Abstract

A coupled-cavity analysis of the resonant loop mirror with a signal flow graph technique is presented. Use of this technique has resulted in simple closed-form expressions for reflectance, transmittance, critical coupling, bandwidth, finesse, gain threshold, and mode splitting. Application of this device to enhance the single longitudinal mode operation of a fiber laser is also proposed. Detailed simulation results show a dramatic reduction in bandwidth and a threefold enhancement of the free spectral range by use of the vernier effect. The formulas and results obtained would also be useful in laser sensors and filters.

© 2005 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. G. A. Ball, W. H. Glenn, “Design of a single-mode linear-cavity erbium fiber laser utilizing Bragg reflectors,” J. Lightwave Technol. 10, 1338–1343 (1992).
    [CrossRef]
  3. S. Yamashita, G. J. Cowle, “Single-polarization operation of fiber distributed feedback (DFB) lasers by injection locking,” J. Lightwave Technol. 17, 509–513 (1999).
    [CrossRef]
  4. Z. Hu, L. Zheng, Y. Zhang, Q. Tang, “Composite cavity semiconductor fiber ring laser,” Opt. Lett. 25, 469–471 (2000).
    [CrossRef]
  5. S. K. Kim, G. Stewart, W. Johnstone, B. Culshaw, “Mode-hop-free single-longitudinal-mode erbium-doped fiber laser frequency scanned with a fiber ring resonator,” Appl. Opt. 38, 5154–5157 (1999).
    [CrossRef]
  6. C.-C. Lee, Y.-K. Chen, S.-K. Liaw, “Single-longitudinal-mode fiber laser with a passive multiple-ring cavity and its application for video transmission,” Opt. Lett. 23, 358–360 (1998).
    [CrossRef]
  7. T. Komukai, M. Nakazawa, “Tunable single frequency erbium doped fiber ring lasers using fiber grating etalons,” Jpn. J. Appl. Phys. 34, 679–680 (1995).
    [CrossRef]
  8. L. Dong, W. H. Loh, J. E. Caplen, J. D. Minelly, K. Hsu, L. Reekie, “Efficient single-frequency fiber lasers with novel photosensitive Er/Yb optical fibers,” Opt. Lett. 22, 694–696 (1997).
    [CrossRef] [PubMed]
  9. R. Paschotta, J. Nilsson, A. C. Trooper, D. C. Hanna, “Single-frequency ytterbium-doped fiber laser stabilized by spatial hole burning,” Opt. Lett. 22, 40–42 (1997).
    [CrossRef] [PubMed]
  10. P. Barnsley, P. Urquhart, C. Millar, M. Brierley, “Fiber Fox–Smith resonators: application to single-longitudinal-mode operation of fiber lasers,” J. Opt. Soc. Am. A 5, 1339–1346 (1988).
    [CrossRef]
  11. R. Paschotta, D. J. B. Brinck, S. G. Farwell, D. C. Hanna, “Resonant loop mirror with narrow-band reflections and its application in single-frequency fiber lasers,” Appl. Opt. 36, 593–596 (1997).
    [CrossRef] [PubMed]
  12. S. Srivastava, “Novel fiber optical resonating structures: applications in lasing, sensing and pulse compression,” Ph.D. dissertation (Department of Physics, Sri Sathya Sai Institute of Higher Learning, Prasanthinilayam, India, 2000), Chap. 3.
  13. F. Sanchez, “Matrix algebra for all-fiber optical resonators,” J. Lightwave Technol. 9, 838–844 (1991).
    [CrossRef]
  14. J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
    [CrossRef]
  15. O. Schwelb, “Characteristics of lattice networks and spectral filters built with 2 × 2 couplers,” J. Lightwave Technol. 17, 1470–1480 (1999).
    [CrossRef]
  16. G. Barbarossa, A. M. Matteo, M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” J. Lightwave Technol. 13, 148–157 (1995).
    [CrossRef]
  17. P. Urquhart, “Compound optical-fiber-based resonators,” J. Opt. Soc. Am. A 5, 803–812 (1988).
    [CrossRef]
  18. Y. Fang, S. Tao, P. Ye, “Analysis of compound cross-coupling fiber resonator by means of signal flow graphs,” Opt. Commun. 93, 87–91 (1992).
    [CrossRef]
  19. L. N. Binh, N. Q. Ngo, S. F. Luk, “Graphical representation and analysis of the Z-shaped double-coupler optical resonator,” J. Lightwave Technol. 11, 1782–1792 (1993).
    [CrossRef]
  20. S. J. Mason, “Feedback theory: further properties of signal flow graphs,” Proc. IRE 44, 920–926 (1956).
    [CrossRef]
  21. B. E. Little, J.-P. Laine, S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. 22, 4–6 (1997).
    [CrossRef] [PubMed]

