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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 5 — Mar. 2, 2009
  • pp: 3531–3542
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Dynamic ultrafast laser spatial tailoring for parallel micromachining of photonic devices in transparent materials

C. Mauclair, G. Cheng, N. Huot, E. Audouard, A. Rosenfeld, I. V. Hertel, and R. Stoian  »View Author Affiliations


Optics Express, Vol. 17, Issue 5, pp. 3531-3542 (2009)
http://dx.doi.org/10.1364/OE.17.003531


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Abstract

Femtosecond laser processing of bulk transparent materials can generate localized positive changes of the refractive index. Thus, by translation of the laser spot, light-guiding structures are achievable in three dimensions. Increasing the number of laser processing spots can consequently reduce the machining effort. In this paper, we report on a procedure of dynamic ultrafast laser beam spatial tailoring for parallel photoinscription of photonic functions. Multispot operation is achieved by spatially modulating the wavefront of the beam with a time-evolutive periodical binary phase mask. The parallel longitudinal writing of multiple waveguides is demonstrated in fused silica. Using this technique, light dividers in three dimensions and wavelength-division demultiplexing (WDD) devices relying on evanescent wave coupling are demonstrated.

© 2009 Optical Society of America

1. Introduction

Ultrafast laser processing of bulk transparent materials [1

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996). [CrossRef] [PubMed]

,2

2. E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Three-dimensional optical storage inside transparent materials,” Opt. Lett. 21, 2023–2025 (1996). [CrossRef] [PubMed]

] experiences an increasing interest as it continuously demonstrates its capability to achieve more and more complex embedded photonic structures in a few-steps process (see [3

3. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31, 620–625 (2006). [CrossRef]

] and references therein). The nonlinear interaction between the infrared femtosecond pulses and the dielectric materials allows localized modifications of the material structural properties. These changes depend on the material’s response to the laser solicitation and on the laser irradiation spatio-temporal characteristics [4

4. A. Mermillod-Blondin, I. M. Burakov, Yu. P. Meshcheryakov, N. M. Bulgakova, E. Audouard, A. Rosenfeld, A. Husakou, I. V. Hertel, and R. Stoian, “Flipping the sign of refractive index changes in ultrafast and temporally shaped laser-irradiated borosilicate crown optical glass at high repetition rates,” Phys. Rev. B 77, 104205/1-8 (2008). [CrossRef]

,5

5. D. Wortmann, M. Ramme, and J. Gottmann, “Refractive index modification using fs-laser double pulses,” Opt. Express 15, 10149–10153 (2007). [CrossRef] [PubMed]

]. These structural modifications may modify several optical properties such as birefringence, absorption, and refractive index [6

6. E. Bricchi, B. G. Klappauf, and P. G. Kazansky, “Form birefringence and negative index change created by femtosecond direct writing in transparent materials,” Opt. Lett. 29, 119–121 (2004). [CrossRef] [PubMed]

], serving as means of optical functionalization. In fused silica, defect centers, densification, and thermo-mechanical effects have been identified in the femtosecond laser irradiated zone accompanied by index changes with emphasis on an increase of the refractive index on a micrometer scale [3

3. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31, 620–625 (2006). [CrossRef]

]. This outcome can be seen as the building block of more complex photonic devices. In other words, by translation of the single laser spot, light-guiding structures are achievable in the three dimensions [3

3. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31, 620–625 (2006). [CrossRef]

]. Relying on this processing principle, several groups were able to photoinscribe rather complex embedded optical components, particularly in fused silica [1–3

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996). [CrossRef] [PubMed]

] but also in other media of optical interest, fabricating gain waveguides [7

7. Y. Sikorski, A.A. Said, P. Bado, R. Maynard, C. Florea, and K.A. Winick,“Optical waveguide amplifier in Nd-doped glass written with near-IR femtosecond laser pulses,” Electron. Lett. 36, 226–227 (2000). [CrossRef]

], frequency-doubling waveguides [8

8. S. Campbell, R. R. Thomson, D. P. Hand, A. K. Kar, D. T. Reid, C. Canalias, V. Pasiskevicius, and F. Lau-rell, “Frequency-doubling in femtosecond laser inscribed periodically-poled potassium titanyl phosphate waveguides,” Opt. Express 15, 17146–17150 (2007). [CrossRef] [PubMed]

] and femtosecond-written lasers [9

9. G. Della Valle, S. Taccheo, R. Osellame, A. Festa, G. Cerullo, and P. Laporta, “1.5 μm single longitudinal mode waveguide laser fabricated by femtosecond laser writing,” Opt. Express 15, 3190–3194 (2007). [CrossRef] [PubMed]

]. Great efforts were carried out to define proper irradiation conditions in order to improve the quality of the laser written structures. Techniques based on controlling the aspect of the focal region at different repetition rates and polarization regimes emerged [10

10. S. M. Eaton, H. Zhang, M. L. Ng, J. Li, W. Chen, S. Ho, and P. R. Herman, “Transition from thermal diffusion to heat accumulation in high repetition rate femtosecond laser writing of buried optical waveguides,” Opt. Express 16, 9443–9458 (2008). [CrossRef] [PubMed]

,11

11. D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford,“Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure,” Opt. Express 16, 20029–20037 (2008). [CrossRef] [PubMed]

]. Among the main characteristics of interest are found the amplitude of the refractive index increase [5

5. D. Wortmann, M. Ramme, and J. Gottmann, “Refractive index modification using fs-laser double pulses,” Opt. Express 15, 10149–10153 (2007). [CrossRef] [PubMed]

,10

10. S. M. Eaton, H. Zhang, M. L. Ng, J. Li, W. Chen, S. Ho, and P. R. Herman, “Transition from thermal diffusion to heat accumulation in high repetition rate femtosecond laser writing of buried optical waveguides,” Opt. Express 16, 9443–9458 (2008). [CrossRef] [PubMed]

