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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 13 — Jun. 22, 2009
  • pp: 10738–10747
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Spectroscopy of a tapered-fiber photonic crystal waveguide coupler

Raymond Sarkissian, Stephen Farrell, and John D. O’Brien  »View Author Affiliations


Optics Express, Vol. 17, Issue 13, pp. 10738-10747 (2009)
http://dx.doi.org/10.1364/OE.17.010738


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Abstract

A spectroscopic technique based on a full four port modeling and measurement is developed and utilized to study a tapered- fiber photonic crystal waveguide coupler. The coupler is made by directly situating the tapered fiber on the defect region of a silicon membrane photonic crystal waveguide. The waveguide is lithographically terminated resulting a Fabry-Perot cavity. It turns out that the line-shape of the resonances is not merely a Lorentzian but can be constructed from that of the unloaded waveguide resonator. By knowing how the resonances broaden the experimental data is then fit and the coupling efficiency is extracted for the entire spectrum.

© 2009 Optical Society of America

1. Introduction

Planar photonic crystal technology has undergone extensive research as a technology platform for integration of optical elements Ref. [1

1. J. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997). [CrossRef]

, 2

2. T. Krauss, “Planar photonic crystal waveguide devices for integrated optics,” Phys. Stat. Sol. (a) 197, 688–702 (2003). [CrossRef]

]. One aspect of this technology is to successfully interface an integrated optical circuit with the outside world. To address this issue, coupling between a tapered fiber and photonic crystal waveguides have been demonstrated. Both theoretical Ref. [3

3. W. Kuang, C. Kim, A. Stapleton, and J. O’Brien, “Grating-assisted coupling of optical fibers and photonic crystal waveguides,” Opt. Lett. 27, 1602–1606 (2002). [CrossRef]

, 4

4. P. Barclay, K. Srinivasan, and O. Painter, “Design of photonic crystal waveguide for evanescent coupling to optical fiber tapers and integration with high-Q cavities,” Opt. Express 20, 2274–2284 (2003).

] and experimental Ref. [5

5. P. Barclay, K. Srinivasan, M. Borselli, and O. Painter “Efficient input and output fiber coupling to a photonic crystal waveguide,” Opt. Lett. 29, 697–699 (2004). [CrossRef] [PubMed]

] studies have been carried out that confirm an efficient interfacing between an integrated circuit and outside environment is possible. Optimization of the coupling has been investigated Ref. [6

6. I.-K. Hwang, G.-H. Kim, and Y.-H. Lee, “Optimization of coupling between photonic crystal resonators and curved microfiber,” IEEE J. Quantum Electron. 42, 131–136 (2006). [CrossRef]

] and an enhanced probing of the resonances of a photonic crystal Fabry-Perot waveguide cavity for the purpose of characterizing the photonic crystal waveguides via a curved tapered fiber has been reported Ref. [7

7. M. lee, C. Grillet, C. Poulton, C. Monat, C. Smith, E. Magi, D. Freeman, S. Madden, B. Luther-Davis, and B. Eggleton, “Characterizing photonic crystal waveguides with an extended k-space evanescent coupling technique,” Opt. Express 16, 13800–13808 (2008). [CrossRef] [PubMed]

]. In this work we also probe the resonances of a photonic crystal Fabry-Perot resonator and it is done based on a full four port modeling of the coupler where the coupling junction is considered to change both the amplitude and phase of the fields. The spectroscopic technique developed throughout this report enables us to characterize the line shape of the resonances beyond that of a simple Fabry-Perot cavity as the Fabry-Perot cavity is allowed to be affected by the fiber coupling in the model. The model can be beneficial when four port measurements are to be conducted or direct extraction of cavity Q is not readily possible. The coupler is fabricated by directly situating the tapered fiber on a silicon membrane photonic crystal waveguide. The tapered fiber partially overlaps with the defect region of the waveguide and the waveguide is lithographically terminated resulting in a Fabry-Perot cavity.

