Abstract

In diamond-machined freeform manufacturing processes, a tool-tip often leaves behind characteristic mid-spatial frequency (MSF) structures on the optical surface. Unwanted movement between the tool-tip and the part results in MSF structures with random variations. Here, we analyze the effects of these MSF structures on the system’s optical performance and derive simple analytic estimates for the optical transfer function in terms of the parameters of these structures. These expressions are expected to aid in MSF tolerancing.

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

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References

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  1. J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49(25), 4825–4835 (2010).
    [Crossref]
  2. G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Proc. SPIE 9525, 95251B (2015).
    [Crossref]
  3. K. Liang and M. A. Alonso, “Effects of defocus and other quadratic errors on OTF,” Opt. Lett. 42(24), 5254–5257 (2017).
    [Crossref]
  4. K. Liang and M. A. Alonso, “Understanding the effects of groove structures on the OTF,” Opt. Express 25(16), 18827–18841 (2017).
    [Crossref]
  5. H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
    [Crossref]
  6. H. Wang, S. To, and C. Y. Chan, “Investigation on the influence of tool-tip vibration on surface roughness and its representative measurement in ultra-precision diamond turning,” Int. J. Mach. Tools Manufact. 69, 20–29 (2013).
    [Crossref]
  7. S. J. Zhang and S. To, “A theoretical and experimental study of surface generation under spindle vibration in ultra-precision raster milling,” Int. J. Mach. Tools Manufact. 75, 36–45 (2013).
    [Crossref]
  8. S. Takasu, M. Masuda, and T. Nishiguchi, “Influence of Study Vibration with Small Amplitude Upon Surface Roughness in Diamond Machining,” CIRP Ann. 34(1), 463–467 (1985).
    [Crossref]
  9. D. L. Martin, A. N. Tabenkin, and F. G. Parsons, “Precision Spindle and Bearing Error Analysis,” Int. J. Mach. Tools Manufact. 35(2), 187–193 (1995).
    [Crossref]
  10. D. Kim, I. Chang, and S. Kim, “Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces,” Precis. Eng. 26(2), 168–174 (2002).
    [Crossref]
  11. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), Chap. 6.

2017 (2)

2015 (1)

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Proc. SPIE 9525, 95251B (2015).
[Crossref]

2013 (2)

H. Wang, S. To, and C. Y. Chan, “Investigation on the influence of tool-tip vibration on surface roughness and its representative measurement in ultra-precision diamond turning,” Int. J. Mach. Tools Manufact. 69, 20–29 (2013).
[Crossref]

S. J. Zhang and S. To, “A theoretical and experimental study of surface generation under spindle vibration in ultra-precision raster milling,” Int. J. Mach. Tools Manufact. 75, 36–45 (2013).
[Crossref]

2010 (2)

J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49(25), 4825–4835 (2010).
[Crossref]

H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
[Crossref]

2002 (1)

D. Kim, I. Chang, and S. Kim, “Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces,” Precis. Eng. 26(2), 168–174 (2002).
[Crossref]

1995 (1)

D. L. Martin, A. N. Tabenkin, and F. G. Parsons, “Precision Spindle and Bearing Error Analysis,” Int. J. Mach. Tools Manufact. 35(2), 187–193 (1995).
[Crossref]

1985 (1)

S. Takasu, M. Masuda, and T. Nishiguchi, “Influence of Study Vibration with Small Amplitude Upon Surface Roughness in Diamond Machining,” CIRP Ann. 34(1), 463–467 (1985).
[Crossref]

Alonso, M. A.

Chan, C. Y.

H. Wang, S. To, and C. Y. Chan, “Investigation on the influence of tool-tip vibration on surface roughness and its representative measurement in ultra-precision diamond turning,” Int. J. Mach. Tools Manufact. 69, 20–29 (2013).
[Crossref]

H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
[Crossref]

Chang, I.

D. Kim, I. Chang, and S. Kim, “Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces,” Precis. Eng. 26(2), 168–174 (2002).
[Crossref]

Cheung, C. F.

