Abstract

In the nano-fabrication field, high precision displacement tracing of the fabricating beam is extensively required. Due to the coherence noise and the sensitivity to environmental disturbances, the commonly used measuring methods base on the laser interferometry are unstable. In this paper, a high-precision measuring method for the three-dimensional displacements is developed based on the low coherence interferometry. The interferogram at a particular location is unique and distinctive, which can be applied as a benchmark for the absolute measurement of positions. Consequently, interferograms are continuously acquired during the movement of the nano-stage, then the quantitative relationship between the stage position/tilt and the interferograms is established by analytic calculation. Besides, the influence of random errors can be suppressed by the averaging effect of the least squares fitting, thereby enhancing the precision by more than an order of magnitude compared with traditional methods. The measuring uncertainty is derived and the impacts of the main influencing factors are investigated. Experiments demonstrate that the measuring repeatability can achieve 1.16 nm. As a result, the proposed method can reliably obtain the absolute position and three dimensional trajectory of the nano-stage, and it is of significance to improve the reliability of nano-measurement and fabrication.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

N. N. Phan, H. H. Le, D. C. Duong, and D. Van Ta, “Measurement of nanoscale displacements using a Mirau white-light interference microscope and an inclined flat surface,” Opt. Eng. 58(06), 064106 (2019).
[Crossref]

2018 (4)

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

E. Zhang, B. Chen, H. Zheng, L. Yan, and X. Teng, “Laser heterodyne interferometer with rotational error compensation for precision displacement measurement,” Opt. Express 26(1), 90–98 (2018).
[Crossref]

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

W. Yang, X. Liu, W. Lu, and X. Guo, “Influence of probe dynamic characteristics on the scanning speed for white light interference based AFM,” Precis. Eng. 51, 348–352 (2018).
[Crossref]

2017 (2)

2016 (1)

2015 (1)

2014 (1)

2013 (2)

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150 m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013).
[Crossref]

J. Lee and G. Jiang, “Displacement measurement using a wavelength- phase-shifting grating interferometer,” Opt. Express 21(21), 25553–25564 (2013).
[Crossref]

2012 (2)

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

2011 (1)

2009 (3)

G. de Vine, D. S. Rabeling, B. J. J. Slagmolen, T. T.-Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express 17(2), 828–837 (2009).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Y. Bitou, “High-accuracy displacement metrology and control using a dual Fabry-Perot cavity with an optical frequency comb generator,” Precis. Eng. 33(2), 187–193 (2009).
[Crossref]

2006 (1)

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246, 62460G (2006).
[Crossref]

2005 (1)

1996 (2)

1992 (1)

W. V. Sorin and D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry - IEEE Photonics Technology Letters,” IEEE Photonics Technol. Lett. 4(12), 1404–1406 (1992).
[Crossref]

1991 (1)

Abidi, B.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246, 62460G (2006).
[Crossref]

Abidi, M.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246, 62460G (2006).
[Crossref]

Baney, D. M.

W. V. Sorin and D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry - IEEE Photonics Technology Letters,” IEEE Photonics Technol. Lett. 4(12), 1404–1406 (1992).
[Crossref]

Berkovic, G.

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

Bitou, Y.

Y. Bitou, “High-accuracy displacement metrology and control using a dual Fabry-Perot cavity with an optical frequency comb generator,” Precis. Eng. 33(2), 187–193 (2009).
[Crossref]

Boisrobert, C. Y.

Chang, S.

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

Chen, B.

Chen, F.

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Chen, J.-C.

Chua, S.

Coddington, I.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Coupland, J. M.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

Dändliker, R.

Danielson, B. L.

Davila, A.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

de Vine, G.

Doggaz, N.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246, 62460G (2006).
[Crossref]

Duong, D. C.

N. N. Phan, H. H. Le, D. C. Duong, and D. Van Ta, “Measurement of nanoscale displacements using a Mirau white-light interference microscope and an inclined flat surface,” Opt. Eng. 58(06), 064106 (2019).
[Crossref]

Ekberg, P.

Fang, Y.

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Gao, F.

Gray, S.

Guo, B.

Guo, X.

W. Yang, X. Liu, W. Lu, and X. Guo, “Influence of probe dynamic characteristics on the scanning speed for white light interference based AFM,” Precis. Eng. 51, 348–352 (2018).
[Crossref]

Häusler, G.

Hou, X.

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Hsieh, H.

Hsieh, H.-L.

Hu, C.

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

Huntley, J. M.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

Huo, J.

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Jiang, G.

Jiang, X.

Lam, T. T.-Y.

Larkin, K. G.

Le, H. H.