2. Fabrication

Figure 1 illustrates the technique we used in the fabrication of the coupler. As can be seen from this figure the first stage is the process of tapering which is done by heating and simultaneously pulling the fiber until diameters of about 1µm was reached. The heat zone is created by a spatially oscillating gas burner torch that utilizes a mixture of Hydrogen and Oxygen gases. The torch and the fiber pulling stages are all motorized and must be carefully controlled to achieve adiabatic tapering. Theoretical and experimental investigation of adiabatic tapering of optical fibers have been thoroughly studied before Ref. [8

8. J. Love, W. Henry, W. Stewart, R. Black, S. Lacroix, and F. Gonthier,“Tapered single-mode fibers and devices; Part1: Adiabaticity criteria,” IEE Proc. J. 138, 343–354 (1991).

, 9

9. R. Black, S. Lacroix, F. Gonthier, and J. Love, “Tapered single-mode fibers and devices; Part2: Experimental and theoretical qualification,” IEE Proc. J. 138, 355–364 (1991).

]. The second step is to drive back the motorized stages along the z axis while they are also moved away from each other along the y axis. This allows for the tapered region to be positioned along the defect region of the waveguide. The third process is to raise the motorized stage carrying the photonic crystal waveguide to the proximity of the fiber where the fiber is then accurately situated on the photonic crystal waveguide defect. We contrast this process with previously reported techniques where the fiber is suspended in the air and a post processing is necessary to mount the fiber on a fixture. Note that the reshaping of the fiber is done in room temperature and can be very quick utilizing the motorized stages. After positioning, the fiber stays in place for days and no glue or such substances were used. The fiber may be reconfigured to meet a particular purpose. One example of this reconfiguration is achieved by moving the stages in such manners that curve the fiber away from the defect region near the facets (see Fig. 2) enabling the effective length of the coupler to be adjusted. This also helps in conducting free space measurements as the far field distribution of the light emanating from the facets is not disturbed. Placing the fiber on the device can cause the light to couple to the top silicon slab and scatter at the point of entry which will reduce the amount of light available for input-output coupling. The couplers made by directly situating the fiber on the waveguide may have limited coupling band width compared to those utilizing an extended k-space coupling scheme Ref. [7

7. M. lee, C. Grillet, C. Poulton, C. Monat, C. Smith, E. Magi, D. Freeman, S. Madden, B. Luther-Davis, and B. Eggleton, “Characterizing photonic crystal waveguides with an extended k-space evanescent coupling technique,” Opt. Express 16, 13800–13808 (2008). [CrossRef] [PubMed]

]. One additional challenge in achieving a controlled positioning of the fiber on the defect region of the waveguide is the air flow and vibration of the fiber as the stages move.

Fig. 1. An illustration of the technique used in fabrication of the photonic coupler under study
Fig. 2. Top microscope view of tapered fiber partially overlapping the waveguide defect region.

Figure 2 displays the actual fabricated device utilizing the technology described here. As can be seen in this figure and more clearly in Fig. 3, the waveguide is lithographically terminated at both sides to allow free space probing of the light. The waveguide is 150µm in length and the lattice constant, hole radius, membrane thickness and the separation between the membrane and the substrate are 420nm, 120nm, 250nm, and 2µm respectively. This waveguide has a transmission window between 1540nm and 1620nm. Note that this technique allows for integration of the entire setup, including the free space optics, and that no transportation of the tapered fiber is necessary. Therefore the tapered fiber fabrication, interfacing with a photonic crystal circuit, and full four port measurements all are possible in one place.

Fig. 3. SEM image of a fabricated photonic crystal waveguide with lithographically defined facets.

3. Discussion

In this section we will derive the transfer functions for all four ports of the coupler using a matrix method Ref. [10

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

]. Figure 4 illustrates our four port device. We assume that the coupler is lossless and instead take all the losses to arise from the coupler’s arms. We also assume identical facet reflectivities considering the fact that the facets have identical termination of photonic crystal lattice. Assuming a contradirectional coupling scheme we start setting up our equations for the case where the input is at the left facet of the waveguide. (See sets of Eq. (1))

Ea=[κ]Ef,Ef=[R]Ea+[tin]Ein,Eout=[T]((I[κ][R])1[κ][tin])Ein
[κ]=(0κ120κ14κ210κ2300κ230κ21κ140κ120),[R]=(re2(α+iβg)lg00000000000000re2(α+iβg)lg)
[tin]=(κine2(α+iβg)lg000),[T]=(te(α+iβg)lg0000eiβflf0000eiβflf0000te(α+iβg)lg)
(1)
Fig. 4. An illustration of the coupler. The black lines represent the fiber and the red lines represent the waveguide

