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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 7, Iss. 3 — Jul. 31, 2000
  • pp: 113–122
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Experimental and scalar beam propagation analysis of an air-silica microstructure fiber

C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler  »View Author Affiliations


Optics Express, Vol. 7, Issue 3, pp. 113-122 (2000)
http://dx.doi.org/10.1364/OE.7.000113


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Abstract

We study the higher order guided modes in an air-silica microstructure fiber comprising a ring of six large air-holes surrounding a Germanium doped core. We characterize the modes experimentally using an intra-core Bragg grating. The experimentally observed modes are then accurately modeled by beam propagation simulations using an index profile similar to the observed fiber cross section. Theory and experiment confirm the presence of “inner cladding” modes with approximate cylindrical symmetry near the core, similar to conventional cladding modes, but which strongly exhibit the symmetry of the microstructure at large radius. Such modes are useful in fabricating robust tunable grating filters and we show that the Bragg grating is a useful diagnostic to measure their effective indices and intensity profiles.

© Optical Society of America

1. Introduction

Air-silica micro-structure fibers (ASMF) [1

1. P. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica,” The Bell System Technical Journal 53, 1021–1039 (1974).

] have attracted wide interest over the past few years due to their unique guidance properties and ability to control light propagation. These fibers are typically all-silica and incorporate air-holes that run along the length of the fiber. A wide range of different air-silica microstructure fibers have been explored. These include: photonic band gap fiber, which guides light because of a bandgap in the cladding [2

2. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

]; crystal fiber [3

3. T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

], which comprises a core surrounded by an array of air-holes in the cladding region and guides light by total internal reflection; air-clad fiber [4

4. A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, and T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photonics Tech. Lett. 11, 445–447 (1999). [CrossRef]

] in which a ring of air-holes is introduced into the cladding that forms an inner cladding region; and high delta microstructure fiber [5

5. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air-silica microstructure optical fibers”, Opt. Lett. 25, 796–798 (2000). [CrossRef]

], which consists of silica core surrounded by closely spaced air-holes. Each fiber has different geometric air-hole shapes with unique optical properties and different potential applications.

In this paper, we use a Bragg grating to measure the effective indices and mode intensity patterns of a fiber with large air-holes in the cladding (Fig. 1) and then compute these same quantities using a scalar beam propagation algorithm applied to a realistic index profile. This fiber, shown in Fig.1, has a variety of interesting potential applications. Specifically, this fiber allows for the infusion of different novel materials, such as polymers, into the large air holes [10

10. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. Burdge, “Control of waveguide properties in hybrid polymer-silica microstructured optical fiber gratings,” OFC 2000, ThI3, Baltimore, Maryland (2000).

]. Moreover, the air-holes create an effective “inner cladding” for the lower order cladding modes, a feature that has been exploited in design of tunable filters [7

7. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-mode resonances in hybrid polymer-silica microstructured optical fiber gratings,” IEEE Photonics Tech. Lett. 12, 495–497 (2000). [CrossRef]

]. For example, Westbrook et al. manipulated the propagation of light thermally in hybrid polymer-silica microstructure fiber gratings by changing the refractive index of the polymer introduced inside the air-hole regions [4

4. A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, and T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photonics Tech. Lett. 11, 445–447 (1999). [CrossRef]

,7

7. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-mode resonances in hybrid polymer-silica microstructured optical fiber gratings,” IEEE Photonics Tech. Lett. 12, 495–497 (2000). [CrossRef]

,8

8. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Technology, In Press, August issue (2000). [CrossRef]

,10

10. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. Burdge, “Control of waveguide properties in hybrid polymer-silica microstructured optical fiber gratings,” OFC 2000, ThI3, Baltimore, Maryland (2000).

]. Further potential applications of such fibers include grating based devices such as dispersion control long period gratings devices [11

11. D. B. Stegall and T. Erdogan, “ Dispersion control with use of long-period fiber gratings,” J. Opt. Soc. Am. A 17, 304–312 (2000). [CrossRef]

].

