Waveguiding Properties of Fiber-Shaped Aggregates Self-Assembled

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J. Phys. Chem. C 2007, 111, 8671-8676

8671

Waveguiding Properties of Fiber-Shaped Aggregates Self-Assembled from Thiacyanine Dye Molecules Ken Takazawa* Tsukuba Magnet Laboratory, National Institute for Materials Science, 3-13 Sakura, Tsukuba 305-0003, Japan ReceiVed: February 21, 2007; In Final Form: April 11, 2007

The waveguiding properties of the fiber-shaped aggregates self-assembled from the thiacyanine (TC) dye in solution were investigated by spatially resolved fluorescence microscopy. The TC fibers transferred onto a glass substrate were excited by a focused laser beam. The spectra of the outcoupled light from the fiber end were recorded as a function of the distance between the laser spot and the fiber end. The spectral images in a false-color intensity scale constructed from the outcoupled light spectra exhibited the waveguiding properties of the fibers well. The optical losses for the wavelength range over the entire emission band (470-620 nm) were quantitatively determined. It was found that the fibers have a cutoff wavelength at ∼480 nm, below which the light is not guided, and these fibers act as low-loss waveguides for the wavelength range of 530600 nm. The spectral images also revealed that the TC fiber functions as an optical cavity (Fabry-Perot cavity) by the recurrence of the guided light due to the reflection at the end facets of the fiber.

Introduction Nano- to micrometer-sized optical waveguides are one of the key building blocks in miniaturized optoelectronic circuits, which have promising applications in high-speed and low-loss integrated circuits. Therefore, dielectric waveguides in such scales have been actively developed by using semiconductors1-11 and organic materials,12-18 and their waveguiding properties, including the optical cavity effect and lasing, have been extensively studied. Recently, we reported that the aggregates of the organic dye molecules thiacyanine (TC, Figure 1a) exhibit a self-waveguiding behavior.19 The TC dyes in solution selfassemble into fiber-shaped aggregates with a length of up to ∼250 µm. The solution consisting of the fibers exhibit an absorption band peaked at ∼395 nm, which is blue-shifted from monomer absorption (Figure 1b), thereby suggesting that the fibers are H-aggregates.20 Both the monomer and fiber exhibit their emission band peaked at ∼510 nm (Figure 1b). The TC fibers in the solution were transferred onto a glass substrate, and their optical properties and morphology were investigated by using microscopic techniques. The fluorescence microscopy image recorded by exciting the fiber with a focused laser beam at 405 nm exhibited bright spots at both the ends of the fiber as well as at the excited position, indicating that the fluorescence generated at the excited position is guided to the fiber ends (Figure 1c). The atomic force microscopy (AFM) image of the fiber showed that the fiber has a rectangular cross section with a typical width and height of ∼700 nm and ∼200 nm, respectively. An analysis of the propagating light modes showed that the fiber with such a geometry functions as a single-mode waveguide for the fluorescence.19 Since the discovery of pseudoisocyanine (PIC) J-aggregates,21-24 a number of organic dye molecules, including the derivatives of TC, have been found to form fiber-shaped J- and Haggregates in solution. The characteristic optical properties of these aggregates, such as the appearance of the intense and * Corresponding author. Phone: +81-29-863-5487. Fax: +81-29-8635599. E-mail: [email protected].

Figure 1. (a) Chemical structure of thiacyanine (TC). (b) Absorption and fluorescence spectra of the solution containing the TC fibers. (c) Fluorescence microscopy image of a 190 µm-long TC fiber on a glass substrate recorded by exciting the fiber with a focused laser beam at 405 nm. The excited position is indicated by an arrow.

narrow absorption bands (J- and H-bands), nonlinear optical response, and fast energy transfer within the aggregates, have been studied from the viewpoints of both fundamental science and device applications.20 In recent years, the fluorescence (photoluminescence) microscopy and scanning near-field optical microscopy (SNOM) have been applied, and the optical properties of individual aggregates have been revealed.25-28 To date, however, the efficient self-waveguiding behavior has been observed only in the TC aggregates.29 This suggests that the TC fiber has unique optical (and geometrical) characteristics that allow them to function as waveguides. Therefore, a detailed investigation of their waveguiding properties is of particular

10.1021/jp071455e CCC: $37.00 © 2007 American Chemical Society Published on Web 05/24/2007

