Abstract

We study optical spectral filter synthesis with arrays of pistonactuated micro-mirrors. We propose two algorithms for the calculation of the positions of the micro-mirrors, giving us control of both the amplitude and phase of the synthetic filter. Both algorithms for filter synthesis are explored in an analytic version and in numerical searches for the least deviations between the target and the synthesized filter. We measure the quality of the filter both in terms of the deviations and in filter transmissivity, and present results of numerical simulations for a wide selection of target filters. We find that numerical searches can sometimes yield considerable improvement in the filter synthesis compared to the analytic approximation.

© 2006 Optical Society of America

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  1. R. Belikov, Diffraction-Based Optical Filtering: Theory and Implementation with MEMS, (Doctoral Dissertation, Stanford University, UMI Dissertation Publishing, 2005).
  2. MichealB. Sinclair, Micheal A. Butler, Anthony J. Ricco, and Stephen D. Senturia, "Synthetic spectra: a tool for correlation spectroscopy," Appl. Opt. 36, 3342-3348 (1997).
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    [CrossRef]
  4. M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
    [CrossRef]
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  6. P. M. Birch, R. Young, D. Budgett, and C. Chatwin, "Two-pixel computer-generated hologram with a zero-twist nematic liquid-crystal spatial light modulator," Opt. Lett. 25, 1013-1015 (2000).
    [CrossRef]
  7. J. M. Florence, and R. D. Juday, "Full Complex Spatial Filtering with a Phase Mostly DMD," Proc. SPIE 1558, 487-498 (1991).
    [CrossRef]
  8. R. Belikov, "Programmable Optical Wavelength Filter Based on Diffraction from a 2-D MEMS Micromirror Arrays," in Proceedings of CLEO 2003, (Baltimore, Maryland, 2003).
  9. X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
    [CrossRef]
  10. G. Zhou, F.E.H. Tay and F.S. Chau, "Design of diffractive optical elements for synthetic spectra," Opt. Express 11, 1392-1399 (2003).
    [CrossRef] [PubMed]
  11. A. M. Wiener, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-60 (2000).
    [CrossRef]
  12. R. Belikov, C. Antoine-Snowden, O. Solgaard, "Femtosecond Direct Space-to-Time Pulse Shaping with MEMS Micromirror Arrays," in IEEE/LEOS International Conference on Optical MEMS, (Hawaii, HI, August 2003).
  13. R. Belikov and O. Solgaard, "Optical wavelength filtering by diffraction from a surface relief," Opt. Lett. 28, 447-49 (2003).
    [CrossRef] [PubMed]
  14. M. Lacolle, Optical spectral filtering with segmented diffractive optical elements, (Dissertation for the Degree of Dr. Scient., Faculty of Mathematics and Natural Sciences, University of Oslo, Unipub, 2006).
  15. O. Solgaard, F.S.A. Sandejas and D.M. Bloom, "Deformable grating optical modulator," Opt. Lett. 17, 688-90 (1992).
    [CrossRef] [PubMed]
  16. J.P.llebach and B. Liu, "Minimax spectrum shaping with a bandwidth constraint," App. Opt. 14, 3062-72 (1975).

2006

X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
[CrossRef]

2005

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

2004

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

2003

2000

1997

1992

1991

J. M. Florence, and R. D. Juday, "Full Complex Spatial Filtering with a Phase Mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[CrossRef]

1975

J.P.llebach and B. Liu, "Minimax spectrum shaping with a bandwidth constraint," App. Opt. 14, 3062-72 (1975).

1970

Antoine, C.

X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
[CrossRef]

Belikov, R.

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

R. Belikov and O. Solgaard, "Optical wavelength filtering by diffraction from a surface relief," Opt. Lett. 28, 447-49 (2003).
[CrossRef] [PubMed]

Birch, P. M.

Bloom, D.M.

Budgett, D.

Chatwin, C.

Chau, F.S.

Florence, J. M.

J. M. Florence, and R. D. Juday, "Full Complex Spatial Filtering with a Phase Mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[CrossRef]

Johansen, I.R.

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

Johansen, I.-R.

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

Juday, R. D.

J. M. Florence, and R. D. Juday, "Full Complex Spatial Filtering with a Phase Mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[CrossRef]

Lacolle, M.

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

Lee, D.

X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
[CrossRef]

Lee, W. H.

Li, X.

X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
[CrossRef]

Løvhaugen, O.