Here all the parameters in [κ] are complex. α is the loss associated with the waveguide, and β f and βg are the fiber and waveguide propagation constants. lg and lf are the lengths of the coupler’s arms for waveguide and fiber respectively. r, t, and κin are reflectivity, transmissivity, and coupling coefficient of the waveguide facets. Solving for the transfer function, [T]((I-[κ] [R])-1[κ] [tin]) yields:

(e3lg(α+iβg)rtκinκ141e4lg(α+iβg)r2κ14e4lg(α+iβg)(κ21eiβflf+eiβflf4lg(α+iβg)r2κ21κ141e4lg(α+iβg)r2κ14)κineiβflf3lg(α+iβg)rκ21κ14κin1e4lg(α+iβg)r2κ14e2lg(α+iβg)tκ14κin1e4lg(α+iβg)r2κ14)
(2)

Had we assigned the input to the port 3 in Fig. 4 we would have obtained:

(eiβflf3lg(α+iβg)rtκ12κ141e4lg(α+iβg)r2κ14eiβflf(κ23eiβflf+eiβflf4lg(α+iβg)r2κ21κ12κ141e4lg(α+iβg)r2κ14)e2iβflf2lg(α+iβg)rκ21κ121e4lg(α+iβg)r2κ14eiβflflg(α+iβg)tκ121e4lg(α+iβg)r2κ14)
(3)

Each term in these matrices can be interpreted. For example the second element in matrix (2) shows that the light emanating from the fiber at port 2 is a combination of the light that is contradirectionally coupled from port 1 and the light that is contradirectionally and repeatedly coupled from the port 1 but after one complete round trip along the waveguide and through the coupler. Now let’s take the fourth element of matrix (2), by setting κ14=1 and Lg=2lg one recovers the transfer function of the unloaded waveguide resonator where Lg is the length of the waveguide resonator.

To develop the spectroscopy tools for analyzing our experimental data we will first start with our unloaded (before the fiber is placed on defect region of the waveguide) Fabry-Perot cavity. The output power detected off of the right facet can be written as:

Iout=t2e2αLgκin1+R22Rcos(x)Iin
(4)

Where R=r2e2αLg and x=2β g L g. Following Ref. [12

12. V. Cooper, “Analysis of Fabry-Perot interferograms by means of their Fourier transforms,” Appl. Opt. 10, 525–530 (1971). [CrossRef] [PubMed]

], the transfer function in Eq. (4) can be recast in the following form (see Eq. [810]):

22πt2e2αLgκin1R2(InRx2+(InR)2)*q=+δ(2πqx)
(5)

Equation (5) reveals that the spectrum of a Fabry-Perot cavity can be described by convolution of Dirac-Comb function and a broadening function that has a Lorentzian line shape. After performing the convolution it can be shown that the transmission spectrum of the waveguide can be fit by Lorentzians where the half-width of the Lorentzians has the following form HWHM=1ngln(1R) where ng is the group index.

The group index can be extracted from the relative positions of the Lorentzians therefore the loss can be extracted from the width of the Lorentzians.

Figure 5 depicts the detected power from the right facet of the unloaded waveguide, when the light was launched through its left facet, and the fit to the data using the above technique. Figure 6 shows the extracted FWHM and the group index. The minimum waveguide propagation loss we have measured for this waveguide, using this technique, is 4.7dB/mm (A mirror reflectivity of 0.4 was assumed). The choice of 4πLλ for the horizantal axes becomes apparent after performing the convolution in Eq. (5) and regrouping the terms. With this choice of units the spacing between the Lorentzians is inversely proportional to the group index with a factor of 2π.

We have compared this technique with the more generally used Hakki-Paoli method Ref. [13

13. B. Hakki and T. Paoli, “Gain spectra in GaAs double-heterostructure injection lasers,” Appl. Phys. 46, 1299–1306 (1975). [CrossRef]

] for both experimental data and that of an artificially generated data, and the two methods are in very good agreement. Characterizing photonic crystal waveguides using Fabry-Perot resonances has been done before Ref. [14

14. M. Natomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902-1–253902-3 (2001).

]. Direct measurement of the group index of photonic crystal waveguides has also been reported Ref. [15

15. A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107-1–261107-3 (2007). [CrossRef]

]. Our method can be beneficial for the instances where there is not enough spectral resolution to confidently determine the values and spectral positions of the peaks and valleys. Resolving all the resonances in the lab can be a very lengthy process and it may take hours to scan the entire spectrum where the optics can significantly shift during the measurements. Going back to the fourth element of the matrix 2 the output power emitting from the right facet can be written as:

Iout=t2e2αLgκ14κin1+R22Rcos(x+θ14)Iin
(6)

Where R=r2e2αLg|κ14|,x=2βgLg, and κ14=κ14eiθ14.

