Abstract

We systematically characterize the Fabry-Pérot resonator. We derive the generic Airy distribution of a Fabry-Pérot resonator, which equals the internal resonance enhancement factor, and show that all related Airy distributions are obtained by simple scaling factors. We analyze the textbook approaches to the Fabry-Pérot resonator and point out various misconceptions. We verify that the sum of the mode profiles of all longitudinal modes is the fundamental physical function that characterizes the Fabry-Pérot resonator and generates the Airy distribution. Consequently, the resonator losses are quantified by the linewidths of the underlying Lorentzian lines and not by the measured Airy linewidth. Therefore, we introduce the Lorentzian finesse which provides the spectral resolution of the Lorentzian lines, whereas the usually considered Airy finesse only quantifies the performance of the Fabry-Pérot resonator as a scanning spectrometer. We also point out that the concepts of linewidth and finesse of the Airy distribution of a Fabry-Pérot resonator break down at low reflectivity. Furthermore, we show that a Fabry-Pérot resonator has no cut-off resonance wavelength. Finally, we investigate the influence of frequency-dependent mirror reflectivities, allowing for the direct calculation of its deformed mode profiles.

© 2016 Optical Society of America

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References

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  1. C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthode de spectroscopie interférentielle,” Ann. de Chim. et de Phys. 16(7), 115–144 (1899).
  2. J. M. Vaughan, The Fabry-Pérot Interferometer (Bristol and Philadelphia, 1989), Ch. 3.2.3., pp. 97−102.
  3. J. M. Vaughan, The Fabry-Pérot Interferometer (Bristol and Philadelphia, 1989), Ch. 3.3.2., p. 105.
  4. A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.4, pp. 428−430.
  5. A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.3, pp. 413–428.
  6. A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.5, pp. 432–440.
  7. A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.1, pp. 398–408.
  8. O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010), Ch. 4.5.1, pp. 142−146.
  9. O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010), Ch. 5.3, pp. 169−171.
  10. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), pp. 571−572.
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Ch. 7.1B, pp. 254−257.
  12. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Ch. 2.5B, pp. 62−66.
  13. M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 486 (2015).
    [Crossref]
  14. J. O. Stoner, “PEPSIOS purely interferometric high-resolution scanning spectrometer. III. Calculation of interferometer characteristics by a method of optical transients,” J. Opt. Soc. Am. 56(3), 370–376 (1966).
    [Crossref]
  15. G. Koppelmann, “Multiple-beam interference and natural modes in open resonators,” in Progress in Optics, Vol. 7, E. Wolf ed. (1969), Ch. 1, pp. 1−66.
  16. F. Bayer-Helms, “Analyse von Linienprofilen. I Grundlagen und Messeinrichtungen,” Z. Angew. Phys. 15, 330–338 (1963).
  17. J. B. Kumer and W. G. Uplinger, “Elsasser-related approximation to the Airy function,” Appl. Opt. 22(23), 3675–3676 (1983).
    [Crossref] [PubMed]
  18. J. M. Llorens, J. Buencuerpo, and P. A. Postigo, “Absorption features of the zero frequency mode in an ultra-thin slab,” Appl. Phys. Lett. 105(23), 231115 (2014).
    [Crossref]
  19. J. W. Strutt, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274, 403–411, 477–486 (1879).
    [Crossref]
  20. I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
    [Crossref]
  21. W. Kahan, “Further remarks on reducing truncation errors,” Appl. Opt. 8(1), 40 (1965).
  22. R. Remmert, “Convergent series of meromorphic functions,” in Theory of Complex Functions, 4th ed., S. Axler and F. W. Gehring, eds. (Springer, 1998), Ch. 11, pp. 321−341.

