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An investigation of a tapered fiber-microsphere coupling system with gain and evanescent-field sensing device
An investigation of a tapered fiber-microsphere coupling system with gain and evanescent-field sensing device Ying Lu1, Jian-Quan Yao1, Peng Wang1, Cun-Zhou Zhang2 1 2
Institute of Laser and Opto-Electronics, College of Opto-Electronics Information, Science and Technology Lab, Tianjin University, Tianjin 300072, China Photonics Center, Institute of Physics, Nankai University, Tianjin 300071, China
Abstract: We consider a coupling system composed of a tapered fiber and a microsphere with gain. Power amplification and strong sensitivity to the absorbent sample molecule, on the surface of the sphere are found when gain is larger than intrinsic losses in this system. The intrinsic losses being in some extent compensated by gain make this system possess more advantage in integrated optics and photonics devices application. Key words: Microsphere –– tapered fiber –– coupling –– gain
1. Introduction Recently, the optical microsphere resonators have generated considerable interest in areas of fundamental physics [1–3] and applicable science [4–6] because of the high-Q values and small mode volumes of whispering-gallery (WG) modes in the dielectric microsphere. To excite and utilize the most interesting WG modes which correspond to photons being most closely confined to the surface and equator of the sphere by repeated total internal reflection from the surface, one requires the use of near-field coupler devices which can couple light into and out of the microsphere beyond the critical angle. In the experimental results reported on efficient coupler devices [7, 8], the efficiency of tapered fiber coupler is one of the best. Furthermore, tapered fiber couplers are easily used with fiber and tapered fiber-microsphere system can serve as versatile units for integrated optics. Most applications that make use of WG modes in the microsphere, such as frequency-stabilization devices for semiconductor lasers, narrow-band channel dropping filters, and sensors, require quality factors Q of coupling system to be as high as possible. In the conventional coupling system made of sphere without
Received 22 December 2000; accepted 1 July 2001. Correspondence to: C. Z. Zhang Fax: ++86-22-23509356 E-mail: [email protected]
Optik 112, No. 10 (2001) 475–478 ª 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
gain, however, Q values are inevitably limited due to the losses introduced by radiate, scattering, material absorption, surface contaminants and coupling. We consider introducing a gain in microsphere and compensating the losses. This coupling system may produce new features and new applications. In this paper we investigated, for the first time to our knowledge, the effect of gain on coupling between the propagating mode in the tapered fiber and the given WG mode in the microsphere with gain by a simple model. It is important that the relation between the gain and intrinsic losses of the sphere is a condition of different application of this coupling system. We find that strong power amplification and much small FWHM of the signal through the tapered fiber appear when gain is lager than intrinsic losses, which can make the coupling system used as optical amplifier and frequency-stabilization devices. In addition, we discuss the possibility of using this system as a sensor or high efficiency photonic switching device.
2. The model and analysis The geometry of the coupling system of a tapered fiber and a microsphere with gain we analysed is shown in fig. 1., where we consider the coupling between the fundamental cladding-guided mode in a tapered fiber and the most interesting WG mode in the microsphere. Because the coupling zone is small, the coupling is considered as lumped at the reference plane shown in fig. 1. The coupling equations for the field transfer are pffiffiffiffiffiffiffiffiffiffiffiffi E3 ¼ 1 t 2 E1 þ itE2 ð1aÞ pffiffiffiffiffiffiffiffiffiffiffiffi ð1bÞ E4 ¼ itE1 þ 1 t 2 E2 where Ei is the complex field amplitude (at the ith port) normalized to that jEi j2 ¼ Pi , the power entering or exiting that port, and t is real amplitude coupling coefficient related to the usual coupling coefficient of the coupling mode formulation, which is given in ref. [9]. 0030-4026/01/112/10-475 $ 15.00/0
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Ying Lu et al., An investigation of a tapered fiber-microsphere coupling system
where t is peak-coupling efficiency which is given by t¼
f1 exp ½ðg aÞ L g t 2 pffiffiffiffiffiffiffiffiffiffiffiffi f1 exp ½ðg aÞ l=2 1 t 2 g2
ð6Þ
F is the finesse which is given by F ¼
Fig. 1. The geometry of the coupling system of the tapered fiber-microsphere with gain.