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

Micheal,

Sagberg, H.

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

Sandejas, F.S.A.

Solgaard, O.

X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
[CrossRef]

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

R. Belikov and O. Solgaard, "Optical wavelength filtering by diffraction from a surface relief," Opt. Lett. 28, 447-49 (2003).
[CrossRef] [PubMed]

O. Solgaard, F.S.A. Sandejas and D.M. Bloom, "Deformable grating optical modulator," Opt. Lett. 17, 688-90 (1992).
[CrossRef] [PubMed]

Sudbø, A.

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

Tay, F.E.H.

Wang, J.-S.

X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
[CrossRef]

Wiener, A. M.

A. M. Wiener, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-60 (2000).
[CrossRef]

Young, R.

Zhou, G.

App. Opt.

J.P.llebach and B. Liu, "Minimax spectrum shaping with a bandwidth constraint," App. Opt. 14, 3062-72 (1975).

Appl. Opt.

IEEE J. Sel. Top. Quantum. Electron.

H. Sagberg, M. Lacolle, I.-R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard and A. Sudbø, "Micromechanical Gratings for Visible and Near-Infrared Spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004).
[CrossRef]

J. Microelectromech. Syst.

X. Li, C. Antoine, D. Lee, J.-S. Wang, O. Solgaard, "Tunbable Blazed Gratings," IEEE/ASMEJ. Microelectromech. Syst. 15, 597-604 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Photon. Technol. Lett.

M. Lacolle, H. Sagberg, I.R. Johansen, O. Løvhaugen, O. Solgaard and A. Sudbø, "Reconfigurable near-infrared optical filter with a micro-mechanical diffractive Fresnel lens," Photon. Technol. Lett. 17, 2622-2624 (2005).
[CrossRef]

Proc. SPIE

J. M. Florence, and R. D. Juday, "Full Complex Spatial Filtering with a Phase Mostly DMD," Proc. SPIE 1558, 487-498 (1991).
[CrossRef]

Rev. Sci. Instrum.

A. M. Wiener, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-60 (2000).
[CrossRef]

Other

R. Belikov, C. Antoine-Snowden, O. Solgaard, "Femtosecond Direct Space-to-Time Pulse Shaping with MEMS Micromirror Arrays," in IEEE/LEOS International Conference on Optical MEMS, (Hawaii, HI, August 2003).

R. Belikov, "Programmable Optical Wavelength Filter Based on Diffraction from a 2-D MEMS Micromirror Arrays," in Proceedings of CLEO 2003, (Baltimore, Maryland, 2003).

M. Lacolle, Optical spectral filtering with segmented diffractive optical elements, (Dissertation for the Degree of Dr. Scient., Faculty of Mathematics and Natural Sciences, University of Oslo, Unipub, 2006).

R. Belikov, Diffraction-Based Optical Filtering: Theory and Implementation with MEMS, (Doctoral Dissertation, Stanford University, UMI Dissertation Publishing, 2005).

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Figures (20)

Fig. 1.
Fig. 1.

(a) Surface relief composed of several array elements positioned with a regular spacing, which is represented by dashed lines separated by a distance D. The array of micromirrors is assumed to be illuminated by a plane wave from a broad-band source, propagating parallel with the optical axis. We consider the reflected light propagating along the optical axis, but in the opposite direction, as the output of the optical filter. (b) The periodical filter transmission synthesised by a phased array. In grey the spectral range corresponding to the 4 th diffraction order, which defines the signal window.

Fig. 2.
Fig. 2.

(a) Time-delay array: the array elements have been moved around their idle positions in order to add small time delays to the light reflected on them. (b) The filter transmission synthesized by a time-delay array. In grey the spectral range corresponding to the 4 th diffraction order, which defines the signal window.

Fig. 3.
Fig. 3.

An array of mirror pairs where each array element is composed of 2 micro-mirrors, that we call sub-elements, so that it is possible to control both the amplitude and phase of the complex reflection function of each array element.

Fig. 4.
Fig. 4.

The 18 test filters used to test the algorithms. Two kinds of filters were tested. 11 filters had a rectangular transmissions with various widths and center positions, and 7 filters consisted of the sum of a random number of Gaussian peaks with random positions, widths and intensities. We simulated an array consisting of 32 mirror pairs, so that the test filters were band limited in order to be sampled at N=32 points. To evaluate algorithms for complex filter synthesis, we have set the phase of the target filters to zero.