2015 (1)

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 486 (2015).
[Crossref]

2014 (1)

J. M. Llorens, J. Buencuerpo, and P. A. Postigo, “Absorption features of the zero frequency mode in an ultra-thin slab,” Appl. Phys. Lett. 105(23), 231115 (2014).
[Crossref]

2006 (1)

I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
[Crossref]

1983 (1)

1966 (1)

1965 (1)

W. Kahan, “Further remarks on reducing truncation errors,” Appl. Opt. 8(1), 40 (1965).

1963 (1)

F. Bayer-Helms, “Analyse von Linienprofilen. I Grundlagen und Messeinrichtungen,” Z. Angew. Phys. 15, 330–338 (1963).

1899 (1)

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthode de spectroscopie interférentielle,” Ann. de Chim. et de Phys. 16(7), 115–144 (1899).

1879 (1)

J. W. Strutt, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274, 403–411, 477–486 (1879).
[Crossref]

Bayer-Helms, F.

F. Bayer-Helms, “Analyse von Linienprofilen. I Grundlagen und Messeinrichtungen,” Z. Angew. Phys. 15, 330–338 (1963).

Buencuerpo, J.

J. M. Llorens, J. Buencuerpo, and P. A. Postigo, “Absorption features of the zero frequency mode in an ultra-thin slab,” Appl. Phys. Lett. 105(23), 231115 (2014).
[Crossref]

Carnicer, A.

I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
[Crossref]

Eichhorn, M.

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 486 (2015).
[Crossref]

Fabry, C.

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthode de spectroscopie interférentielle,” Ann. de Chim. et de Phys. 16(7), 115–144 (1899).

Ferré-Borrull, J.

I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
[Crossref]

Juvells, I.

I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
[Crossref]

Kahan, W.

W. Kahan, “Further remarks on reducing truncation errors,” Appl. Opt. 8(1), 40 (1965).

Kumer, J. B.

Llorens, J. M.

J. M. Llorens, J. Buencuerpo, and P. A. Postigo, “Absorption features of the zero frequency mode in an ultra-thin slab,” Appl. Phys. Lett. 105(23), 231115 (2014).
[Crossref]

Martín-Badosa, E.

I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
[Crossref]

Montes-Usategui, M.

I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
[Crossref]

Pérot, A.

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthode de spectroscopie interférentielle,” Ann. de Chim. et de Phys. 16(7), 115–144 (1899).

Pollnau, M.

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 486 (2015).
[Crossref]

Postigo, P. A.

J. M. Llorens, J. Buencuerpo, and P. A. Postigo, “Absorption features of the zero frequency mode in an ultra-thin slab,” Appl. Phys. Lett. 105(23), 231115 (2014).
[Crossref]

Stoner, J. O.

Strutt, J. W.

J. W. Strutt, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274, 403–411, 477–486 (1879).
[Crossref]

Uplinger, W. G.

Ann. de Chim. et de Phys. (1)

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthode de spectroscopie interférentielle,” Ann. de Chim. et de Phys. 16(7), 115–144 (1899).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. M. Llorens, J. Buencuerpo, and P. A. Postigo, “Absorption features of the zero frequency mode in an ultra-thin slab,” Appl. Phys. Lett. 105(23), 231115 (2014).
[Crossref]

Eur. J. Phys. (1)

I. Juvells, A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui, “Understanding the concept of resolving power in the Fabry-Perot interferometer using a digital simulation,” Eur. J. Phys. 27(5), 1111–1119 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Eichhorn and M. Pollnau, “Spectroscopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 486 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

Philos. Mag. (1)

J. W. Strutt, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274, 403–411, 477–486 (1879).
[Crossref]

Z. Angew. Phys. (1)

F. Bayer-Helms, “Analyse von Linienprofilen. I Grundlagen und Messeinrichtungen,” Z. Angew. Phys. 15, 330–338 (1963).

Other (13)

R. Remmert, “Convergent series of meromorphic functions,” in Theory of Complex Functions, 4th ed., S. Axler and F. W. Gehring, eds. (Springer, 1998), Ch. 11, pp. 321−341.

G. Koppelmann, “Multiple-beam interference and natural modes in open resonators,” in Progress in Optics, Vol. 7, E. Wolf ed. (1969), Ch. 1, pp. 1−66.

J. M. Vaughan, The Fabry-Pérot Interferometer (Bristol and Philadelphia, 1989), Ch. 3.2.3., pp. 97−102.

J. M. Vaughan, The Fabry-Pérot Interferometer (Bristol and Philadelphia, 1989), Ch. 3.3.2., p. 105.

A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.4, pp. 428−430.

