Abstract

It is shown that an aberrated wavefront incident upon a Fabry–Perot optical cavity excites higher order spatial modes in the cavity and that the spectral width and distribution of these modes is indicative of the type and magnitude of the aberration. The cavities are purely passive, and therefore frequency content is limited to that provided by the original light source. To illustrate this concept, spatial mode decomposition and transmission spectrum calculation are simulated on an example cavity; the effects of various phase delays, in the form of two basic Seidel aberrations and a composite of Zernike polynomial terms, are shown using both Laguerre–Gaussian and plane wave incident beams. The aggregate spectral width of the cavity modes excited by the aberrations is seen to widen as the magnitude of the aberrations’ phase delay increases.

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

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References

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2018 (1)

A. T. Watnik and D. F. Gardner, “Wavefront sensing in deep turbulence,” Opt. Photon. News 29(10), 38–45 (2018).
[Crossref]

2014 (1)

2013 (1)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7, 861–867 (2013).
[Crossref]

2012 (1)

N. Ji, T. R. Sato, and E. Betzig, “Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex,” Proc. Natl. Acad. Sci. USA 109, 22–27 (2012).
[Crossref]

2011 (3)

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D 84, 102002 (2011).
[Crossref]

K. Takeno, N. Ohmae, N. Mio, and T. Shirai, “Determination of wavefront aberrations using a Fabry–Perot cavity,” Opt. Commun. 284, 3197–3201 (2011).
[Crossref]

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[Crossref]

2008 (1)

2007 (2)

A. D. Corbett, T. D. Wilkinson, J. J. Zhong, and L. Diaz-Santana, “Designing a holographic modal wavefront sensor for the detection of static ocular aberrations,” J. Opt. Soc. Am. A 24, 1266–1275 (2007).
[Crossref]

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

2006 (2)

W. Liu and J. J. Talghader, “Spatial-mode analysis of micromachined optical cavities using electrothermal mirror actuation,” J. Microelectromech. Syst. 15, 777–785 (2006).
[Crossref]

S. Zamek and Y. Yitzhaky, “Turbulence strength estimation from an arbitrary set of atmospherically degraded images,” J. Opt. Soc. Am. A 23, 3106–3113 (2006).
[Crossref]

2003 (2)

G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite–Gaussian decomposition,” J. Opt. Soc. Am. B 20, 670–676 (2003).
[Crossref]

N. Trappe, J. A. Murphy, and S. Withington, “The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems,” Eur. J. Phys. 24, 403–412 (2003).
[Crossref]

2001 (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

1996 (1)

1995 (2)

1993 (1)

D. H. Martin and J. W. Bowen, “Long-wave optics,” IEEE Trans. Microwave Theory Tech. 41, 1676–1690 (1993).
[Crossref]

1987 (1)

1980 (1)

1976 (1)

1973 (1)

1967 (1)

Allen, L.

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[Crossref]

Andersen, G. P.

Betzig, E.

N. Ji, T. R. Sato, and E. Betzig, “Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex,” Proc. Natl. Acad. Sci. USA 109, 22–27 (2012).
[Crossref]

Bond, C.

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D 84, 102002 (2011).
[Crossref]

Boreman, G. D.

Bowen, J. W.

D. H. Martin and J. W. Bowen, “Long-wave optics,” IEEE Trans. Microwave Theory Tech. 41, 1676–1690 (1993).
[Crossref]

Carbone, L.

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D 84, 102002 (2011).
[Crossref]

Corbett, A. D.

Creath, K.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds., Vol. 11 (Academic, 1992), pp. 2–53.

Dainty, C.

Danzmann, K.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Diaz-Santana, L.

Ellerbroek, B. L.

Fakis, M.

Fang, Q.

Fleck, A.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[Crossref]

Freise, A.

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D 84, 102002 (2011).
[Crossref]

Fried, D. L.

Fulda, P.

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D 84, 102002 (2011).
[Crossref]

Gardner, D. F.

