Abstract

Tunable lenses are widely applied in imaging systems, as they provide inertia-free axial scanning. Their proper placement in the imaging system is critical to maximize tuning range and to limit broadening of the focal spot size. We introduce a purely analytic description of optical systems employing tunable lenses as a toolset to find application-specific optimal trade-offs between these opposing goals. The proposed method is applied to selected configurations of axial scanning systems to derive analytic expressions for their effective focal length and spot size. On this basis, we provide practical guidelines for the design of axial scanning systems.

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

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Axial scanning in confocal microscopy employing adaptive lenses (CAL)

Nektarios Koukourakis, Markus Finkeldey, Moritz Stürmer, Christoph Leithold, Nils C. Gerhardt, Martin R. Hofmann, Ulrike Wallrabe, Jürgen W. Czarske, and Andreas Fischer
Opt. Express 22(5) 6025-6039 (2014)

Optical axial scanning in confocal microscopy using an electrically tunable lens

Joey M. Jabbour, Bilal H. Malik, Cory Olsovsky, Rodrigo Cuenca, Shuna Cheng, Javier A. Jo, Yi-Shing Lisa Cheng, John M. Wright, and Kristen C. Maitland
Biomed. Opt. Express 5(2) 645-652 (2014)

References

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  1. N. Koukourakis, M. Finkeldey, M. Stürmer, C. Leithold, N. C. Gerhardt, M. R. Hofmann, U. Wallrabe, J. W. Czarske, and A. Fischer, “Axial scanning in confocal microscopy employing adaptive lenses (CAL),” Opt. Express 22(5), 6025–6039 (2014).
    [Crossref]
  2. M. Duocastella, G. Vicidomini, and A. Diaspro, “Simultaneous multiplane confocal microscopy using acoustic tunable lenses,” Opt. Express 22(16), 19293–19301 (2014).
    [Crossref]
  3. K. Szulzycki, V. Savaryn, and I. Grulkowski, “Rapid acousto-optic focus tuning for improvement of imaging performance in confocal microscopy [Invited],” Appl. Opt. 57(10), C14 (2018).
    [Crossref]
  4. K. Philipp, F. Lemke, S. Scholz, U. Wallrabe, M. C. Wapler, N. Koukourakis, and J. W. Czarske, “Diffraction-limited axial scanning in thick biological tissue employing an aberration correcting adaptive lens,” arXiv 1811.11457 (2018).
  5. B. F. Grewe, F. F. Voigt, M. van ’t Hoff, and F. Helmchen, “Fast two-layer two-photon imaging of neuronal cell populations using an electrically tunable lens,” Biomed. Opt. Express 2(7), 2035 (2011).
    [Crossref]
  6. S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
    [Crossref]
  7. K. Philipp, A. Smolarski, N. Koukourakis, A. Fischer, M. Stürmer, U. Wallrabe, and J. W. Czarske, “Volumetric HiLo microscopy employing an electrically tunable lens,” Opt. Express 24(13), 15029 (2016).
    [Crossref]
  8. T. Hinsdale, B. H. Malik, C. Olsovsky, J. A. Jo, and K. C. Maitland, “Volumetric structured illumination microscopy enabled by a tunable-focus lens,” Opt. Lett. 40(21), 4943–4946 (2015).
    [Crossref]
  9. F. O. Fahrbach, F. F. Voigt, B. Schmid, F. Helmchen, and J. Huisken, “Rapid 3D light-sheet microscopy with a tunable lens,” Opt. Express 21(18), 21010–21026 (2013).
    [Crossref]
  10. Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
    [Crossref]
  11. M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
    [Crossref]
  12. G. Lan, T. F. Mauger, and G. Li, “Design of high-performance adaptive objective lens with large optical depth scanning range for ultrabroad near infrared microscopic imaging,” Biomed. Opt. Express 6(9), 3362 (2015).
    [Crossref]
  13. M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70(2), 150 (1980).
    [Crossref]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), 2nd ed.