A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.3, pp. 413–428.

A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.5, pp. 432–440.

A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11.1, pp. 398–408.

O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010), Ch. 4.5.1, pp. 142−146.

O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010), Ch. 5.3, pp. 169−171.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), pp. 571−572.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Ch. 7.1B, pp. 254−257.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, 2007), Ch. 2.5B, pp. 62−66.

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Figures (9)

Fig. 1
Fig. 1 Fabry-Pérot resonator with electric-field mirror reflectivities r1 and r2. Indicated are the characteristic electric fields produced by an electric field Einc incident upon mirror 1: Erefl,1 initially reflected at mirror 1, Elaun launched through mirror 1, Ecirc and Eb-circ circulating inside the resonator in forward and backward propagation direction, respectively, ERT propagating inside the resonator after one round trip, Etrans transmitted through mirror 2, Eback transmitted through mirror 1, and the total field Erefl propagating backward. Interference occurs at the left- and right-hand sides of mirror 1 between Erefl,1 and Eback, resulting in Erefl, and between Elaun and ERT, resulting in Ecirc, respectively.
Fig. 2
Fig. 2 Generic Airy distribution Acirc, equaling the spectrally dependent internal resonance enhancement which the resonator provides to light that is launched into it. For the curve with R1 = R2 = 0.9, the peak value is at Acirc(νq) = 100, outside the scale of the ordinate.
Fig. 3
Fig. 3 (a) Integral resonance enhancement AFSR (integrated over one free spectral range) and (b) fraction of the free spectral range over which enhancement occurs, i.e., Acirc ≥ 1, as a function of the product of mirror reflectivities, R1R2.
Fig. 4
Fig. 4 Airy distribution A trans (solid lines), corresponding to light transmitted through a Fabry-Pérot resonator, calculated from Eq. (31) for different values of the reflectivities R1 = R2, and comparison with a single Lorentzian line (dashed lines) calculated from Eq. (17) for the same R1 = R2. At half maximum (black line), with decreasing reflectivities the FWHM linewidth ΔνAiry of the Airy distribution broadens compared to the FWHM linewidth Δνc of its corresponding Lorentzian line: R1 = R2 = 0.9, 0.6, 0.32, 0.172 results in ΔνAiry / Δνc = 1.001, 1.022, 1.132, 1.717, respectively.
Fig. 5
Fig. 5 Airy distribution A trans of a Fabry-Pérot micro-resonator with a length of ℓ = 1 µm versus wavelength, calculated from Eq. (31) for different values of the reflectivities R1 = R2. The resonance peak with q = 1, in our example occurring at a wavelength of 2 µm, which is often considered as a “cut-off wavelength”. At wavelengths beyond 4 µm, however, the Airy distribution increases towards the resonance peak with q = 0, located at infinite wavelength.
Fig. 6