A. T. Watnik and D. F. Gardner, “Wavefront sensing in deep turbulence,” Opt. Photon. News 29(10), 38–45 (2018).
[Crossref]

Ghebremichael, F.

Giannetas, V.

Goldsmith, P. F.

P. F. Goldsmith, Quasioptical Systems (IEEE, 1998).

Gurley, K. S.

Izquierdo, M.

Jauregui, C.

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7, 861–867 (2013).
[Crossref]

Ji, N.

N. Ji, T. R. Sato, and E. Betzig, “Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex,” Proc. Natl. Acad. Sci. USA 109, 22–27 (2012).
[Crossref]

Keister, M. P.

Kokeyama, K.

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D 84, 102002 (2011).
[Crossref]

Kwee, P.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Lakshminarayanan, V.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[Crossref]

Limpert, J.

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7, 861–867 (2013).
[Crossref]

Liu, W.

W. Liu and J. J. Talghader, “Spatial-mode analysis of micromachined optical cavities using electrothermal mirror actuation,” J. Microelectromech. Syst. 15, 777–785 (2006).
[Crossref]

W. Liu, “Electrically tunable micromirrors and microcavities,” Ph.D. thesis (University of Minnesota, 2004).

Martin, D. H.

D. H. Martin and J. W. Bowen, “Long-wave optics,” IEEE Trans. Microwave Theory Tech. 41, 1676–1690 (1993).
[Crossref]

McDonald, C.

Mevers, G. E.

Mio, N.

K. Takeno, N. Ohmae, N. Mio, and T. Shirai, “Determination of wavefront aberrations using a Fabry–Perot cavity,” Opt. Commun. 284, 3197–3201 (2011).
[Crossref]

Morris, G. J.

Murphy, J. A.

N. Trappe, J. A. Murphy, and S. Withington, “The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems,” Eur. J. Phys. 24, 403–412 (2003).
[Crossref]

Noll, R. J.

Norwood, R. A.

Ogloza, A.

A. Ogloza, private communication (2018).

Ohmae, N.

K. Takeno, N. Ohmae, N. Mio, and T. Shirai, “Determination of wavefront aberrations using a Fabry–Perot cavity,” Opt. Commun. 284, 3197–3201 (2011).
[Crossref]

Padgett, M. J.

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[Crossref]

Pennington, T. L.

Persephonis, P.

Peyghambarian, N.

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

Polyzos, I.

Roggemann, M. C.

Sato, T. R.

N. Ji, T. R. Sato, and E. Betzig, “Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex,” Proc. Natl. Acad. Sci. USA 109, 22–27 (2012).
[Crossref]

Seifert, F.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

Shi, W.

Shirai, T.

K. Takeno, N. Ohmae, N. Mio, and T. Shirai, “Determination of wavefront aberrations using a Fabry–Perot cavity,” Opt. Commun. 284, 3197–3201 (2011).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Silva, D. E.

Smith, J.

Takeno, K.

K. Takeno, N. Ohmae, N. Mio, and T. Shirai, “Determination of wavefront aberrations using a Fabry–Perot cavity,” Opt. Commun. 284, 3197–3201 (2011).
[Crossref]

Talghader, J. J.

W. Liu and J. J. Talghader, “Spatial-mode analysis of micromachined optical cavities using electrothermal mirror actuation,” J. Microelectromech. Syst. 15, 777–785 (2006).
[Crossref]

Tippie, A. E.

A. E. Tippie, “Aberration correction in digital holography,” Ph.D. thesis (University of Rochester, 2012).

Trappe, N.

N. Trappe, J. A. Murphy, and S. Withington, “The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems,” Eur. J. Phys. 24, 403–412 (2003).
[Crossref]

Tsibouri, M.

Tsigaridas, G.

Tünnermann, A.

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7, 861–867 (2013).
[Crossref]

Wang, J. Y.

Watnik, A. T.

A. T. Watnik and D. F. Gardner, “Wavefront sensing in deep turbulence,” Opt. Photon. News 29(10), 38–45 (2018).
[Crossref]

Welsh, B. M.

