Abstract

We propose to use a Nomarski imaging interferometer to measure the out- of-plane displacement field of micro-electro-mechanical systems. It is shown that the measured optical phase arises from both height and slope gradients. By using four integrating buckets, a more efficient approach to unwrap the measured phase is presented, thus making the method well suited for highly curved objects. Slope and height effects are then decoupled by expanding the displacement field on a functions basis, and the inverse transformation is applied to get a displacement field from a measured optical phase map change with a mechanical loading. A measurement reproducibility of approximately 10 pm is achieved, and typical results are shown on a microcantilever under thermal actuation, thereby proving the ability of such a setup to provide a reliable full-field kinematic measurement without surface modification.

© 2006 Optical Society of America

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References

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  1. N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, "Cantilever transducers as a platform for chemical and biological sensors," Rev. Sci. Instrum. 75, 2229-2253 (2004).
    [CrossRef]
  2. T. P. Weihs, S. Hong, J. C. Bravman, and W. D. Nix, "Mechanical deflection of cantilever micro-beams: a new technique for testing the mechanical properties of thin films," J. Mater. Res. 3, 931-942 (1988).
    [CrossRef]
  3. D. T. Read and J. W. Dally, "A new method for measuring the strength and ductility of thin films," J. Mater. Res. 8, 1542-1549 (1993).
    [CrossRef]
  4. M. A. Haque and M. T. A. Saif, "A review of MEMS-based microscale and nanoscale tensile and bending testing," Exp. Mech. 43, 248-255 (2003).
    [CrossRef]
  5. M. Gad-El-Hak, The MEMS Handbook (CRC, 2002).
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. M. Françon, "Polarization interference microscopes," Appl. Optics 3, 1033-1036 (1964).
    [CrossRef]
  9. G. Nomarski, "Microinterféromètre différentiel à ondes polarisées," J. Phys. Radium 16, 9S-11S (1955).
  10. D. L. Lessor, J. S. Hartman, and R. L. Gordon, "Quantitative surface topography determination by Nomarski reflection microscopy. I. Theory," J. Opt. Soc. Am. 69, 357-365 (1979).
    [CrossRef]
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    [CrossRef]
  12. P. Gleyzes, F. Guernet, and A. C. Boccara, "Profilométrie picométrique. II. L'approche multi-détecteur et la détection synchrone multiplexée," J. Opt. (Paris) 26, 251-265 (1995).
    [CrossRef]
  13. P. Gleyzes, A. C. Boccara, and H. Saint-Jalmes, "Multichannel Nomarski microscope with polarization modulation: performance and applications," Opt. Lett. 22, 1529-1531 (1997).
    [CrossRef]
  14. C. Montarou and T. K. Gaylord, "Analysis and design of modified Wollaston prisms," Appl. Opt. 38, 6604-6616 (1999).
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  15. A. Dubois, "Phase-map measurements by interferometry with sinuoidal phase modulation and four integrating buckets," J. Opt. Soc. Am. A 18, 1972-1979 (2001).
    [CrossRef]
  16. R. Cusack, J. M. Huntley, and H. T. Goldrein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 34, 781-789 (1995).
    [CrossRef] [PubMed]
  17. X. Y. He, X. Kang, C. J. Tay, C. Quan, and H. M. Shang, "Proposed algorithm for phase unwrapping," Appl. Opt. 41, 7422-7428 (2002).
    [CrossRef] [PubMed]
  18. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, 1970).
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    [CrossRef]
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    [CrossRef] [PubMed]
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2004 (1)

N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, "Cantilever transducers as a platform for chemical and biological sensors," Rev. Sci. Instrum. 75, 2229-2253 (2004).
[CrossRef]

2003 (2)

M. A. Haque and M. T. A. Saif, "A review of MEMS-based microscale and nanoscale tensile and bending testing," Exp. Mech. 43, 248-255 (2003).
[CrossRef]

P. Aswendt, C.-D. Schmidt, D. Zielke, and S. Schubert, "ESPI solution for noncontacting MEMS-on-wafer testing," Opt. Lasers Eng. 40, 501-515 (2003).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