Fig. 6 (a) Relative Lorentzian linewidth ΔνcνFSR, with Δνc from Eq. (16) (blue curve), relative Airy linewidth ΔνAiryνFSR, with ΔνAiry from Eq. (48) (green curve), and its approximation of Eq. (51) (red curve), and (b) Lorentzian finesse Fc of Eq. (45) (blue curve), Airy finesse FAiry of Eq. (50) (green curve), and its approximation of Eq. (52) (red curve) as a function of reflectivity value R1R2. (c), (d) Zoom into the low-reflectivity region. The exact solutions of the Airy linewidth and finesse (green lines) correctly break down at ΔνAiry = ΔνFSR, equivalent to FAiry = 1, whereas their approximations (red lines) incorrectly do not break down. (e) Ratio between the Airy linewidth ΔνAiry of Eq. (48) and the Lorentzian linewidth Δνc of Eq. (16), equaling the ratio between the Lorentzian finesse Fc of Eq. (45) and the Airy finesse FAiry of Eq. (50), as a function of reflectivity value R1R2.
Fig. 7
Fig. 7 Illustration of the physical meaning of the Lorentzian finesse Fc of a Fabry-Pérot resonator. Displayed is the situation for R1 = R2 ≈4.32%, at which Δνc = ΔνFSR and Fc = 1, i.e., two adjacent Lorentzian lines (dashed colored lines, only 5 lines are shown for clarity) cross at half maximum (solid black line) and the Taylor criterion for spectrally resolving two peaks in the resulting Airy distribution (solid purple line) is reached.
Fig. 8
Fig. 8 Illustration of the physical meaning of the Airy finesse FAiry of a Fabry-Pérot resonator. When scanning the Fabry-Pérot length (or alternatively the angle of incident light), Airy distributions (solid lines) are created by signals at individual frequencies. If the signals occur at frequencies νm = νq + mΔνAiry, where m is an integer starting at q, the Airy distributions at adjacent frequencies are separated from each other by the linewidth ΔνAiry, thereby fulfilling the Taylor criterion for the spectroscopic resolution of two adjacent peaks. The maximum number of signals that can be resolved is FAiry. Since in this specific example the reflectivities R1 = R2 = 0.59928 have been chosen such that FAiry = 6 is an integer, the signal for m = FAiry at the frequency νq + FAiryΔνAiry = νq + ΔνFSR coincides with the signal for m = q at νq. In this example, a maximum of FAiry = 6 peaks can be resolved when applying the Taylor criterion. However, the sum of two adjacent “resolvable” peaks (dashed gray line) exhibits a deeper dip between the adjacent peaks to be resolved, i.e., a better resolution, than the sum of all “resolvable” peaks (dashed black line), see the difference highlighted in the red circle.
Fig. 9
Fig. 9 Spectral investigation of resonators with their nearest resonance frequency (dashed green vertical line) coinciding with (left), or offset by 35 GHz (right) from the reflectivity maximum (dashed black vertical line). Dashed red and blue vertical lines: next resonance frequencies. (a), (b) Mirror reflectivity of the investigated resonators. (c), (d) Airy distributions Aemit calculated analytically (solid black line) and numerically as the sum of mode profiles (magenta dots). Red, green, and blue solid lines: the three nearest resonator modes γq,emit. (e), (f) Airy distributions Acirc and Atrans. (g), (h) Airy distributions Aemit, calculated analytically (solid black line) and numerically as the sum of mode profiles (magenta dots). Red, green, and blue solid lines: mode profiles γ q,trans of the three nearest resonator modes. (i), (j) Airy distributions A circ and A emit , calculated analytically.