Let us define H(x)=11+R22Rcos(x+θ14) and expand H (x) in powers of θ 14 so that H(x)=A 0+A 1 θ 14+A 2 θ 2 14 +⋯. Comparing the expansion coefficients reveals that they have the following regenerative form:

A0=11+R22Rcos(x)
A1=2Rcos(x+π2)1!(1+R22Rcos(x))A0
A2=2Rcos(x+2π2)2!(1+R22Rcos(x))A0+2Rcos(x+π2)1!(1+R22Rcos(x))A1
A3=2Rcos(x+3π2)3!(1+R22Rcos(x))A0+2Rcos(x+2π2)2!(1+R22Rcos(x))A1+2Rcos(x+π2)1!(1+R22Rcos(x))A2
(7)
Fig. 5. Fig. 5. Transmission spectrum of the unloaded waveguide and the fit obtained via expansion by Lorentzians
Fig. 6. Fig. 6. FWHM and group index extracted by fitting the spectrum of the unloaded waveguide resonator

Ignoring all the terms with higher powers than one of quantity R in the numerator and by substituting the coefficients back in the expanded version of H (x) one realizes that H (x) can be rewritten as:

H(x)=11+R22Rcos(x)2Rsin(x)(1+R22Rcos(x))2(θ141!θ1433!+θ1455!)
+2Rcos(x)(1+R22Rcos(x))2(θ1422!+θ1444!θ1466!)
(8)

The terms involving θ 14 can be summed up to produce, sin (θ 14), and (cos (θ 14)-1) respectively. Now let’s turn our attention to the terms multiplying sin (θ 14), (cos (θ 14)-1), and A 0. We can write by expanding to geometric Fourier series:

A0=11R2(1+2Rcos(x)+2R2cos(2x)+2R3cos(3x)+)
sin(x)(1+R22Rcos(x))2=0+11R2sin(x)+2R1R2sin(2x)+3R21R2sin(3x)
cos(x)(1+R22Rcos(x))2=2R(1R2)3+1(1R4)+4R2(1R2)3cos(x)+2(1R4)+4R2(1R2)3Rcos(2x)+
(9)

Substituting Eq. (9) back into H (x) and taking an inverse Fourier transform to the conjugate coordinate space of x (x̃) yields (after regrouping):

h(x˜)=2π(11R24R2(1cos(θ14))(1R2)3)Rx˜Σn=+δ(x˜n)
2π(1R4)(1cos(θ14))(1R2)3x˜Rx˜Σn=+δ(x˜n)
2πisin(x˜)1R2x˜Rx˜Σn=+δ(x˜n)
(10)

Now we can map h(x̃) back to H (x) and obtain:

(η1[R,θ14](2ln(1R)x2+(ln(1R))2)+η2[R,θ14](2ln(1R)x2+(ln(1R))2)2+η3[R,θ14]x(2ln(1R)x2+(ln(1R))2))
*(2πΣq=+δ(2πqx))
(11)

where,

η1[R,θ14]=11R21cos(θ14)(1R2)3(4R2+1R4ln(1R)),η2[R,θ14]=1cos(θ14)(1R2)3(1R4)
η3[R,θ14]=sin(θ14)1R2
(12)