1999 (1)

1997 (1)

1995 (3)

1994 (1)

P. Gleyzes and A. C. Boccara, "Profilométrie picométrique par interférométrie de polarisation. I. L'approche monodétecteur," J. Opt. (Paris) 25, 207-224 (1994).
[CrossRef]

1993 (1)

D. T. Read and J. W. Dally, "A new method for measuring the strength and ductility of thin films," J. Mater. Res. 8, 1542-1549 (1993).
[CrossRef]

1991 (1)

1988 (1)

T. P. Weihs, S. Hong, J. C. Bravman, and W. D. Nix, "Mechanical deflection of cantilever micro-beams: a new technique for testing the mechanical properties of thin films," J. Mater. Res. 3, 931-942 (1988).
[CrossRef]

1979 (2)

1964 (1)

M. Françon, "Polarization interference microscopes," Appl. Optics 3, 1033-1036 (1964).
[CrossRef]

1955 (1)

G. Nomarski, "Microinterféromètre différentiel à ondes polarisées," J. Phys. Radium 16, 9S-11S (1955).

Aswendt, P.

P. Aswendt, C.-D. Schmidt, D. Zielke, and S. Schubert, "ESPI solution for noncontacting MEMS-on-wafer testing," Opt. Lasers Eng. 40, 501-515 (2003).
[CrossRef]

Boccara, A. C.

A. Dubois, J. Selb, L. Vabre, and A. C. Boccara, "Phase measurements with wide-aperture interferometers," Appl. Opt. 39, 2326-2331 (2000).
[CrossRef]

P. Gleyzes, A. C. Boccara, and H. Saint-Jalmes, "Multichannel Nomarski microscope with polarization modulation: performance and applications," Opt. Lett. 22, 1529-1531 (1997).
[CrossRef]

P. Gleyzes, F. Guernet, and A. C. Boccara, "Profilométrie picométrique. II. L'approche multi-détecteur et la détection synchrone multiplexée," J. Opt. (Paris) 26, 251-265 (1995).
[CrossRef]

P. Gleyzes and A. C. Boccara, "Profilométrie picométrique par interférométrie de polarisation. I. L'approche monodétecteur," J. Opt. (Paris) 25, 207-224 (1994).
[CrossRef]

Bravman, J. C.

T. P. Weihs, S. Hong, J. C. Bravman, and W. D. Nix, "Mechanical deflection of cantilever micro-beams: a new technique for testing the mechanical properties of thin films," J. Mater. Res. 3, 931-942 (1988).
[CrossRef]

Cusack, R.

Dally, J. W.

D. T. Read and J. W. Dally, "A new method for measuring the strength and ductility of thin films," J. Mater. Res. 8, 1542-1549 (1993).
[CrossRef]

Datskos, P. G.

N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, "Cantilever transducers as a platform for chemical and biological sensors," Rev. Sci. Instrum. 75, 2229-2253 (2004).
[CrossRef]

Dubois, A.

Elssner, K.-E.

Françon, M.

M. Françon, "Polarization interference microscopes," Appl. Optics 3, 1033-1036 (1964).
[CrossRef]

Gad-El-Hak, M.

M. Gad-El-Hak, The MEMS Handbook (CRC, 2002).

Gaylord, T. K.

Gleyzes, P.

P. Gleyzes, A. C. Boccara, and H. Saint-Jalmes, "Multichannel Nomarski microscope with polarization modulation: performance and applications," Opt. Lett. 22, 1529-1531 (1997).
[CrossRef]

P. Gleyzes, F. Guernet, and A. C. Boccara, "Profilométrie picométrique. II. L'approche multi-détecteur et la détection synchrone multiplexée," J. Opt. (Paris) 26, 251-265 (1995).
[CrossRef]

P. Gleyzes and A. C. Boccara, "Profilométrie picométrique par interférométrie de polarisation. I. L'approche monodétecteur," J. Opt. (Paris) 25, 207-224 (1994).
[CrossRef]

Goldrein, H. T.