H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
[Crossref]

Dallas, W.

Forbes, G. W.

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Proc. SPIE 9525, 95251B (2015).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), Chap. 6.

Kim, D.

D. Kim, I. Chang, and S. Kim, “Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces,” Precis. Eng. 26(2), 168–174 (2002).
[Crossref]

Kim, S.

D. Kim, I. Chang, and S. Kim, “Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces,” Precis. Eng. 26(2), 168–174 (2002).
[Crossref]

Lee, W. B.

H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
[Crossref]

Liang, K.

Martin, D. L.

D. L. Martin, A. N. Tabenkin, and F. G. Parsons, “Precision Spindle and Bearing Error Analysis,” Int. J. Mach. Tools Manufact. 35(2), 187–193 (1995).
[Crossref]

Masuda, M.

S. Takasu, M. Masuda, and T. Nishiguchi, “Influence of Study Vibration with Small Amplitude Upon Surface Roughness in Diamond Machining,” CIRP Ann. 34(1), 463–467 (1985).
[Crossref]

Milster, T. D.

Nishiguchi, T.

S. Takasu, M. Masuda, and T. Nishiguchi, “Influence of Study Vibration with Small Amplitude Upon Surface Roughness in Diamond Machining,” CIRP Ann. 34(1), 463–467 (1985).
[Crossref]

Parsons, F. G.

D. L. Martin, A. N. Tabenkin, and F. G. Parsons, “Precision Spindle and Bearing Error Analysis,” Int. J. Mach. Tools Manufact. 35(2), 187–193 (1995).
[Crossref]

Tabenkin, A. N.

D. L. Martin, A. N. Tabenkin, and F. G. Parsons, “Precision Spindle and Bearing Error Analysis,” Int. J. Mach. Tools Manufact. 35(2), 187–193 (1995).
[Crossref]

Takasu, S.

S. Takasu, M. Masuda, and T. Nishiguchi, “Influence of Study Vibration with Small Amplitude Upon Surface Roughness in Diamond Machining,” CIRP Ann. 34(1), 463–467 (1985).
[Crossref]

Tamkin, J. M.

To, S.

H. Wang, S. To, and C. Y. Chan, “Investigation on the influence of tool-tip vibration on surface roughness and its representative measurement in ultra-precision diamond turning,” Int. J. Mach. Tools Manufact. 69, 20–29 (2013).
[Crossref]

S. J. Zhang and S. To, “A theoretical and experimental study of surface generation under spindle vibration in ultra-precision raster milling,” Int. J. Mach. Tools Manufact. 75, 36–45 (2013).
[Crossref]

H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
[Crossref]

Wang, H.

H. Wang, S. To, and C. Y. Chan, “Investigation on the influence of tool-tip vibration on surface roughness and its representative measurement in ultra-precision diamond turning,” Int. J. Mach. Tools Manufact. 69, 20–29 (2013).
[Crossref]

H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
[Crossref]

Zhang, S. J.

S. J. Zhang and S. To, “A theoretical and experimental study of surface generation under spindle vibration in ultra-precision raster milling,” Int. J. Mach. Tools Manufact. 75, 36–45 (2013).
[Crossref]

Appl. Opt. (1)

CIRP Ann. (1)

S. Takasu, M. Masuda, and T. Nishiguchi, “Influence of Study Vibration with Small Amplitude Upon Surface Roughness in Diamond Machining,” CIRP Ann. 34(1), 463–467 (1985).
[Crossref]

Int. J. Mach. Tools Manufact. (4)

D. L. Martin, A. N. Tabenkin, and F. G. Parsons, “Precision Spindle and Bearing Error Analysis,” Int. J. Mach. Tools Manufact. 35(2), 187–193 (1995).
[Crossref]

H. Wang, S. To, C. Y. Chan, C. F. Cheung, and W. B. Lee, “A theoretical and experimental investigation of the tool-tip vibration and its influence upon surface generation in single-point diamond turning,” Int. J. Mach. Tools Manufact. 50(3), 241–252 (2010).
[Crossref]