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Figure 2. (a) Experimental setup. (b) A straight TC fiber is positioned so that it is imaged between the entrance slits of the monochromator and is excited by a focused laser beam at 405 nm. Fluorescence emitted from the excited position and the fiber ends passes through the slits and is recorded by a CCD camera. X represents the distance between the excited position and the fiber ends. (c) Cross section along the dashed line in (b), showing the spectrum of the outcoupled light from the fiber end. (d) Spectra of the outcoupled light recorded by moving the fiber parallel to the slits. (e) A false-color representation of the spectra in panel d.

importance to reveal such unique properties and to further develop and improve the organic waveguides in the nanoscale. Barrelet et al. demonstrated the quantitative measurement of the waveguiding properties of individual semiconductor nanowires.2 In their experiment, a self-waveguiding CdS nanowire was excited by a diffraction-limited laser spot. The laser spot was scanned an area that covered the nanowire, and the intensity of the outcoupled light from the wire end was recorded as a function of the position of the laser spot. This procedure yielded an image of the nanowire that quantitatively exhibited the waveguiding efficiency (optical loss). On the basis of this technique, we have developed a technique to measure the waveguiding efficiency of the TC fibers as a function of wavelength. We recorded the spectra at the fiber end by scanning the laser spot along the straight fiber. The set of the spectra recorded by exciting different positions of the fiber were arranged in the order of the distance between the laser spot and the fiber end and then represented with a false-color intensity scale. By using this technique, we quantitatively determined the waveguiding efficiency of the TC fibers for the wavelength range over the entire emission band (470-620 nm). Furthermore, we found that the TC fiber functions as an optical cavity (Fabry-Perot cavity) by the recurrence of the guided light due to the reflection at the end facets of the fiber.

Experimental Section The synthesis of the TC fiber has been described elsewhere.19 Briefly, 0.15 mM of a TC solution was obtained by dissolving TC in hot water at ∼50 °C by stirring the solution for 24 h. Then, the solution was cooled down to room temperature; this resulted in the self-assembly of the TC molecules into fibershaped aggregates. After ∼3 days of the growth time, the fibers grew up to ∼70 µm in length. The sample was prepared by depositing the solution containing the fibers onto a glass substrate (microscope cover glass, 22 mm × 22 mm). The sample was dried under ambient conditions in order to evaporate the solvent. The waveguiding properties of the fibers were measured by using a setup for fluorescence microscopy, which consists of an epi-illuminated fluorescence microscope (Olympus, BX-51) with a motorized translation stage (Prior, H-101), an imaging monochromator (Acton Research Corp., SpectraPro 2150), and a liquid nitrogen-cooled back-illuminated CCD camera (Princeton Instruments, Spect10, 1340 pixel × 400 pixel) (Figure 2a). Most of the fibers transferred onto the glass substrate exhibited a bent shape. A straight fiber with a maximum deviation of less than ∼300 nm from the straight line was selected and positioned so that the fiber was imaged between

