Open Access
Add to BibSonomy
Abstract
We report single scan transverse writing of depressed cladding waveguides inside ZBLAN glass with the longitudinally oriented annular ring-shaped focal intensity distribution of the femtosecond laser. The entire region of depressed cladding at the cross section, where a negative change of refraction index is induced, can be modified simultaneously with the ring-shaped focal intensity profile. The fabricated waveguides exhibit good single guided mode.
© 2017 Optical Society of America
1. Introduction
Femtosecond laser provides a powerful and flexible tool for three dimensional (3D) direct writing of waveguides in transparent optical materials [1–10]. The high-order nonlinear process makes focused femtosecond laser pulses to induce a permanent refractive index change, which could be either positive or negative, in interacted media [11–19]. A refractive-index increase requires to irradiate the material directly at the focal volume in order to form a waveguide core, which sometimes brings detrimental effect upon the material [20–23]. To preserve the pristine properties of the core, stress-induced “dual-line” waveguides [24,25] and depressed cladding waveguides surrounded by a depressed cladding with a localized reduction in refractive index [26,27] are more attractive for active devices. Depressed cladding waveguides stand out for the features such as geometric flexibility of cross section, polarization independence and low propagation loss. They were fabricated with longitudinal writing scheme in which the sample was helically moved to realize a circular cladding with high-speed. However, the footprints of waveguides were severely limited by working distances of lenses [28–30]. Alternatively, transverse writing scheme is more flexible. Since the first report of transverse writing of depressed cladding waveguides by Okhrimchuk et al. in 2005 [31], numerous studies have been performed [26–30,32–34]. Lancaster et al. wrote the cladding composed of 24 cylinders formed from two partially overlapping rings of 12 cylinders [32]. Okhrimchuk et al. and Liu et al. produced dozens of parallel damage filaments to form a circular cladding (Fig. 1(a)) [33,34]. It is worth noting that this non-closed cladding can still confine light very well, but the writing of many tracks one by one is time-consuming. More recently, Liao et al. have effectively reduced the scan times to four using a slit-shaped beam (Fig. 1(b)) [35].
Fig. 1 Schematic diagrams for transverse writing of depressed cladding waveguides. (a) Circular cladding composed of dozens of parallel filaments inscribed line by line using an ellipsoidal focal spot. (b) Square cladding with four sides inscribed one by one using a slit-shaped focus. (c) Circular cladding formed in a single scan using the tilted longitudinal annular ring-shaped focal intensity distribution. Insets: phase masks used for focal field engineering. The sample is translated along x-axis.
Here we report the single scan femtosecond laser transverse inscription of depressed cladding waveguides, for the first time to our knowledge. We design special phase masks and load them onto a spatial light modulator (SLM) to generate 3D focal fields with an intensity profile of longitudinal annular ring. The engineered focal field allows for generation of an annular ring-shaped depressed cladding in ZBLAN glass with a single scan scheme as illustrated in Fig. 1(c), which is time-saving compared with the multiple-scan schemes in Figs. 1(a) and 1(b). The fabricated waveguides exhibit good single guided mode and low propagation loss.
2. Focal field engineering
The annular ring-shaped focal intensity distribution in the transverse plane can be readily realized using a vortex beam [36]. However, the generation of the longitudinal annular ring-shaped focal intensity distribution oriented along the laser propagation direction becomes much more challenging due to the lack of an analytical expression to describe this focal field and thus iterative methods have to be adopted to solve the problem. The widely used Gerchberg-Saxton algorithm for phase retrieval is generally only applicable for a unitary transform in which diffraction loss is negligible or the paraxial approximation is satisfied. In contrast, the Yang-Gu algorithm can be extended to a non-unitary transform system in which either the diffraction loss exists or the paraxial approximation is violated, but the iterative process in the Yang-Gu algorithm can sometimes result in local extremum [37–41]. In our experiment, the tight focusing of vectorial optical fields is achieved under the non-paraxial conditions with a high NA (1.25) objective lens, thus the system is non-unitary. Therefore, we apply the weighted Yang-Gu algorithm with better convergence [39, 40] for the design of phase masks according to targeted 3D intensity distribution in the focal region. For simplicity, we assume a linearly polarized monochromatic continuous wave incident field at 800 nm in the calculation. In order to make the intensity on the annular ring more uniform, we dynamically adjust the weighted coefficient α of the weighted Yang-Gu algorithm [39], i.e., the bigger the difference between the actual and the targeted intensity distribution, the larger the α. The computer-generated phase mask is loaded onto a pixelated phase-only SLM to successfully transform the ellipsoidal focal spot of the Gaussian beam into the longitudinal annular ring-shaped 3D focal intensity distribution according to our design.
3. Experimental
ZBLAN (ZrF4-BaF2-LaF3-AlF3-NaF) glass, a kind of heavy-metal fluoride glass, is well known for its high IR transparency. In addition, its refractive index change induced by kHz or MHz repetition rate femtosecond laser irradiation can reach a value between ~-3.9 × 10−3 to ~-1 × 10−3 in the athermal regime or between ~-1.5 × 10−3 to ~-1 × 10−3 in the thermal regime [32, 42], thereby making it an attractive host for depressed cladding waveguides fabrication. ZBLAN sample of a size of 10 × 5 × 2 mm3 is used in our experiment.