When the intensity of the light in the microsphere is not so large that the action of the light on gain coefficient can be neglected, the relation of E2 and E4 is given by [10]: ðg aÞ L ð2Þ E2 ¼ exp exp ðifÞ E4 ; 2 where g is gain coefficient, a is average loss coefficient (absorption, scattering, etc.), L ¼ 2pa is the circumference of the sphere, and f ¼ bL is the total phase shift acquired by the wave during one round trip, where b is the propagation constant of the most interesting mode to be excited, which is approximately given by [11]: k0 ns 1=3 p zn a2=3 þ a1 b k0 ns 2 2 ðns 1Þ1=2 1 2 1=3 2 4=3 þ zn a ð3Þ 6 k0 ns where 8 < ns p¼ 1 : ns
TE modes TM modes ;
zn denotes the nth zero of the Airy function, n is the radial mode numbers, k0 is the wavenumber of the free space. Eqs. (1) and (2) can be solved to give pffiffiffiffiffiffiffiffiffiffiffiffi 1 t 2 exp ½if ða gÞ L=2
E3 pffiffiffiffiffiffiffiffiffiffiffiffi ; ð4Þ ¼ E1 1 1 t 2 exp ½if ða gÞ L=2
then we get 2 E3 ¼1 E 1
t 2 2F f sin 1þ p 2
ð5Þ
pfexp ½ðg aÞ L ð1 t 2 Þg1=4 pffiffiffiffiffiffiffiffiffiffiffiffi : 1 exp ½ðg aLÞ=2 1 t 2
ð7Þ
From eqs. (5) and (6), we can clearly observe that the coupling system have three kinds of fundamental effect on the input light power at resonance: amplifying, decreasing (or vanishing) and not changing, depending on whether the gain of the sphere is larger, or lower than or equal to the intrinsic losses of the sphere. This implies that parameters of the coupling system need to be chosen suitable to different application.
3. Discussion Due to the importance of the relation between the gain and intrinsic losses of the sphere, we study the feature of the coupling system in three cases, separately.
i) T h e c a s e o f g < a When g is smaller than a, we find from eqs. (5) and (6) that output power is equal to 0 at resonance wavelength for t 2 ða gÞ L, while output power equal almost to input power at nonresonant wavelength. This means that the coupling system in this case can offer the possibility of realizing filter characteristics, which make it possible to apply it as narrow-bandwidth filter, optical switch etc, as the conventional coupling system without gain. In these applications, however, there is no doubt that the highest peak-coupling efficiency t ¼ 1 at resonance is desired. In ref. [9], we showed that the necessary condition achieving t ¼ 1 is t02 ða gÞ L. Clearly, losses being partly compensated by gain make it more easily achieved. This exhibits the advantage for these applications.
ii) T h e c a s e o f g ¼ a When g ¼ a, output power is equal to the input power. In this case, we can consider the application of this coupling system from respect of phase of the field transmission factor. According to ref. [12], if the sphere possess nonlinear phase-transfer characteristics which leads effective phase sensitivity with respect to input power to a finesse-squared enhance, the coupling system with very high finesse can replace one arm of a standard Mach-Zehnder to become high efficient optical switches.