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the entrance slits of the monochromator by translating and rotating the sample (Figure 2b). An output of a diode laser (Coherent, Radius405, λ ) 405 nm) was coupled into the fluorescence microscope, and the laser beam was focused onto the fiber by a 20× objective (spot size: ∼500 nm). The laser power at the sample was ∼20 µW. Since the fiber was imaged between the entrance slits, the fluorescence spots at both the fiber ends (the outcoupled light) and at the excited position (direct emission from the fiber body) passed through the slits and were recorded by the CCD camera. The image obtained is spectrally and spatially resolved along the horizontal and vertical axes of the CCD camera, respectively (Figure 2b). The spectral and spatial resolutions are ∼0.4 nm and ∼1 µm, respectively. The fluorescence spectrum at one of the fiber ends was obtained by extracting a horizontal (perpendicular to the fiber axis) cross section at the fiber end (Figure 2c). The spectrum at the fiber end was recorded by moving the excited position from one end to the other end of the fiber by translating the sample in a direction parallel to the slits (parallel to the long axis of the fiber) with a step width of 500 nm. Then, the spectra were arranged in the order of the distance between the fiber end and the excited position X (Figure 2d). The obtained twodimensional spectrum was converted into a false-color image in which the spectral intensities are represented in a color scale (Figure 2e). Results and Discussion In the present work, we investigated the fibers with lengths ranging from 45-70 µm, which were grown in the same sample solution, by the abovementioned technique. Although the TC fibers can be grown up to ∼250 µm in length after a growth time of ∼10 days,19 all the fibers longer than ∼70 µm on the substrate exhibited a bent shape and therefore could not be investigated by using this technique. Following the spectral measurements, the fluorescence and transmitted-illumination microscope images of the identical fibers were recorded by using a 100× objective to measure the widths of the fibers. The widths were homogeneous throughout the investigated fibers within the resolution of the optical microscope ( 5 µm, which is represented by the yellow dashed curve, is slanted toward the long-wavelength side. The slanted edge is indicative of the optical losses in the wavelength region near the edge. On the other hand, the spectrum of the guided light broadly spans up to ∼650 nm (Figure 3c) and therefore a distinct edge is not seen in the longwavelength side. In the intensity scale used in Figure 3a, a vague edge is seen at ∼545 nm, which is represented by the white dashed curve. In contrast to the short-wavelength edge (the yellow dashed curve), this edge is parallel to the vertical axis, indicating lower optical losses in the long-wavelength side. In order to quantitatively analyze the wavelength dependence of the optical losses, vertical cross sections at different wavelengths were extracted from the image (Figure 4). The curves clearly show that the optical losses near the shortwavelength edge are large and decrease with the wavelength. These curves were well-fitted by a first-order exponential decay function expressed by I(X) ) I0 exp(-RX), where I0 is a normalized intensity of the coupled light and R is a fitting parameter (Figure 4). If we assume that the guided light is not reflected by the end facets of the fiber, the optical loss for a guided distance of X µm is given by LO(X)) -10 log[I(X)/I0] dB/X µm ) -10 log[exp(-RX)] dB/X µm. The optical losses for a guided distance of 100 µm (LO(100)) were estimated using R, and they were plotted as a function of the wavelength (Figure 5). All the fibers investigated by us showed a wavelength dependence similar to that shown in Figure 5b; this is summarized as follows. The fibers have the cutoff wavelength at ∼480 nm. LO(100) is 100-50 dB/100 µm at λ ) 485 nm and is nearly constants0.1-0.7 dB/100 µmsfor 530 nm < λ < 590 nm.31 Approximately half of the investigated fibers

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Figure 6. Spectral image of a 70 µm-long TC fiber, which exhibits vertical stripes due to the optical cavity effect.

propagate in the TM modes and the number of possible modes, m, is given by

m< Figure 4. Cross sections of the image shown in Figure 3a at various wavelengths. The curves were fitted by an exponential decay function I(X) ) I0 exp[-RX] (dotted curves).

2a λ

x

⊥  - S || x ||

where a is the width of the waveguide, λ is the wavelength, and s is the dielectric constant of the substrate.13,14 By assuming that the fiber is optically isotropic (iso) || ) ⊥), the cutoff wavelength above which no propagating mode can exist (m < 1) is given as follows:

λcutoff ) 2a xiso - S

Figure 5. Plot of the optical losses for a guided distance of 100 µm (dB/100 µm) as a function of the wavelength. The curve is linearly correlated with the absorption spectrum of the fiber when the optical losses are due to reabsorption by molecules.

exhibited a slight increase in the optical losses (∼0.2 dB/100 µm) for λ > 600 nm, as can be seen in Figure 5. The large optical losses were observed only on the shortwavelength side and the losses could be fitted by a first-order exponential decay function. These facts indicate that the losses are dominantly due to the reabsorption by molecules. Thus, the cutoff at ∼480 nm is also due to the reabsorption, i.e., the coupled light with λ < ∼480 nm is completely reabsorbed within a guided distance of a few micrometers. It should be noted that LO(X) dB/X µm is linearly correlated with the absorbance (A) because A is defined as A ) log(I0/I), where I0 and I are the input and transmitted light intensities, respectively. Therefore, the curve shown in Figure 5 corresponds to the absorption spectrum of the TC fiber in an arbitrary unit when the optical losses originate from the reabsorption by molecules. Here, we consider the increase of the optical losses for λ > ∼600 nm in terms of the propagating light mode. When the fiber has a rectangular cross section and is optically uniaxial with the dielectric tensor components || (along the fiber) and ⊥ (perpendicular to the fiber), light in the waveguide can

(1)