2018 (2)

K. Szulzycki, V. Savaryn, and I. Grulkowski, “Rapid acousto-optic focus tuning for improvement of imaging performance in confocal microscopy [Invited],” Appl. Opt. 57(10), C14 (2018).
[Crossref]

S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
[Crossref]

2016 (1)

2015 (4)

T. Hinsdale, B. H. Malik, C. Olsovsky, J. A. Jo, and K. C. Maitland, “Volumetric structured illumination microscopy enabled by a tunable-focus lens,” Opt. Lett. 40(21), 4943–4946 (2015).
[Crossref]

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

G. Lan, T. F. Mauger, and G. Li, “Design of high-performance adaptive objective lens with large optical depth scanning range for ultrabroad near infrared microscopic imaging,” Biomed. Opt. Express 6(9), 3362 (2015).
[Crossref]

2014 (2)

2013 (1)

2011 (1)

1980 (1)

Bianchini, P.

S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
[Crossref]

Boilot, V.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Clark, J.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Czarske, J. W.

Diaspro, A.

S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
[Crossref]

M. Duocastella, G. Vicidomini, and A. Diaspro, “Simultaneous multiplane confocal microscopy using acoustic tunable lenses,” Opt. Express 22(16), 19293–19301 (2014).
[Crossref]

Doblas, A.

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

Draviam, V. M.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Duocastella, M.

S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
[Crossref]

M. Duocastella, G. Vicidomini, and A. Diaspro, “Simultaneous multiplane confocal microscopy using acoustic tunable lenses,” Opt. Express 22(16), 19293–19301 (2014).
[Crossref]

Fahrbach, F. O.

Finkeldey, M.

Fischer, A.

Funahashi, A.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Gerhardt, N. C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), 2nd ed.

Grewe, B. F.

Grulkowski, I.

Helmchen, F.

Hinsdale, T.

Hiraiwa, T.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Hiroi, N.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Hofmann, M. R.

Hsieh, P.-Y.

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

Huang, Y.-P.

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

Huisken, J.

Jo, J. A.

Koukourakis, N.

Lan, G.

Leithold, C.

Lemke, F.

K. Philipp, F. Lemke, S. Scholz, U. Wallrabe, M. C. Wapler, N. Koukourakis, and J. W. Czarske, “Diffraction-limited axial scanning in thick biological tissue employing an aberration correcting adaptive lens,” arXiv 1811.11457 (2018).

Li, G.

Maitland, K. C.

Malik, B. H.

Martínez-Corral, M.

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

Mauger, T. F.

Nakai, Y.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Nazarathy, M.

Nonaka, S.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Oku, H.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Olsovsky, C.

Ozeki, M.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Philipp, K.

K. Philipp, A. Smolarski, N. Koukourakis, A. Fischer, M. Stürmer, U. Wallrabe, and J. W. Czarske, “Volumetric HiLo microscopy employing an electrically tunable lens,” Opt. Express 24(13), 15029 (2016).
[Crossref]

K. Philipp, F. Lemke, S. Scholz, U. Wallrabe, M. C. Wapler, N. Koukourakis, and J. W. Czarske, “Diffraction-limited axial scanning in thick biological tissue employing an aberration correcting adaptive lens,” arXiv 1811.11457 (2018).

Piazza, S.

S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
[Crossref]

Saavedra, G.

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

Sánchez-Ortiga, E.

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

Savaryn, V.

Schmid, B.

Scholz, S.

K. Philipp, F. Lemke, S. Scholz, U. Wallrabe, M. C. Wapler, N. Koukourakis, and J. W. Czarske, “Diffraction-limited axial scanning in thick biological tissue employing an aberration correcting adaptive lens,” arXiv 1811.11457 (2018).

Shamir, J.

Sheppard, C.

S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
[Crossref]

Shrestha, R.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Smolarski, A.

Stürmer, M.

Szulzycki, K.

Tamura, N.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Taniguchi, A.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Tanimoto, R.