Fig. 5.
Fig. 5.

(a) The NMS error for the array of mirror pairs ξ˜ (AMP) as a function of the relative bandwidth of the filter Δν/νc . On all the figures, the solid line is the mean value over all the test filters. The heights of the vertical bars are twice the corresponding standard deviation. (b) The ratio of the predicited over computed NMS error (T* φ +T* r )(νcν)2/ξ˜ (AMP), as a function of the relative bandwidth of the filter Δν/νc .

Fig. 6.
Fig. 6.

The improvement on the NMS error of the array of mirror pairs, obtained running a numerical search, ξ˜ (AMP,NS)/ξ˜ (AMP), as a function of the relative bandwidth of the filter Δν/νc .

Fig. 7.
Fig. 7.

(a) The ratio (��(AMP)+��(AMP)ν/νc )2)/C (AMP) is plotted as a function of the relative bandwidth of the filter Δν/νc , to check the quality of the fit. (b) The fit coefficients ��(AMP) (represented as “+”) and ��(AMP) (represented as “*”) as a function of the filter characteristic T* φ +T* r .

Fig. 8.
Fig. 8.

Power efficiency <, > for the array of mirror pairs as a function of the relative bandwidth of the filter Δν/νc . Triangles: with the phase and amplitude emulation method. Crosses: after a numerical search.

Fig. 9.
Fig. 9.

Quantized surface relief: In grey the ideal surface relief where the area of each array element is defined by r (ref), in black the quantized surface relief, where the area covered by each array element r (QSR) is proportional to the number of micro-mirrors situated at a given position along the optical axis.

Fig. 10.
Fig. 10.

(a) The power efficiency of the quantized surface relief, obtained from simulations on the set of test spectra, as a function of the relative bandwidth of the filter Δν/νc . Here for time-delay arrays. (b) The ratio of the predicted over computed power efficiency N/(T* N < (QSR), (QSR)>) is plotted as a function of the relative bandwidth of the filter Δν/νc . Here for time-delay arrays.

Fig. 11.
Fig. 11.

(a) The quantization error of the quantized surface relief, ξ (QSR)=Q, as a function of the relative bandwidth of the filter Δν/νc . (b) In order to check the validity of (37), we have plotted the ratio of the predicted over computed NMS error NT* N /(12P 2)/ξ (QSR).

Fig. 12.
Fig. 12.

(a) The total NMS error of the quantized surface relief ξ˜ (QSR)=Q+Wφ as a function of the relative bandwidth of the filter Δν/νc . (b) In order to check the validity of (41), we have plotted the ratio of the predicted over computed NMS error (T* φν/νc )2+ NT* N /(12P 2))/ξ˜ (QSR) as a function of the relative bandwidth of the filter Δν/νc .

Fig. 13.
Fig. 13.

(a) Improvement resulting from a numerical search for the quantized surface relief, ξ˜QSR,NS/ξ˜QSR, as a function of the relative bandwidth of the filter Δν/νc . Diamonds: the mirrors of each array element were moved together, and as a consequence we see that only the phase emulation error Wφ is reduced (no improvement on narrow spectral ranges). Stars: the positions of the P mirrors could be adjusted independently, thus reducing the quantization error Q as well as the phase emulation error Wφ . (b) The decrease in power efficiency < (QSR,NS), (QSR,NS)>/< (QSR), (QSR)> due to the numerical search for the quantized surface relief, as a function of the relative bandwidth of the filter Δν/νc . Diamonds: moving all the mirrors of each array element together causes virtually no change in the power efficieny of the device. Stars: Letting the algorithm adjusting the positions of the P mirrors independently results in a decrease in the power efficiency.

Fig. 14.
Fig. 14.

The ratio (��(QSR,NS)+��(QSR,NS)ν/νc )2)/ξ˜ (QSR),NS as a function of the relative bandwidth of the filter Δν/νc . Diamonds: moving all the mirrors of each array element together. Stars: Letting the algorithm misaligning the mirrors.

Fig. 15.
Fig. 15.

(a) The fit coefficients of the NMS error of the quantized surface relief after a numerical search, ��(QSR,NS), as a function of the filter characteristic NT* N /12/P 2. Diamonds: forcing the mirrors to be aligned. Stars: Letting the algorithm misaligning the mirrors. (b) The fit coefficients of the NMS error of the quantized surface relief after a numerical search, ��(QSR,NS) as a function of the filter characteristic T* φ . Diamonds: forcing the mirrors to be aligned. Stars: Letting the algorithm misaligning the mirrors.