Figure 2 illustrates the experimental setup. Waveguides are inscribed by a Ti: Sapphire linearly polarized laser (Libra-HE, Coherent) producing 100 fs pulses at 800 nm, with a maximum pulse energy of 3.5 mJ at a repetition rate of 1 kHz. The incident pulse energy is controlled by rotating a half-wave plate in front of a Glan-Taylor polarizer. The laser beam is first expanded using a Galilean telescope composed of a concave lens (L1) with a focal length of 75 mm and a convex lens (L2) with a focal length of 150 mm and then impinged on a reflective phase-only SLM (X10468-02, Hamamatsu). To separate the phase-modulated beam from the zero-order diffracted beam imposed by the pixilation of SLM [43], we add a blazed grating on the phase mask and load it onto the SLM. After being focused by a lens (L3) with a focal length of 1200 mm, only the first-order diffracted beam is selected using an aperture. The engineered beam is then imaged onto the back aperture of the oil immersion objective lens (UPLAN, Olympus, NA = 1.25) by a lens (L4) with a focal length of 500 mm, and focused into the glass sample to inscribe waveguides. The size of the laser beam is reduced from 12 mm to 5 mm through the combination of L3 and L4 to match the back aperture of the objective lens. In order to reduce the spherical aberration impact on the focal volume shape, the non-fluorescent immersion oil (Olympus, refractive index n = 1.518) is used during the fabrication. The sample is translated along x-axis at a constant velocity (60 μm/s) by a computer-controlled XYZ stage (H101A, Prior) with a translation resolution of 1 μm. The laser polarization is also in the x-direction.
Fig. 2 Schematic illustration of the experimental setup. HW: half-wave plate. P: Glan-Taylor polarizer. L1, L2, L3, L4: lenses with different focal lengths described in the text. SLM: spatial light modulator. M1, M2, M3, M4: mirrors. A: aperture. T: tube lens. CCD: charge coupled device. Insets: the intensity distribution of focal fields with a shape of ellipsoidal spot and oblique longitudinal annular ring as a result of a uniform and a specially designed phase mask loaded onto a SLM, respectively. Sample is translated along x-axis.
For waveguide characterization, two fiber-coupled lasers with wavelengths of 785 nm and 1550 nm (SIFC, Thorlabs) are used for end-fire coupling into the waveguide. The near-field mode profile at the exit facet is imaged onto an InGaAs camera (XC-130, Xeva) with an objective lens of NA = 0.45.
To estimate the propagation loss of the waveguide, we capture the top-view image of the waveguide under a microscope when it is carrying the propagating laser beam. From the exponential decay of the scattering light along the waveguide, the loss in the waveguide is calculated. Insertion loss measurements are also conducted on the polished samples via direct butt-couple between the laser beam and the waveguides without using refractive-index matching liquid. The 1/e2 mode field diameter of the coupling fiber at 1550 nm is ~10.5 μm.
4. Results
Two types of engineered 3D focal fields have been designed. In the first case, the ideal uniform intensity profile in the focal region consists of four quadrantal arcs, separated from each other along the scanning direction (Fig. 3(a)) to reduce the crosstalk between different parts. The calculated phase mask is presented in Fig. 3(b). The simulated 3D isosurface image (the isosurface is given by the intensity at 30% of the peak value) and the 2D intensity distribution are shown in Fig. 3(c) and 3(d), respectively. In the second case, the targeted focal intensity profile is a continuous uniform annular ring (Fig. 3(e)). Figures 3(f) and 3(g) present the calculated phase mask and the corresponding 3D intensity isosurface.
Fig. 3 Side views of targeted focal profiles with the form of piecewise annular ring (a) and continuous annular ring (e). Phase masks are presented in (b) and (f). 3D isosurface images (the isosurface is given by the intensity at 30% of the peak value) are shown in (c) and (g). The corresponding simulated 2D intensity distribution images in the planes viewed in the directions of the yellow arrows are shown in (d) and (h). It’s worth noting that in (d) only the lower left arc should be visible, but we move other three arcs to the same view plane to facilitate observation of the entire intensity annular ring. The sample is translated along x-axis. Scale bar: 5 μm.
When calculating the phase mask, the smaller the sampling interval in the focal region, the longer the computing time. In our calculation of the 25.6x25.6x102.4 μm3 volume, the sampling interval in the x–y plane is 200 nm (i.e. 128x128 sampling points) and the z-slice number is 512. It takes about one week until the algorithm converges using serial computation with a workstation (HP Z820). However, the preliminary results using parallel computation indicate that the computing time can be reduced to one day.