(2)

By using values iso ) n⊥2 ) 1.622 ) 2.62 and S ) nS2 ) 1.532 ) 2.34 (nS: refractive index of the glass substrate),19 λcutoff is calculated to be 847 and 634 nm when a ) 800 and 600 nm, respectively. This mode analysis implies that due to the cutoff effect, the fibers with a width of ∼600 nm may exhibit an optical loss increase for λ > ∼600 nm. This is consistent with the experimental observations in which approximately half of the investigated fibers exhibited the optical loss increase. The cutoff effect can be observed only for the fibers with a relatively narrow width (∼600 nm) because λcutoff is out of the wavelength range of the emission band for the fibers wider than ∼650 nm. B. Optical Cavity Effects. In Figure 3a, vertical stripes are observed across the entire length of the fiber with a spacing of a few nanometers. Most of the investigated fibers exhibited such stripes though their contrasts and spacing were different for each fiber. Figure 6 shows the spectral image of a 70 µm-long fiber that exhibits the stripes with a relatively high contrast. The appearance of these stripes suggests that the fibers function as optical cavities (Fabry-Perot cavities) by the recurrence of the guided light by reflection at the end facets of the fiber and the cavity modes are observed as vertical stripes. The optical cavity effects have also been observed in the semiconductor and organic nanowire waveguides.3-5,7-10,17,18 In these waveguides, however, the cavity modes were observed in the fluorescence (photoluminescence) spectra measured by an intense pulsed laser excitation, which led to lasing, and the cavity modes were not observed in the spontaneous emission spectra. The appearance of cavity modes in the spontaneous emission spectra may suggest that the TC fiber functions as a good optical cavity. In addition, our technique is advantageous to observe the cavity modes.

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Figure 7. Cross sections of the spectral images of a 70 µm-long fiber at X ) 60 µm (a) and that of a 45 µm-long fiber at X ) 30 µm (b). Solid lines indicate the cavity mode positions. Arrows indicate the longwavelength edge of the absorption band (530 nm).

Figure 7 compares the cavity modes of the fibers with lengths of 70 µm (a) and 45 µm (b). In the both spectra, the spacing between the cavity modes is nearly constant for λ > ∼530 nm and irregularly varies for λ < ∼530 nm. The spacing between the Fabry-Perot cavity modes is given by

λFSR ) λ2/2L[n - λ(dn/dλ)]

(3)

where λ is the wavelength, L is a cavity length, n is the refractive index, and dn/dλ is the first-order dispersion of the refractive index. The average mode spacings for λ > 530 nm were 3.7 nm for the 70 µm-long fiber and 6.7 nm for the 45 µm-long fiber. For most of the investigated fibers (45 µm < L < 70 µm), the mode spacing and the fiber length approximately followed the relation given by eq 3 (λFSR ∝ 1/L), supporting the fact that the fibers function as a Fabry-Perot cavity.32 The irregular mode spacings observed for λ < ∼530 nm suggest that the refractive index of the fiber largely varies with the wavelength for λ < ∼530 nm. In general, dielectric materials exhibit anomalous dispersion in the wavelength region of their absorption bands. For organic dye aggregates, the refractive indexes of several kinds of J-aggregates in thin films were measured using reflection spectroscopy.33-35 These aggregates were also observed to exhibit anomalous dispersion in the wavelength region of their J-bands (absorption bands associated with the J-aggregates). For example, the refractive index of merocyanine J-aggregates show a moderate decrease with wavelength above the J-band peaked at 590 nm (normal dispersion). However, in the J-band (450-650 nm), the index exhibits rapid increase and decrease with wavelength (anomalous dispersion).35 Such rapid changes in the index in the J-band were commonly observed in other J-aggregates.33-35 The curve of the optical losses shown in Figure 5 corresponds to the absorption spectrum of the TC fiber. The spectrum shows that the edge of the absorption band is at ∼530 nm, and the cavity mode exhibited irregular spacings below ∼530 nm. Therefore, it is suggested that the TC fibers also exhibit an anomalous dispersion in the absorption band and the mode spacings irregularly vary due to the rapid changes in the refractive index in the absorption band. Similar to the mode spacings, the linewidths of the cavity modes were nearly constant for λ < ∼530 nm and irregular for λ < ∼530 nm, as shown in Figure 7. The irregular linewidths