2000 (1)

1999 (3)

1998 (1)

1997 (4)

1995 (2)

G. Barbarossa, A. M. Matteo, M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

T. Komukai, M. Nakazawa, “Tunable single frequency erbium doped fiber ring lasers using fiber grating etalons,” Jpn. J. Appl. Phys. 34, 679–680 (1995).
[CrossRef]

1993 (1)

L. N. Binh, N. Q. Ngo, S. F. Luk, “Graphical representation and analysis of the Z-shaped double-coupler optical resonator,” J. Lightwave Technol. 11, 1782–1792 (1993).
[CrossRef]

1992 (2)

G. A. Ball, W. H. Glenn, “Design of a single-mode linear-cavity erbium fiber laser utilizing Bragg reflectors,” J. Lightwave Technol. 10, 1338–1343 (1992).
[CrossRef]

Y. Fang, S. Tao, P. Ye, “Analysis of compound cross-coupling fiber resonator by means of signal flow graphs,” Opt. Commun. 93, 87–91 (1992).
[CrossRef]

1991 (1)

F. Sanchez, “Matrix algebra for all-fiber optical resonators,” J. Lightwave Technol. 9, 838–844 (1991).
[CrossRef]

1990 (1)

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

1989 (1)

1988 (2)

1956 (1)

S. J. Mason, “Feedback theory: further properties of signal flow graphs,” Proc. IRE 44, 920–926 (1956).
[CrossRef]

Armenise, M. N.

G. Barbarossa, A. M. Matteo, M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

Ball, G. A.

G. A. Ball, W. H. Glenn, “Design of a single-mode linear-cavity erbium fiber laser utilizing Bragg reflectors,” J. Lightwave Technol. 10, 1338–1343 (1992).
[CrossRef]

Barbarossa, G.

G. Barbarossa, A. M. Matteo, M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

Barnsley, P.

Binh, L. N.

L. N. Binh, N. Q. Ngo, S. F. Luk, “Graphical representation and analysis of the Z-shaped double-coupler optical resonator,” J. Lightwave Technol. 11, 1782–1792 (1993).
[CrossRef]

Brierley, M.

Brinck, D. J. B.

Caplen, J. E.

Capmany, J.

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

Chen, Y.-K.

Chu, S. T.

Cowle, G. J.

Culshaw, B.

Dong, L.

Fang, Y.

Y. Fang, S. Tao, P. Ye, “Analysis of compound cross-coupling fiber resonator by means of signal flow graphs,” Opt. Commun. 93, 87–91 (1992).
[CrossRef]

Farwell, S. G.

Glenn, W. H.

G. A. Ball, W. H. Glenn, “Design of a single-mode linear-cavity erbium fiber laser utilizing Bragg reflectors,” J. Lightwave Technol. 10, 1338–1343 (1992).
[CrossRef]

Hanna, D. C.

Hsu, K.

Hu, Z.

Johnstone, W.

Kim, S. K.

Komukai, T.

T. Komukai, M. Nakazawa, “Tunable single frequency erbium doped fiber ring lasers using fiber grating etalons,” Jpn. J. Appl. Phys. 34, 679–680 (1995).
[CrossRef]

Laine, J.-P.

Lee, C.-C.

Liaw, S.-K.

Little, B. E.

Loh, W. H.

Luk, S. F.

L. N. Binh, N. Q. Ngo, S. F. Luk, “Graphical representation and analysis of the Z-shaped double-coupler optical resonator,” J. Lightwave Technol. 11, 1782–1792 (1993).
[CrossRef]

Mason, S. J.

S. J. Mason, “Feedback theory: further properties of signal flow graphs,” Proc. IRE 44, 920–926 (1956).
[CrossRef]

Matteo, A. M.

G. Barbarossa, A. M. Matteo, M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

Millar, C.

Minelly, J. D.

Muriel, M. A.

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

Nakazawa, M.

T. Komukai, M. Nakazawa, “Tunable single frequency erbium doped fiber ring lasers using fiber grating etalons,” Jpn. J. Appl. Phys. 34, 679–680 (1995).
[CrossRef]

Ngo, N. Q.

L. N. Binh, N. Q. Ngo, S. F. Luk, “Graphical representation and analysis of the Z-shaped double-coupler optical resonator,” J. Lightwave Technol. 11, 1782–1792 (1993).
[CrossRef]

Nilsson, J.

