Abstract

We theoretically study the three-dimensional behavior of nanoparticles in an active optical conveyor. To do this, we solved the Langevin equation when the forces are generated by a focusing system at the near field. Analytical expressions for the optical forces generated by the optical conveyor were obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form when a mask of two annular pupils is illuminated by a radially polarized Hermite-Gauss beam. Trajectories, in both the transverse plane and the longitudinal direction, are analyzed showing that the behavior of the optical conveyor can be optimized by conveniently choosing the configuration of the mask of the two annular pupils (inner and outer radius of the two rings) in order to trap and transport all particles at the focal plane.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007).
    [CrossRef] [PubMed]
  5. M. Siler, P. Jakl, O. Brzobohaty, P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012).
    [CrossRef] [PubMed]
  6. N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  17. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (3)

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38, 28–30 (2013).
[CrossRef] [PubMed]

2012 (3)

2010 (1)

2008 (2)

T. A. Nieminen, N. R. Heckenberg, H. Rubinstein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
[CrossRef] [PubMed]

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

2007 (1)

2006 (1)

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

2005 (1)

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

2004 (1)

2002 (1)

P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002).
[CrossRef]

2000 (2)

1998 (1)

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation presure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Ashkin, A.

A. Ashkin, “Acceleration and trapping of particles by radiation presure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Borromeo, M.

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

Brown, T.

Brzobohaty, O.

Chantada, L.

Chaumet, P. C.

Chen, J.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Chen, Z.

Choudhury, A.

Cizmar, T.

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Dholakia, H.

Dholakia, K.

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Ding, W.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Dogariu, A.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Garces-Chavez, V.

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Gómez-Medina, R.

Grier, D. G.

D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

Heckenberg, N. R.

Iati, M.

Jakl, P.

Jonas, A.

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

Jones, P. H.

Kajorndenukul, V.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Lin, Z.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Liu, S.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Marago, O.

Marchesoni, F.

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

McGloin, D.

Milne, G.

Nieminen, T. A.

Nieto-Vesperinas, M.

Qiu, C.-W.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).

Reimann, P.

P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Rubinstein-Dunlop, H.

Ruffner, D. B.

D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

Sáenz, J.

Saija, R.

Sergides, M.

Sheppard, C. J. R.

Siler, M.

M. Siler, P. Jakl, O. Brzobohaty, P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012).
[CrossRef] [PubMed]

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

Skelton, S. E.

Sukhov, S.

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

Volke-Sepulveda, K.

Wang, N.

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Youngworth, K. S.

Zemanek, P.

M. Siler, P. Jakl, O. Brzobohaty, P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012).
[CrossRef] [PubMed]

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007).
[CrossRef] [PubMed]

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Zhao, D.

Appl. Opt. (1)

Appl. Phys. B (1)

T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Nat. Photonics (1)

V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013).
[CrossRef]

New J. Phys. (1)

M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Phys. Lett. A (1)

M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998).
[CrossRef]

Phys. Rep. (1)

P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002).
[CrossRef]

Phys. Rev. A (1)

N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[CrossRef]

Phys. Rev. Lett. (2)

D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation presure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Proc. R. Soc. A (1)

B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Other (1)

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).

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Figures (8)

Fig. 1
Fig. 1

Main parameters used in Eqs. (1)(3). The inset, shows |lo(θ)T (θ)| (continuous line) compared to linearly polarized Gaussian beam as apodization function (dashed lines). Values of the used parameters are given in Table 1

Fig. 2
Fig. 2

Normalized Intensity I = ir + iz at time t = 0 as a function of coordinates r and z for the three conveyors configurations (T1,T2 and T3) described in Table 1

Fig. 3
Fig. 3

Normalized Axial Irradiance I(r = 0) = iz at time t = 0s versus z-coordinate for the three conveyors configurations (T1 (blue),T2 (red) and T3 (green)) described in Table 1.