Wilkinson, T. D.

Willke, B.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Withington, S.

N. Trappe, J. A. Murphy, and S. Withington, “The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems,” Eur. J. Phys. 24, 403–412 (2003).
[Crossref]

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds., Vol. 11 (Academic, 1992), pp. 2–53.

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2006).

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2006).

Yitzhaky, Y.

Zamek, S.

Zhong, J. J.

Zhu, X.

Appl. Opt. (4)

Eur. J. Phys. (1)

N. Trappe, J. A. Murphy, and S. Withington, “The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems,” Eur. J. Phys. 24, 403–412 (2003).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

D. H. Martin and J. W. Bowen, “Long-wave optics,” IEEE Trans. Microwave Theory Tech. 41, 1676–1690 (1993).
[Crossref]

J. Microelectromech. Syst. (1)

W. Liu and J. J. Talghader, “Spatial-mode analysis of micromachined optical cavities using electrothermal mirror actuation,” J. Microelectromech. Syst. 15, 777–785 (2006).
[Crossref]

J. Mod. Opt. (1)

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

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

J. Refractive Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

Nat. Photonics (1)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7, 861–867 (2013).
[Crossref]

Opt. Commun. (2)

K. Takeno, N. Ohmae, N. Mio, and T. Shirai, “Determination of wavefront aberrations using a Fabry–Perot cavity,” Opt. Commun. 284, 3197–3201 (2011).
[Crossref]

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995).
[Crossref]

Opt. Photon. News (1)

A. T. Watnik and D. F. Gardner, “Wavefront sensing in deep turbulence,” Opt. Photon. News 29(10), 38–45 (2018).
[Crossref]

Phys. Rev. D (1)

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D 84, 102002 (2011).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

N. Ji, T. R. Sato, and E. Betzig, “Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex,” Proc. Natl. Acad. Sci. USA 109, 22–27 (2012).
[Crossref]

Rev. Sci. Instrum. (1)

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Other (7)

A. E. Tippie, “Aberration correction in digital holography,” Ph.D. thesis (University of Rochester, 2012).

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2006).

P. F. Goldsmith, Quasioptical Systems (IEEE, 1998).

A. E. Siegman, Lasers (University Science Books, 1986).

W. Liu, “Electrically tunable micromirrors and microcavities,” Ph.D. thesis (University of Minnesota, 2004).

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds., Vol. 11 (Academic, 1992), pp. 2–53.

A. Ogloza, private communication (2018).

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Figures (13)