N. N. Phan, H. H. Le, D. C. Duong, and D. Van Ta, “Measurement of nanoscale displacements using a Mirau white-light interference microscope and an inclined flat surface,” Opt. Eng. 58(06), 064106 (2019).
[Crossref]

Leach, R.

Lee, J.

Lee, J.-Y.

Lerondel, G.

Liu, X.

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

W. Yang, X. Liu, W. Lu, and X. Guo, “Influence of probe dynamic characteristics on the scanning speed for white light interference based AFM,” Precis. Eng. 51, 348–352 (2018).
[Crossref]

Lu, W.

W. Yang, X. Liu, W. Lu, and X. Guo, “Influence of probe dynamic characteristics on the scanning speed for white light interference based AFM,” Precis. Eng. 51, 348–352 (2018).
[Crossref]

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

Lu, Y.

Martin, H.

Matsumoto, H.

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150 m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013).
[Crossref]

McClelland, D. E.

Nenadovic, L.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Newbury, N. R.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Pallikarakis, C.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

Pan, S.

Pavlícek, P.

Phan, N. N.

N. N. Phan, H. H. Le, D. C. Duong, and D. Van Ta, “Measurement of nanoscale displacements using a Mirau white-light interference microscope and an inclined flat surface,” Opt. Eng. 58(06), 064106 (2019).
[Crossref]

Rabeling, D. S.

Ruiz, P. D.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

Schnell, U.

Shaddock, D. A.

Shafir, E.

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

Slagmolen, B. J. J.

Sorin, W. V.

W. V. Sorin and D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry - IEEE Photonics Technology Letters,” IEEE Photonics Technol. Lett. 4(12), 1404–1406 (1992).
[Crossref]

Su, R.

Swann, W. C.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Takahashi, S.

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150 m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013).
[Crossref]

Takamasu, K.

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150 m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013).
[Crossref]

Tan, Y.

Tang, D.

Teng, X.

Van Ta, D.

N. N. Phan, H. H. Le, D. C. Duong, and D. Van Ta, “Measurement of nanoscale displacements using a Mirau white-light interference microscope and an inclined flat surface,” Opt. Eng. 58(06), 064106 (2019).
[Crossref]

Wang, X.

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150 m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013).
[Crossref]

Williamson, J.

Wuchenich, D. M.

Yan, L.

Yang, Q.

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Yang, W.

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

W. Yang, X. Liu, W. Lu, and X. Guo, “Influence of probe dynamic characteristics on the scanning speed for white light interference based AFM,” Precis. Eng. 51, 348–352 (2018).
[Crossref]

Yao, Y.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246, 62460G (2006).
[Crossref]

Yong, J.

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Yu, N.

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

Zhang, E.

Zhang, J.

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Zhang, S.

Zheng, H.

Zhu, K.

Zygo,

Zygo, “Absolute Position Sensors,” https://www.zygo.com/?/met/absolutepositionmeasurement/ .

Adv. Mater. Interfaces (1)

Y. Fang, J. Yong, F. Chen, J. Huo, Q. Yang, J. Zhang, and X. Hou, “Bioinspired Fabrication of Bi/Tridirectionally Anisotropic Sliding Superhydrophobic PDMS Surfaces by Femtosecond Laser,” Adv. Mater. Interfaces 5(6), 1701245 (2018).
[Crossref]

Adv. Opt. Photonics (1)

G. Berkovic and E. Shafir, “Optical methods for distance and displacement measurements,” Adv. Opt. Photonics 4(4), 441–471 (2012).
[Crossref]

Appl. Opt. (3)

IEEE Photonics Technol. Lett. (1)

W. V. Sorin and D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry - IEEE Photonics Technology Letters,” IEEE Photonics Technol. Lett. 4(12), 1404–1406 (1992).
[Crossref]

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

Nat. Photonics (1)

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Opt. Eng. (1)

N. N. Phan, H. H. Le, D. C. Duong, and D. Van Ta, “Measurement of nanoscale displacements using a Mirau white-light interference microscope and an inclined flat surface,” Opt. Eng. 58(06), 064106 (2019).
[Crossref]

Opt. Express (7)

Opt. Lasers Eng. (2)

C. Hu, X. Liu, W. Yang, W. Lu, N. Yu, and S. Chang, “Improved zero-order fringe positioning algorithms in white light interference based atomic force microscopy,” Opt. Lasers Eng. 100, 71–76 (2018).
[Crossref]

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

Opt. Lett. (1)

Optica (1)

Precis. Eng. (3)

W. Yang, X. Liu, W. Lu, and X. Guo, “Influence of probe dynamic characteristics on the scanning speed for white light interference based AFM,” Precis. Eng. 51, 348–352 (2018).
[Crossref]

X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150 m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013).
[Crossref]

Y. Bitou, “High-accuracy displacement metrology and control using a dual Fabry-Perot cavity with an optical frequency comb generator,” Precis. Eng. 33(2), 187–193 (2009).
[Crossref]

Proc. SPIE (1)

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246, 62460G (2006).
[Crossref]

Other (2)

Zygo, “Absolute Position Sensors,” https://www.zygo.com/?/met/absolutepositionmeasurement/ .