Ying Lu et al., An investigation of a tapered fiber-microsphere coupling system
477
iii) T h e c a s e o f g > a We now deal with the case where the gain of the sphere is larger than the intrinsic losses of the sphere. To show the feature of the system more vividly, we perform some special calculations for sphere (with radius a ¼ 50 mm, refractive index of 1.44, and average loss coefficient a ¼ 105/mm). The radius of the waist b0 ¼ 2 mm, tapered angle d ¼ 0.04, and refractive index nc ¼ 1.44 of the tapered fiber, and the gap distance between the microsphere and the waist of the taper fiber d ¼ 0 are chosen so that the optimal coupling for g ¼ 0 at resonance [9] is achieved. So advantage of the introduction of gain may be more clearly revealed by comparison between with gain and without gain. Here, we consider the excitation of TE mode with n ¼ 1 only. Fig. 2 compares the output power characteristics of the coupling systems comprised of the sphere with gain g ¼ 1:3 105 =mm (curve 1) and without gain (curve 2) by eq. (5). The evident difference between two systems is the appearance of an amplified power ðjE3 j2 > jE1 j2 Þ in the coupling systems comprised of the sphere with gain. This feature implies that the coupling system with gain have the possibility to become a new optical amplifier as far as the sphere could be made of the active material corresponding to the resonance wavelength. In addition, we can find the significant decreasing of the linewidth in the coupling systems comprised of the sphere with gain. Evidently, this feature means that the coupling systems comprised of the sphere with gain possess more advantage for the frequency-stabilization device of semiconductor lasers than the coupling systems comprised of the sphere without gain, although it can also be used due to its narrow resonance linewidth. Now we discuss an other possible application: sensors. It is well known that chemists have long used absorption spectroscopy to measure small amounts of a substance. Our coupling system may make the technique more sensitive. For this purpose, the system is slightly
Fig. 3. The geometry of the altered coupling system of the tapered fiber-microsphere with gain for sensor applications.
altered as shown in fig. 3. If a sample molecule is brought to the surface and absorbs the evanescent wave outside the sphere, light is coupled out of the sphere, which is described by small coupling coefficient D clearly related with the concentration and the other parameter of the sample. As the analysis before, the relations between Ei with each other can be given by: pffiffiffiffiffiffiffiffiffiffiffiffi ð8aÞ E3 ¼ itE1 þ 1 t 2 E2 E4 ¼ E3 exp ½ðg aÞL=4 exp ðif=2Þ pffiffiffiffiffiffiffiffiffiffiffiffi E6 ¼ 1 t 2 E4
ð8bÞ
E5 ¼ itE4
ð8dÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 ¼ E6 exp ½ðg aÞL=4 exp ðif=2Þ 1 D2 :
ð8cÞ
ð8eÞ Then we get: E5 ¼
t 2 exp ½if=2 ða gÞ L=4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E1 : 1 ð1 t 2 Þ exp ½if ða gÞ L=2 1 D2 ð9Þ
At resonance ðf ¼ 2npÞ, considering jða gÞLj; t2, D2 1, the output power is 2 t 2 exp ½ðg aÞL=4
2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E1 jE5 j ¼ 1 ð1 t 2 Þ exp ½ðg aÞL=2 1 D2 2t 2 ð10Þ j2 jE1 j2 : ða gÞL þ 2t 2 þ D2
Fig. 2. A comparison of the output power characteristics of the coupling system comprised of a sphere with gain g ¼ 1:3 105 =mm (curve 1) and without gain (curve 2).
To make the sensivitiy of the output power to coupling coefficient more evident, we rewrite eq. (10) with intrinsic quality factor of the coupling system without 2pns and assume sample Q¼ ½ða gÞL þ 2t 2 l
478
Ying Lu et al., An investigation of a tapered fiber-microsphere coupling system
D2 ða gÞL þ t 2 . Then we get 2 2 2 4 E5 2t Q 8t Q3 D2 : E 3 k0 ns k0 n3s 1
References ð11Þ
The sensitivity of the output power to the change of 8t 4 the small D2 evidently depends on the term 3 3 Q3 , k0 ns which is named sensitivity factor as below. In the case of g ¼ 0 and even g a, because value of t is in practice usually in the order of 103 and in turn that the highest value of Q is in the order of 106, we can derive that the highest sensitivity factor is in the order of 106. In the case of g > a, however, sensitivity factor much larger than 1012 can be achieved as long as the value of Q is much larger than 108. Such high value of sensitivity factor indicates that our new coupling system with g > a is well suited e.g. as sensor for sniffing out explosives or for finding traces of chemical and biological weapons, and etc. [13]. In conclusion, we have presented theoretical predictions of the characteristic and possible application of the fiber-microsphere coupling element with gain. These results indicate the potential of the coupler in integrated optics and photonic devices applications. Furthermore, the experiments in which the practical coupler is constructed by fluoride glass microsphere, doped with rare earth irons and a fiber taper formed by heating and pulled standard single-mode fiber are in progress. Acknowledgement. This work was projection 69678015 supported by National Natural Science Foundation of China.