], propagation losses [12

12. H. Zhang, S. M. Eaton, and P. R. Herman, “Low-loss Type II waveguide writing in fused silica with single picosecond laser pulses,” Opt. Express 14, 4826–4834 (2006). [CrossRef] [PubMed]

], cylindrical symmetry [13

13. Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama,“Control of the cross-sectional shape of a hollow microchannel embedded in photostructurable glass by use of a femtosecond laser,” Opt. Lett. 28, 55–57 (2003). [CrossRef] [PubMed]

,14

14. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerullo,“Femtosecond writing of active ptical waveguides with astigmatically shaped beams,” J. Opt. Soc. Am. B 20, 1559–1567 (2003). [CrossRef]

] and compensation of the wavefront distortion during the photoinscription process of deep structures [15

15. C. Mauclair, A. Mermillod-Blondin, N. Huot, E. Audouard, and R. Stoian, “Ultrafast laser writing of homogeneous longitudinal waveguides in glasses using dynamic wavefront correction,” Opt. Express 16, 5481–5492 (2008). [CrossRef] [PubMed]

,16

16. A. Mermillod-Blondin, C. Mauclair, A. Rosenfeld, J. Bonse, I. V. Hertel, E. Audouard, and R. Stoian, “Size correction in ultrafast laser processing of fused silica by temporal pulse shaping,” Appl. Phys. Lett. 93, 021921–21924 (2008). [CrossRef]

]. Whereas the three-dimensional photoinscription access constitutes a clear advantage over multilayer lithographic techniques, the single laser focal spot must undergo potentially complicated movements with respect to the sample in order to draw the whole photonic structure. This single spot operation (SSO) may involve long processing times when the writing of several complex structures is envisaged.

A binary phase mask, when placed at the object plane of a focusing lens, enables the generation of multiple laser spots [17

17. S. Li, G. Yu, C. Zheng, and Q. Tan, “Quasi-Dammann grating with proportional intensity array spots,” Opt. Lett. 33, 2023–2025 (2008). [CrossRef] [PubMed]

]. Static multispot machining of glass relying on this technique was recently reported [18

18. S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31, 1705–1707 (2006). [CrossRef] [PubMed]

]. We present in the following a time-saving method giving access to dynamic multispot operation (DMSO) with variable and reconfigurable patterns and we successfully apply it to the parallel fabrication of light guiding structures in the bulk. Dynamic multispot operation based on laser wavefront modulation constitutes a step forward as it enables the simultaneous processing of several embedded structures, thus lowering the process time and reducing the complexity of the mechanical support. The paper is organized as follows. After a brief analytical outline pointing out the DMSO through spatial phase modulation details of the experimental apparatus will be given. Then, the demonstration of parallel longitudinal writing of waveguides is presented followed by examples of photonic structures based on DMSO in the two and three dimensions (2D and 3D).

2. Description of dynamic multispot operation

In the object plane of a focusing objective, an optically addressed spatial light modulator (SLM) imprints a binary phase-only mask (BPM) ϕ (x) of period T and amplitude Δϕ on the laser wave-front as shown on Fig. 1. For simplicity, it is considered infinite in the x direction and invariant in the y direction, z being the laser propagation axis. In the frame of the Fraunhofer approximation and supposing that the SLM reproduces perfectly the BPM, the diffraction efficiency η of the m -th order is given by [19

19. J. W. Goodman, Introduction to Fourier Optics (2nd ed. McGraw-Hill, Singapore, 1996).

] :

ηm0=1cos(Δϕ)2sinc2(πm2).
(1)

And for the 0-th order:

η0=1+cos(Δϕ)2.
(2)
Fig. 1. (color online): Scheme of double spot operation. BPM: Binary phase mask. Insert: Double spot intensity profile inside the sample at the focus of the objective generated by a simple step grating phase at very low power captured by direct imaging on a CCD camera. The blue arrow shows the direction of motion of the sample for longitudinal photoinscription.

For the ideal binary grating, the even orders disappear regardless of the phase step value Δϕ. Moreover, when Δϕ = π, not only does no diffraction occur at the 0-th order but the first order efficiency becomes much higher than the other orders, thus constituting two available laser spots for laser processing. In that case, adding the positive and negative first order efficiency shows that these two peaks gather 82% of the incoming light. By varying the period T of the BPM, the separation of the two spots becomes an accessible parameter to be controlled during photoinscription through the SLM.

From the experimentally-obtained double spot beam profile depicted in Fig. 1(insert), it is worth noting that the spots corresponding to the +1 and -1 orders show an expected circular symmetry. As foreseen, these two peaks widely surpass other diffracted orders. Evaluating the relative portion of power contained in these two processing spots by summing the corresponding intensity levels permits to realize that approximately one third of the total power is left into the non-processing orders. This additional spreading out of usable power compared to the above-mentioned 18% is inherent to the incapacity of the continuous liquid crystal layer of the SLM to reproduce perfectly the sharp steps of the desired BPM [20

20. C. Yu, J. Park, J. Kim, M. Jung, and S. Lee, “Design of Binary Diffraction Gratings of Liquid Crystals in a Linearly Graded Phase Model,” Appl. Opt. 43, 1783–1788 (2004). [CrossRef] [PubMed]

]. These losses can be reduced through the design of more complex phase masks with iterative optimization algorithms [17

17. S. Li, G. Yu, C. Zheng, and Q. Tan, “Quasi-Dammann grating with proportional intensity array spots,” Opt. Lett. 33, 2023–2025 (2008). [CrossRef] [PubMed]

]. However, in view of the nonlinear response of the interaction, only the high intensity peaks contribute to the structuring process and in the frame of the present work, the simple step grating phase mask is sufficient for the DMSO demonstration purpose. Moreover, this phase modulation offers a straightforward way to control the position of the two spots by varying its period; a feature more difficult to obtain in the case of more complex phase masks.