Paschotta, R.

Reekie, L.

Sanchez, F.

F. Sanchez, “Matrix algebra for all-fiber optical resonators,” J. Lightwave Technol. 9, 838–844 (1991).
[CrossRef]

Schwelb, O.

Srivastava, S.

S. Srivastava, “Novel fiber optical resonating structures: applications in lasing, sensing and pulse compression,” Ph.D. dissertation (Department of Physics, Sri Sathya Sai Institute of Higher Learning, Prasanthinilayam, India, 2000), Chap. 3.

Stewart, G.

Tang, Q.

Tao, S.

Y. Fang, S. Tao, P. Ye, “Analysis of compound cross-coupling fiber resonator by means of signal flow graphs,” Opt. Commun. 93, 87–91 (1992).
[CrossRef]

Trooper, A. C.

Urquhart, P.

Yamashita, S.

Ye, P.

Y. Fang, S. Tao, P. Ye, “Analysis of compound cross-coupling fiber resonator by means of signal flow graphs,” Opt. Commun. 93, 87–91 (1992).
[CrossRef]

Zhang, Y.

Zheng, L.

Appl. Opt. (3)

J. Lightwave Technol. (7)

S. Yamashita, G. J. Cowle, “Single-polarization operation of fiber distributed feedback (DFB) lasers by injection locking,” J. Lightwave Technol. 17, 509–513 (1999).
[CrossRef]

O. Schwelb, “Characteristics of lattice networks and spectral filters built with 2 × 2 couplers,” J. Lightwave Technol. 17, 1470–1480 (1999).
[CrossRef]

F. Sanchez, “Matrix algebra for all-fiber optical resonators,” J. Lightwave Technol. 9, 838–844 (1991).
[CrossRef]

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

G. Barbarossa, A. M. Matteo, M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

G. A. Ball, W. H. Glenn, “Design of a single-mode linear-cavity erbium fiber laser utilizing Bragg reflectors,” J. Lightwave Technol. 10, 1338–1343 (1992).
[CrossRef]

L. N. Binh, N. Q. Ngo, S. F. Luk, “Graphical representation and analysis of the Z-shaped double-coupler optical resonator,” J. Lightwave Technol. 11, 1782–1792 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

Jpn. J. Appl. Phys. (1)

T. Komukai, M. Nakazawa, “Tunable single frequency erbium doped fiber ring lasers using fiber grating etalons,” Jpn. J. Appl. Phys. 34, 679–680 (1995).
[CrossRef]

Opt. Commun. (1)

Y. Fang, S. Tao, P. Ye, “Analysis of compound cross-coupling fiber resonator by means of signal flow graphs,” Opt. Commun. 93, 87–91 (1992).
[CrossRef]

Opt. Lett. (5)

Proc. IRE (1)

S. J. Mason, “Feedback theory: further properties of signal flow graphs,” Proc. IRE 44, 920–926 (1956).
[CrossRef]

Other (1)

S. Srivastava, “Novel fiber optical resonating structures: applications in lasing, sensing and pulse compression,” Ph.D. dissertation (Department of Physics, Sri Sathya Sai Institute of Higher Learning, Prasanthinilayam, India, 2000), Chap. 3.

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Figures (14)

Fig. 1
Fig. 1

(a) 2 × 2 coupler with four ports: 1, 2, 3, and 4; (b) planar SFG for the coupler; (c) SFG for the coupler (flow directions reversed).

Fig. 2
Fig. 2

Single coupler resonator with an embedded reflector.

Fig. 3
Fig. 3

SFG of the SCRWR.

Fig. 4
Fig. 4

Reflectance of the SCRWR with coupling losses of (a) γ = 0 and (b) γ = 0.01 and propagation losses of (a) α(11 + 12) = 0 and (b) α(11 + 12) = 0.02.

Fig. 5
Fig. 5

Effects of the embedded reflector on the (a) output reflectance and (b) output reflectance with loss included.

Fig. 6
Fig. 6

Critical reflectance of the SCRWR obtained with Eq. (6), the lossless condition.

Fig. 7
Fig. 7

Mode splitting obtained with Eq. (8).

Fig. 8
Fig. 8

FWHM of the SCRWR. The change in FWHM by including loss is not observable on the same scale and is therefore not shown.

Fig. 9
Fig. 9

Effects of the embedded reflector on the FWHM of the SCRWR: (a) R = Rcritical by adjustment of the coupling coefficient for each R in the plot and (b) loss included.

Fig. 10
Fig. 10

Plot of reflectance versus round-trip phase for the SCRWR with gain.

