) [ sin ( ε ¯ ) + sin ( 3 ε ¯ ) ] , D 4 = I 1 I 3 = I 0 V c γ cos ( Φ ) [ 1 + cos ( 2 ε ¯ ) ] + I 0 V c γ sin ( Φ ) sin ( 2 ε ¯ ) , I 04 = 2 I 0 + I 0 V c γ cos ( Φ ) [ 1 cos ( 2 ε ¯ ) ] I 0 V c γ sin ( Φ ) sin ( 2 ε ¯ ) .
N 4 = 2 I 0 V c γ [ sin ( Φ ) ( 1 5 2 ε ¯ 2 ) 2 ε ¯ cos ( Φ ) ] , D 4 = 2 I 0 V c γ [ cos ( Φ ) ( 1 ε ¯ 2 ) + ε ¯ sin ( Φ ) ] , I 04 = 2 I 0 + 2 I 0 V c γ ¯ ε ¯ 2 cos ( Φ ) 2 I 0 V c γ ε ¯ sin ( Φ ) .
Δ Φ 4 arctan [ 3 2 ε ¯ + ε ¯ 2 cos ( 2 Φ ) + 2 ε ¯ 2 sin ( 2 Φ ) ] + O ( ε ¯ 3 and higher ) .
V 4 V c γ ( 1 ε ¯ 2 ) [ 1 ε ¯ 2 ε ¯ ( 1 2 sin ( 2 Φ ) V c γ sin ( Φ ) ) + ε ¯ 2 ( 3 2 cos ( 2 Φ ) V c γ cos ( Φ ) V c γ 2 sin ( Φ ) sin ( 2 Φ ) ) ] + O ( ε ¯ 3 and higher ) .
Φ 5 = arctan ( N 5 D 4 ) = arctan ( 2 I 2 2 I 4 I 1 2 I 3 + I 5 ) = arctan ( I 2 I 4 I 1 + I 5 2 I 3 ) , V 5 = N 5 2 + D 5 2 I 05 = ( 2 I 2 2 I 4 ) 2 + ( I 1 2 I 3 + I 5 ) 2 I 1 + 2 I 3 + I 5 .
N 5 = 2 I 0 V c γ sin ( Φ ) [ cos ( ε ¯ ) + cos ( 3 ε ¯ ) ] 2 I 0 V c γ cos ( Φ ) [ sin ( ε ¯ ) + sin ( 3 ε ¯ ) ] , D 5 = I 0 V c γ cos ( Φ ) [ 1 + 2 cos ( 2 ε ¯ ) + cos ( 4 ε ¯ ) ] + I 0 V c γ sin ( Φ ) [ 2 sin ( 2 ε ¯ ) + sin ( 4 ε ¯ ) ] , I 05 = 4 I 0 + I 0 V c γ cos ( Φ ) [ 1 2 cos ( 2 ε ¯ ) + cos ( 4 ε ¯ ) ] + I 0 V c γ sin ( Φ ) [ 2 sin ( 2 ε ¯ ) + sin ( 4 ε ¯ ) ] ,
N 5 = 4 I 0 V c γ [ sin ( Φ ) [ 1 5 2 ε ¯ 2 ] 2 ε ¯ cos ( Φ ) ] , D 5 = 4 I 0 V c γ [ cos ( Φ ) [ 1 3 ε ¯ 2 ] + 2 ε ¯ sin ( Φ ) ] , I 05 = 4 I 0 [ 1 V c γ ε ¯ 2 cos ( Φ ) ] .
Δ Φ 5 arctan ( ε ¯ 2 4 sin ( 2 Φ ) 2 ε ¯ ) + O ( ε ¯ 3 and higher ) .
V 5 V c γ [ 1 3 ε ¯ 2 4 + ε ¯ 2 ( V c γ cos ( Φ ) 1 4 cos ( 2 Φ ) ) ] + O ( ε ¯ 3 and higher ) .
σ = x tan 2 ( α ) = x sin 2 ( α ) cos ( α ) = x ( λ p ) 2 1 ( λ p ) 2 ,
OPD = n AA ¯ n A A ¯ ,
OPD = n [ x sin ( α ) cos 2 ( α ) x tan ( α ) ] = n x tan ( α ) ( 1 cos ( α ) cos ( α ) ) ,

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