[1] Barber PW, Chang RK: Optical effects associated with small particles. World Scientific, Singapore 1988 [2] Collot L, Lefevre-Seguin V, Brune M, Raimond JM, Haroche S: Very high-Q whispering gallery modes resonances observed on fused silica microspheres. Europhys. Lett. 23 (1993) 327–333 [3] Braginsky VB, Gorodetsky ML, Ilchenko VS: Quality-factor and nonlinear properties of optical whispering-gallery modes. Phys. Lett. A137 (1989) 393–396 [4] Arnold S, Liu CT, Whitten WB, Ramsey JM: Room-temperature microparticle-based persistent spectral hole buring memory. Opt. Lett. 16 (1991) 420–422 [5] Lu Y, Wang JY, Xu XX, Pan SH, Zhang CZ: Optical bistability and differential amplification in a tapered fiber coupler microsphere resonator with nonlinear medium. Opt. Laser Technol. 32 (2000) 245–248 [6] Little BE, Chu ST, Haus HA, Foresi J, Laine J-P: Microring resonator channel dropping filters. J. Lightwave Technol. 15 (1997) 998–1005 [7] Knight JC, Cheung JC, Jacques F, Birks TA: Phasematched excitation of whispering-gallery-mode resonances by a fiber taper. Opt. Lett. 22 (1997) 1129–1131 [8] Ilchenko VS, Yao XS, Maleki L: Pigtailing the high-Q microsphere cavity: a simple fiber coupler for optical whispering-gallery. Opt. Lett. 24 (1999) 723–725 [9] Lu Y, Wang JY, Xu XX, Pan SH, Zhang CZ: Optimal conditions of coupling between the propagating mode in a tapered fiber and the given WG mode in a high-Q microsphere. Optik 112 (2001) 109–113 [10] Maitland A, Dunn MH: Laser Physics. North-Holland, Amsterdam 1969 [11] Lock JA: Improved Gaussian beam-scattering algorithm. Appl. Opt. 34 (1995) 559 [12] Heebner JE, Boyd RW: Enhanced all-optical switching by use of a nonlinear fiber ring resonator. Opt. Lett. 24 (1999) 847–850 [13] Savage N: Optical cavity opens door to better detecters. Laser Focus World 35 (12) (1999) 26–27
Institute of Laser and Opto-Electronics, College of Opto-Electronics Information, Science and Technology Lab, Tianjin University, Tianjin 300072, China Photonics Center, Institute of Physics, Nankai University, Tianjin 300071, China
Abstract: We consider a coupling system composed of a tapered fiber and a microsphere with gain. Power amplification and strong sensitivity to the absorbent sample molecule, on the surface of the sphere are found when gain is larger than intrinsic losses in this system. The intrinsic losses being in some extent compensated by gain make this system possess more advantage in integrated optics and photonics devices application. Key words: Microsphere –– tapered fiber –– coupling –– gain
1. Introduction Recently, the optical microsphere resonators have generated considerable interest in areas of fundamental physics [1–3] and applicable science [4–6] because of the high-Q values and small mode volumes of whispering-gallery (WG) modes in the dielectric microsphere. To excite and utilize the most interesting WG modes which correspond to photons being most closely confined to the surface and equator of the sphere by repeated total internal reflection from the surface, one requires the use of near-field coupler devices which can couple light into and out of the microsphere beyond the critical angle. In the experimental results reported on efficient coupler devices [7, 8], the efficiency of tapered fiber coupler is one of the best. Furthermore, tapered fiber couplers are easily used with fiber and tapered fiber-microsphere system can serve as versatile units for integrated optics. Most applications that make use of WG modes in the microsphere, such as frequency-stabilization devices for semiconductor lasers, narrow-band channel dropping filters, and sensors, require quality factors Q of coupling system to be as high as possible. In the conventional coupling system made of sphere without
Received 22 December 2000; accepted 1 July 2001. Correspondence to: C. Z. Zhang Fax: ++86-22-23509356 E-mail: [email protected]
Optik 112, No. 10 (2001) 475–478 ª 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
gain, however, Q values are inevitably limited due to the losses introduced by radiate, scattering, material absorption, surface contaminants and coupling. We consider introducing a gain in microsphere and compensating the losses. This coupling system may produce new features and new applications. In this paper we investigated, for the first time to our knowledge, the effect of gain on coupling between the propagating mode in the tapered fiber and the given WG mode in the microsphere with gain by a simple model. It is important that the relation between the gain and intrinsic losses of the sphere is a condition of different application of this coupling system. We find that strong power amplification and much small FWHM of the signal through the tapered fiber appear when gain is lager than intrinsic losses, which can make the coupling system used as optical amplifier and frequency-stabilization devices. In addition, we discuss the possibility of using this system as a sensor or high efficiency photonic switching device.