Fig. 1. (a) Cross-section of the air-silica microstructure fiber. (b) Schematic diagram of simulated fiber

2. Bragg grating characterization

The interaction of light with a Bragg grating written into the core of a fiber is well-understood [12

12. R. Kayshap, Fiber Bragg Gratings, (1st edition ed: Academic Press, 1999).

,13

13. Turan Erdogan, “Fiber grating spectra,” J. Lightwave Technology 15, 1277–1294 (1997). [CrossRef]

]. Light propagating in the core of a fiber incident upon a Bragg grating couples to a backward propagating core mode in addition to other counter-propagating cladding modes. Fig. 2 (a) shows a schematic of a Bragg grating in a conventional fiber and illustrates grating facilitated coupling to cladding modes that are confined by the silica-air interface. Coupling to these cladding modes manifests as sharp peaks in the transmission spectrum as shown in Fig.2 (b). Each peak corresponds to a particular mode.

Fig. 2. (a) Schematic diagram of Bragg grating in a fiber. Incident core mode is coupled to counter-propagating higher order cladding modes. (b) Transmission spectrum of a Bragg grating written in the core of a conventional fiber. The simulated profiles of the modes shown correspond to the first three order cladding modes (LP01, LP02, and LP03)

A resonance occurs when the core is phase matched to a given cladding mode. The phase matching conditions, expressed in term of wave vectors are: βcoreclad,i=K, where βcore, βclad,i are respectively, the propagation constants of the incident (forward) core mode and the backward coupled ith cladding modes, K=2π/L is the wave vector of the grating, and Λ is the period of the grating. When expressed in terms of resonance wavelengths, the conditions are [8

8. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Technology, In Press, August issue (2000). [CrossRef]

]:

λcore=2ncoreΛ
(1)
λclad,i=(ncore+nclad,i)Λ
(2)

where λcore, λcore, λclad,i and nclad,i are the resonance wavelengths and the effective indices of the core and the ith cladding modes respectively.

3. Fiber description and modal characterization

The ASMF is shown in Fig. 1(a) with six approximately cylindrical air-holes surrounding an inner cladding region of ~30µm in diameter. The fiber has a Germanium (Ge) doped core, with diameter ~8µm and Δ=(n1-n2)/n1~0.35%, where n1 and n2 are the refractive indices of the Ge core and the silica respectively. The outer diameter is 125µm and the outer cladding ring thickness, the region of silica between the outer surface of the fiber and the air-holes, is about 14mm. The interstitial region between the holes is about 5.8µm.

The fiber was deuterium loaded to enhance the photosensitivity of the Germanium core for writing the Bragg grating. Periodic modulation in the refractive index of the core was achieved by exposing a length of 4 cm of the fiber to ultraviolet light (242nm) through a phase mask [15

15. Andreas Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, (1997). [CrossRef]

] of period 1.075mm (Λ=1.075µm/2) at a fluence of 240mJ/cm2 from a frequency-doubled excimer-pumped dye laser. The amplitude of the index-modulation (Δn) is of the order of ~8×10-4.

Fig. 3 shows the transmission spectrum of the Bragg grating written into the core of the ASMF with the polymer jacket stripped off. The ASMF reveals a dramatically different mode spectrum when compared to the conventional fiber grating spectrum, shown in Fig. 2 (b).

Fig. 3. (a) Transmission Spectrum of Bragg grating in the ASMF. (b) First four order modes (A, B, C, D) confined to the inner cladding region.

The first peak on the right side of the transmission spectrum, labeled A in Fig. 3, corresponds to excitation of the backward propagating core mode. The other resonances on the shorter wavelength side of the main peak correspond to coupling to higher order modes (cladding modes) with smaller coupling strength, and are analogous to, but qualitatively different from the conventional cladding modes discussed in section II. Only the four lowest order cladding modes (labeled B, C, D and E) in the transmission spectrum of the ASMF in Fig. 3(b) have large coupling with the core mode. As we show below, these modes correspond to those confined primarily in the inner cladding region.