3. Experiment

The choice of fused silica glass is motivated by its numerous potential applications in the fields of integrated optics and microfluidics. Moreover, this well-known material has been thoroughly investigated in order to identify several femtosecond waveguide writing regimes [1

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996). [CrossRef] [PubMed]

,12

12. H. Zhang, S. M. Eaton, and P. R. Herman, “Low-loss Type II waveguide writing in fused silica with single picosecond laser pulses,” Opt. Express 14, 4826–4834 (2006). [CrossRef] [PubMed]

, 21

21. R. Graf, A. Fernandez, M. Dubov, H.J. Brueckner, B.N. Chichkov, and A. Apolonski, “Pearl-chain waveguides written at megahertz repetition rate,” Appl. Phys. B 87, 21–27 (2007). [CrossRef]

]. Polished fused silica parallelepipedic samples are irradiated with 150 fs pulses from an 800 nm 100 kHz Ti:Sapphire ultrafast regenerative amplifier laser system. Used at 10 kHZ, the system delivers an usable power of 30mW. A computer driven electromechanical shutter, synchronized with the movements of the positioning system (Physik Instrumente M-126.DG) permits the writing of longitudinal structures in the bulk of the sample. A long working distance microscope objective (numerical aperture NA=0.3) is employed to focus the femtosecond beam in the silica glass. The diameter Db of the beam at 1/e 2 is 1.5 mm and is adjusted with respect to the objective aperture diameter Da = 3 mm resulting in a truncation ratio of 0.5.

As a rapid observation mean, a Zernike-type positive optical phase-contrast microscopy (PCM) system is set to study the laser-irradiated areas. The observation axis is perpendicular to the laser propagation direction and presents a sub-μm depth of field. A precise optical alignment permits the real-time imaging of the laser-exposed zones and subsequent optical transformations. PCM is particularly adapted for the study of femtosecond-induced phase objects as it optically translates a two-dimension map of optical path variations into a gray-level picture captured in our case by a high-resolution Charge-Coupled Device (CCD) camera. In the frame of positive PCM and for relatively small phase shift values, a darker region relative to the background corresponds to an increase of the refractive index while white regions indicate its lowering or the onset of absorbing/diffusing regions. Consequently, this qualitative mapping of the refractive index allows for rapid estimations of the index variation subsequent to various irradiation conditions. The structures were imaged in PCM in a side view geometry, perpendicular to the pulse propagation axis.

4. Results and discussion

We mention here that all the photowritten structures presented below were achieved in the so-called longitudinal configuration, where the sample is translated parallel to the laser propagation (see Fig. 1). One of the main advantages of this technique is found in the cylindrical symmetry of the photoinscribed structure while transverse waveguide writing generally requires additional strategies to preserve this symmetry [13

13. Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama,“Control of the cross-sectional shape of a hollow microchannel embedded in photostructurable glass by use of a femtosecond laser,” Opt. Lett. 28, 55–57 (2003). [CrossRef] [PubMed]

, 14

14. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerullo,“Femtosecond writing of active ptical waveguides with astigmatically shaped beams,” J. Opt. Soc. Am. B 20, 1559–1567 (2003). [CrossRef]

]. However, the major challenge to be faced in the longitudinal configuration concerns the spherical aberrations due the air-glass interface through which the laser passes to reach its target [23

23. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98, 013517/1-5 (2005). [CrossRef]

]. The subsequent degradation of the intensity profile at the focus becomes more important when high NAs and deep focusing are considered. Using the formalism depicted in [23

23. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98, 013517/1-5 (2005). [CrossRef]

], it is possible to estimate the decrease of the focused spot peak intensity with the depth of the spot in the sample for a given NA. In our case, the relatively low effective NA = 0.2 resulting from the small beam truncation preserved 84% of the non-aberrated axial peak intensity at 8 mm deep spot in the bulk, allowing us to write longitudinal structures of similar size without prohibitive optical degradation. In addition, it is of interest to point out that, at constant sample velocity, the accumulated spatial rate of deposited energy remains constant during longitudinal writing regardless of the position of the spot in the sample. Those considerations plead in favor of a rather restricted spherical aberration influence in the photowriting conditions presented in this work. Even though the devices depicted hereafter were achieved in the longitudinal configuration, it is worth noting that the extent of the DMSO technique to the transversal case for bulk parallel waveguide writing is straightforward.

Fig. 2. (color online): Comparison between single and double spot longitudinal photoin-scription of waveguides in fused silica. Up: Single spot operation, down double spot operation. (a) Side view PCM picture of SSO photowritten waveguides written at different translation speeds at 10 kHz, 8 mW, superposed with transversal cross section of the relative gray scale decrease indicating a positive change of the refractive index (see text body). (b) corresponding nearfield profile of 633nm injected guide written at 5μm/s, 10kHz, 8mW in single spot operation. (c) PCM picture of two simultaneously photowritten waveguides through double spot operation at 5μm/s, 10 kHz, 24 mW. (d) Optical transmission microscope picture of the cross section of one of the waveguides in (c) in white light illumination. (e) nearfield profile of one of the waveguides pictured in (c) at 633 nm.