R. Leach, Foundamental Principles of Engineering Nanometrology (Elsevier, 2014).

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Figures (10)

Fig. 1.
Fig. 1. The real time displacement tracing system for the femto-second laser fabrication. (a) The LCI system. (b) The interference intensity of LCI. (c) The recorded interferograms.
Fig. 2.
Fig. 2. The flow chart of the system calibration.
Fig. 3.
Fig. 3. Relationship between the measuring uncertainty and OPD.
Fig. 4.
Fig. 4. Influence of the coherence length and OPD to the measuring uncertainty. The central wavelength is λ0=500 nm.
Fig. 5.
Fig. 5. Numerical demonstration results of different system parameters. To make the plots clearer, the red dot lines presenting the solved displacements in (a) and (b) are shifted by 5 µm, and those in (c) and (d) by 0.5 µm.
Fig. 6.
Fig. 6. Monte Carlo testing results. The random errors added are 5% and 10%, respectively.
Fig. 7.
Fig. 7. Three-dimensional displacement tracing. The ideal trajectory is depicted in red, and the actual trajectory in black.
Fig. 8.
Fig. 8. Measurement results of tilts. To make the plots clearer, the black dot line presenting actual tilts are shifted by 0.1 mrad from the actual values. (a) and (b) are the results of triangular mode, and (c) and (d) for sinusoidal mode.
Fig. 9.
Fig. 9. System calibration results. (a) Calculated surface (b) deviation.
Fig. 10.
Fig. 10. Experimental results. (a) presents the displacement measurement result. The black dot line presenting actual displacement is shifted by 0.1 µm to make the plots clearer; (b) illustrates the tilt measurement results; (c) depicts the positioning repeatability.

Tables (3)

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Table 1. Parameter setting

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Table 2. System calibration results and displacement measurement results

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Table 3. Specifications of Monte-Carlo results

Equations (11)

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I ( x , y , h ) = A 1 + A 2 { 1 + e [ 2 π z ( x , y ) + h L c ] 2 cos [ 4 π z ( x , y ) + h λ 0 ] } ,
h ( t ) = k x ( t ) x + k y ( t ) y + d ( t ) ,
x ( s + 1 ) = x ( s ) ( J T J + k E ) 1 J T r ( s ) with r ( s ) = I m I ( x ( s ) ) and J = I x ,
V a r = 1 X Y x = 1 X y = 1 Y [ I ( x , y ) I ¯ ] 2 .
Δ h = Δ I λ 0 L c 2 4 π A 2 e ( 2 π L c h ) 2 | L c 2 sin ( 4 π λ 0 h ) + 8 π λ 0 h cos ( 4 π λ 0 h ) | .
h = arg min k = 1 m | | I k f ( z k + h ) | | 2 with f ( x ) = A 1 + A 2 [ 1 + e ( 2 π x L c ) 2 cos ( 4 π x λ 0 ) ] .
Q = k = 1 m | | I k f ( z k + h ) | | 2 h = k = 1 m 2 [ I k f ( z k + h ) ] f ( z k + h ) d Q | h = k = 1 m 2 f ( z k + h ) d I k + k = 1 m { 2 f 2 ( z k + h ) 2 f ( z k + h ) [ I k f ( z k + h ) ] } d h m = 0 .
d h = k = 1 m f ( z k + h ) d I k k = 1 m f 2 ( z k + h ) .
Δ h m = Δ I H H f 2 ( z + h ) d z m 2 H H H f 2 ( z + h ) d z m 2 H = p L c H H f 2 ( z + h ) d z m 2 H with f 2 ( x ) = 32 π 2 e 2 ( 2 π L c x ) 2 A 2 2 [ ( 4 π 2 L c 4 x 2 1 λ 2 ) cos ( 8 π λ x ) + 4 π L c 2 λ x sin ( 8 π λ x ) + 2 π 2 L c 4 x 2 + 1 λ 2 ] .
Δ h m = λ 0 p L c H m ( 2 π ) 3 4 A 2 e r f ( 8 π h + H L c ) e r f ( 8 π h H L c ) .
I ( x , y , h ) = INT[ I t h e o r e t i c a l ( x , y , h ) + n ( x , y , h ) ] ,

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