may also reflect the anomalous dispersion and large optical losses for λ < ∼530 nm. The average widths of the spectral modulations due to the cavity modes for λ > 530 nm were measured to be 1.8 nm for the 70 µm-long fiber and 3.3 nm for the 45 µm fiber. Spectral simulations using a superposition of Lorentzians with a width of ∆λ separated by λFSR showed that the observed modulation widths were reproduced when ∆λ ) 4.5 nm for the 70 µm-long fiber (λFSR ) 3.7 nm) and ∆λ ) 7.5 nm for 45 µm-long fiber (λFSR ) 6.7 nm).36 If we consider the ∆λ as the line width of the Fabry-Perot cavity mode, the finesses (F) of the 70 and 45 µm-long fibers are estimated to be F ) λFSR/∆λ ) ∼0.8 and F ) ∼0.9, respectively. The average modulation width of the investigated fibers (45 µm < L < 70 µm) ranged from 1.5 to 5.2 nm and were not explicitly correlated with the fiber lengths. The line width of the FabryPerot cavity mode is proportional to 1/L.36 The absence of correlation between the line width and L indicates that the reflectivity of the two end mirrors differs for each fiber, probably reflecting the geometrical differences between the fibers, such as the flatness and parallelity of the two end facets. Conclusions The waveguiding properties of self-assembled TC fibers in solution were investigated by spatially resolved fluorescence microscopy. The spectra of the outcoupled light from the fiber end were recorded by exciting the fiber with a focused laser beam. The spectral image in a false-color intensity scale constructed from the outcoupled light spectra clearly exhibited the waveguiding properties of the fiber. It was found that the fibers have a cutoff wavelength of ∼480 nm below which the light is not guided in the fiber. The cutoff effect was explained by the reabsorption of the guided light by molecules. The optical losses for the wavelength range over the entire emission band (470-620 nm) were quantitatively determined. The optical losses were measured to be 100-50 dB/100 µm at λ ) 485 nm and were nearly constant at 0.1-0.7 dB/100 µm for 530 nm < λ < 590 nm. A slight increase in the optical losses for λ > ∼600 nm was observed in some of the fibers and were explained by the cutoff effect due to the propagating light mode. From the vertical stripes observed in the spectral images, it was found that the TC fiber function as an optical cavity (Fabry-Perot cavity) by the recurrence of the guided fluorescence due to the reflection at the end facets. The analysis of the cavity mode spacings suggested that the TC fiber exhibits anomalous dispersion in the absorption band (λ > ∼530 nm). Acknowledgment. This work was supported by Grant-inAid for Scientific Research (No. 18550135), Japan Society for the Promotion of Science. References and Notes (1) Duan, X. F.; Huang, Y.; Agarwal, R.; Lieber, C. M. Nature 2003, 421, 241. (2) Barrelet, C. J.; Greytak, A. B.; Lieber, C. M. Nano Lett. 2004, 4, 1981. (3) Agarwal, R.; Barrelet, C. J.; Lieber, C. M. Nano Lett. 2005, 5, 917. (4) Greytak, A. B.; Barrelet, C. J.; Li, Y.; Lieber, C. M. Appl. Phys. Lett. 2005, 87, 151103. (5) Gradecak. S.; Qian, F.; Li, Y.; Park, H. G.; Lieber, C. M. Appl. Phys. Lett. 2005, 87, 173111. (6) Barrelet, C. J.; Bao, J. M.; Loncar, M.; Park, H. G.; Capasso, F.; Lieber, C. M. Nano Lett. 2006, 6, 11. (7) Johnson, J. C.; Choi, H. J.; Knutsen, K. P.; Schaller, R. D.; Yang, P. D.; Saykally, R. J. Nature Mat. 2002, 1, 106. (8) Johnson, J. C.; Yan, H. Q.; Yang, P. D.; Saykally, R. J. J. Phys. Chem. B 2003, 107, 8816.