Equations (72)

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t RT = 2 c .
r i 2 = R i =1 t out,i 2 =1 T out,i = e t RT / τ out,i = e δ out,i 1 τ out,i = ln( R i ) t RT = ln( 1 T out,i ) t RT = δ out,i t RT .
1 τ c = i 1 τ out,i = i δ out,i t RT = i ln( R i ) t RT .
d dt φ( t )= R decay ( t )= 1 τ c φ( t ).
φ( t )= φ s e t/ τ c .
φ( t+ t RT )=φ( t ) e t RT / τ c =φ( t ) R 1 R 2 .
R decay = cw 1 τ c φ s .
φ decay ( Δt ) = transient t=0 t=Δt R decay ( t )dt = t=0 t=Δt 1 τ c φ s e t/ τ c dt = φ s ( 1 e Δt / τ c ) = Δt= τ c φ s ( 1 e 1 ),
φ decay ( Δt ) = cw t=0 t=Δt R decay dt = Δt τ c φ s = Δt= τ c φ s ,
2ϕ( ν )=2πν t RT =2πν 2 c .
d dν ( 2ϕ )= d dν ( 2πν t RT )=2π t RT 2π Δ ν FSR =2π t RT Δ ν FSR = 1 t RT .
ν q =qΔ ν FSR =q/ t RT k q = 2πqΔ ν FSR c .
E q (t)={ E q,s e i2π ν q t e t/( 2 τ c ) t0 0 t<0 .
E ˜ q (ν)= + E q (t) e i2πνt dt = E q,s 0 + e [ 1/( 2 τ c )+i2π( ν ν q ) ]t dt = E q,s 1 ( 2 τ c ) 1 +i2π(ν ν q ) .
γ ˜ q ( ν )= 1 τ c | E ˜ q (ν) E q,s | 2 = 1 τ c | 1 ( 2 τ c ) 1 +i2π( ν ν q ) | 2 = 1 τ c 1 ( 2 τ c ) 2 +4 π 2 ( ν ν q ) 2 , = 1 π 1/ ( 4π τ c ) 1/ ( 4π τ c ) 2 + ( ν ν q ) 2 with γ ˜ q ( ν )dν =1
Δ ν c = 1 2π τ c γ ˜ q ( ν )= 1 π Δ ν c /2 ( Δ ν c /2 ) 2 + ( ν ν q ) 2 with γ ˜ q ( ν )dν =1.
γ q,L ( ν )= π 2 Δ ν c γ ˜ q ( ν )= ( Δ ν c ) 2 ( Δ ν c ) 2 +4 ( ν ν q ) 2 with γ q,L ( ν q )=1.
E circ = E laun + E RT = E laun + r 1 r 2 e i2ϕ E circ E circ E laun = 1 1 r 1 r 2 e i2ϕ .
A circ = I circ I laun = | E circ | 2 | E laun | 2 = 1 | 1 r 1 r 2 e i2ϕ | 2 = 1 ( 1 R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ ) .
A circ ( ν q )= 1 ( 1 R 1 R 2 ) 2 .
A FSR = 1 π π/2 +π/2 A circ dϕ = 1 1 R 1 R 2 .
A circ 1 sin( ϕ ) 1 2 2 R 1 R 2 Δ ν A circ 1 Δ ν FSR = 2 π arcsin( 1 2 2 R 1 R 2 ).
I laun =( 1 R 1 ) I inc ,
I trans =( 1 R 2 ) I circ , I bcirc = R 2 I circ , I back =( 1 R 1 ) I bcirc ,
A b-circ = I b-circ I laun = R 2 A circ ,
A trans = I trans I laun =( 1 R 2 ) A circ ,
A back = I back I laun =( 1 R 1 ) R 2 A circ ,
A emit = A trans + A back = I trans + I back I laun =( 1 R 1 R 2 ) A circ ,
A circ = I circ I inc =( 1 R 1 ) A circ ,
A b-circ = I b-circ I inc =( 1 R 1 ) R 2 A circ ,
A trans = I trans I inc =( 1 R 1 )( 1 R 2 ) A circ ,
A back = I back I inc = ( 1 R 1 ) 2 R 2 A circ ,
A emit = A trans + A back = I trans + I back I inc =( 1 R 1 )( 1 R 1 R 2 ) A circ .