Equation (11) contains all the spectroscopic information we need. It tells us how the resonances of the Fabry-Perot cavity change when the tapered fiber is present. As can be seen the Dirac- Comb function no longer can be assumed to be broadened by a simple Lorentzian instead the square of a Lorentzian and its derivative also take part. HWHM of the Lorentzians in the new line shape function are wider by 1ngln(1κ14) when compared to that of unloaded waveguide, since ln(1r2e2αLgκ14)=ln(1r2e2αLg)+ln(1κ14). Other fitting parameters like the area under Lorentzians and their spectral position can be taken from the fit that is previously obtained from the spectrum of the unloaded waveguide. The quantity R can be recast in terms of the width of the Lorentzians. This leaves |κ 14|, and θ 14 the only fitting parameters to be obtained from the measurements performed on the coupler. Note that no assumptions are made in regard to reflectivity or losses of the facets and of the waveguide (except for that we assumed identical reflectivities for the waveguide facets), and that all the physics of the coupling mechanism is embodied in the matrix [κ]. To further relate the elements of that matrix to such parameters as the overlap integrals and phase matching of the fields coupled mode equations have to be solved with proper boundary conditions dictated by the coupling scheme Ref. [11

11. K. Okamoto, Fundamentals of optical waveguides (Academic Press, 2000)

, 4

4. P. Barclay, K. Srinivasan, and O. Painter, “Design of photonic crystal waveguide for evanescent coupling to optical fiber tapers and integration with high-Q cavities,” Opt. Express 20, 2274–2284 (2003).

].

4. Experiment and analysis

Measurements were made by launching a free space laser beam with wavelengths within 1540nm–1620nm through the left facet of the waveguide and the output power was detected for all other ports.

Knowing how the resonances broaden enables us to properly fit the output power measured from all ports of the coupler. The results can be seen in Fig. 7 and Fig. 8.

Fig. 7. Measured output power from right waveguide facet when the input light is launched through the left facet.
Fig. 8. Measured output power from the right waveguide facet when the input light is launched via port 3

The coupling efficiency was extracted from the Fig. 7 and can be seen in Fig. 9. It is apparent that the coupling efficiency spectrally matches with the measured optical power from port 2. An examination of the extracted coupling efficiency and the fit reveals that the coupling efficiency reaches to 97.3% at its highest point and the coupling FWHM is about 16.9nm.

Fig. 9. Coupling efficiency and the measured output power from port 2 when the input light is launched through the left facet

5. Summary

In summary we discussed the fabrication technique used in making of the photonic coupler under study and presented the possible benefits and drawbacks. We also demonstrated a spectroscopic technique based on full four port modeling and measurement and showed that by knowing how the resonances of the waveguide resonator, embedded in our photonic coupler, are changed it is possible to extract the coupling efficiency for the entire spectrum and that no assumptions are necessary for the values of the losses and facet reflectivities of the waveguide resonator.

Acknowledgements

This work was supported by US NSF under award number ECS-0636677.

References

1.

J. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997). [CrossRef]

2.

T. Krauss, “Planar photonic crystal waveguide devices for integrated optics,” Phys. Stat. Sol. (a) 197, 688–702 (2003). [CrossRef]

3.

W. Kuang, C. Kim, A. Stapleton, and J. O’Brien, “Grating-assisted coupling of optical fibers and photonic crystal waveguides,” Opt. Lett. 27, 1602–1606 (2002). [CrossRef]

4.

P. Barclay, K. Srinivasan, and O. Painter, “Design of photonic crystal waveguide for evanescent coupling to optical fiber tapers and integration with high-Q cavities,” Opt. Express 20, 2274–2284 (2003).

5.

P. Barclay, K. Srinivasan, M. Borselli, and O. Painter “Efficient input and output fiber coupling to a photonic crystal waveguide,” Opt. Lett. 29, 697–699 (2004). [CrossRef] [PubMed]

6.

I.-K. Hwang, G.-H. Kim, and Y.-H. Lee, “Optimization of coupling between photonic crystal resonators and curved microfiber,” IEEE J. Quantum Electron. 42, 131–136 (2006). [CrossRef]

7.

M. lee, C. Grillet, C. Poulton, C. Monat, C. Smith, E. Magi, D. Freeman, S. Madden, B. Luther-Davis, and B. Eggleton, “Characterizing photonic crystal waveguides with an extended k-space evanescent coupling technique,” Opt. Express 16, 13800–13808 (2008). [CrossRef] [PubMed]

8.

J. Love, W. Henry, W. Stewart, R. Black, S. Lacroix, and F. Gonthier,“Tapered single-mode fibers and devices; Part1: Adiabaticity criteria,” IEE Proc. J. 138, 343–354 (1991).

9.

R. Black, S. Lacroix, F. Gonthier, and J. Love, “Tapered single-mode fibers and devices; Part2: Experimental and theoretical qualification,” IEE Proc. J. 138, 355–364 (1991).

