Abstract

We have developed a full-field high resolution quantitative phase imaging technique for observing dynamics of transparent biological samples. By using a harmonically matched diffraction grating pair (600 and 1200 lines/mm), we were able to obtain non-trivial phase difference (other than 0° or 180°) between the output ports of the gratings. Improving upon our previous design, our current system mitigates astigmatism artifacts and is capable of high resolution imaging. This system also employs an improved phase extraction algorithm. The system has a lateral resolution of 1.6 µm and a phase sensitivity of 62 mrad. We employed the system to acquire high resolution phase images of onion skin cells and a phase movie of amoeba proteus in motion.

© 2007 Optical Society of America

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References

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  1. F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
    [CrossRef]
  2. F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
    [CrossRef]
  3. R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted light microscopy," Z. wiss.Mikr. 69, 193-221 (1969).
    [PubMed]
  4. K. Creath, "Phase-measurement interferometry techniques," Prog. Opt. 26, 349-393 (1988).
    [CrossRef]
  5. K. J. Chalut, W. J. Brown, and A. Wax, "Quantitative phase microscopy with asynchronous digital holography," Opt. Lett. 15, 3047-3052 (2007).
  6. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
    [CrossRef] [PubMed]
  7. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005).
    [CrossRef] [PubMed]
  8. B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, "Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy," Opt. Express 13, 9361-9373 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30,1165-1167 (2005).
    [CrossRef] [PubMed]
  11. G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
    [CrossRef] [PubMed]
  12. M. V. Sarunic, S. Weinberg, and J. A. Izatt, "Full-field swept-source phase microscopy," Opt. Lett. 31, 1462-1464 (2006).
    [CrossRef] [PubMed]
  13. D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, "Three-dimensional images generated by quadrature interferometry," Opt. Lett. 23, 783-785 (1998).
    [CrossRef]
  14. J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, 151123 (2007).
    [CrossRef]
  15. Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, "Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements," Opt. Express 14, 8127-8137 (2006).
    [CrossRef] [PubMed]
  16. M. A. Choma, C. Yang, and J. A. Izatt, "Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers," Opt. Lett. 28, 2162-2164 (2003).
    [CrossRef] [PubMed]
  17. Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, "Homodyne en face optical coherence tomography," Opt. Lett. 31, 1815-1817 (2006)
    [CrossRef] [PubMed]
  18. M. Pilu, A. W. Fitzgibbon, and R. B. Fisher, "Ellipse-specific direct least-square fitting," in Proceedings of IEEE International Conference on Image Processing (Lausanne, 1998), vol. 3, pp. 599-602.
  19. D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.
  20. C. Yang, "Molecular Contrast Optical Coherence Tomography: A Review," Photochemistry and Photobiology 81, 215-237 (2005)
    [CrossRef]

2007 (4)

K. J. Chalut, W. J. Brown, and A. Wax, "Quantitative phase microscopy with asynchronous digital holography," Opt. Lett. 15, 3047-3052 (2007).

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, 151123 (2007).
[CrossRef]

J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, "Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition," Opt. Express 15, 7231-7242 (2007).
[CrossRef] [PubMed]

2006 (4)

2005 (4)

2003 (1)

1998 (1)

1988 (1)

K. Creath, "Phase-measurement interferometry techniques," Prog. Opt. 26, 349-393 (1988).
[CrossRef]

1969 (1)

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted light microscopy," Z. wiss.Mikr. 69, 193-221 (1969).
[PubMed]

1942 (2)

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
[CrossRef]

Allen, R. D.

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted light microscopy," Z. wiss.Mikr. 69, 193-221 (1969).
[PubMed]

Badizadegan, K.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Best-Popescu, C. A.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Brown, W. J.

K. J. Chalut, W. J. Brown, and A. Wax, "Quantitative phase microscopy with asynchronous digital holography," Opt. Lett. 15, 3047-3052 (2007).

Chalut, K. J.

K. J. Chalut, W. J. Brown, and A. Wax, "Quantitative phase microscopy with asynchronous digital holography," Opt. Lett. 15, 3047-3052 (2007).

Charriere, F.

Choi, W.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Choma, M. A.

Colomb, T.

Creath, K.

K. Creath, "Phase-measurement interferometry techniques," Prog. Opt. 26, 349-393 (1988).
[CrossRef]

Cuche, E.