A general feature of the cladding mode spectrum of the ASMF is that the wavelength spacing Δλ(=λp+1-λp) between adjacent modes of the air-silica microstructure fiber is larger than that of the conventional fiber cladding resonance of Fig. 2(b). This is because Δλ, the wavelength spacing between two adjacent cladding mode resonances, scales as the inverse square of cladding diameter [7

7. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-mode resonances in hybrid polymer-silica microstructured optical fiber gratings,” IEEE Photonics Tech. Lett. 12, 495–497 (2000). [CrossRef]

,8

8. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Technology, In Press, August issue (2000). [CrossRef]

]. Because the ASMF cladding modes are confined primarily in the “inner cladding” region, they have a correspondingly larger wavelength separation than conventional cladding modes, which are confined in the entire silica diameter [7

7. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-mode resonances in hybrid polymer-silica microstructured optical fiber gratings,” IEEE Photonics Tech. Lett. 12, 495–497 (2000). [CrossRef]

]. The spacing between the core (main resonance) mode and the first cladding mode depends on the core-cladding index difference (Δ) as well as on the location and size of the air-holes.

Because the lower order cladding modes are well confined in the inner cladding region, we expect that they are insensitive [14

14. R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T . A. Strasser, and D. J. DiGiovanni, “External refractive index insensitive air-clad Long Period Fiber Grating,” Electron. Lett. 35, 327–328 (1999). [CrossRef]

] to the external refractive index surrounding the fiber. The transmission spectrum of the ASMF immersed in an index matching fluid is shown in dashed line in Fig. 6. The solid line represents the transmission spectrum of the ASMF grating in air. Note that the modes confined in the inner cladding region are mostly insensitive to the outer environment. On the other hand, the higher mode, as shown in the inset, is slightly affected by the external refractive index for reasons discussed later.

Fig. 6. Transmission spectrum of Bragg grating in ASMF with (blue) and without (red) index matching gel surrounding the fiber. The inset shows the cladding mode slightly affected by external index.

We also recorded images of the ASMF cladding mode power distributions using the setup shown in Fig. 7. Light incident from a tunable laser is coupled into the core of the fiber by means of a 40× objective after being collimated by means of a 10× objective When the wavelength of the incident light satisfies the Bragg conditions, Eqs. (1) & (2), it is coupled to backward propagating core and cladding modes. These modes are observed by means of a camera on a screen.

Fig. 7. Setup experiment. Light incident from a tunable laser on the grating written in the core of the fiber is coupled back into different modes which are observed on the screen.

Fig. 8(a) below shows the experimental transmission spectrum and the corresponding mode images (bottom). The lower order modes (A, B, C, D and E) are mainly confined inside the inner-cladding region. They are surrounded by the holes and their propagation is governed by the total internal reflection at the interface of the cladding-holes. However, the higher order mode (F) spreads throughout the fiber through the interstitial in the cladding between the holes.

4. Beam propagation simulation

Fig. 8. (a) Experimental and (b) simulated mode spectrum. (a) The measured spectrum, shown in black, is plotted as transmission loss (in arbitrary units) versus wavelength. (b) The simulated plot shows the relative power versus the wavelength calculated from the effective indices of each mode.

Table 1. The effective index differences between the core and the ith cladding mode, calculated using Eqs.(1) and (2), are compared to the simulated values. The wavelength values correspond to those at which the modes are observed

table-icon
View This Table

The simulated modes, shown in Fig. 10, are very similar to those observed experimentally. The profile and the distribution of the energy of the modes are clearly affected by the presence of the holes. The circular shapes of the modes of a conventional fiber are lost in this ASMF. Instead the images exhibit symmetry of the air-hole geometry. However, at small radius near the core, the inner cladding modes have approximate cylindrical symmetry and hence have large overlap with the core mode.