H. Wang, S. To, and C. Y. Chan, “Investigation on the influence of tool-tip vibration on surface roughness and its representative measurement in ultra-precision diamond turning,” Int. J. Mach. Tools Manufact. 69, 20–29 (2013).
[Crossref]

S. J. Zhang and S. To, “A theoretical and experimental study of surface generation under spindle vibration in ultra-precision raster milling,” Int. J. Mach. Tools Manufact. 75, 36–45 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Precis. Eng. (1)

D. Kim, I. Chang, and S. Kim, “Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces,” Precis. Eng. 26(2), 168–174 (2002).
[Crossref]

Proc. SPIE (1)

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Proc. SPIE 9525, 95251B (2015).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), Chap. 6.

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Figures (11)

Fig. 1.
Fig. 1. (a) Diagram of the tool-tip (orange) and the rotating optical part (purple) in a diamond-turning process, and the three directions of possible vibrations. (b) Nominal MSF structure (black) with height $h$ and period $T$. The size of the pupil is $2R$, twice the radius. An example of a MSF structure with random variations is also shown (gray). A sample cycle from (b) is shown in (c) with the probability densities $K_{\textrm T}$ (red, with uncertainty $\mu$) governing thrust (and cutting) vibrations, and $K_{\textrm F}$ (blue, with uncertainty $\sigma$) governing feed vibrations.
Fig. 2.
Fig. 2. (a) Two oppositely shifted copies of $W$, along with their difference $\Delta W$, where $t_i$ and $t_j$ denote the vertical shift of each parabolic segment for $W(q_{\textrm n}-\rho _{\textrm n}/2)$ and $W(q_{\textrm n}+\rho _{\textrm n}/2)$, respectively, due to thrust vibration. (b) Zoom for one quasi-period, where the effect of thrust vibration is seen to correspond to a vertical shift defined by the difference $t_i - t_j$.
Fig. 3.
Fig. 3. OTF sections for various values of $\mu /h$, for $kh = 1$ and 30 cycles across the aperture. In each part, the nominal (black) OTF corresponds to the case of $\mu = 0$, the numerical (blue) OTF is the average OTF from 30 randomly generated [with $K_{\textrm {T}}$ given by Eq. (8)] MSF surfaces, and the theoretical models are calculated from Eq. (9) (red) and Eq. (14) (green).
Fig. 4.
Fig. 4. (a) Two oppositely shifted copies of $W$, along with their difference $\Delta W$, where $t_i$ and $t_j$ denote the vertical shift of each parabolic segment for $W(q_{\textrm n}-\rho _{\textrm n}/2)$ and $W(q_{\textrm n}+\rho _{\textrm n}/2)$, respectively, due to thrust vibration. (b) Zoom for one quasi-period, where the effect of thrust vibration is seen to correspond to a vertical shift $s$ of the intersection points, which is defined in terms of $\rho$, $t_i$, $t_{i+1},$ and $t_j$.
Fig. 5.