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

van ’t Hoff, M.

Vicidomini, G.

Voigt, F. F.

Wallrabe, U.

Wapler, M. C.

K. Philipp, F. Lemke, S. Scholz, U. Wallrabe, M. C. Wapler, N. Koukourakis, and J. W. Czarske, “Diffraction-limited axial scanning in thick biological tissue employing an aberration correcting adaptive lens,” arXiv 1811.11457 (2018).

Appl. Opt. (1)

Biomed. Opt. Express (2)

J. Biophotonics (1)

S. Piazza, P. Bianchini, C. Sheppard, A. Diaspro, and M. Duocastella, “Enhanced volumetric imaging in 2-photon microscopy via acoustic lens beam shaping,” J. Biophotonics 11(2), e201700050 (2018).
[Crossref]

J. Disp. Technol. (1)

M. Martínez-Corral, P.-Y. Hsieh, A. Doblas, E. Sánchez-Ortiga, G. Saavedra, and Y.-P. Huang, “Fast Axial-Scanning Widefield Microscopy With Constant Magnification and Resolution,” J. Disp. Technol. 11(11), 913–920 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Express (4)

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

Y. Nakai, M. Ozeki, T. Hiraiwa, R. Tanimoto, A. Funahashi, N. Hiroi, A. Taniguchi, S. Nonaka, V. Boilot, R. Shrestha, J. Clark, N. Tamura, V. M. Draviam, and H. Oku, “High-speed microscopy with an electrically tunable lens to image the dynamics of in vivo molecular complexes,” Rev. Sci. Instrum. 86(1), 013707 (2015).
[Crossref]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), 2nd ed.

K. Philipp, F. Lemke, S. Scholz, U. Wallrabe, M. C. Wapler, N. Koukourakis, and J. W. Czarske, “Diffraction-limited axial scanning in thick biological tissue employing an aberration correcting adaptive lens,” arXiv 1811.11457 (2018).

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Figures (6)

Fig. 1.
Fig. 1. Telecentric setup in forward direction. The tunable and the objective lens are placed in the front and back focal plane of a 4f system, respectively. The whole telecentric system can be described using the effective focal length $f_{\mathrm {tc}}$.
Fig. 2.
Fig. 2. Effective refractive power $P_{\mathrm {qtc}}$ of the a) quasi-telecentric and b) reversed quasi-telecentric configuration employing the weak approximation according Eqs. (28) and (29), respectively.
Fig. 3.
Fig. 3. Effective focal lengths for axial scanning systems employing a tunable lens in a) forward and b) backwards quasi-telecentric configuration.
Fig. 4.
Fig. 4. Tuning range for a) positive and b) symmetric tunable lenses in forward (solid lines) and backwards quasi-telecentric configuration (dashed lines).
Fig. 5.
Fig. 5. Tuning range depending on distance $d$ for a) positive and b) symmetric tunable lenses in forward (solid lines) and backwards quasi-telecentric configuration (dashed lines).
Fig. 6.
Fig. 6. Beam waists in units of (a) local and (b) global beam waist, Solid lines correspond to forward and dashed lines to backward propagation through the system. b) The effective beam waist in units of the global initial beam waist $w_{0, \mathrm {global}}$ is independent from the propagation direction through the system.

Equations (45)