Cui, X.

J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, 151123 (2007).
[CrossRef]

Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, "Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements," Opt. Express 14, 8127-8137 (2006).
[CrossRef] [PubMed]

Dasari, R. R.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30,1165-1167 (2005).
[CrossRef] [PubMed]

David, G. B.

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted light microscopy," Z. wiss.Mikr. 69, 193-221 (1969).
[PubMed]

Depeursinge, C.

Devaney, A. J.

DiMarzio, C. A.

Emery, Y.

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Feld, M. S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30,1165-1167 (2005).
[CrossRef] [PubMed]

Fingler, J.

Gaudette, T. J.

Goda, K.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Heng, X.

Hogenboom, D. O.

Ikeda, T.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30,1165-1167 (2005).
[CrossRef] [PubMed]

Izatt, J. A.

Kuhn, J.

Laposata, M.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Lee, L. M.

J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, 151123 (2007).
[CrossRef]

Lindberg, S. C.

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Magistretti, P. J.

Manley, S.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Marquet, P.

Montfort, F.

Nomarski, G.

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted light microscopy," Z. wiss.Mikr. 69, 193-221 (1969).
[PubMed]

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Popescu, G.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30,1165-1167 (2005).
[CrossRef] [PubMed]

Rappaz, B.

Sarunic, M. V.

Wax, A.

K. J. Chalut, W. J. Brown, and A. Wax, "Quantitative phase microscopy with asynchronous digital holography," Opt. Lett. 15, 3047-3052 (2007).

Weinberg, S.

Wu, J.

J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, 151123 (2007).
[CrossRef]

Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, "Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements," Opt. Express 14, 8127-8137 (2006).
[CrossRef] [PubMed]

Yang, C.

Yaqoob, Z.

Zernike, F.

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
[CrossRef]

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

Appl. Phys. Lett. (1)

J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, 151123 (2007).
[CrossRef]

Mikr. (1)

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted light microscopy," Z. wiss.Mikr. 69, 193-221 (1969).
[PubMed]

Nature methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (7)

Photochemistry and Photobiology (1)

C. Yang, "Molecular Contrast Optical Coherence Tomography: A Review," Photochemistry and Photobiology 81, 215-237 (2005)
[CrossRef]

Phys. Rev. Lett. (1)

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Physica (2)

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects," Physica 9, 686-698 (1942).
[CrossRef]

F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects Part II," Physica 9, 974-986 (1942).
[CrossRef]

Prog. Opt. (1)

K. Creath, "Phase-measurement interferometry techniques," Prog. Opt. 26, 349-393 (1988).
[CrossRef]

Other (2)

M. Pilu, A. W. Fitzgibbon, and R. B. Fisher, "Ellipse-specific direct least-square fitting," in Proceedings of IEEE International Conference on Image Processing (Lausanne, 1998), vol. 3, pp. 599-602.

D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.

Supplementary Material (2)

» Media 1: AVI (360 KB)     
» Media 2: AVI (1710 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

(a). Simple beamsplitter based interferometer: 180o phase shift between output ports; (b) Single shallow grating based interferometer: still 180o phase shift between output ports; (c) G1G2 grating based interferometer: non-trivial phase shift can be obtained.

Fig. 2.
Fig. 2.

Experimental setup for phase imaging. BS: beam splitter; O1 and O2: objective lenses 1 and 2; P: pinhole; L1-4: lens 1-4; S: sample; G1G2: the harmonically matched grating pair (G1G2 grating) on a holographic plate.

Fig. 3.
Fig. 3.

Geometric aberration induced by the grating. (a) Previous imaging setup; (b) Astigmatism of the focal spots of previous setup; (c) Current imaging setup; (d) No aberration in the focal spot of current setup; (e) Image of letter “C” acquired by the previous setup; (f) Image of letter “C” acquired by the current setup; (g) Aberration of illumination beam caused by the grating diffraction in current setup.

Fig. 4.
Fig. 4.

“Stretch” and “compression” distortion caused by the diffraction of the grating in the imaging system.

Fig. 5.
Fig. 5.

Compare unwrap algothms. (a) Wrapped image; (b) Unwrapped image by simple unwrap algorithm; (c) Unwrapped image by Flynn’s algorithm.