Fig. 4
Fig. 4

Conveyor configuration T1. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

Fig. 5
Fig. 5

Conveyor configuration T2. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

Fig. 6
Fig. 6

Conveyor configuration T2 for spherical particles with radius of a = λ/30. (a), (b) and (c) show the temporal dependence of the axial uncertainty for 3 different starting positions. (d), (e) and (f) represent the corresponding temporal dependence of radial position uncertainty for 3 different starting positions. (a) and (d) correspond to an initial radial positions ((r(0) = 0)), (b) and (e) to (r(0) = 1λ) and (c) and (f) correspond to an initial position (r(0) = 1.5λ). The three different intensities were pointed out by color code (13.27 (mW/cm2) in blue, 132.7 (mW/cm2) in green and 1327 (mW/cm2) in red).

Fig. 7
Fig. 7

Conveyor configuration T2 for spherical particles with radius of a = λ/60. (a), (b) and (c) show the temporal dependence of the axial uncertainty for 3 different starting positions. (d), (e) and (f) represent the corresponding temporal dependence of radial position uncertainty for 3 different starting positions. (a) and (d) correspond to an initial radial positions ((r(0) = 0)), (b) and (e) to (r(0) = 1λ) and (c) and (f) correspond to an initial position (r(0) = 1.5λ). The three different intensities were pointed out by color code (13.27 (mW/cm2) in blue, 132.7 (mW/cm2) in green and 1327 (mW/cm2) in red).

Fig. 8
Fig. 8

Conveyor configuration T3. (a) and (c) show the temporal dependence of the axial and radial position respectively for spherical particles with radius of a = λ/30 as a function of light intensity for 3 different starting positions. (b) and (d) represent the temporal dependence of axial and radial position respectively for spherical particles with radius a = λ/60 as a function of light intensity for 3 different starting positions. The different initial radial positions were pointed out by color code ((r(0) = 0) in blue, (r(0) = 1λ) in green and (r(0) = 1.5λ) in red), while the three different intensities simulated were indicated by dotted lines (13.27 (mW/cm2)), dashed lines (132.7 (mW/cm2)) and continuous lines (1327 (mW/cm2)).

Tables (1)

Tables Icon

Table 1 Values of the parameters used in the numerical simulations and theoretical axial range

Equations (23)