Goodier, J. N.

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, 1970).

Gordon, R. L.

Guernet, F.

P. Gleyzes, F. Guernet, and A. C. Boccara, "Profilométrie picométrique. II. L'approche multi-détecteur et la détection synchrone multiplexée," J. Opt. (Paris) 26, 251-265 (1995).
[CrossRef]

Haque, M. A.

M. A. Haque and M. T. A. Saif, "A review of MEMS-based microscale and nanoscale tensile and bending testing," Exp. Mech. 43, 248-255 (2003).
[CrossRef]

Hartman, J. S.

He, X. Y.

Hong, S.

T. P. Weihs, S. Hong, J. C. Bravman, and W. D. Nix, "Mechanical deflection of cantilever micro-beams: a new technique for testing the mechanical properties of thin films," J. Mater. Res. 3, 931-942 (1988).
[CrossRef]

Hung, Y. Y.

Huntley, J. M.

Kang, X.

Larkin, K. G.

Lavrik, N. V.

N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, "Cantilever transducers as a platform for chemical and biological sensors," Rev. Sci. Instrum. 75, 2229-2253 (2004).
[CrossRef]

Lessor, D. L.

Liang, C. Y.

Montarou, C.

Nix, W. D.

T. P. Weihs, S. Hong, J. C. Bravman, and W. D. Nix, "Mechanical deflection of cantilever micro-beams: a new technique for testing the mechanical properties of thin films," J. Mater. Res. 3, 931-942 (1988).
[CrossRef]

Nomarski, G.

G. Nomarski, "Microinterféromètre différentiel à ondes polarisées," J. Phys. Radium 16, 9S-11S (1955).

Quan, C.

Read, D. T.

D. T. Read and J. W. Dally, "A new method for measuring the strength and ductility of thin films," J. Mater. Res. 8, 1542-1549 (1993).
[CrossRef]

Saif, M. T. A.

M. A. Haque and M. T. A. Saif, "A review of MEMS-based microscale and nanoscale tensile and bending testing," Exp. Mech. 43, 248-255 (2003).
[CrossRef]

Saint-Jalmes, H.

Schmidt, C.-D.

P. Aswendt, C.-D. Schmidt, D. Zielke, and S. Schubert, "ESPI solution for noncontacting MEMS-on-wafer testing," Opt. Lasers Eng. 40, 501-515 (2003).
[CrossRef]

Schubert, S.

P. Aswendt, C.-D. Schmidt, D. Zielke, and S. Schubert, "ESPI solution for noncontacting MEMS-on-wafer testing," Opt. Lasers Eng. 40, 501-515 (2003).
[CrossRef]

Schulz, G.

Selb, J.

Sepaniak, M. J.

N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, "Cantilever transducers as a platform for chemical and biological sensors," Rev. Sci. Instrum. 75, 2229-2253 (2004).
[CrossRef]

Shang, H. M.

Sheppard, C. R. J.

Tay, C. J.

Timoshenko, S. P.

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, 1970).

Vabre, L.

Weihs, T. P.

T. P. Weihs, S. Hong, J. C. Bravman, and W. D. Nix, "Mechanical deflection of cantilever micro-beams: a new technique for testing the mechanical properties of thin films," J. Mater. Res. 3, 931-942 (1988).
[CrossRef]

Zielke, D.

P. Aswendt, C.-D. Schmidt, D. Zielke, and S. Schubert, "ESPI solution for noncontacting MEMS-on-wafer testing," Opt. Lasers Eng. 40, 501-515 (2003).
[CrossRef]

Appl. Opt. (7)

Appl. Optics (1)

M. Françon, "Polarization interference microscopes," Appl. Optics 3, 1033-1036 (1964).
[CrossRef]

Exp. Mech. (1)

M. A. Haque and M. T. A. Saif, "A review of MEMS-based microscale and nanoscale tensile and bending testing," Exp. Mech. 43, 248-255 (2003).
[CrossRef]