10.

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

11.

K. Okamoto, Fundamentals of optical waveguides (Academic Press, 2000)

12.

V. Cooper, “Analysis of Fabry-Perot interferograms by means of their Fourier transforms,” Appl. Opt. 10, 525–530 (1971). [CrossRef] [PubMed]

13.

B. Hakki and T. Paoli, “Gain spectra in GaAs double-heterostructure injection lasers,” Appl. Phys. 46, 1299–1306 (1975). [CrossRef]

14.

M. Natomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902-1–253902-3 (2001).

15.

A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller, and T. Krauss, “Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry,” Appl. Phys. Lett. 90, 261107-1–261107-3 (2007). [CrossRef]

OCIS Codes
(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers
(130.3120) Integrated optics : Integrated optics devices

ToC Category:
Integrated Optics

History
Original Manuscript: April 23, 2009
Revised Manuscript: May 27, 2009
Manuscript Accepted: May 30, 2009
Published: June 11, 2009

Citation
Raymond Sarkissian, Stephen Farrell, and John D. O'Brien, "Spectroscopy of a tapered-fiber photonic crystal waveguide coupler," Opt. Express 17, 10738-10747 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10738


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References

  1. J. Joannopoulos, P. Villeneuve, and S. Fan, "Photonic crystals: Putting a new twist on light," Nature 386,143-149 (1997). [CrossRef]
  2. T. Krauss, "Planar photonic crystal waveguide devices for integrated optics," Phys. Stat. Sol. (a) 197, 688-702 (2003). [CrossRef]
  3. W. Kuang, C. Kim, A. Stapleton, and J. O’Brien, "Grating-assisted coupling of optical fibers and photonic crystal waveguides," Opt. Lett. 27, 1602-1606 (2002). [CrossRef]
  4. P. Barclay, K. Srinivasan, and O. Painter, "Design of photonic crystal waveguide for evanescent coupling to optical fiber tapers and integration with high-Q cavities," Opt. Express 20, 2274-2284 (2003).
  5. P. Barclay, K. Srinivasan, M. Borselli, and O. Painter "Efficient input and output fiber coupling to a photonic crystal waveguide," Opt. Lett. 29, 697-699 (2004). [CrossRef] [PubMed]
  6. I.-K. Hwang, G.-H. Kim, and Y.-H. Lee, "Optimization of coupling between photonic crystal resonators and curved microfiber," IEEE J. Quantum Electron. 42, 131-136 (2006). [CrossRef]
  7. M. Lee, C . Grillet, C . Poulton, C. Monat, C . Smith, E. Magi, D . Freeman, S . Madden, B. Luther-Davis, and B . Eggleton, "Characterizing photonic crystal waveguides with an extended k-space evanescent coupling technique," Opt. Express 16, 13800-13808 (2008). [CrossRef] [PubMed]
  8. J. Love,W. Henry,W. Stewart, R. Black, S. Lacroix, F. Gonthier,"Tapered single-mode fibers and devices; Part1: Adiabaticity criteria," IEE Proc. J. 138, 343-354 (1991).
  9. R. Black, S. Lacroix, F. Gonthier, J. Love, "Tapered single-mode fibers and devices; Part2: Experimental and theoretical qualification," IEE Proc. J. 138, 355-364 (1991).
  10. F. Sanchez, "Matrix algebra for all-fiber optical resonators," J. Lightwave Technol. 9, 838-844 (1991). [CrossRef]
  11. K. Okamoto, Fundamentals of optical waveguides (Academic Press, 2000)
  12. V. Cooper, "Analysis of Fabry-Perot interferograms by means of their Fourier transforms," Appl. Opt. 10, 525-530 (1971). [CrossRef] [PubMed]
  13. B. Hakki, T. Paoli, "Gain spectra in GaAs double-heterostructure injection lasers," Appl. Phys. 46, 1299-1306 (1975). [CrossRef]
  14. M. Natomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, " Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett.  87, 253902-1-253902-3 (2001).
  15. A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller and T. Krauss, "Direct measurement of the group index of photonic crystal waveguides via Fourier transform spectral interferometry," Appl. Phys. Lett.  90, 261107-1-261107-3 (2007). [CrossRef]

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