Fig. 6.
Fig. 6.

Measurement of the temporal phase stability. (a) Phase image of a cover glass, the two spots that are used to measure the phase stability are indicated; (b) Fluctuation of the phase of spot 2 with respect to spot 1 versus time, the standard deviation is 62 mrad.

Fig. 7.
Fig. 7.

Images of “CIT” logo by our iamging system. (a) Intensity image; (b) Phase image; (c) 3D reconstruction of the phase image; (d) Step-height measurement.

Fig. 8.
Fig. 8.

Images of onion skin cells. (a) Intensity image; (b) Phase image; (c) 3D reconstruction of the phase image.

Fig. 9.
Fig. 9.

(a). (360 KB) MOVIE: phase movie of movement of amoeba proteus. The nucleus and contractile vacuole can be clearly seen. In the movie, the food vacuoles are moving inside the amoeba. The size of one frame is 147 µm (width)×123 µm (height) (372×312 pixels) [Media 1] (b) (1.66 MB) Movie: intensity movie of movement of amoeba proteus [Media 2].

Fig.B1.
Fig.B1.

1. Schematic of the phase noise assessment.

Equations (43)

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

Port 1 : P 1 = P r 2 + P s 2 + P r P s cos ( ψ s ψ r + θ )
Port 2 : P 2 = P r 2 + P s 2 P r P s cos ( ψ s ψ r + θ )
ϕ ( x 0 ) = m [ sgn ( m ) 2 π x 0 Λ + π 2 ]
Port 1 : P 1 = P r 1 + P s 1 + 2 P r 1 P s 1 cos ( 2 π x 0 Λ + π 2 + θ )
Port 2 : P 2 = P r 2 + P s 2 + 2 P r 2 P s 2 cos ( 2 π x 0 Λ π 2 + θ )
ϕ R , 1 = ϕ 0 , G 1 G 2 = 0 , ϕ S , 1 = ϕ + 1 , G 1 = 2 π x 1 Λ 1 + π 2 ,
ϕ R , 2 = ϕ 1 , G 2 = 2 π x 2 Λ 2 + π 2 , ϕ S , 2 = ϕ 1 , G 1 = 2 π x 1 Λ 1 + π 2
Δ ϕ 1 = ϕ S , 1 ϕ R , 1 = 2 π x 1 Λ 1 + π 2 , Δ ϕ 2 = ϕ S , 2 ϕ R , 2 = 2 π x 1 Λ 1 + 2 π x 2 Λ 2
Δ ϕ = Δ ϕ 2 Δ ϕ 1 = 4 π ( x 2 x 1 ) Λ 1 π 2
x 2 = f 2 tan [ sin 1 ( λ d sin ( tan 1 x 1 f 1 ) ) sin 1 λ d ]
x 2 = f 2 f 1 1 λ 2 d 2 x 1
P 1 ( i , j ) = P r 1 ( i , j ) + P s 1 ( i , j ) + A P r 1 ( i , j ) P s 1 ( i , j ) cos ( Δ ψ ( i , j ) )
P 2 ( i , j ) = P r 2 ( i , j ) + η ( i , j ) P s 1 ( i , j )
+ A P r 2 ( i , j ) η ( i , j ) P s 1 ( i , j ) cos ( Δ ψ ( i , j ) + Δ ϕ )
[ 1 P r 1 + η P r 2 2 cos ( Δ ϕ ) η P r 1 P r 2 ] P s 1 2 [ 2 P 1 P r 1 P r 1 + 2 P 2 P r 2 P r 2
2 cos ( Δ ϕ ) ( P 1 P r 1 ) η + ( P 2 P r 2 ) η P r 1 P r 2 + A 2 sin 2 ( Δ ϕ ) ] P s 1