4.1. Processing window for parallel longitudinal waveguide writing

To demonstrate the capability of the DMSO to generate similar bulk photoinscribed devices as SSO, longitudinal waveguides were written using both techniques in a regime which usually delivers smooth and nonbirefringent traces. A first set of SSO investigations were carried out with various scan velocities for the glass sample at a power level of 8 mW (measured after the objective) with a repetition rate of 10 kHz (see Fig. 2(a)) i.e; slightly below the single pulse modification threshold. The PCM side image along with its gray scale cross section denotes a relative gray scale decrease after the passage of the femtosecond laser for translation speeds below 10μm/s. These darker structures under positive PCM mainly indicate an increase of the refractive index. They are not readily detectable under normal transmission microscopy (not shown) and present light guiding properties when injected with HeNe radiation at 633 nm. We mention here that in those scanning conditions, moving away from this relatively soft interaction regime by using higher laser power results in strongly scattering structures with poor or no guiding property. Based on this processing window, several 8 mm-long buried waveguides were written at 5μm/s with 8mW for characterization. Fig. 2(b) depicts a typical nearfield mode under HeNe laser injection at 633nm imaged on a CCD camera with a 20x objective associated with a 200 mm positive lens. The light is guided along the center of the modified area and the nearfield mode presents a size of 7 μm at 1/e 2, larger than the transverse dimension of the structure (3.5 μm) evaluated at 1/e 2 on the PCM image. The averaged amplitude of the refractive index change was estimated to be of approximately Δn = 1.6× 10−4 through the waveguide NA measurement in the farfield. This rather low value [3

3. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31, 620–625 (2006). [CrossRef]

] was confirmed using a mode solving software assuming a step index profile. The guides show an excellent cylindrical symmetry, visible on the nearfield mode and the structure itself. Using the method reported in [24

24. Z. Wang, K. Sugioka, Y. Hanada, and K. Midorikawa, “Optical waveguide fabrication and integration with a micro-mirror inside photosensitive glass by femtosecond laser direct writing,” Appl. Phys. A 88, 699–704 (2007). [CrossRef]

], propagation losses of less than 0.7 dB/cm were evaluated, which approaches the best values reported so far for bulk photowritten waveguides in fused silica [3

3. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31, 620–625 (2006). [CrossRef]

].

Using the DMSO, identical light guides were simultaneously photoinscribed by pairs in one pass. Taking advantage of the two spots available at the focus of the objective, dual structures with a separation of s = 56 μm were achieved in a single motor scan at 5 μm/s under the full available power of 24 mW, after the focusing objective. For this purpose, a step grating phase mask of π amplitude at 800 nm with ν = 14 periods across the clear aperture of the objective was displayed with the SLM during the sample displacement. Imaging the focusing plane on a CCD camera, the linear relationship between ν and s was experimentally determined. The PCM side image and nearfield mode of the waveguides are pictured in Fig. 2(c) and Fig. 2(e), respectively. The amplitude of the refractive index change as well as the propagation losses were found to be identical to the SSO written waveguides at 8 mW, same speed (Fig. 2(b)), verifying our estimation of the power spread in the other diffraction orders mentioned in Section 2. The clean cylindrical symmetry is also observable on the nearfield mode as well as on the cross section microscope image (Fig. 2(d)). These results clearly demonstrate the possibility to write buried photonic devices in parallel with a very simple phase function. Observing the processing conditions and comparing them to the available output of the laser, it appeared necessary to restrict the DMSO to two spots in our case in order to preserve enough energy to trigger the proper waveguide writing regime in fused silica. However, there is no a priori restriction to increase this number to higher values in the absence of this limiting factor. Having identified this processing window for parallel photoinscription, efforts were carried out in order to achieve more complex photonic devices based on this principle.

4.2. 2D photonic components

Embedded waveguide arrays have been recently investigated pointing out to novel behavioral aspects [25

25. D. Christodoulides, N. Demetrios, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature (London) 424, 817–823, (2003). [CrossRef]

] when light propagates in discrete propagation spaces due to the evanescent light coupling between neighboring guides. In particular, linear effects such as Bloch oscillations [26

26. T. Pertsch, P. Dannberg, W. Elflein, A. Braüer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]

], light localization, and quasi-incoherent propagation [27

27. A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tuennermann, and F. Lederer “Quasi-incoherent propagation in waveguide arrays,” Appl. Phys. Lett. 90, 241113–241116 (2007). [CrossRef]

] were theoretically predicted and experimentally verified in femtosecond photowritten waveguides arrays. Nonlinear behaviours in the form of spatial solitons were underlined as well in such structures [28

28. A. Szameit, D. Bloemer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, “Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica,” Opt. Express 13, 10552–10557 (2005). [CrossRef] [PubMed]

] indicating the possibility of obtaining new-types of nonlinear components written by femtosecond lasers. We aim to demonstrate here that complex 2D and 3D arrays and matrices of light guiding structures can be efficiently written using DMSO. In the following, examples of 2D photonic devices relying on evanescent coupling achieved with DMSO are presented, performing various optical functions. Evanescent coupling between parallel waveguides is supported by a precise formalism [29

29. A. Yariv, Optical Electronics (4th ed. Saunders College Publ., 1991).

] and has been recently studied in the case of transversal waveguides written in fused silica by femtosecond optical irradiation [30

30. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, “Control of directional evanescent coupling in fs laser written waveguides,”. Opt. Express 15, 1579–1587 (2007), [CrossRef] [PubMed]

]. Due to the relative asymmetry of such waveguides, investigations were carried out underlining the coupling efficiency variations between two waveguides when their relative position changes. Using an additional static beam shaping technique, transverse waveguides with rotational symmetry were achieved fully counteracting this asymmetry feature [30

30. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, “Control of directional evanescent coupling in fs laser written waveguides,”. Opt. Express 15, 1579–1587 (2007), [CrossRef] [PubMed]

]. In the longitudinal configuration for waveguide writing, the resulting guides keep the circular symmetry of the irradiation beam permitting to set aside those considerations when the processing spot is aberration-free. It is therefore justified to adopt in a first approach the formalism proposed by Yariv [29

29. A. Yariv, Optical Electronics (4th ed. Saunders College Publ., 1991).

] for planar waveguides to design embedded light dividers based on evanescent coupling. Several main concepts are given below.