Fig. 11 shows a simulation of the profile of a higher order mode. The energy of this mode tunnels to the outer silica region. As we discussed above, this higher order mode (F), in Fig. 11(b), is slightly affected by the outer refractive index. So the mode is confined by the inner (silica—air-hole) interface and the outer (silica-air) interface.

Fig. 11. Mode (F) simulated (a) and observed (b). Some of the energy of the mode tunnels to the outer cladding.

We note that our simulations do not show all of the observed modes. Some of these, such as the resonance falling between B and C, in Fig. 8 (b), are found to be odd order modes, which we have neglected due to the symmetric launch in our beam propagation simulations. The presence of odd mode resonances in the Bragg grating is consistent with ultraviolet induced asymmetry across the core [12

12. R. Kayshap, Fiber Bragg Gratings, (1st edition ed: Academic Press, 1999).

]. In launching an off-axis rectangular profile, the odd modes are observed on the simulated spectrum, in Fig. 12(a), which correspond to those measured in the transmission spectrum, Fig.12 (b).

Fig. 12. (a) Experimental and (b) simulated mode spectrum. (a) The measured spectrum is plotted as transmission loss in arbitrary units versus wavelength. (b) The simulated mode spectrum, with an off-axis launch, is plotted as relative power versus wavelength. The simulated plot shows the excited odd modes which correspond to those observed on the transmission spectrum.

5. Comparison with conventional fiber

In order to compare our results with the conventional fiber, a simulated mode spectrum was computed for a conventional fiber with cladding diameter equal to that of the inner cladding of the air-silica microstructure fiber, which is approximately 40µm.

The three main cladding modes that are confined in the inner cladding region, as was discussed above, are observed again, Fig. 13. So we can conclude that the air-silica microstructure fiber can be compared to a conventional fiber with a cladding diameter the same size as the inner cladding diameter of the ASMF. This is important, for example, in applications that exploit a small cladding diameter to enhance the tunability of filters [7

7. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-mode resonances in hybrid polymer-silica microstructured optical fiber gratings,” IEEE Photonics Tech. Lett. 12, 495–497 (2000). [CrossRef]

], but which also require the cladding modes to be isolated from the environment and hence insensitive to the condition of the outer surface of the fiber [11

11. D. B. Stegall and T. Erdogan, “ Dispersion control with use of long-period fiber gratings,” J. Opt. Soc. Am. A 17, 304–312 (2000). [CrossRef]

].

Fig. 13. Simulated mode spectra of the ASMF under study (top) and of conventional fiber with a cladding diameter (D) of 40 mm (bottom), which is inverted in order to emphasize the location of the peaks.

6. Conclusion

The modes of an air-silica microstructure fiber comprising six large air-holes in the cladding have been observed and compared to beam propagation simulations using a realistic index profile of the fiber cross section (Fig. 1(b)). A Bragg grating was written into the core of the fiber to characterize the cladding modes. The measured mode spectrum is in very good agreement with the simulated results. The propagating modes in the inner cladding region of the air-silica microstructure fiber can be compared to those of a conventional fiber with the same cladding diameter and the same optical waveguide properties, a feature that can be exploited in the design of tunable filters and grating devices.

References and links

1.

P. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica,” The Bell System Technical Journal 53, 1021–1039 (1974).

2.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic bandgap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

3.

T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

4.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, and T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photonics Tech. Lett. 11, 445–447 (1999). [CrossRef]

5.

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air-silica microstructure optical fibers”, Opt. Lett. 25, 796–798 (2000). [CrossRef]

6.

B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spalter, and T. A. Strasser, “Grating resonances in air-silica microstructures,” Opt. Lett. 24, 1460–1462 (1999). [CrossRef]

7.

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-mode resonances in hybrid polymer-silica microstructured optical fiber gratings,” IEEE Photonics Tech. Lett. 12, 495–497 (2000). [CrossRef]

8.

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. Burdge, “Cladding mode resonances in air-silica microstructure fiber,” J. Lightwave Technology, In Press, August issue (2000). [CrossRef]

9.