2. The model and analysis The geometry of the coupling system of a tapered fiber and a microsphere with gain we analysed is shown in fig. 1., where we consider the coupling between the fundamental cladding-guided mode in a tapered fiber and the most interesting WG mode in the microsphere. Because the coupling zone is small, the coupling is considered as lumped at the reference plane shown in fig. 1. The coupling equations for the field transfer are pffiffiffiffiffiffiffiffiffiffiffiffi E3 ¼ 1 t 2 E1 þ itE2 ð1aÞ pffiffiffiffiffiffiffiffiffiffiffiffi ð1bÞ E4 ¼ itE1 þ 1 t 2 E2 where Ei is the complex field amplitude (at the ith port) normalized to that jEi j2 ¼ Pi , the power entering or exiting that port, and t is real amplitude coupling coefficient related to the usual coupling coefficient of the coupling mode formulation, which is given in ref. [9]. 0030-4026/01/112/10-475 $ 15.00/0
476
Ying Lu et al., An investigation of a tapered fiber-microsphere coupling system
where t is peak-coupling efficiency which is given by t¼
f1 exp ½ðg aÞ L g t 2 pffiffiffiffiffiffiffiffiffiffiffiffi f1 exp ½ðg aÞ l=2 1 t 2 g2
ð6Þ
F is the finesse which is given by F ¼
Fig. 1. The geometry of the coupling system of the tapered fiber-microsphere with gain.
When the intensity of the light in the microsphere is not so large that the action of the light on gain coefficient can be neglected, the relation of E2 and E4 is given by [10]: ðg aÞ L ð2Þ E2 ¼ exp exp ðifÞ E4 ; 2 where g is gain coefficient, a is average loss coefficient (absorption, scattering, etc.), L ¼ 2pa is the circumference of the sphere, and f ¼ bL is the total phase shift acquired by the wave during one round trip, where b is the propagation constant of the most interesting mode to be excited, which is approximately given by [11]: k0 ns 1=3 p zn a2=3 þ a1 b k0 ns 2 2 ðns 1Þ1=2 1 2 1=3 2 4=3 þ zn a ð3Þ 6 k0 ns where 8 < ns p¼ 1 : ns
TE modes TM modes ;
zn denotes the nth zero of the Airy function, n is the radial mode numbers, k0 is the wavenumber of the free space. Eqs. (1) and (2) can be solved to give pffiffiffiffiffiffiffiffiffiffiffiffi 1 t 2 exp ½if ða gÞ L=2
E3 pffiffiffiffiffiffiffiffiffiffiffiffi ; ð4Þ ¼ E1 1 1 t 2 exp ½if ða gÞ L=2
then we get 2 E3 ¼1 E 1
t 2 2F f sin 1þ p 2
ð5Þ
pfexp ½ðg aÞ L ð1 t 2 Þg1=4 pffiffiffiffiffiffiffiffiffiffiffiffi : 1 exp ½ðg aLÞ=2 1 t 2
ð7Þ
From eqs. (5) and (6), we can clearly observe that the coupling system have three kinds of fundamental effect on the input light power at resonance: amplifying, decreasing (or vanishing) and not changing, depending on whether the gain of the sphere is larger, or lower than or equal to the intrinsic losses of the sphere. This implies that parameters of the coupling system need to be chosen suitable to different application.