In this frame, when only one of a pair of parallel and homogeneous identical waveguides is injected with light, the coupling length lc after which the light is entirely evanescently coupled in the adjacent waveguide writes (also known as the half beat length):

lc=π2κ,
(3)

κ being the coupling coefficient defined by:

κ=2h2p×exp(ps)β(w+2/p)(h2+p2).
(4)

In the above expression, the parameters can be written as

h=(n12k02β2)1/2
p=(β2n02k02)1/2
β=k0neff.
(5)

n 0 and n 1 being respectively the refractive index of the surrounding medium and of the guides, neff the effective index of the guided mode which obeys n 0 < neff < n 1. w is the waveguide width, s the separation distance between the two parallel waveguides, k 0 the wave number of the injected light and β the propagation constant. The coupling coefficient κ was experimentally evaluated for several waveguide pairs using the method described in [30

30. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, “Control of directional evanescent coupling in fs laser written waveguides,”. Opt. Express 15, 1579–1587 (2007), [CrossRef] [PubMed]

]. Having access to all the other terms of Eq. (4), the effective index neff could then be estimated.

In the case of a cylindrical guide surrounded by N other identical waveguides having equal distance to the main one at vertices of a polygon [31

31. A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972). [CrossRef]

], the coupling coefficient κN for the light to be transmitted to the surrounding structures reads:

κN=κN.
(6)

The corresponding coupling lcN length is then:

lcN=lcN.
(7)

In that case, the maximum power transfer F between the center and the surrounding waveguides writes:

F=1if1N<3
=[1+(κssκcsN)2]1otherwise.
(8)

where κss is the coupling coefficient between the adjacent external waveguides and κCS from the central waveguide to one of the surrounding ones.

For example, in the case of two parallel waveguides with the above-mentioned characteristics (Δn = 1.6 × 10-4 and w = 3.5 μm), the coupling length lc becomes 1.6 mm for a separation s of 9 μm and 633 nm light injection. Similarly, an excited waveguide placed in between two identical guides generates a coupling length l c2 of 1.1 mm; s,w, Δn and k 0 being the same. In these two cases, we have F = 1, meaning that all the light is evanescently coupled after the coupling length. Based on these results, it is then possible to design the concept of a light divider reposing on a successive combination of parallel waveguides with defined lengths and separation allowing evanescent coupling from a seed waveguide to two or more output waveguides. The DMSO technique appears as an ideal processing solution as it offers the flexibility to photoinscribe a pair of coupled waveguides simultaneously.

Fig. 3. (color online): Bulk photowritten light divider based on evanescent coupling of injected waves in partial arrays achieved through double spot operation at 24 mW total power and 5μm/s scan velocity: (a) schematic view of the structure: The conditions followed by lc and l c2 enabling evanescent coupling from the central to the external guides are given in the text. (b) Assemblage of PCM side-pictures of the device, its total length is 8.2 mm and the lateral separation between tracks is 9μm. (c) Nearfield profile under HeNe injection at 633 nm.

Considering a pair of identical waveguides injected in only one arm, a variation of the exciting wavelength modifies the coupling coefficient and length according to Eq. (3) and (4). More precisely, for a given structure, increasing the wavelength will shorten the coupling length. In the case of two available wavelengths λ 1 and λ 2 with λ 1 > λ 2, it is possible to create a device for which the coupling length l 2 for λ 2 is twice as big as the coupling length l 1 for λ 1. Then, if the total overlapping length lo of the waveguides verifies lo = l 2 = 2 + l 1, exciting one arm with both λ 1 and λ 2 generates total evanescent coupling of the λ 2 light in the other arm whereas the λ 1 excitation is be back coupled to the initial arm demultiplexing the input frequencies.

Fig. 4. (color online): Bulk photowritten WDD device achieved through double spot operation at 24 mW total power and 5 μm/s scan velocity. (a) Schematic view of the structure (condition followed by lo is given in the text). (b) Nearfield profile under 633 nm (solid) and 800 nm (dashed) simultaneous injection in the top arm. (c) PCM side-image of the overlapping region. (d) Theoretical prediction of the 633 nm (solid) and 800 nm (dashed) intensity variations in the excited waveguide according to [31,32] taking into account the wavelength dependance of the coupling coefficient. The total length of the structure is 7.4 mm.

Figure 4 shows a device performing the wavelength-division demultiplexing (WDD) operation. The entire device was photowritten with DMSO in a single motor scan in similar exposure conditions as already described. At the beginning of the irradiation, the laser beam was spatially modulated with a BPM having ν = 20 periods resulting in two 84 μm separated spots. The number of periods ν was dynamically decreased while the sample was moving thus lowering the separation to s = 16μm in synchronization with the photoinscription procedure. At 633 nm, the coupling length for this separation is l c633 = 4.5 mm. For injection with 800 nm CW laser radiation, the length becomes l c800 = 2.4 mm. To take into account the weak coupling in the non-parallel part of the structure, we used the formalism proposed by McIntyre and Snyder [32

32. P. D. McIntyre and A. W. Snyder, “Power transfer between nonparallel and tapered optical fibers,” J. Opt. Soc. Am. 64, 286–288 (1974). [CrossRef]

] for non-parallel structures. Consequently, the overlapping length lo = 4,4 mm is slightly smaller than l c633. Calculation results for the structure are depicted in Fig. 4(d) showing clean WDD between 633 nm and 800 nm. The experimental nearfield modes of the excited device at 633 nm and 800 nm are presented in Fig. 4(b), in agreement with the expected behavior.

4.3. 3D photonic components

The beam shaping properties allow easy access to the 3D space. In order to underline further three-dimensional capabilities as well as the high flexibility of the DMSO technique, two optical components are presented hereafter. First, a simple twisted X-coupler is depicted on Fig. 5 where a structure rotation from the vertical to the horizontal plane occurs. This device was pho- towritten in a single scan of the sample during which the wavefront was continuously controlled in order to draw simultaneously the two arms of the structure. At the beginning of the move, the spatial phase mask was driven the same way as for the above-mentioned WDD device i.e by changing the BPM cycling frequency. Then, it was gradually rotated by 90° during the writing of the central part (Fig. 5(b)). Finally, the number ν of periods was increased to augment the separation of the two arms. The structure dimensions (separation, overlapping length lo) were designed with the help of the above-mentioned formalism in order to transmit the totality of the 633 nm light injected in the bottom arm to the top one. Fig. 5(c) depicts the experimentally obtained nearfield mode at the output of the device.