B. J. Eggleton, P. S. +, R. S. Windeler, T. A. Strasser, and G. Burdge, “Grating spectra in air-silica microstructure fibers,” OFC 2000, ThI2, Baltimore, Maryland (2000).

10.

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. Burdge, “Control of waveguide properties in hybrid polymer-silica microstructured optical fiber gratings,” OFC 2000, ThI3, Baltimore, Maryland (2000).

11.

D. B. Stegall and T. Erdogan, “ Dispersion control with use of long-period fiber gratings,” J. Opt. Soc. Am. A 17, 304–312 (2000). [CrossRef]

12.

R. Kayshap, Fiber Bragg Gratings, (1st edition ed: Academic Press, 1999).

13.

Turan Erdogan, “Fiber grating spectra,” J. Lightwave Technology 15, 1277–1294 (1997). [CrossRef]

14.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T . A. Strasser, and D. J. DiGiovanni, “External refractive index insensitive air-clad Long Period Fiber Grating,” Electron. Lett. 35, 327–328 (1999). [CrossRef]

15.

Andreas Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, (1997). [CrossRef]

16.

M. D. Feit and J. A. Fleck Jr., “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” Appl. Opt. 19, 2240–2246 (1980). [CrossRef] [PubMed]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2310) Fiber optics and optical communications : Fiber optics
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Research Papers

History
Original Manuscript: June 15, 2000
Published: July 31, 2000

Citation
Charles Kerbage, Benjamin Eggleton, Paul Westbrook, and Robert Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-3-113


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References

  1. P. Kaiser and H. W. Astle, "Low-loss single-material fibers made from pure fused silica," The Bell System Technical Journal 53, 1021-1039 (1974).
  2. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic bandgap guidance of light in air," Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  3. T. A. Birks, J. C. Knight, and P. S. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  4. A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, and T. A. Strasser, "Electrically tunable efficient broad-band fiber filter," IEEE Photonics Tech. Lett. 11, 445-447 (1999). [CrossRef]
  5. J. K. Ranka, R. S. Windeler, and A. J. Stentz, "Optical properties of high-delta air-silica microstructure optical fibers", Opt. Lett. 25, 796-798 (2000). [CrossRef]
  6. B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spalter, and T. A. Strasser, "Grating resonances in air-silica microstructures," Opt. Lett. 24, 1460-1462 (1999). [CrossRef]
  7. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, "Cladding-mode resonances in hybrid polymer-silica microstructured optical fiber gratings," IEEE Photonics Tech. Lett. 12, 495-497 (2000). [CrossRef]
  8. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, G. Burdge, "Cladding mode resonances in air-silica microstructure fiber," J. Lightwave Technology, In Press, August issue (2000). [CrossRef]
  9. B. J. Eggleton, P. S. Westbrook, R. S. Windeler, T. A. Strasser, G. Burdge, "Grating spectra in air-silica microstructure fibers," OFC 2000, ThI2, Baltimore, Maryland (2000).
  10. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. Burdge, "Control of waveguide properties in hybrid polymer-silica microstructured optical fiber gratings," OFC 2000, ThI3, Baltimore, Maryland (2000).
  11. D. B. Stegall and T. Erdogan, " Dispersion control with use of long-period fiber gratings," J. Opt. Soc. Am. A 17, 304-312 (2000). [CrossRef]
  12. R. Kayshap, Fiber Bragg Gratings, (1st edition ed: Academic Press, 1999).
  13. Turan Erdogan, "Fiber grating spectra," J. Lightwave Technology 15, 1277-1294 (1997). [CrossRef]
  14. R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T . A. Strasser, and D. J. DiGiovanni, "External refractive index insensitive air-clad Long Period Fiber Grating," Electron. Lett. 35, 327-328 (1999). [CrossRef]
  15. Andreas Othonos, "Fiber Bragg gratings," Rev. Sci. Instrum. 68, (1997). [CrossRef]
  16. M. D. Feit and J. A. Fleck, Jr. "Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method," Appl. Opt. 19, 2240-2246 (1980). [CrossRef] [PubMed]

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