Abstract

We analyze an optical three-port reflection grating by means of a scattering matrix formalism. Amplitude and phase relations among the three ports, i.e., the three orders of diffraction, are derived. Such a grating can be used as an all-reflective, low-loss coupler to Fabry–Perot cavities. We derive the input–output relations of a three-port grating coupled cavity and find distinct properties that are not present in two-port coupled cavities. The cavity relations further reveal that the three-port coupler can be designed such that the additional cavity port interferes destructively. In this case the all-reflective, low-loss, single-end Fabry–Perot cavity becomes equivalent to a standard transmissive, two-port coupled cavity.

© 2005 Optical Society of America

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References

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  1. A. Bunkowksi, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. Tünnermann, Opt. Lett. 29, 2342 (2004).
    [CrossRef]
  2. R. W.P. Drever, in Proceedings of the Seventh Marcel Grossman Meeting on General Relativity, M. Keiser and R. T. Jantzen, eds. (World Scientific, Singapore, 1995), p. 1401.
  3. S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
    [CrossRef]
  4. K.-X. Sun and R. L. Byer, Opt. Lett. 23, 567 (1997).
    [CrossRef]
  5. A. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

2004

2003

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

1997

Beyersdorf, P.

Bunkowksi, A.

Burmeister, O.

Byer, R.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Byer, R. L.

Cagnoli, G.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Clausnitzer, T.

Crooks, D.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Danzmann, K.

Drever, R. W.P.

R. W.P. Drever, in Proceedings of the Seventh Marcel Grossman Meeting on General Relativity, M. Keiser and R. T. Jantzen, eds. (World Scientific, Singapore, 1995), p. 1401.

Fejer, M.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Hough, J.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Kley, E.-B.

Route, R.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Rowan, S.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Schnabel, R.

Siegman, A.

A. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

Sneddon, P.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Sun, K.-X.

Tünnermann, A.

Winkler, W.

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Opt. Lett.

Proc. SPIE

S. Rowan, R. Byer, M. Fejer, R. Route, G. Cagnoli, D. Crooks, J. Hough, P. Sneddon, and W. Winkler, Proc. SPIE 4856, 292 (2003).
[CrossRef]

Other

A. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

R. W.P. Drever, in Proceedings of the Seventh Marcel Grossman Meeting on General Relativity, M. Keiser and R. T. Jantzen, eds. (World Scientific, Singapore, 1995), p. 1401.

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Figures (4)

Fig. 1
Fig. 1

Three-port reflection grating: (a) labeling of the input and output ports, (b) amplitudes of reflection coefficients for normal incidence, (c) amplitudes of reflection coefficients for second-order Littrow incidence.

Fig. 2
Fig. 2

Fabry–Perot cavity with a three-port grating coupler and a conventional end mirror. The amplitudes of the fields of interest ( c 1 , c 2 , c 3 , t ) are indicated by arrows.

Fig. 3
Fig. 3

Power c 1 2 of cavity backreflecting port for gratings of different values of η 2 . Left, power as a function of ϕ and η 2 , right, power as a function of ϕ for (a) η 2 = η 2 , max , (b) η 2 = [ ( η 2 , max 2 + η 2 , min 2 ) 2 ] 1 2 , (c) η 2 = η 2 , min . Cavity parameters: ρ 0 2 = 0.5 , ρ 1 = 1 .

Fig. 4
Fig. 4

Powers of the two reflected ports and the transmitting port as a function of end mirror transmittance τ 1 2 for a coupler with ρ 0 2 = 0.99 and η 2 = η 2 , min for a tuning of ϕ = 0 .

Equations (18)

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b = S × a .
S 2 p = [ ρ τ τ ρ ] , S 2 p = [ ρ i τ i τ ρ ] .
r FP = [ ρ 0 ρ 1 exp ( 2 i ϕ ) ] d ,
t FP = τ 0 τ 1 exp ( i ϕ ) d ,
d = [ 1 ρ 0 ρ 1 exp ( 2 i ϕ ) ] 1 .
g FP = τ 0 d 2 .
S 3 p = [ η 2 exp ( i ϕ 2 ) η 1 exp ( i ϕ 1 ) η 0 exp ( i ϕ 0 ) η 1 exp ( i ϕ 1 ) ρ 0 exp ( i ϕ 0 ) η 1 exp ( i ϕ 1 ) η 0 exp ( i ϕ 0 ) η 1 exp ( i ϕ 1 ) η 2 exp ( i ϕ 2 ) ] .
ρ 0 2 + 2 η 1 2 = 1 ,
η 0 2 + η 1 2 + η 2 2 = 1 .
ϕ 0 = 0 ,
ϕ 1 = ( 1 2 ) arccos [ ( η 1 2 2 η 0 2 ) ( 2 ρ 0 η 0 ) ] ,
ϕ 2 = arccos [ η 1 2 ( 2 η 2 η 0 ) ] .
η 0 , min max = η 2 , min max = ( 1 ± ρ 0 ) 2 .
( c 1 c 2 c 3 ) = S 3 p × ( 1 ρ 1 c 2 exp ( 2 i ϕ ) 0 ) .
c 1 = η 2 exp ( i ϕ 2 ) + η 1 2 exp [ 2 i ( ϕ 1 + ϕ ) ] d ,
c 2 = η 1 exp ( i ϕ 1 ) d ,
c 3 = η 0 + η 1 2 exp [ 2 i ( ϕ 1 + ϕ ) ] d ,
t = i τ 1 c 2 exp ( i ϕ ) .

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