Fig. 5. (color online): (a) Schematic view of a bulk photowritten twisted X coupler achieved through DMSO in a single scan. (b) Optical transmission microscopy pictures of the central region, showing 90° rotation. (c) Nearfield profile at the output of the top arm under 633 nm injection in the bottom waveguide.
Fig. 6. (color online): Schematic view of a bulk photowritten 3D hexagonal light divider achieved through DMSO. The condition followed by lo is given in the text. Insert: Nearfield profile under 633 nm injection in the central waveguide.

The second structure of interest achieved by DMSO consists of a hexagonal light divider for 633 nm (Fig. 6). Hexagonal waveguide arrays written by femtosecond SSO were recently achieved and light propagation in such structures was investigated [33

33. A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B 82, 507–512 (2006). [CrossRef]

]. The light divider presented here was photoinscribed taking advantage of the flexibility and the 3D access of DMSO. Four sample scans were necessary to achieve the device in similar exposure conditions as already described. We recall here that DMSO was restricted to two processing spots in order to preserve sufficient enough energy to trigger the proper photowriting regime in fused silica. First, the sample was irradiated with the wavefront modulated laser beam offering two processing spots separated by 30μm. The two subsequent scans were performed with the same mask rotated by an angle of 60° and 120°, successively. Finally, the central guide was photowrit-ten without any wavefront modulation at lower power. The distance sAA between two adjacent guides equals the separation sCA between the central guide and the six others having s = 15μm, the whole structure forming a regular hexagon.

In order to obtain optimal coupling, the overlapping length lo (Fig. 6) verifies lo = l c6 = 1.5 mm obtained from Eq. (7). Having sAA = sCA, it is straightforward to show that κss = κcs. Thus, using Eq.(8), it is expected to obtain F = 0.85, meaning that 85% of the injected light is theoretically retrieved at the output of the six external guides (~ 14% per guide), the rest remaining in the seed waveguide. In other words, each of the seven outputs is expected to have the same nearfield mode peak intensity. The insert of (Fig. 6) depicts the experimental nearfield intensity distribution, being in good agreement with the foreseen distribution.

5. Conclusion

An innovative technique is proposed to photoinscribe optical 3D devices in the bulk of transparent materials using spatial ultrafast beam shaping. The parallel photowriting technique uses time-evolutive modulation of femtosecond laser wavefront, producing two laser processing spots for parallel photoinscription of light guiding structures. The method relies on the employment of a periodical binary phase mask where a variation of the period (cycling frequency) results in a modification of the spots separation, a feature permitting dynamic positioning of the processing spots during laser exposure. Provided that sufficient energy is available, the technique can be scaled up to a higher number of machining foci. Several devices are demonstrated in the longitudinal writing configuration, thus underlining the high processing flexibility of this technique easily applicable for more demanding photowritten optical components. Examples are given emphasizing waveguide arrays where parallel processing appears as a natural choice. Two and three dimensional photonic structures relying on evanescent coupling were achieved with this method performing functions of light coupling, division and wavelength demultiplexing.

Acknowledgments

Support of ANR and PICS programs is gratefully acknowledged. We also thank J.-P. Meunier for useful discussions.

References and links

1.

K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996). [CrossRef] [PubMed]

2.

E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Three-dimensional optical storage inside transparent materials,” Opt. Lett. 21, 2023–2025 (1996). [CrossRef] [PubMed]

3.

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31, 620–625 (2006). [CrossRef]

4.

A. Mermillod-Blondin, I. M. Burakov, Yu. P. Meshcheryakov, N. M. Bulgakova, E. Audouard, A. Rosenfeld, A. Husakou, I. V. Hertel, and R. Stoian, “Flipping the sign of refractive index changes in ultrafast and temporally shaped laser-irradiated borosilicate crown optical glass at high repetition rates,” Phys. Rev. B 77, 104205/1-8 (2008). [CrossRef]

5.

D. Wortmann, M. Ramme, and J. Gottmann, “Refractive index modification using fs-laser double pulses,” Opt. Express 15, 10149–10153 (2007). [CrossRef] [PubMed]

6.

E. Bricchi, B. G. Klappauf, and P. G. Kazansky, “Form birefringence and negative index change created by femtosecond direct writing in transparent materials,” Opt. Lett. 29, 119–121 (2004). [CrossRef] [PubMed]

7.

Y. Sikorski, A.A. Said, P. Bado, R. Maynard, C. Florea, and K.A. Winick,“Optical waveguide amplifier in Nd-doped glass written with near-IR femtosecond laser pulses,” Electron. Lett. 36, 226–227 (2000). [CrossRef]

8.

S. Campbell, R. R. Thomson, D. P. Hand, A. K. Kar, D. T. Reid, C. Canalias, V. Pasiskevicius, and F. Lau-rell, “Frequency-doubling in femtosecond laser inscribed periodically-poled potassium titanyl phosphate waveguides,” Opt. Express 15, 17146–17150 (2007). [CrossRef] [PubMed]

9.

G. Della Valle, S. Taccheo, R. Osellame, A. Festa, G. Cerullo, and P. Laporta, “1.5 μm single longitudinal mode waveguide laser fabricated by femtosecond laser writing,” Opt. Express 15, 3190–3194 (2007). [CrossRef] [PubMed]

10.