8676 J. Phys. Chem. C, Vol. 111, No. 24, 2007 (9) Yan, H. Q.; Johnson, J.; Law, M.; He, R. R.; Knutsen, K.; McKinney, J. R.; Pham, J.; Saykally, R.; Yang, P. D. AdV. Mater. 2003, 15, 1907. (10) Law, M.; Sirbuly, D. J.; Johnson, J. C.; Goldberger, J.; Saykally, R. J.; Yang, P. D. Science 2004, 305, 1269. (11) Sirbuly, D. J.; Law, M.; Yan, H. Q.; Yang, P. D. J. Phys. Chem. B 2005, 109, 15190. (12) Yanagi, H.; Morikawa, T. Appl. Phys. Lett. 1999, 75, 187. (13) Balzer, F.; Bordo, V. G.; Simonsen, A. C.; Rubahn, H. G. Appl. Phys. Lett. 2003, 82, 10. (14) Balzer, F.; Bordo, V. G.; Simonsen, A. C.; Rubahn, H. G. Phys. ReV. B 2003, 67, 115408. (15) Balzer, F.; Beermann, J.; Bozhevolnyi, S. I.; Simonsen, A. C.; Rubahn, H. G. Nano Lett. 2003, 3, 1311. (16) Balzer, F.; Rubahn, H. G. AdV. Func. Mat. 2005, 15, 17. (17) Quochi, F.; Cordella, F.; Mura, A.; Bongiovanni, G.; Balzer, F.; Rubahn, H. G. J. Phys. Chem. B 2005, 109, 21690. (18) Quochi, F.; Cordella, F.; Mura, A.; Bongiovanni, G.; Balzer, F.; Rubahn, H. G. Appl. Phys. Lett. 2006, 88, 041106. (19) Takazawa, K.; Kitahama, Y.; Kimura, Y.; Kido, G. Nano Lett. 2005, 5, 1293. (20) Kobayashi, T., Ed. In J-aggregates; World Scientific Publishing: Singapore, 1996. (21) Jelley, E. E. Nature 1936, 138, 1009. (22) Jelley, E. E. Nature 1937, 139, 631. (23) Scheibe, G. Angew. Chem. 1936, 49, 563. (24) Scheibe, G. Angew. Chem. 1937, 50, 212. (25) Vacha, M.; Takei, S.; Hashizume, K.; Sakakibara, Y.; Tani, T. Chem. Phys. Lett. 2000, 331, 387. (26) VandenBout, D. A.; Kerimo, J.; Higgins, D. A.; Barbara, P. F. J. Phys. Chem. 1996, 100, 11843.

Takazawa (27) Higgins, D. A.; Kerimo, J.; VandenBout, D. A.; Barbara, P. F. J. Am. Chem. Soc. 1996, 118, 4049. (28) Higgins, D. A.; Reid, P. J.; Barbara, P. F. J. Phys. Chem. 1996, 100, 1174. (29) Recently, the waveguiding behavior of the rod-like crystallites of pseudoisocyanine (PIC) was reported. Lebedenko, A. N.; Guralchuk, G. Y.; Sorokin, A. V.; Yefimova, S. L.; Malyukin, Y. V. J. Phys. Chem. B 2006, 110, 17772. (30) Although more accurate measurement of the fiber width is possible by the AFM, the fluorescence microscopy and AFM measurement on identical fibers cannot be conducted with our setup. (31) As described in the next section, the TC fibers exhibit the optical cavity effect due to the reflection of the guided light at the end facets. Therefore, the actual values of the optical losses can be different from these values. The reflection at the end facet reduces I(X) by RI0 exp(-RX), where R is the reflectivity of the end facet. On the other hand, the reflection at the other end facet increases I(X) by I0 exp[-R(L - X)]R exp(-RL). However, R of the Fresnel reflection at the fiber (nfiber) 1.62) and air (nair) 1) interface is expected to be smaller than ∼0.06, and therefore these values are negligible in the optical loss estimations. (32) From eq 3, the values of dn/dλ is estimated to be +0.002 nm-1 for λ ) 540 nm and n )1.62. This value yields the group index ng ) n λ(dn/dλ) ) 0.55 < 1. A satisfactory explanation for this discrepancy has not been found yet. The presence of subcavities with shorter lengths may be suggested. (33) Wakamatsu, T.; Odauchi, S. Appl. Opt. 2003, 42, 6929. (34) Wakamatsu, T.; Watanabe, K.; Saito, K. Appl. Opt. 2005, 44, 906. (35) Wakamatsu, T.; Toyoshima, S.; Saito, K. J. Opt. Soc. Am., B 2006, 23, 1859. (36) Svelto, O. Principles of Lasers; Plenum: New York, 1989; p 148161.