Fig. 5. (a) Two oppositely shifted copies of $W$, along with their difference $\Delta W$, where $f_i$ and $f_j$ denote the horizontal shift for each parabolic segment of $W(q_{\textrm n}-\rho _{\textrm n}/2)$ and $W(q_{\textrm n}+\rho _{\textrm n}/2)$, respectively. (b) Zoom for one quasi-period, where the effect of feed vibration is to alter both the vertical position of the intersection ($w/2$ instead of the nominal $w_0/2$) and the weight of the contribution to the probability distribution ($\alpha /w$ instead of the nominal $T/w_0$).
Fig. 6.
Fig. 6. OTFs for various values of $\sigma /T$, for $kh = 1$ and 30 cycles across the aperture. In each part, the nominal OTF (black) corresponds to the case of $\sigma = 0$, the numerical OTF (blue) is generated by averaging the OTFs of 30 randomly generated MSF surfaces [with $K_{\textrm {T}}$ given by Eq. (15)], and the theoretical models (red and green) are calculated from Eqs. (22) and (24), respectively. The insets are zooms of the sections in the dotted black borders, and show the failure of Eq. (22) (red) within a region of width $2\sigma$.
Fig. 7.
Fig. 7. OTFs for various values of $\mu /h$ and $\sigma /T$, for $kh = 1$ and 30 cycles across the aperture. In each, the nominal OTF (black) corresponds to the case of $\mu = \sigma = 0$, the numerical OTF (blue) is generated by averaging the OTFs of 75 randomly generated MSF surfaces [with $K_{\textrm {T}}$ given by Eq. (15)], and the theoretical model (red) is calculated from Eq. (27).
Fig. 8.
Fig. 8. Estimated (red) and numerically calculated (blue) standard deviation of the OTF for the indicated values of $\mu /h$, $\sigma /T$ and $kh$, with 30 cycles across the aperture, based on 50 randomly generated MSF surfaces [with $K_{\textrm {T}}$ given by Eq. (15)]. The difference of the OTF of each realization with the average OTF is shown as a translucent gray curve.
Fig. 9.
Fig. 9. OTFs for various values of $\mu /h$ and $\sigma /T$, for $kh = 1$ with 20 cycles across the aperture. In each, the nominal OTF (black) corresponds to the case of $\mu = \sigma = 0$, the numerical OTF (blue) is generated by averaging the OTFs of 75 randomly generated MSF surfaces (with Gaussian statistics), and the theoretical model (red) is calculated from Eq. (31).
Fig. 10.
Fig. 10. Estimated (red) and numerically calculated (blue) standard deviation of the OTF for the indicated values of $\mu /h$, $\sigma /T$ and $kh$, with 20 cycles across the aperture, based on 50 randomly generated MSF surfaces (with Gaussian statistics). The difference of the OTF of each realization with the average OTF is shown as a translucent gray curve.
Fig. 11.
Fig. 11. The overlap area between two shifted copies of a circular pupil is divided into $A,B,$ and $E$ (blue, red, and green). (a) and (b) show the cases of $\rho \le R$ and $\rho\;>\;R$, respectively.