Fig. 16.
Fig. 16.

(a) The convergence error of the gradient search algorithm, as a function of the relative bandwidth of the filter Δν/νc . (b) The ratio (��(GSA)+��(GSA)ν/νc )2)/ ζ ˜ (GSA), as a function of the relative bandwidth of the filter Δν/νc .

Fig. 17.
Fig. 17.

The power efficiencies for amplitude filter synthesis <, >, as a function of the relative bandwidth of the filter Δν/νc .

Fig. 18.
Fig. 18.

(a) Mean NMS error computed on the amplitude of the filter transmissions, using analytical algorithms to find the position of the array elements. Triangles: Array of mirror pairs. Squares: Quantized surface relief. (b) Mean NMS error computed on the amplitude of the filter transmissions, using numerical algorithms. Circles: GSA. Crosses: Array of mirror pairs after a numerical search. Diamonds and stars: Quantized surface relief after a numerical search, forcing the mirror to remain aligned and letting the algorithm misaligning the mirrors respectively.

Fig. 19.
Fig. 19.

(a) Mean of the NMS error computed on the complex filter transmissions, using analytical algorithms to find the position of the array elements. Triangles: Array of mirror pairs. Squares: Quantized surface relief. (b) Mean of the NMS error computed on the complex filter transmissions, using numerical algorithms. Crosses: Array of mirror pairs after a numerical search. Diamonds and stars: Quantized surface relief after a numerical search, forcing the mirror to remain aligned and letting the algorithm misaligning the mirrors respectively.

Fig. 20.
Fig. 20.

(a) Mean power efficiency computed on the test filters using analytical algorithms to find the position of the array elements. Triangles: Array of mirror pairs. Squares: Quantized surface relief. (b) Mean power efficiency computed on the test filters using numerical algorithms. Circles: GSA. Crosses: Array of mirror pairs after a numerical search. Diamonds and stars: Quantized surface relief after a numerical search, forcing the mirror to remain aligned and letting the algorithm misaligning the mirrors respectively.

Tables (1)

Tables Icon

Table 1. Complex filter synthesis: NMS errors and power efficiencies using analytical methods for the calculation of the positions of the micro-mirrors.

Equations (59)