J. Mater. Res. (2)

T. P. Weihs, S. Hong, J. C. Bravman, and W. D. Nix, "Mechanical deflection of cantilever micro-beams: a new technique for testing the mechanical properties of thin films," J. Mater. Res. 3, 931-942 (1988).
[CrossRef]

D. T. Read and J. W. Dally, "A new method for measuring the strength and ductility of thin films," J. Mater. Res. 8, 1542-1549 (1993).
[CrossRef]

J. Opt. (2)

P. Gleyzes and A. C. Boccara, "Profilométrie picométrique par interférométrie de polarisation. I. L'approche monodétecteur," J. Opt. (Paris) 25, 207-224 (1994).
[CrossRef]

P. Gleyzes, F. Guernet, and A. C. Boccara, "Profilométrie picométrique. II. L'approche multi-détecteur et la détection synchrone multiplexée," J. Opt. (Paris) 26, 251-265 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. Radium (1)

G. Nomarski, "Microinterféromètre différentiel à ondes polarisées," J. Phys. Radium 16, 9S-11S (1955).

Opt. Lasers Eng. (1)

P. Aswendt, C.-D. Schmidt, D. Zielke, and S. Schubert, "ESPI solution for noncontacting MEMS-on-wafer testing," Opt. Lasers Eng. 40, 501-515 (2003).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, "Cantilever transducers as a platform for chemical and biological sensors," Rev. Sci. Instrum. 75, 2229-2253 (2004).
[CrossRef]

Other (2)

M. Gad-El-Hak, The MEMS Handbook (CRC, 2002).

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, 1970).

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Figures (12)

Fig. 1
Fig. 1

(Color online) Schematic view of the basic interferential microscopy imaging setup.

Fig. 2
Fig. 2

Typical interference pattern obtained in water with two 70   μm × 20   μm × 0.84   μm microcantilevers and a shear distance d 50   μm   ( NA = 0.3 ) .

Fig. 3
Fig. 3

(Color online) Schematic view of a Wollaston prism.

Fig. 4
Fig. 4

(Color online) Ray tracing for a plane object.

Fig. 5
Fig. 5

(Color online) Ray tracing in the case of a tilted and stepped sample (height Δ z ).

Fig. 6
Fig. 6

(Color online) Ray tracing for a sample subjected to slope variations.

Fig. 7
Fig. 7

(Color online) Calibration measurement of the mean phase induced by a sample as a function of its tilt.

Fig. 8
Fig. 8

(Color online) Typical wrapped phase map obtained in water with two 70   μm × 20   μm × 0 .84   μm microcantilevers and a shear distance d = 53.4   μm ( NA = 0.3 ) .

Fig. 9
Fig. 9

Phase unwrapping principle.

Fig. 10
Fig. 10

Typical unwrapped phase map obtained in water with two 70   μm × 20   μm × 0 .84   μm microcantilevers and a shear distance d = 53.4   μm   ( NA = 0.3 ) .

Fig. 11
Fig. 11

(Color online) Estimation of the reproducibility on the measurement of a differential topography as a function of the exposure time.

Fig. 12
Fig. 12

(Color online) (a) Measured phase map change when the cantilever is subjected to a bimaterial effect. (b) Displacement field calculated from the measured phase map change shown in (a).

Equations (69)