Fig. 1.
Fig. 1. Conceptual diagram of the measurement setup emulated by the simulations below. A laser provides a source beam that has some phase aberrations imposed upon it. The light is fed—optionally with the help of collection optics—into an optical cavity, and the transmitted output is spectrally analyzed using some kind of selectable spectral filter and a detector. A small portion of the light may be sampled to measure the total beam power.
Fig. 2.
Fig. 2. Phase fronts and difference in resulting imaginary field components I ( E aberrated E source ) of the Seidel aberrations of distortion (left) and field curvature (right). Each aberration is scaled such that a maximum of one wave of phase delay occurs within the area where the source field magnitude | E source | > 1 / e 2 . Periodic folding limits the phase delay displayed to ± 0.5 wave. (a) Distortion phase delay. (b)  I ( E withdistortion E source ) . (c) Field curvature phase delay. (d)  I ( E withfieldcurvature E source ) .
Fig. 3.
Fig. 3. Transverse cavity mode power distribution caused by (a) distortion and (b) field curvature as the aberration strength increases from 0.5 to 3 waves of maximum phase delay.
Fig. 4.
Fig. 4. Intensity spectrum transmitted by the cavity as a result of transverse mode excitation by an input Laguerre–Gaussian, with and without varying amounts of (a) distortion or (b) field curvature. Each spatial mode’s transmission peak contribution is taken to have Lorentzian line shape, approximating the typical response of dielectric mirrors.
Fig. 5.
Fig. 5. Spectral power distribution, or the portion of light intensity within a certain spectral bandwidth above the fundamental mode frequency, resulting from various strengths of (a) distortion and (b) field curvature.
Fig. 6.
Fig. 6. The (a) cavity mode decomposition and (b) resulting transmission spectrum for a plane wave with a flat phase front, bounded by a circular pupil with negligible diffraction effects.
Fig. 7.
Fig. 7. Transverse cavity mode power distribution caused by 0.5–3 waves of maximum phase delay, shaped as (a) distortion and (b) field curvature. In the case of the largest strengths of aberration, power begins to escape the sampled range of modes.
Fig. 8.
Fig. 8. Intensity spectrum transmitted by the cavity as a result of transverse mode excitation by a pupiled plane wave, with and without varying amounts of (a) distortion and (b) field curvature. It should be noted that the frequencies shown have accounted for all (degenerately) contributing modes, which is a larger range of mode indices than shown in Fig. 7.
Fig. 9.
Fig. 9. Spectral power distribution plots introduced in Fig. 5, now with a pupiled plane wave source beam instead of a cavity-matched Laguerre–Gaussian. (a) Distortion. (b) Field curvature.
Fig. 10.
Fig. 10. Phase delay, defined by (a) randomly weighting the first 10 Zernike terms, (b) summing them, and (c) scaling the resulting surface to achieve a phase front as previously with the Seidels, is (d) applied (seen as I ( E aberrated E source ) ) to a plane wave bounded by a circular pupil function with negligible frequency component loss from diffraction. (a) Zernike term weightings. (b) Phase surface. (c) Field with one wave of phase delay. (d) Difference in fields.
Fig. 11.
Fig. 11. Transmitted cavity spectrum resulting from a pupiled plane wave with varying strengths of phase delay in the shape of 10 randomly-weighted Zernike terms.
Fig. 12.
Fig. 12. Intensity spectrum transmitted by the cavity for a plane wave with a phase front derived from 10 randomly weighted Zernike terms.
Fig. 13.
Fig. 13. Spectral power distribution of optical cavity output when stimulated by a pupiled plane wave with varying amounts of phase delay, shaped by arbitrarily weighting 10 Zernike polynomial terms.

Tables (1)

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Table 1. Five Seidel Aberrations and Their Functional Forms, in the Context of a Lens or Imaging System a

Equations (9)

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E n , α ( r , ϕ , z ) = E 0 w 0 w ( z ) 2 n ! π ( | α | + n ) ! ( 2 r w ( z ) ) | α | L n | α | ( 2 r 2 w 2 ( z ) ) exp ( r 2 w 2 ( z ) i k r 2 2 R ( z ) i k z i α ϕ i ( 2 n + | α | + 1 ) tan 1 z z 0 ) ,
ν q , n , α = c 2 n medium d ( q + 2 n + | α | + 1 π c o s 1 ( ( 1 d r 1 ) ( 1 d r 2 ) ) ) ,
c n , α = 0 r = r pupil 0 ϕ = 2 π E in ( r , ϕ , z ) E n , α ( r , ϕ , z ) * r d r d ϕ ,
E aberrated ( r , ϕ ) = E source ( r , ϕ , z = z cavity ) · exp ( i W r cos ϕ )
E aberrated ( r , ϕ ) = E source ( r , ϕ , z = z cavity ) · exp ( i W r 2 )
Z n m ( ρ , θ ) = ( 2 ( n + 1 ) 1 + δ m , 0 ) 1 / 2 R m n ( ρ ) cos m θ ,
Z n m ( ρ , θ ) = ( 2 ( n + 1 ) 1 + δ m , 0 ) 1 / 2 R m n ( ρ ) sin m θ ,
R n m ( ρ ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! ( n + m ) / 2 s ) ! ( ( n m ) / 2 s ) ! ρ ( n 2 s ) .
E aberrated ( r , ϕ ) = E source ( r , ϕ , z = z cavity ) · exp ( i W · n = 0 n = n max a = 0 a = n Z n m = n + 2 a ) .

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