A circ ( ν q )= ( 1 R 1 ) ( 1 R 1 R 2 ) 2 =( 1 R 1 ) A circ ( ν q ) and A FSR =( 1 R 1 ) A FSR .
A trans ( ν q )= ( 1 R 1 )( 1 R 2 ) ( 1 R 1 R 2 ) 2 = R 1 = R 2 1.
E circ =i t 1 E inc + r 1 r 2 e i2ϕ E circ E circ E inc = i t 1 1 r 1 r 2 e i2ϕ ,
E trans =i t 2 E circ e iϕ E trans E inc = t 1 t 2 e iϕ 1 r 1 r 2 e i2ϕ ,
A trans = I trans I inc = | E trans | 2 | E inc | 2 = | t 1 t 2 e iϕ | 2 | 1 r 1 r 2 e i2ϕ | 2 = ( 1 R 1 )( 1 R 2 ) ( 1 R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ ) ,
E trans,1 = E inc i t 1 i t 2 e iϕ = E inc t 1 t 2 e iϕ and E trans,m+1 = E trans,m r 1 r 2 e i2ϕ ,
m=0 x m = 1 1x E trans = m=1 E trans,m = E inc t 1 t 2 e iϕ m=0 ( r 1 r 2 ) m e im2ϕ = E inc t 1 t 2 e iϕ 1 r 1 r 2 e i2ϕ .
γ q,emit ( ν )= 1 t RT γ ˜ q ( ν ),
q= γ q,emit ( ν ) = 1 R 1 R 2 ( 1 R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ ) = A emit ,
A emit ( ν q )= 1 R 1 R 2 ( 1 R 1 R 2 ) 2 = 1+ R 1 R 2 1 R 1 R 2 >1,
γ q,circ = 1 R 2 γ q,b-circ = 1 1 R 2 γ q,trans = 1 1 R 1 γ q,back = 1 1 R 1 R 2 γ q,emit γ q,circ = 1 R 2 γ q,b-circ = 1 1 R 2 γ q,trans = 1 1 R 1 γ q,back = 1 1 R 1 R 2 γ q,emit =( 1 R 1 ) γ q,circ .
F c := Δ ν FSR Δ ν c = 2π ln( R 1 R 2 ) .
Δ ν c =Δ ν FSR R 1 R 2 = e 2π 0.001867,
4 R 1 R 2 sin 2 ( Δϕ )= ( 1 R 1 R 2 ) 2 Δϕ=arcsin( 1 R 1 R 2 2 R 1 R 2 4 ).
Δ ν Airy =Δ ν FSR 2 π arcsin( 1 R 1 R 2 2 R 1 R 2 4 ).
Δ ν Airy =Δ ν FSR 1 R 1 R 2 2 R 1 R 2 4 =1 R 1 R 2 =1712 2 0.02944.
F Airy := ! Δ ν FSR Δ ν Airy = π 2 [ arcsin( 1 R 1 R 2 2 R 1 R 2 4 ) ] 1 .
4 R 1 R 2 ( Δϕ ) 2 ( 1 R 1 R 2 ) 2 Δ ν Airy Δ ν FSR 2 π Δϕ=Δ ν FSR 1 π 1 R 1 R 2 R 1 R 2 4 .
F Airy := ! Δ ν FSR Δ ν Airy π R 1 R 2 4 1 R 1 R 2 .
F Airy := ? π R 1 R 2 4 1 R 1 R 2 Δ ν FSR Δ ν Airy ,
E back =i t 1 r 2 E circ e i2ϕ E back E inc = t 1 2 r 2 e i2ϕ 1 r 1 r 2 e i2ϕ .
E refl E inc = E refl,1 + E back E inc = r 1 + t 1 2 r 2 e i2ϕ 1 r 1 r 2 e i2ϕ = r 1 r 2 e i2ϕ 1 r 1 r 2 e i2ϕ .
| e iϕ | 2 = | cos( ϕ )isin( ϕ ) | 2 = cos 2 ( ϕ )+ sin 2 ( ϕ )=1, cos( 2ϕ )=12 sin 2 ( ϕ )
| 1 r 1 r 2 e i2ϕ | 2 = | 1 r 1 r 2 cos( 2ϕ )+i r 1 r 2 sin( 2ϕ ) | 2 = [ 1 r 1 r 2 cos( 2ϕ ) ] 2 + r 1 2 r 2 2 sin 2 ( 2ϕ ) =1+ R 1 R 2 2 R 1 R 2 cos( 2ϕ )= ( 1 R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ ),
| r 1 r 2 e i2ϕ | 2 = | r 1 r 2 cos( 2ϕ )+i r 2 sin( 2ϕ ) | 2 = [ r 1 r 2 cos( 2ϕ ) ] 2 + [ r 2 sin( 2ϕ ) ] 2 = R 1 + R 2 2 R 1 R 2 cos( 2ϕ )= R 1 + R 2 2 R 1 R 2 [ 12 sin 2 ( ϕ ) ]. = ( R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ )
A refl = I refl I inc = | E refl | 2 | E inc | 2 = | r 1 r 2 e i2ϕ | 2 | 1 r 1 r 2 e i2ϕ | 2 = ( R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ ) ( 1 R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ ) , A trans + A refl = I trans + I refl I inc =1
A FSR = 1 π π/2 π/2 A circ dϕ = 1 π π/2 π/2 1 ( 1 R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ ) dϕ , =a π/2 π/2 1 b+ sin 2 ( ϕ ) dϕ
a= 1 4π R 1 R 2 , b= ( 1 R 1 R 2 ) 2 4 R 1 R 2 ,
sin(ϕ)= tan(ϕ) 1+ tan 2 (ϕ) = x 1+ x 2 , cos(ϕ)= 1 1+ tan 2 (ϕ) = 1 1+ x 2 . dx dϕ = dtan(ϕ) dϕ = 1 cos 2 (ϕ) =1+ x 2 dϕ= dx 1+ x 2
A FSR =a π/2 π/2 1 b+ sin 2 ( ϕ ) dϕ =a tan( π/2 ) tan( π/2 ) 1 b+ x 2 / ( 1+ x 2 ) 1 1+ x 2 dx = a ( b+1 ) tan( π/2 ) tan( π/2 ) 1 b/ ( b+1 ) + x 2 dx = a/ ( b+1 ) b/ ( b+1 ) [ arctan( x b/ ( b+1 ) ) ] tan( π/2 ) tan( π/2 ) . = a ( b+1 ) π b/ ( b+1 ) = 1 1 R 1 R 2
R:= R 1 R 2 δ out :=ln( R )= 1 2 ln( R 1 R 2 )= t RT 2 τ c ,
γ ˜ q ( ν )= 1 π 1/ ( 4π τ c ) 1/ ( 4π τ c ) 2 + ( ν ν q ) 2 = 1 π δ out / ( 2π t RT ) [ δ out / ( 2π t RT ) ] 2 + ( νqΔ ν FSR ) 2 = 2π t RT δ out 1 π δ out 2 δ out 2 + ( 2πν t RT 2π t RT qΔ ν FSR ) 2 = 2 t RT δ out δ out 2 δ out 2 + ( 2ϕ2πq ) 2 .
q= γ q,emit ( ν ) = q= 1 t RT γ ˜ q ( ν ) = 1 t RT 2 t RT δ out q= + δ out 2 δ out 2 + ( 2ϕ2πq ) 2 = 2 δ out δ out 2 q= + [ δ out +i( 2ϕ2πq ) δ out 2 + ( 2ϕ2πq ) 2 + δ out i( 2ϕ2πq ) δ out 2 + ( 2ϕ2πq ) 2 ] = 1 i2π q= + 1 ( δ out i2ϕ ) / ( i2π ) q + 1 i2π q= + 1 ( δ out +i2ϕ ) / ( i2π ) q .
q= + 1 xq =πcot( πx ) q= γ q,emit ( ν ) = i 2 [ cot( 2ϕ+i δ out 2 )cot( 2ϕi δ out 2 ) ].
cot( α )cot( β )= sin( βα ) sin( α )sin( β ) q= γ q,emit ( ν ) = i 2 sin( i δ out ) sin[ ϕ+i δ out /2 ]sin[ ϕi δ out /2 ] .
sin( α+β )sin( αβ ) =[ sin( α )cos( β )+cos( α )sin( β ) ][ sin( α )cos( β )cos( α )sin( β ) ] q= γ q,emit ( ν ) = i 2 sin( i δ out ) sin 2 ( ϕ ) cos 2 ( i δ out /2 ) cos 2 ( ϕ ) sin 2 ( i δ out /2 )
cos 2 ( ϕ )=1 sin 2 ( ϕ ) q= γ q,emit ( ν ) = i 2 sin( i δ out ) sin 2 ( ϕ ) sin 2 ( i δ out /2 ) .
q= γ q,emit ( ν ) = i 2 sin[ i ln( R 1 R 2 ) /2 ] sin 2 ( ϕ ) sin 2 [ i ln( R 1 R 2 ) /4 ] .
sin(x)= e ix e ix 2i q= γ q,emit ( ν ) = i 2 e ln( R 1 R 2 ) /2 e ln( R 1 R 2 ) /2 2i{ sin 2 ( ϕ ) [ e ln( R 1 R 2 )/4 e ln( R 1 R 2 )/4 ] 2 / [ 2i ] 2 } . = 1 R 1 R 2 ( 1 R 1 R 2 ) 2 +4 R 1 R 2 sin 2 ( ϕ )

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