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D = M λ c 2 ,
U out ( v ) = U in ( v ) C v n = 0 N 1 s n ( v ) exp ( j 2 π v n M λ c ) .
H ( v ) = n = 0 N 1 s n ( v ) exp ( j 2 π v n M λ c ) .
r n = s n ( v ) = A n A tot n = 0 N 1 r n = 1 ,
A tot = n = 0 N 1 A n .
s n = r n exp ( j φ n ) ,
H ( v ) = n = 0 N 1 s n exp ( j 2 π v n M λ c ) .
v [ 1 λ c 1 2 M λ c , 1 λ c + 1 2 M λ c ] .
Δ v v c = 1 M ,
v m = 1 λ c + m N 2 N M λ c ,
H m = n = 0 N 1 s n exp ( j 2 π ( m N 2 ) n N ) H = 𝒟 𝓕 𝒯 ( s ) .
s ~ n ( v ) = r ~ n exp ( j 2 π v d n ) ,
H ~ m = n = 0 N 1 r ~ n exp ( j 2 π v m d n ) exp ( j 2 π ( m N 2 ) n N ) ,
H ~ ( v ) = n = 0 N 1 r ~ n exp ( j 2 π v d n ) exp ( j 2 π v n M λ c ) ,
γ H = T + e ,
ξ = m = 0 N 1 e m 2 m = 0 N 1 T m 2 = e , e T , T = 1 ( H , T ) 2 T , T H , H ,
H , H = m = 0 N 1 H m 2 .
m = 0 N 1 H m 2 = N n = 0 N 1 r n 2 .
s n ( AMP ) = 1 2 N ( exp ( j Φ n ) + exp ( j Ψ n ) ) = r n ( AMP ) exp ( j φ n ( AMP ) ) ,
{ r n ( AMP ) = 1 N cos ( Φ n Ψ n 2 ) φ n ( AMP ) = Φ n + Ψ n 2 { Φ n = φ n ( AMP ) + arccos ( N r n ( AMP ) ) Ψ n = φ n ( AMP ) arccos ( N r n ( AMP ) ) .
Δ d max 2 = 5 8 λ c ,
s ~ n ( AMP ) = r ~ n ( AMP ) ( v ) exp ( j φ ~ n ( AMP ) ( v ) )
= 1 N cos ( 2 π v d n ( 1 ) d n ( 2 ) 2 ) exp ( j 2 π v d n ( 1 ) + d n ( 2 ) 2 ) .
{ d n ( 1 ) = 1 2 π v c Φ n d n ( 2 ) = 1 2 π v c Ψ n ,
ξ ~ ( AMP ) = W φ + W r ( T φ ٭ + T r ٭ ) ( Δ v v c ) 2 ,
W r W φ ξ ~ ( AMP ) W r .
γ H ~ ( AMP , NS ) = T + c ( AMP ) .
ξ ~ ( AMP , NS ) = C ( AMP ) = c ( AMP ) , c ( AMP ) T , T 𝓑 ( AMP ) ( Δ v v c ) 2 .
H ( AMP ) , H ( AMP ) = m = 0 N 1 H m ( AMP ) 2 = N n = 0 N 1 ( r n ( AMP ) ) 2 1 .
p n = P r n ( QSR ) = round ( r n ( ref ) ρ ) ,
n = 0 N 1 r n ( QSR ) = 1 .
Δ d max 2 = N 2 Δ v .
Δ d typical 2 N Δ v .
H ( QSR ) , H ( QSR ) N T N ٭ ,
T N ٭ = 1 n = 0 N 1 ( r n ( ref ) ) 2 .
s ( QSR ) = r ( QSR ) exp ( j φ ( ref ) ) .
γ H ( QSR ) = T + q .
ξ ( QSR ) = Q = q , q T , T N T N ٭ 12 P 2 .
d ( QSR ) = 1 2 π v c φ ( ref ) .
s ~ ( QSR ) = r ( QSR ) exp ( j 2 π v d ( QSR ) ) .
γ H ~ ( QSR ) = T + q + w φ .
ξ ~ ( QSR ) W φ + Q T φ ٭ ( Δ v v c ) 2 + N T N ٭ 12 P 2 .
ζ = 1 ( H , T ) 2 T , T H , H ,
{ H m | = H m T m | = T m
Δ d max 2 = λ c 2 .
γ H ~ | GSA = T | + c ( GSA ) γ H ~ m ( GSA ) = T m + c m ( GSA ) .
ζ ~ ( GSA ) = C ( GSA ) = c ( GSA ) , c ( GSA ) T , T 𝒜 ( GSA ) + 𝓑 ( GSA ) ( Δ v v c ) 2 .
H ( GSA ) , H ( GSA ) = 1 .
a , b = n = 0 N 1 Re ( a n ) Re ( b n ) + Im ( a n ) Im ( b n ) ,
e ~ ( AMP ) = γ H ~ ( AMP ) T .
γ H ( ref ) = T e ~ ( AMP ) = γ ( H ~ ( AMP ) H ( ref ) ) ,
e ~ m ( AMP ) γ
w m φ γ { j v m v c v c n = 0 N 1 [ r n ( ref ) exp ( j φ n ( ref ) ) φ n ( ref ) exp ( j 2 π n m N 2 N ) ]
w m r γ { v m v c N v c n = 0 N 1 [ 2 π v c d n ( 1 ) d n ( 2 ) 2 sin ( 2 π v c d n ( 1 ) d n ( 2 ) 2 ) exp ( j φ n ( ref ) ) exp ( j 2 π n m N 2 N ) ] .
e ~ ( AMP ) = w φ + w r .
ξ ~ ( AMP ) = w φ + w r , w φ + w r T , T w φ , w φ + w r , w r γ 2 H ( AMP ) , H ( AMP )
W φ + W r ( T φ ٭ + T r ٭ ) ( Δ v v c ) 2 ,
T φ ٭ = m = 0 N 1 m N 2 N 𝒟 𝓕 𝒯 ( r n ( ref ) exp ( j φ n ( ref ) ) φ n ( ref ) ) m 2 N n = 0 N 1 ( r n ( ref ) ) 2 ,
T r ٭ = m = 0 N 1 m N 2 N 2 𝒟 𝓕 𝒯 ( acos ( N r n ( ref ) ) 1 N ( r n ( ref ) ) 2 exp ( j φ n ( ref ) ) ) m 2 N n = 0 N 1 ( r n ( ref ) ) 2 ,

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