Equations (61)

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P ( η , ρ ) O ( ρ ) δ { η [ W ( q ρ / 2 ) W ( q + ρ / 2 ) ] } d 2 q O ( ρ ) d 2 q ,
OTF perf ( ρ ) = O ( ρ ) d 2 q O ( 0 ) d 2 q = 2 π [ cos 1 ( ρ 2 R ) ρ 2 R 1 ( ρ 2 R ) 2 ] .
OTF ( k , ρ ) = OTF perf ( ρ ) P ~ ( k , ρ ) = OTF perf ( ρ ) P ( η , ρ ) exp ( i k η ) d η ,
P 1 ( η , ρ n ) = 1 w 0 ( h , T , ρ n ) rect [ η w 0 ( h , T , ρ n ) ] and P ~ 1 ( k , ρ n ) = sinc [ k w 0 ( h , T , ρ n ) / 2 ] ,
T ρ n ( f , g ) Max ( 1 ρ n T , 0 ) f + Min ( ρ n T , 1 ) g ,
P 1 , T ( η , ρ n ) = K T ( t i ) K T ( t j ) P 1 [ η ( t i t j ) ] d t i d t j .
P ~ 1 , T ( k , ρ n ) = K ~ T 2 ( k ) P ~ 1 ( k , ρ n ) .
K T ( t ) = Re ( 1 π 2 μ 2 t 2 ) and K ~ T ( k ) = J 0 ( 2 k μ ) ,
P ~ 1 , T ( k , ρ n ) = J 0 2 ( 2 k μ ) P ~ 1 ( k , ρ n ) .
s ( t i , t i + 1 , t j , ρ ) = ( t i t j ) ρ ^ + + ( t i + 1 t j ) ρ ^ ,
P 1 , T ( η , ρ n ) = K T ( t i ) K T ( t j ) K T ( t i + 1 ) P 1 [ η s ( t i , t i + 1 , t j , ρ n ) , ρ n ] d t i d t i + 1 d t j .
P 1 , T ( η , ρ n ) = [ K T ( a ) K T ( b ) K T ( a ρ ^ + + b ρ ^ s ) d a d b ] P 1 ( η s , ρ n ) d s .
P ~ 1 , T ( k , ρ n ) = K ~ T ( k ) K ~ T ( k ρ ^ + ) K ~ T ( k ρ ^ ) P ~ 1 ( k , ρ n ) .
P ~ 1 , T ( k , ρ n ) = J 0 ( 2 k μ ) J 0 ( 2 k μ ρ ^ + ) J 0 ( 2 k μ ρ ^ ) P ~ 1 ( k , ρ n ) .
K T ( t ) = 1 2 π μ exp ( t 2 2 μ 2 ) and K ~ T ( k ) = exp ( k 2 μ 2 2 ) ,
P ~ 1 , T ( k , ρ n ) = exp [ k 2 μ 2 2 ( ρ ^ + 2 + ρ ^ 2 + 1 ) ] P ~ 1 ( k , ρ n ) .
w ( f i , f i + 1 , f j , ρ n ) = 8 h ( ρ ^ + + f j f i T ) ( ρ ^ + f i + 1 f j T ) ,
α ( f i , f i + 1 ) = T + f i + 1 f i ,
P 1 , F ( η , ρ n ) = 1 T K F ( f i ) K F ( f i + 1 ) K F ( f j ) α ( f i , f i + 1 ) w ( f i , f i + 1 , f j ) rect [ η w ( f i , f i + 1 , f j ) ] d f i d f i + 1 d f j .
P ~ 1 , F ( k , ρ n ) = 1 T K F ( f i ) K F ( f i + 1 ) K F ( f j ) α ( f i , f i + 1 ) sinc [ k w ( f i , f i + 1 , f j ) 2 ] d f i d f i + 1 d f j .
K F ( f ) = 1 2 π σ exp ( f 2 2 σ 2 ) ,
P ~ 1 , F ( k , ρ n ) = T π 8 k h σ Re { exp ( T 2 ρ ^ + 2 4 σ 2 ) erf [ γ + ( ρ n ) ] + exp ( T 2 ρ ^ 2 4 σ 2 ) erf [ γ ( ρ n ) ] } ,
γ ± ( ρ n ) 4 k h σ 2 ( 1 + ρ ^ ) + i T 2 ρ ^ ± 2 σ T 2 8 i k h σ 2 + 48 k 2 h 2 σ 4 / T 2 .