+ [ ( P 1 P r 1 ) 2 P r 1 + ( P 2 P r 2 ) 2 P r 2 η 2 cos ( Δ ϕ ) ( P 1 P r 1 ) ( P 2 P r 2 ) η P r 1 P r 2 ] = 0
P s 1 P r 1 + P s 2 P r 2 2 P s 1 P r 1 P s 2 P r 2 cos ( Δ ϕ ) < A 2 4 sin 2 ( Δ ϕ )
P s 1 P r 1 + P s 2 P r 2 < 1
P s 1 = b ± Δ 2 a , where Δ = b 2 4 ac
δ P s 1 = P s 1 P 1 δ P 1 + P s 1 P 2 δ P 2
= b P 1 ± ( Δ P 1 ) 2 Δ 2 a δ P 1 + b P 2 ± ( Δ P 2 ) 2 Δ 2 a δ P 2
s 0 + s 1 = 2 P 1 P r 1 P r 1 + 2 P 2 P r 2 P r 2 2 cos ( Δ ϕ ) ( P 1 P r 1 ) η + ( P 2 P r 2 ) η P r 1 P r 2 + A 2 sin 2 ( Δ ϕ ) 1 P r 1 + η P r 2 2 cos ( Δ ϕ ) η P r 1 P r 2
2 s 0 < 2 P 1 P r 1 P r 1 + 2 P 2 P r 2 P r 2 2 cos ( Δ ϕ ) ( P 1 P r 1 ) η + ( P 2 P r 2 ) η P r 1 P r 2 + A 2 sin 2 ( Δ ϕ ) 1 P r 1 + η P r 2 2 cos ( Δ ϕ ) η P r 1 P r 2
s 0 ( 1 P r 1 + η P r 2 2 cos ( Δ ϕ ) η P r 1 P r 2 )
< P 1 P r 1 P r 1 + P 2 P r 2 P r 2 cos ( Δ ϕ ) ( P 1 P r 1 ) η + ( P 2 P r 2 ) η P r 1 P r 2 + A 2 2 sin 2 ( Δ ϕ )
= s 0 + A P r 1 s 0 cos ( Δ ψ ) P r 1 + η s 0 + A P r 2 η s 0 cos ( Δ ψ + Δ ϕ ) P r 2
cos Δ ϕ [ s 0 + A P r 1 s 0 cos ( Δ ψ ) ] η + [ η s 0 + A P r 2 η s 0 cos ( Δ ψ + Δ ϕ ) ] η P r 1 P r 2 + A 2 2 sin 2 ( Δ ϕ )
A s 0 P r 1 sin ( Δ ψ ) sin ( Δ ψ + Δ ϕ ) A η s 0 P r 2 sin Δ ϕ sin ( Δ ψ ) + A 2 2 sin 2 ( Δ ϕ ) > 0
A s 0 P r 1 sin ( Δ ψ + Δ ϕ ) A η s 0 P r 2 sin ( Δ ψ ) + A 2 2 sin ( Δ ϕ ) > 0
[ A s 0 P r 1 cos ( Δ ϕ ) A η s 0 P r 2 ] sin ( Δ ψ ) + A s 0 P r 1 sin ( Δ ϕ ) cos ( Δ ψ ) + A 2 2 sin ( Δ ϕ ) > 0
[ A s 0 P r 1 cos ( Δ ϕ ) A η s 0 P r 2 ] 2 + [ A s 0 P r 1 sin ( Δ ϕ ) ] 2 + A 2 2 sin ( Δ ϕ ) > 0
P s 1 P r 1 + P s 2 P r 2 2 P s 1 P r 1 P s 2 P r 2 cos ( Δ ϕ ) < A 2 4 sin 2 ( Δ ϕ )
P 1 = P r + P s + 2 P r P s cos Δ ψ
P 2 = P r + P s + 2 P r P s sin Δ ψ
2 P r P s cos Δ ψ = P 1 P r P s
2 P r P s sin Δ ψ = P 2 P r P s
σ x 1 = h ν η τ P 1 , σ x 2 = h ν η τ P 2
δ ψ σ x 1 2 + σ x 2 2 2 P r P s = 1 2 h ν η τ P 1 + P 2 P r P s = 2 2 h ν η τ P r + P s + P r P s ( cos Δ ψ + sin Δ ψ ) P r P s
2 2 h ν η τ P r + P s + 2 P r P s P r P s
δ ψ σ x 1 2 + σ x 2 2 2 P r P s = 0.015 P 2 1 + P 2 2 P r P s
= 0.015 2 ( P r + P s ) 2 + 4 P r P + 4 ( P r + P s ) P r P s ( cos Δ ψ + sin Δ ψ ) P r P s
0.015 2 ( P r + P s ) 2 + 4 P r P + 4 ( P r + P s ) 2 P r P s P r P s

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