S. M. Eaton, H. Zhang, M. L. Ng, J. Li, W. Chen, S. Ho, and P. R. Herman, “Transition from thermal diffusion to heat accumulation in high repetition rate femtosecond laser writing of buried optical waveguides,” Opt. Express 16, 9443–9458 (2008). [CrossRef] [PubMed]

11.

D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford,“Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure,” Opt. Express 16, 20029–20037 (2008). [CrossRef] [PubMed]

12.

H. Zhang, S. M. Eaton, and P. R. Herman, “Low-loss Type II waveguide writing in fused silica with single picosecond laser pulses,” Opt. Express 14, 4826–4834 (2006). [CrossRef] [PubMed]

13.

Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama,“Control of the cross-sectional shape of a hollow microchannel embedded in photostructurable glass by use of a femtosecond laser,” Opt. Lett. 28, 55–57 (2003). [CrossRef] [PubMed]

14.

R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerullo,“Femtosecond writing of active ptical waveguides with astigmatically shaped beams,” J. Opt. Soc. Am. B 20, 1559–1567 (2003). [CrossRef]

15.

C. Mauclair, A. Mermillod-Blondin, N. Huot, E. Audouard, and R. Stoian, “Ultrafast laser writing of homogeneous longitudinal waveguides in glasses using dynamic wavefront correction,” Opt. Express 16, 5481–5492 (2008). [CrossRef] [PubMed]

16.

A. Mermillod-Blondin, C. Mauclair, A. Rosenfeld, J. Bonse, I. V. Hertel, E. Audouard, and R. Stoian, “Size correction in ultrafast laser processing of fused silica by temporal pulse shaping,” Appl. Phys. Lett. 93, 021921–21924 (2008). [CrossRef]

17.

S. Li, G. Yu, C. Zheng, and Q. Tan, “Quasi-Dammann grating with proportional intensity array spots,” Opt. Lett. 33, 2023–2025 (2008). [CrossRef] [PubMed]

18.

S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31, 1705–1707 (2006). [CrossRef] [PubMed]

19.

J. W. Goodman, Introduction to Fourier Optics (2nd ed. McGraw-Hill, Singapore, 1996).

20.

C. Yu, J. Park, J. Kim, M. Jung, and S. Lee, “Design of Binary Diffraction Gratings of Liquid Crystals in a Linearly Graded Phase Model,” Appl. Opt. 43, 1783–1788 (2004). [CrossRef] [PubMed]

21.

R. Graf, A. Fernandez, M. Dubov, H.J. Brueckner, B.N. Chichkov, and A. Apolonski, “Pearl-chain waveguides written at megahertz repetition rate,” Appl. Phys. B 87, 21–27 (2007). [CrossRef]

22.

N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. 30, 1479–1481 (2005). [CrossRef] [PubMed]

23.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. 98, 013517/1-5 (2005). [CrossRef]

24.

Z. Wang, K. Sugioka, Y. Hanada, and K. Midorikawa, “Optical waveguide fabrication and integration with a micro-mirror inside photosensitive glass by femtosecond laser direct writing,” Appl. Phys. A 88, 699–704 (2007). [CrossRef]

25.

D. Christodoulides, N. Demetrios, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature (London) 424, 817–823, (2003). [CrossRef]

26.

T. Pertsch, P. Dannberg, W. Elflein, A. Braüer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]

27.

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tuennermann, and F. Lederer “Quasi-incoherent propagation in waveguide arrays,” Appl. Phys. Lett. 90, 241113–241116 (2007). [CrossRef]

28.

A. Szameit, D. Bloemer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, “Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica,” Opt. Express 13, 10552–10557 (2005). [CrossRef] [PubMed]

29.

A. Yariv, Optical Electronics (4th ed. Saunders College Publ., 1991).

30.

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, “Control of directional evanescent coupling in fs laser written waveguides,”. Opt. Express 15, 1579–1587 (2007), [CrossRef] [PubMed]

31.

A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972). [CrossRef]

32.

P. D. McIntyre and A. W. Snyder, “Power transfer between nonparallel and tapered optical fibers,” J. Opt. Soc. Am. 64, 286–288 (1974). [CrossRef]

33.

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B 82, 507–512 (2006). [CrossRef]

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(140.3390) Lasers and laser optics : Laser materials processing
(230.7370) Optical devices : Waveguides

ToC Category:
Laser Micromachining

History
Original Manuscript: December 1, 2008
Revised Manuscript: January 12, 2009
Manuscript Accepted: January 13, 2009
Published: February 23, 2009

Citation
C. Mauclair, G. Cheng, N. Huot, E. Audouard, A. Rosenfeld, I. V. Hertel, and R. Stoian, "Dynamic ultrafast laser spatial tailoring for parallel micromachining of photonic devices in transparent materials," Opt. Express 17, 3531-3542 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3531