P ~ 1 , F ( k , ρ n ) C 0 ( σ ) + C 1 ( σ ) cos ( 2 π ρ n T ) ,
C 0 ( σ ) a 0 ( k h ) 2 exp ( 3 k 2 h 2 σ 2 T 2 ) and C 1 ( σ ) a 1 ( k h ) exp ( 22 k 2 h 2 σ 2 T 2 ) ,
a 0 ( k h ) 2 8 45 k 2 h 2 and a 1 ( k h ) 8 π 4 k 2 h 2 .
P ~ 1 , C ( k , ρ n ) = exp [ k 2 μ 2 2 ( ρ ^ + 2 + ρ ^ 2 + 1 ) ] P ~ 1 , F ( k , ρ n ) .
P ~ 1 , C ( k , ρ n ) G 0 ( μ , σ ) + G 1 ( μ , σ ) cos ( 2 π ρ n T ) ,
G 0 ( μ , σ ) C 0 ( σ ) exp ( 3 k 2 μ 2 4 ) and G 1 ( μ , σ ) C 1 ( σ ) exp ( k 2 μ 2 ) ,
Δ P ~ 1 , C ( k , ρ n ) ( k h ) 2 2 [ 30 T 2 R ( 2 σ 2 T 2 + μ 2 2 h 2 ) ] 1 / 2 P ~ 1 , C ( k , ρ n ) .
P ~ 2 , C ( k , ρ ) { G 0 ( μ , σ ) + G 1 ( μ , σ ) κ 2 F ( ρ ) [ A ( ρ ) cos ( 2 π ρ T ϕ 0 ) + B ( ρ ) cos ( 2 π ρ T + ϕ 0 ) ] } × { [ 1 G 0 ( 0 , 0 ) G 1 ( 0 , 0 ) C ( 2 ) / 2 ] exp ( 16 ρ 2 / T 2 ) + 1 } ,
Δ P ~ 2 , C ( k , ρ ) ( k h ) 3 / 2 2 3 [ 30 2 R / T ( { 8 σ 2 T 2 + σ T } 2 + μ 2 h 2 ) ] 1 / 2 P ~ 2 , C ( k , ρ ) .
OTF ( k , ρ ) OTF perf ( ρ ) B ( ρ ) { Max ( 1 x T , 0 ) G 0 ( 0 , 0 ) + Min ( x T , 1 ) G 0 ( μ , σ ) + [ Max ( 1 x T , 0 ) G 1 ( 0 , 0 ) + Min ( x T , 1 ) G 1 ( μ , σ ) ] C ( x ) } ,
G 0 ( μ , σ ) ( 1 4 45 k 2 h 2 ) exp ( 3 k 2 h 2 σ 2 T 2 3 k 2 μ 2 4 ) ,
G 1 ( μ , σ ) 8 π 4 k 2 h 2 exp ( 22 k 2 h 2 σ 2 T 2 k 2 μ 2 ) ,
C ( x ) { cos ( 2 π x T ) milled , 0.3 F ( x ) [ A ( x ) cos ( 2 π x T π / 5 ) + B ( x ) cos ( 2 π x T + π / 5 ) ] turned ,
Δ OTF ( k , x ) = OTF ( k , x ) Min ( x T , 1 ) { ( k h ) 2 2 30 T 2 R ( 2 σ 2 T 2 + μ 2 2 h 2 ) milled , ( k h ) 3 / 2 2 3 30 T 2 R [ ( 8 σ 2 T 2 + σ T ) 2 + μ 2 h 2 ] turned .
K F ( f i ) = 1 2 π K ~ F ( p i ) e i p i f i d p i .
P ~ 1 , F ( k , ρ ) = 1 T ( 2 π ) 3 K ~ F ( p i ) K ~ F ( p j ) K ~ F ( p i + 1 ) e i ( p i f i + p j f j + p i + 1 f i + 1 ) × α ( f i , f i + 1 ) sinc [ k w ( f i , f j , f i + 1 ) 2 ] d f i d f j d f i + 1 d p i d p j d p i + 1 .
P ~ 1 , F ( k , ρ ) = 1 T ( 2 π ) 3 K ~ F ( p i ) K ~ F ( p i + p i + 1 ) K ~ F ( p i + 1 ) e i p i ρ e i c ( T ρ ) × [ e i p i q e i p i + 1 r ( q r ) sinc ( 4 k h T 2 q r ) d q d r ] d p i d p i + 1 .
P ~ 1 , F ( k , ρ ) = i T 8 π k h K ~ F ( p i ) K ~ F ( p i + p i + 1 ) K ~ F ( p i + 1 ) e i p i ρ e i c ( T ρ ) × cos ( p i p i + 1 T 2 4 k h ) ( 1 p i 1 p i + 1 ) d p i d p i + 1 .