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e r = A 0 α h r ( θ ) l 0 ( θ ) T ( θ ) J 1 ( k r Sin ( θ ) ) Exp ( i k z Cos ( θ ) ) d θ e ψ = 0 e z = i A 0 α h z ( θ ) l 0 ( θ ) T ( θ ) J 0 ( k r Sin ( θ ) ) Exp ( i k z Cos ( θ ) ) d θ
l 0 ( θ ) = Exp ( β 2 Sin ( θ ) 2 Sin ( α ) 2 ) β Sin ( θ ) Sin ( α )
T ( θ ) = { g 1 θ 1 δ 1 2 θ θ 1 + δ 1 2 g 2 θ 2 δ 2 2 θ θ 2 + δ 2 2 0 otherwise
Exp ( i k z Cos ( θ ) ) Exp ( i k z Cos ( θ l ) Sin ( θ l ) ( θ θ l ) ) )
e r ( r , z ) = A l = 1 2 g l h ^ r l J 1 ( k s l r ) Exp ( i k c l z ) Sin c ( k s l z δ l 2 ) δ l e ψ = 0 e z ( r , z ) = i A l = 1 2 g l h ^ z l J 0 ( k s l r ) Exp ( i k c l z ) Sin c ( k s l z δ l 2 ) δ l
e r ( r , z ) = i r ( r , z ) Exp ( i φ r ( r , z ) ) e ψ = 0 e z ( r , z ) = i z ( r , z ) Exp ( i φ z ( r , z ) ) ,
I = | e r | 2 + | e z | 2 = i r + i z
i r = ( δ 1 2 J 1 ( k s 1 r ) 2 h ^ r 1 2 Sin c ( δ 1 k s 1 z 2 ) 2 + δ 2 2 J 1 ( k s 2 r ) 2 h ^ r 2 2 Sin c ( δ 2 k s 2 z 2 ) 2 ) + 2 δ 1 δ 2 J 1 ( k s 1 r ) J 1 ( k s 2 r ) Cos ( ξ t + ( k s 2 k s 1 ) z ) h ^ r 1 h ^ r 2 Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 k s 2 z 2 )
i z = ( δ 1 2 J 0 ( k s 1 r ) 2 h ^ z 1 2 Sin c ( δ 1 k s 1 z 2 ) 2 + δ 2 2 J 0 ( k s 2 r ) 2 h ^ z 2 2 Sin c ( δ 2 k s 2 z 2 ) 2 ) + 2 δ 1 δ 2 J 0 ( k s 1 r ) J 0 ( k s 2 r ) Cos ( ξ t + ( k s 2 k s 1 ) z ) h ^ z 1 h ^ z 2 Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 k s 2 z 2 )
φ r = Tan 1 ( δ 1 J 1 ( k s 1 r ) h ^ r 1 Sin ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 1 ( k s 2 r ) h ^ r 2 Sin ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) δ 1 J 1 ( k s 1 r ) h ^ r 1 Cos ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 1 ( k s 2 r ) h ^ r 2 Cos ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) )
φ z = Tan 1 ( δ 1 J 0 ( k s 1 r ) h ^ z 1 Cos ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 0 ( k s 2 r ) h ^ z 2 Cos ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) δ 1 J 0 ( k s 1 r ) h ^ z 1 Sin ( k c 1 z ) Sin c ( δ 1 k s 1 z 2 ) + δ 2 J 0 ( k s 2 r ) h ^ z 2 Sin ( ξ t + k c 2 z ) Sin c ( δ 2 k s 2 z 2 ) )
F r = 1 2 [ α ^ ( e r e r * r + e ψ e ψ * r + e z e z * r ) ] F ψ = 1 2 [ α ^ ( e r e r * r ψ + e ψ e ψ * r ψ + e z e z * r ψ ) ] F z = 1 2 [ α ^ ( e r e r * z + e ψ e ψ * z + e z e z * z ) ]
α R = 9 a 3 ( ε p ε ) ε ( ε p + 2 ε ) ( 9 + 4 a 6 k 0 6 ) ε p 2 + 4 ( 9 2 a 6 k 0 6 ) ε ε p + 4 ( 9 + a 6 k 0 6 ) ε 2 α I = 6 a 6 k 0 3 ( ε p ε ) 2 ε ( 9 + 4 a 6 k 0 6 ) ε p 2 + 4 ( 9 2 a 6 k 0 6 ) ε ε p + 4 ( 9 + a 6 k 0 6 ) ε 2
F r = 1 4 α R I r + α I 2 ( φ r r i r + φ ψ r i ψ + φ z r i z ) F ψ = 0 F z = 1 4 α R I z + α I 2 ( φ r z i r + φ ψ z i ψ + φ z z i z )
F = α R 4 I + α I 2 u i u φ u
m d 2 R d t 2 = F ( R ( t ) ) γ d R d t + ( t )
I ( z ) = ( δ 1 2 h ^ z 1 2 Sin c ( δ 1 k s 1 z 2 ) 2 + δ 2 2 h ^ z 2 2 Sin c ( δ 2 k s 2 z 2 ) 2 ) + 2 Cos ( ξ t + ( k s 2 k s 1 ) z ) h ^ z 1 h ^ z 2 Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 k s 2 z 2 )
δ 2 e = h ^ z 1 δ 1 h ^ z 2 = δ 1 l 0 ( θ 1 ) Cos ( θ 1 ) Sin ( θ 1 ) 2 l 0 ( θ 2 ) Cos ( θ 2 ) Sin ( θ 2 ) 2
I ( z ) = δ 1 2 h ^ z 1 2 ( Sin c ( δ 1 k s 1 z 2 ) 2 + Sin c ( δ 2 e k s 2 z 2 ) 2 ) + δ 1 2 h ^ z 1 2 ( 2 Cos ( ξ t + ( k s 2 k s 1 ) z ) Sin c ( δ 1 k s 1 z 2 ) Sin c ( δ 2 e k s 2 z 2 ) )
Δ z = λ Max ( δ 2 e Sin ( θ 2 ) , δ 1 Sin ( θ 1 ) )
θ 1 = Sin 1 ( Sin ( α ) 2 β )
β = Sin ( α ) 2 Sin ( α δ 1 2 )
Δ z = λ δ 2 e Sin ( θ 2 )

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