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E p p = E 0 t p ,
E p q = E 0 ϵ p t p .
I = I p + I q .
I = I 0 + A cos ( ϕ ) ,
I 0 = | E 0 t a t n t p | 2 2 ( 1 + ϵ a 2 ) ( 1 + ϵ p 2 ) ,
A = | E 0 t a t n t p | 2 2 ( 1 - ϵ a 2 ) ( 1 - ϵ p 2 ) .
A I 0 = ( 1 ϵ a 2 ) ( 1 ϵ p 2 ) ( 1 + ϵ a 2 ) ( 1 + ϵ p 2 )
ϕ = ϕ 0 + ϕ m .
n E sin ( β TM ) = n o sin ( θ ) ,
n o sin ( β TE ) = n E sin ( θ ) ,
sin ( β air TM ) = n E sin ( β TM θ ) ,
sin ( β air TE ) = n o sin ( β TE θ ) .
β air TM β air TE 2 ( n o n E ) θ .
l TE ( y , α 1 ) n E ( h 2 + y θ ) + n o ( h 2 y θ ) + h α 1 n 1 θ 2 × ( n o n E + n E n o )
l TM ( y , α 1 ) n E ( h 2 y θ ) + n o ( h 2 + y θ ) + h α 1 n 1 θ 2 × [ n E n o ( 2 n E n o ) + 1 ] .
ϕ W 0 = 2 π λ ( l a r TE l a r TM ) = 2 π λ θ [ 4 ( n E n o ) T w + h α 1 n 1 × ( n o n E + n E n o ( n E n o 1 ) 1 ) ] .
ϕ PAS 8 π λ θ α 1 δ PAS ( n E n o ) ,
ϕ h = 2 π λ n × ( A B + B C ) = 4 π λ n Δ z cos ( α ) ,
ϕ h 4 π n Δ z λ .
ϕ h = 4 π n Δ z ι λ ,
y R , TE = y + f tan ( 2 γ TE ) 1 + y f tan ( 2 γ TE ) ,
y R , TM = y + f tan ( 2 γ TM ) 1 + y f tan ( 2 γ TM ) ,
l a r TE l TE ( T w + y , 0 ) + l TE ( T w + y R , TE , 0 ) ,
l a r TM l TM ( T w + y , 0 ) + l TM ( T w + y R , TM , 0 ) ,
ϕ W = 2 π λ ( l a r TE l a r TM ) = 4 π λ ( n E n o ) tan ( θ ) [ 1 + ( y f ) 2 ] × f ( γ TE + γ TM ) .
ϕ W = ϕ ori + ϕ tilt = ϕ W γ ( γ TE γ TM ) + ϕ tilt ,
ϕ tilt = 2 ϕ W γ γ TM .
2 ϕ W γ = 1.1 × 10 3 .
ϕ = ϕ W 0 ( T w ) + ϕ PAS ( δ PAS ) + ϕ h ( Δ z ) + ϕ W ( γ TE , γ TM ) ,
I ( l , m , t ) = I 0 + A cos [ ϕ ( l , m ) + ψ mod ( t ) ] ,
ψ mod ( t ) = ψ 0 sin ( 2 π f mod t + θ mod ) .
E p = ( p 1 ) T / 4 p T / 4 I ( t ) d t .
E p = T 4 [ I 0 + A J 0 ( ψ ) cos ( ϕ ) ] + T A cos ( ϕ ) π n = 1 J 2 n ( ψ 0 ) 2 n × { sin [ n p π + 2 n θ mod ] sin [ n ( p 1 ) π + 2 n θ mod ] } T A sin ( ϕ ) π n = 0 J 2 n + 1 ( ψ 0 ) 2 n + 1 × { cos [ π 2 ( 2 n + 1 ) ( p 1 ) + ( 2 n + 1 ) θ mod ] cos [ π 2 ( 2 n + 1 ) p + ( 2 n + 1 ) θ mod ] } ,
Σ s = ( E 1 E 2 E 3 + E 4 ) = 4 T A π Γ s sin ( ϕ ) , Σ c = ( E 1 E 2 + E 3 E 4 ) = 4 T A π Γ c cos ( ϕ ) .
Γ s = n = 0 ( 1 ) n J 2 n + 1 ( ψ 0 ) 2 n + 1 sin [ ( 2 n + 1 ) θ mod ] , Γ c = n = 0 J 4 n + 2 ( ψ 0 ) 2 n + 1 sin [ 2 ( 2 n + 1 ) θ mod ] .