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References

  1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, "Writing waveguides in glass with a femtosecond laser," Opt. Lett. 21, 1729-1731 (1996). [CrossRef] [PubMed]
  2. E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, "Three-dimensional optical storage inside transparent materials," Opt. Lett. 21, 2023-2025 (1996). [CrossRef] [PubMed]
  3. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast processes for bulk modification of transparent materials," MRS Bull. 31,620-625 (2006). [CrossRef]
  4. A. Mermillod-Blondin, I. M. Burakov, Yu. P. Meshcheryakov, N. M. Bulgakova, E. Audouard, A. Rosenfeld, A. Husakou, I. V. Hertel, and R. Stoian, "Flipping the sign of refractive index changes in ultrafast and temporally shaped laser-irradiated borosilicate crown optical glass at high repetition rates," Phys. Rev. B 77, 104205/1-8 (2008). [CrossRef]
  5. D. Wortmann, M. Ramme, and J. Gottmann, "Refractive index modification using fs-laser double pulses," Opt. Express 15, 10149-10153 (2007). [CrossRef] [PubMed]
  6. E. Bricchi, B. G. Klappauf, and P. G. Kazansky, "Form birefringence and negative index change created by femtosecond direct writing in transparent materials," Opt. Lett. 29, 119-121 (2004). [CrossRef] [PubMed]
  7. Y. Sikorski, A. A. Said, P. Bado, R. Maynard, C. Florea, and K. A. Winick, "Optical waveguide amplifier in Nddoped glass written with near-IR femtosecond laser pulses," Electron. Lett. 36, 226-227 (2000). [CrossRef]
  8. S. Campbell, R. R. Thomson, D. P. Hand, A. K. Kar, D. T. Reid, C. Canalias, V. Pasiskevicius, and F. Laurell, "Frequency-doubling in femtosecond laser inscribed periodically-poled potassium titanyl phosphate waveguides," Opt. Express 15, 17146-17150 (2007). [CrossRef] [PubMed]
  9. G. Della Valle, S. Taccheo, R. Osellame, A. Festa, G. Cerullo, and P. Laporta, "1.5 μm single longitudinal mode waveguide laser fabricated by femtosecond laser writing," Opt. Express 15, 3190-3194 (2007). [CrossRef] [PubMed]
  10. S. M. Eaton, H. Zhang, M. L. Ng, J. Li,W. Chen, S. Ho, and P. R. Herman, "Transition from thermal diffusion to heat accumulation in high repetition rate femtosecond laser writing of buried optical waveguides," Opt. Express 16, 9443-9458 (2008). [CrossRef] [PubMed]
  11. D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford,"Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure," Opt. Express 16, 20029-20037 (2008). [CrossRef] [PubMed]
  12. H. Zhang, S. M. Eaton, and P. R. Herman, "Low-loss Type II waveguide writing in fused silica with single picosecond laser pulses," Opt. Express 14, 4826-4834 (2006). [CrossRef] [PubMed]
  13. Y. Cheng, K. Sugioka, K. Midorikawa, M. Masuda, K. Toyoda, M. Kawachi, and K. Shihoyama,"Control of the cross-sectional shape of a hollow microchannel embedded in photostructurable glass by use of a femtosecond laser," Opt. Lett. 28, 55-57 (2003). [CrossRef] [PubMed]
  14. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerullo,"Femtosecond writing of active optical waveguides with astigmatically shaped beams," J. Opt. Soc. Am. B 20, 1559-1567 (2003). [CrossRef]
  15. C. Mauclair, A. Mermillod-Blondin, N. Huot, E. Audouard, and R. Stoian, "Ultrafast laser writing of homogeneous longitudinal waveguides in glasses using dynamic wavefront correction," Opt. Express 16, 5481-5492 (2008). [CrossRef] [PubMed]
  16. A. Mermillod-Blondin, C. Mauclair, A. Rosenfeld, J. Bonse, I. V. Hertel, E. Audouard, and R. Stoian, "Size correction in ultrafast laser processing of fused silica by temporal pulse shaping," Appl. Phys. Lett. 93, 021921-021924 (2008). [CrossRef]
  17. S. Li, G. Yu, C. Zheng, and Q. Tan, "Quasi-Dammann grating with proportional intensity array spots," Opt. Lett. 33, 2023-2025 (2008). [CrossRef] [PubMed]
  18. S. Hasegawa, Y. Hayasaki, and N. Nishida, "Holographic femtosecond laser processing with multiplexed phase Fresnel lenses," Opt. Lett. 31, 1705-1707 (2006). [CrossRef] [PubMed]
  19. J. W. Goodman, Introduction to Fourier Optics (2nd ed. McGraw-Hill, Singapore, 1996).
  20. C. Yu, J. Park, J. Kim, M. Jung, and S. Lee, "Design of Binary Diffraction Gratings of Liquid Crystals in a Linearly Graded Phase Model," Appl. Opt. 43, 1783-1788 (2004). [CrossRef] [PubMed]
  21. R. Graf, A. Fernandez, M. Dubov, H. J. Brueckner, B. N. Chichkov, and A. Apolonski, "Pearl-chain waveguides written at megahertz repetition rate," Appl. Phys. B 87, 21-27 (2007). [CrossRef]
  22. N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, "Programmable focal spot shaping of amplified femtosecond laser pulses," Opt. Lett. 30, 1479-1481 (2005). [CrossRef] [PubMed]
  23. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, "High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations," J. Appl. Phys. 98, 013517/1-5 (2005). [CrossRef]
  24. Z. Wang, K. Sugioka, Y. Hanada, and K. Midorikawa, "Optical waveguide fabrication and integration with a micro-mirror inside photosensitive glass by femtosecond laser direct writing," Appl. Phys. A 88, 699-704 (2007). [CrossRef]
  25. D. Christodoulides, N. Demetrios, F. Lederer, and Y. Silberberg, " Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003). [CrossRef]
  26. T. Pertsch, P. Dannberg, W. Elflein, A. Br¨auer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999). [CrossRef]
  27. A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tuennermann, and F. Lederer "Quasi-incoherent propagation in waveguide arrays," Appl. Phys. Lett. 90, 241113-241116 (2007). [CrossRef]
  28. A. Szameit, D. Bloemer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Express 13, 10552-10557 (2005). [CrossRef] [PubMed]
  29. A. Yariv, Optical Electronics (4th ed. Saunders College Publ., 1991).
  30. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides,". Opt. Express 15, 1579-1587 (2007), [CrossRef] [PubMed]
  31. A. W. Snyder, "Coupled mode theory for optical fibers," J. Opt. Soc. Am. 62, 1267-1277 (1972). [CrossRef]
  32. P. D. McIntyre, and A. W. Snyder, "Power transfer between nonparallel and tapered optical fibers," J. Opt. Soc. Am. 64, 286-288 (1974). [CrossRef]
  33. A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B 82, 507-512 (2006). [CrossRef]

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