3. Discussion Due to the importance of the relation between the gain and intrinsic losses of the sphere, we study the feature of the coupling system in three cases, separately.
i) T h e c a s e o f g < a When g is smaller than a, we find from eqs. (5) and (6) that output power is equal to 0 at resonance wavelength for t 2 ða gÞ L, while output power equal almost to input power at nonresonant wavelength. This means that the coupling system in this case can offer the possibility of realizing filter characteristics, which make it possible to apply it as narrow-bandwidth filter, optical switch etc, as the conventional coupling system without gain. In these applications, however, there is no doubt that the highest peak-coupling efficiency t ¼ 1 at resonance is desired. In ref. [9], we showed that the necessary condition achieving t ¼ 1 is t02 ða gÞ L. Clearly, losses being partly compensated by gain make it more easily achieved. This exhibits the advantage for these applications.
ii) T h e c a s e o f g ¼ a When g ¼ a, output power is equal to the input power. In this case, we can consider the application of this coupling system from respect of phase of the field transmission factor. According to ref. [12], if the sphere possess nonlinear phase-transfer characteristics which leads effective phase sensitivity with respect to input power to a finesse-squared enhance, the coupling system with very high finesse can replace one arm of a standard Mach-Zehnder to become high efficient optical switches.
Ying Lu et al., An investigation of a tapered fiber-microsphere coupling system
477
iii) T h e c a s e o f g > a We now deal with the case where the gain of the sphere is larger than the intrinsic losses of the sphere. To show the feature of the system more vividly, we perform some special calculations for sphere (with radius a ¼ 50 mm, refractive index of 1.44, and average loss coefficient a ¼ 105/mm). The radius of the waist b0 ¼ 2 mm, tapered angle d ¼ 0.04, and refractive index nc ¼ 1.44 of the tapered fiber, and the gap distance between the microsphere and the waist of the taper fiber d ¼ 0 are chosen so that the optimal coupling for g ¼ 0 at resonance [9] is achieved. So advantage of the introduction of gain may be more clearly revealed by comparison between with gain and without gain. Here, we consider the excitation of TE mode with n ¼ 1 only. Fig. 2 compares the output power characteristics of the coupling systems comprised of the sphere with gain g ¼ 1:3 105 =mm (curve 1) and without gain (curve 2) by eq. (5). The evident difference between two systems is the appearance of an amplified power ðjE3 j2 > jE1 j2 Þ in the coupling systems comprised of the sphere with gain. This feature implies that the coupling system with gain have the possibility to become a new optical amplifier as far as the sphere could be made of the active material corresponding to the resonance wavelength. In addition, we can find the significant decreasing of the linewidth in the coupling systems comprised of the sphere with gain. Evidently, this feature means that the coupling systems comprised of the sphere with gain possess more advantage for the frequency-stabilization device of semiconductor lasers than the coupling systems comprised of the sphere without gain, although it can also be used due to its narrow resonance linewidth. Now we discuss an other possible application: sensors. It is well known that chemists have long used absorption spectroscopy to measure small amounts of a substance. Our coupling system may make the technique more sensitive. For this purpose, the system is slightly
Fig. 3. The geometry of the altered coupling system of the tapered fiber-microsphere with gain for sensor applications.
altered as shown in fig. 3. If a sample molecule is brought to the surface and absorbs the evanescent wave outside the sphere, light is coupled out of the sphere, which is described by small coupling coefficient D clearly related with the concentration and the other parameter of the sample. As the analysis before, the relations between Ei with each other can be given by: pffiffiffiffiffiffiffiffiffiffiffiffi ð8aÞ E3 ¼ itE1 þ 1 t 2 E2 E4 ¼ E3 exp ½ðg aÞL=4 exp ðif=2Þ pffiffiffiffiffiffiffiffiffiffiffiffi E6 ¼ 1 t 2 E4
ð8bÞ
E5 ¼ itE4
ð8dÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 ¼ E6 exp ½ðg aÞL=4 exp ðif=2Þ 1 D2 :
ð8cÞ
ð8eÞ Then we get: E5 ¼
t 2 exp ½if=2 ða gÞ L=4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E1 : 1 ð1 t 2 Þ exp ½if ða gÞ L=2 1 D2 ð9Þ
At resonance ðf ¼ 2npÞ, considering jða gÞLj; t2, D2 1, the output power is 2 t 2 exp ½ðg aÞL=4
2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E1 jE5 j ¼ 1 ð1 t 2 Þ exp ½ðg aÞL=2 1 D2 2t 2 ð10Þ j2 jE1 j2 : ða gÞL þ 2t 2 þ D2
Fig. 2. A comparison of the output power characteristics of the coupling system comprised of a sphere with gain g ¼ 1:3 105 =mm (curve 1) and without gain (curve 2).