ϒ 2 = Σ c 2 + Σ s 2
P = E ,
P t = [ T I 0 4 ,  T A π cos ( ϕ ) ,  T A π sin ( ϕ ) ] .
E t = [ E 1 ,  E 2 ,  E 3 ,  E 4 ] .
= [ 1 c ( 1 , ψ 0 , θ mod ) s ( 1 , ψ 0 , θ mod ) 1 c ( 2 , ψ 0 , θ mod ) s ( 2 , ψ 0 , θ mod ) 1 c ( 3 , ψ 0 , θ mod ) s ( 3 , ψ 0 , θ mod ) 1 c ( 4 , ψ 0 , θ mod ) s ( 4 , ψ 0 , θ mod ) ] ,
c ( p , ψ 0 , θ mod ) = n = 1 J 2 n ( ψ 0 ) 2 n { sin ( n p π + 2 n θ mod ) sin [ n ( p 1 ) π + 2 n θ mod ] }
s ( p , ψ 0 , θ mod ) = n = 0 J 2 n + 1 ( ψ 0 ) 2 n + 1 × { cos [ π 2 ( 2 n + 1 ) ( p 1 ) + ( 2 n + 1 ) θ mod ] cos [ π 2 ( 2 n + 1 ) p + ( 2 n + 1 ) θ mod ] } .
η 2 ( P ) = ( P E ) t ( P E ) ,
t ℳP sol = t E .
( X , Y ) = C [ cos ( ϕ ) , sin ( ϕ ) ] ,
ϕ p = arctan 2 ( Y , X ) .
tan [ ϕ 2 ϕ 1 ] = X 2 sin ( ϕ 1 ) + Y 2 cos ( ϕ 1 ) X 2 cos ( ϕ 1 ) + Y 2 sin ( ϕ 1 )
ϕ ( l , m ) = ϕ d ( l , m ) + b ( l , m ) ,
ϕ + ( l , m ) = ϕ d ( l , m ) + b + ( l , m ) ,
( ϕ + ϕ - ) ( l , m ) = ( b + b ) ( l , m ) .
ϕ ( x , y ) = ϕ 0 ( x , y ) + ϕ tilt + Φ ( x , y ) .
Φ ( x , y ) = 4 π n ι λ [ z ( x , y + d 2 ) z ( x , y d 2 ) ] + ϕ W γ × [ z y ( x , y + d 2 ) z y ( x , y d 2 ) ] ,
z ( x , y ) = s μ s z s ( x , y ) ,
ϕ ref ( x , y ) = ϕ 0 ( x , y ) + ϕ tilt + s μ s Φ s ( x , y ) .
z ( x , y ) + w ( x , y ) = s ( μ s + v s ) z s ( x , y ) ,
w ( x , y ) = s v s z s ( x , y ) .
η v 2 ( v ) = [ s v s Φ s ( x , y ) ( ϕ w ϕ ref ) ( x , y ) ] 2 d x d y
N v = F ,
N s i = Φ s ( x , y ) Φ i ( x , y ) d x d y ,
F s = Φ s ( x , y ) ( ϕ w ϕ ref ) d x d y .
dI ( α ) = ( dI 0 + d A cos [ 4 n π Δ z λ cos ( α ) + ψ ] ) d α ,
I = I 0 + A 2 sin 2 ( α max ) 0 α max cos [ 4 n π Δ z λ cos ( α ) + ψ ] P ( α ) 2 × sin ( α ) d α .
N A = n sin ( α max ) ,
P ( α ) = [ cos ( α ) ] m ,
I = I 0 + A m F N A , m ( Δ z , ψ ) .
F N A , m ( Δ z , ψ ) = 2 sin 2 ( α max ) 0 α max cos ( 2 k Δ z cos ( α ) + ψ ) × P ( α ) 2 m sin ( α ) d α cos ( 2 k Δ z + ψ ) + [ 1 - 2 m 4 cos ( 2 k Δ z + ψ ) + 1 2 sin ( 2 k Δ z + ψ ) k Δ z ] α max 2 +
ϵ inter   f = 1 i inter   f π .
r i = i 10 10 i inter   f ,
ϕ h = 4 π n Δ z ι λ ,

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