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V [ b ] { U ( x ) } = | b | 1 2 U ( b x ) ,
Q [ c ] { U ( x ) } = e i k 2 c x 2 U ( x )
R [ d ] { U ( x 1 ) } = 1 i λ d U ( x 1 ) e i k 2 d ( x 2 x 1 ) 2 d x 1 .
Q [ c 1 ] Q [ c 2 ] = Q [ c 1 + c 2 ]
Q [ c ] R [ d ] = R [ ( d 1 + c ) 1 ] V [ 1 + c d ] Q [ ( c 1 + d ) 1 ]
R [ d 1 ] R [ d 2 ] = R [ d 1 + d 2 ]
V [ t 1 ] V [ t 2 ] = V [ t 1 t 2 ]
V [ t ] R [ d ] = R [ d / t 2 ] V [ t ]
R [ d ] V [ t ] = V [ t ] R [ t 2 d ]
S t e l e c e n t r i c = Q [ f O L 1 ] S 4 f Q [ f T L 1 ]
S 4 f = R [ f ] Q [ f 1 ] R [ 2 f ] Q [ f 1 ] R [ f ] .
S 4 f = R [ f ] R [ 2 f ] V [ 1 ] Q [ f 1 ] Q [ f 1 ] R [ f ]
= R [ f ] V [ 1 ] R [ f ]
= V [ 1 ] ,
S t e l e c e n t r i c = V [ 1 ] Q [ ( f O L 1 + f T L 1 ) ] .
f t c = f O L f T L f O L + f T L
P t c = 1 f t c = P T L + P O L .
Δ f t c ( f 1 , f 2 ) = f O L 2 | f 2 f 1 | ( f O L + f 1 ) ( f O L + f 2 ) ,
Δ f T L , p o s = f O L 2 f O L + f 1  for  f 2
Δ f T L , s y m = 2 f 1 f O L 2 f 1 2 f O L 2  for  f 2 = f 1 .
S = Q [ f O L 1 ] S 4 f R [ d ] Q [ f T L 1 ]
= V [ 1 ] Q [ f O L 1 ] R [ d ] Q [ f T L 1 ]
S = V [ d f O L f O L ] R [ ( f O L d ) d f O L ] Q [ 1 f q t c ]
f q t c = f T L ( f O L d ) f T L + f O L d , P q t c = P T L + 1 f O L d .
S r e v = V [ d f T L f T L ] R [ ( f T L d ) d f T L ] Q [ 1 f q t c , r e v ]
f q t c , r e v = ( f T L d ) f O L f T L d + f O L , P q t c , r e v = 1 f T L d + P O L .
S q t c , s t r o n g = S q t c , r e v , s t r o n g = V [ 1 ] R [ d ] Q [ 1 f t c ]
S q t c = V [ 1 ] R [ Δ s q t c ] Q [ P q t c ] ,
S q t c , r e v = V [ 1 ] R [ Δ s q t c , r e v ] Q [ P q t c , r e v ]
Δ s q t c = ( f O L d ) d f O L  and  Δ s q t c , r e v = ( f T L d ) d f T L .
E F L q t c = f q t c Δ s q t c
E F L q t c , r e v = f q t c , r e v Δ s q t c , r e v .
S = V [ f O L d f O L ] R [ ( f O L d ) d f O L ] Q [ 1 f q t c ] ,
S r e v = V [ f T L d f T L ] R [ ( f T L d ) d f T L ] Q [ 1 f q t c , r e v ] .
U i n ( x , z ) = U 0 w 0 w ( z ) exp ( x 2 w ( z ) 2 ) exp ( i ( k z + k x 2 2 R ( z ) ψ ( z ) ) )
U i n ( x , 0 ) = U 0 exp ( x 2 w 0 2 )
U o u t ( x ) = ( R [ E F L ] V [ f O L d f O L ] ) { U i n ( x , z = E F L q t c ) }
= V [ f O L d f O L ] U i n ( x , z f o c u s )
z f o c u s = ( 1 d f O L ) 2 E F L E F L q t c
U o u t ( x ) = V [ d f O L f O L ] U i n ( x , 0 )
= U i n , 0 | d f O L f O L | 1 / 2 exp { ( d f O L f O L ) 2 x 2 w 0 2 }
I o u t ( x ) = I o u t , 0 exp { 2 ( 1 d f O L ) 2 x 2 w 0 2 }
w 0 , out = w 0 1 d f O L
w 0 , out, rev = w 0 1 d f T L .
w 0 , out = w 0 , g l o b a l 1 d f O L f q t c f O L = w 0 , out, rev = w 0 , g l o b a l 1 d f T L f q t c , r e v f O L

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