To make the sensivitiy of the output power to coupling coefficient more evident, we rewrite eq. (10) with intrinsic quality factor of the coupling system without 2pns and assume sample Q¼ ½ða gÞL þ 2t 2 l
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Ying Lu et al., An investigation of a tapered fiber-microsphere coupling system
D2 ða gÞL þ t 2 . Then we get 2 2 2 4 E5 2t Q 8t Q3 D2 : E 3 k0 ns k0 n3s 1
References ð11Þ
The sensitivity of the output power to the change of 8t 4 the small D2 evidently depends on the term 3 3 Q3 , k0 ns which is named sensitivity factor as below. In the case of g ¼ 0 and even g a, because value of t is in practice usually in the order of 103 and in turn that the highest value of Q is in the order of 106, we can derive that the highest sensitivity factor is in the order of 106. In the case of g > a, however, sensitivity factor much larger than 1012 can be achieved as long as the value of Q is much larger than 108. Such high value of sensitivity factor indicates that our new coupling system with g > a is well suited e.g. as sensor for sniffing out explosives or for finding traces of chemical and biological weapons, and etc. [13]. In conclusion, we have presented theoretical predictions of the characteristic and possible application of the fiber-microsphere coupling element with gain. These results indicate the potential of the coupler in integrated optics and photonic devices applications. Furthermore, the experiments in which the practical coupler is constructed by fluoride glass microsphere, doped with rare earth irons and a fiber taper formed by heating and pulled standard single-mode fiber are in progress. Acknowledgement. This work was projection 69678015 supported by National Natural Science Foundation of China.
[1] Barber PW, Chang RK: Optical effects associated with small particles. World Scientific, Singapore 1988 [2] Collot L, Lefevre-Seguin V, Brune M, Raimond JM, Haroche S: Very high-Q whispering gallery modes resonances observed on fused silica microspheres. Europhys. Lett. 23 (1993) 327–333 [3] Braginsky VB, Gorodetsky ML, Ilchenko VS: Quality-factor and nonlinear properties of optical whispering-gallery modes. Phys. Lett. A137 (1989) 393–396 [4] Arnold S, Liu CT, Whitten WB, Ramsey JM: Room-temperature microparticle-based persistent spectral hole buring memory. Opt. Lett. 16 (1991) 420–422 [5] Lu Y, Wang JY, Xu XX, Pan SH, Zhang CZ: Optical bistability and differential amplification in a tapered fiber coupler microsphere resonator with nonlinear medium. Opt. Laser Technol. 32 (2000) 245–248 [6] Little BE, Chu ST, Haus HA, Foresi J, Laine J-P: Microring resonator channel dropping filters. J. Lightwave Technol. 15 (1997) 998–1005 [7] Knight JC, Cheung JC, Jacques F, Birks TA: Phasematched excitation of whispering-gallery-mode resonances by a fiber taper. Opt. Lett. 22 (1997) 1129–1131 [8] Ilchenko VS, Yao XS, Maleki L: Pigtailing the high-Q microsphere cavity: a simple fiber coupler for optical whispering-gallery. Opt. Lett. 24 (1999) 723–725 [9] Lu Y, Wang JY, Xu XX, Pan SH, Zhang CZ: Optimal conditions of coupling between the propagating mode in a tapered fiber and the given WG mode in a high-Q microsphere. Optik 112 (2001) 109–113 [10] Maitland A, Dunn MH: Laser Physics. North-Holland, Amsterdam 1969 [11] Lock JA: Improved Gaussian beam-scattering algorithm. Appl. Opt. 34 (1995) 559 [12] Heebner JE, Boyd RW: Enhanced all-optical switching by use of a nonlinear fiber ring resonator. Opt. Lett. 24 (1999) 847–850 [13] Savage N: Optical cavity opens door to better detecters. Laser Focus World 35 (12) (1999) 26–27