Fig. 11
Fig. 11

(a) Bandwidth reduction in the SCRWR as end reflector (SCRWRER) of a Fabry–Perot cavity for a lossless case. (b) FWHM of the SCRWRER versus the reflectance of the embedded reflector.

Fig. 12
Fig. 12

Plot of the phase of the reflected light superimposed on the reflected intensity under lossless conditions. R = Rcritical = 0.04.

Fig. 13
Fig. 13

Side-mode suppression for a 10-cm loop and (a) a Fabry–Perot length of 150 cm under lossless conditions and (b) with threefold vernier enhancement of the FSR by use of a Fabry–Perot length of 51.6 cm.

Fig. 14
Fig. 14

Mode spacing of the SCRWR and the SCRWRER (a) without vernier enhancement and (b) with FSR enhancement by use of P = 31/3.

Equations (26)

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E a 3 = x E t 1 + y E t 2 ,             E a 4 = y E t 1 + x E t 2 .
E a 1 = x E t 3 + y E t 4 ,             E a 2 = y E t 3 + x E t 4 .
Δ = 1 - ( a + b + c ) + ( a b ) ,             Δ 1 = 1 , P 1 = r x 2 L 1 2 .
E a 1 E t 1 = P 1 Δ Δ 1 = r x 2 ( L 1 2 1 - 2 y t ( L 1 L 2 ) + y 2 ( L 1 L 2 ) 2 = r 1 = - r ( 1 - y 2 ) exp ( 2 j φ 1 ) 1 - 2 y t exp ( j φ loop ) + y 2 exp ( 2 j φ loop ) ,
t 1 = E a 4 E t 1 = P 1 Δ 1 + P 2 Δ 2 + P 3 Δ 3 Δ = y [ 1 - 2 y t exp ( j φ loop ) + y 2 exp ( 2 j φ loop ) ] - t ( 1 - y ) 2 exp ( j φ loop ) - y ( 1 - y 2 ) exp ( 2 j φ loop ) 1 - 2 y t exp ( j φ loop ) + y 2 exp ( 2 j φ loop ) .
R 1 = r 1 2 = 1 = r critical 2 [ 1 - y 2 ] 2 ( 1 - p cos φ + q cos 2 φ ) 2 + ( p sin φ - q sin 2 φ ) 2 .
r critical = x 2 2 - x 2 = k 2 - k             or             k critical = 2 R 1 + R ,
φ = cos - 1 [ t ( 1 + y 2 ) 2 y ] ,
Δ ν mode split = cos - 1 [ t ( 1 + y 2 ) 2 y ] c π n eff ( l 1 + l 2 ) ,
I max = r 2 ( 1 - y 2 ) 2 [ 1 - ( p - q ) ] 2 .
I = I max 2 = r 2 ( 1 - y 2 ) 2 ( 1 - p cos φ + q cos 2 φ ) 2 + ( p sin φ - q sin 2 φ ) 2 .
± φ = cos - 1 ( 1 - R 1 - R ) .
φ = [ R ( R + 2 ) ] 1 / 2 .
Δ ν 1 / 2 = FSR [ R critical ( R critical + 2 ) ] 1 / 2 π = c [ R critical ( R critical + 2 ) ] 1 / 2 n eff ( l 1 + l 2 ) π .
E a 1 E t 1 = r ( 1 - y 2 ) exp ( 2 j φ 1 ) Δ = .
( a + b ) - ( a b / t 2 ) = 1 ,
exp ( G th ) = 1 y ( threshold condition ) ,
cos φ = t = ( 1 - r 2 ) 1 / 2 ( oscillation phase condition ) .
± φ r = R .
Δ ν mode split = FSR R π .
E out E in = t 1 t 2 exp ( j φ FP ) 1 - r 1 r 2 exp ( 2 j φ FP ) exp ( G ) ,
Δ = 1 - 2 y t exp ( j φ loop ) + y 2 exp ( 2 j φ loop ) - r r 2 ( 1 - y 2 ) exp ( 2 j φ 11 ) exp ( 2 j φ FP ) exp ( G ) .
G th = - ln ( r 1 r 2 ) = - ln [ r ( 1 - y 2 ) r 2 1 - 2 y t + y 2 ] .
( M - N / N ) l = l 3 .
H j k = i N P j k i Δ j k i Δ ,
Δ = 1 - ( - 1 ) r + 1 m r P mr = 1 - m P m 1 + m P m 2 - m P m 3 + = 1 - ( sum of all loop gains ) + ( sum of all gain products of pairs of nontouching loops ) - ( sum of all gain products of triplets of nontouching loops ) + ,

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