C 0 ( σ ) = T π ( 1 + 2 ) 8 k h σ ( 2 Re { exp ( T 2 256 σ 2 ) erf [ γ + ( T 8 ) ] + exp ( 49 T 2 256 σ 2 ) erf [ γ ( T 8 ) ] } + 2 exp ( T 2 16 σ 2 ) Re [ erf ( γ 0 ) ] ) ,
C 1 ( σ ) = ( 2 2 ) T π 8 k h σ ( Re { exp ( T 2 256 σ 2 ) erf [ γ + ( T 8 ) ] + exp ( 49 T 2 256 σ 2 ) erf [ γ ( T 8 ) ] } 2 exp ( T 2 16 σ 2 ) Re [ erf ( γ 0 ) ] ) ,
G 0 ( μ , σ ) = T π ( 1 + 2 ) 8 k h σ ( 2 Re { exp ( T 2 256 σ 2 ) erf [ γ + ( T 8 ) ] + exp ( 49 T 2 256 σ 2 ) erf [ γ ( T 8 ) ] } + 2 exp ( 9 k 2 μ 2 64 ) exp ( T 2 16 σ 2 ) Re [ erf ( γ 0 ) ] ) exp ( 57 k 2 μ 2 64 ) ,
G 1 ( μ , σ ) = ( 2 2 ) T π 8 k h σ ( Re { exp ( T 2 256 σ 2 ) erf [ γ + ( T 8 ) ] + exp ( 49 T 2 256 σ 2 ) erf [ γ ( T 8 ) ] } 2 exp ( 9 k 2 μ 2 64 ) exp ( T 2 16 σ 2 ) Re [ erf ( γ 0 ) ] ) exp ( 57 k 2 μ 2 64 ) .
P ~ 2 ( k , ρ ) = 1 F ( ρ ) [ A ( ρ ) 2 Q ^ t P ~ 1 ( k , ρ ) + B ( ρ ) 2 Q ^ τ P ~ 1 ( k , ρ ) + E ( ρ ) P ~ 0 ( k ) ] ,
Q ^ x f ( ρ ) = 0 1 f ( ρ x v ) v d v ,
P ~ 1 ( k , ρ ) = a 0 2 + m = 1 a m cos ( 2 π m ρ T ) ,
Q ^ x P ~ 1 ( k , ρ ) = a 0 + T | x | m = 1 a m m [ C ( 2 m | x | T ) cos ( 2 π m ρ T ) + sgn ( x ) S ( 2 m | x | T ) sin ( 2 π m ρ T ) ] ,
P ~ 2 ( k , ρ ) a 0 2 + a 1 [ P C ( ρ ) cos ( 2 π ρ T ) + P S ( ρ ) sin ( 2 π ρ T ) ] ,
P C ( ρ ) 1 2 F ( ρ ) [ A ( ρ ) T t C ( 2 t T ) + B ( ρ ) T τ C ( 2 τ T ) ] ,
P S ( ρ ) 1 2 F ( ρ ) [ A ( ρ ) T t S ( 2 t T ) B ( ρ ) T τ S ( 2 τ T ) ] .
P C ( ρ ) A ( ρ ) C ( 2 ) + B ( ρ ) C ( 2 ) 2 F ( ρ ) and P S ( ρ ) A ( ρ ) S ( 2 ) B ( ρ ) S ( 2 ) 2 F ( ρ ) .
P ~ 2 ( k , ρ ) a 0 2 + a 1 κ 2 F ( ρ ) [ A ( ρ ) cos ( 2 π ρ T ϕ 0 ) + B ( ρ ) cos ( 2 π ρ T + ϕ 0 ) ] ,
a m = π 8 k h | erfi [ ( 1 + i ) ( 2 k h m π ) 4 k h / 2 ] + erfi [ ( 1 + i ) ( 2 k h + m π ) 4 k h / 2 ] | 2 ,
A ( ρ ) = 1 8 { 4 R 2 cos 1 [ ρ ( V 2 + X + ) 2 R U + 2 ] ρ V cosh 1 ( X + U + 2 ) ρ V U + 2 X + 2 U + 4 }
B ( ρ ) = π ρ V / 16 ,
A ( ρ ) = 0
B ( ρ ) = 1 8 { 2 π R 2 4 R 2 sin 1 ( 2 R ρ V 2 + X ) ρ V sin 1 ( X U 2 ) + ρ V U 2 U 4 X 2 } ,
V = T ( T + 2 ρ ) , U ± = V 2 ± ρ 2 , X ± = V 4 ± 4 R 2 U ± 2 , X ± = X ± ρ 2 .
F ( ρ ) = 1 2 [ R 2 arccos ( ρ 2 R ) ρ 2 R 2 ( ρ 2 ) 2 ] = π R 2 4 OTF perf ( ρ ) .

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