The simulated 2D intensity distribution, shown in Fig. 3(h), is not as uniform and continuous as the ideal target (Fig. 3(e)). This non-uniformity might be caused by the transverse elongation of the tight focus along the incident linear polarization direction and the longitudinal elongation along z-axis, which can be eliminated or mitigated by using vectorial beams with circularly symmetric polarization distribution [44] and by adding the amplitude modulation [45]. The inner and external diameters for the annular rings in both cases are 16 μm and 19 μm, respectively. Since the pixel pitch of the SLM is 20 μm, the maximum spatial frequency provided by the SLM is 50 mm−1 and the corresponding focal field size is about 170 μm, which is much larger than the computed region size of 25.6 μm. Therefore, the limited spatial frequency imposed by the pixilation of SLM can be ignored. In order to prevent perturbation of the beam passing through the previously modified areas, depressed cladding waveguides were written from the bottom to the top in the multiple-scan schemes [32–35]. Similarly, the intensity annular rings here are tilted with respect to the x-axis or the scanning direction. By comparison of experimental results with five tilt angles (i.e. 0°, 10°, 20°, 30°, 40°), we find that the 10° tilt is enough to avoid this perturbation.
The fabricated straight waveguides with a length of 10 mm written 100 μm beneath the surface of ZBLAN sample using the two phase masks are presented. Figures 4(a) and 4(b) show the top view and the cross-section of a waveguide written with the piecewise annular ring-shaped focal field at the pulse energy of 8 μJ, respectively. As the actual volume of the annular ring-shaped focal field is larger than that of an incident Gaussian beam [42], higher pulse energy is required. Although some undesired tracks appear outside the cladding, the guided region is well enclosed as shown within the yellow dotted circle (Fig. 4(b)). These outside tracks are induced by extra peripheral intensity distribution surrounding the annular ring, which can be reduced using more slices along z-axis in the calculation at the cost of the computing time. The 2D intensity distribution of the guided mode presents single-mode at 785 nm (Fig. 4(c)) and exhibits a symmetrical Gaussian shape in the orthogonal directions at 1550 nm as shown in Fig. 4(d). Moreover, experimental results also indicate that there is little difference between TE and TM guided modes at 1550 nm. The propagation loss at 785 nm is measured to be 1.3 dB/cm, which can be further reduced by designing an intensity annular ring with a wider cladding at higher pulse energy. It was demonstrated in [32] that the confinement loss greatly decreased from 2 dB/cm to 0.018 dB/cm when the cladding width increased from 10 μm to 23 μm. The coupling loss at 785 nm is estimated to be 0.68 dB/facet calculated using the mode field profiles of both the single-mode fiber and the fabricated waveguide. The Fresnel loss at 785 nm is estimated to be 0.22 dB/facet. The insertion loss at 1550 nm for the waveguide with a length of 9 mm is measured to be 2.3 dB. The relatively high value might be caused by a large coupling loss (1.1 dB/facet at 1550 nm) owing to its large mode-field size as shown in Fig. 4(d) compared with that of the coupling fiber. The Fresnel loss at 1550 nm is estimated to be 0.2 dB/facet. The propagation loss at 1550 nm is estimated to be 1 dB/cm.
Fig. 4 Optical micrographs and guided modes of a waveguide written through the piecewise annular ring-shaped focal field. (a) Top view of the waveguide. (b) Cross-section of the depressed cladding. The guided region is indicated within the yellow dotted circle. (c), (d) 2D intensity distribution images of the guided modes at 785 nm and 1550 nm, respectively. The mode profile in horizontal and vertical patterns are presented in (d). The resulting 1/e2 mode field diameters are ∆h = 17.63 μm and ∆v = 17.25 μm, respectively. (e) Top-view image of a 10-mm-long waveguide showing the decay of scattered light along the waveguide. The He-Ne laser is incident on the waveguide from left as indicated by the red arrow.
The cross-section of a waveguide written with the continuous annular ring-shaped focal field is shown in Fig. 5(a) with the guided region surrounded by an almost closed circular cladding indicated by the yellow dotted circle. The intensity on the ring itself is non-uniform as shown in Fig. 3(h) and this non-uniformity is aggravated by the modification/damage of the glass via the Kerr self-focusing and other nonlinear effects. The deformation of the focal intensity profile is more evident at the part with higher pulse energies [43]. Therefore, the resulting depressed index structures surrounding the guiding region are discontinuous. Since the gaps are small, the waveguide can still confine light well [33, 34]. The pulse energy is set to 5 μJ in the fabrication because the focal volume in this case is smaller than that of the piecewise ring, therefore the fabrication pulse energy is lower. The propagation loss at 785 nm is measured to be 2.5 dB/cm. The coupling and Fresnel losses at 785 nm are estimated to be 1.52 dB/facet and 0.22 dB/facet, respectively. In the case of the linear polarized incident beam and the phase only modulation, the intensity distribution of the continuous annular ring is less uniform than that of the piecewise annular ring, so larger gaps appear on the depressed cladding as shown in Fig. 5(a) compared with Fig. 4(b), resulting in higher propagation loss. The insertion loss at 1550 nm for the waveguide with a length of 9 mm is measured to be 3.2 dB. The coupling and Fresnel losses at 1550 nm are estimated to be 1.43 dB/facet and 0.2 dB/facet, respectively. The propagation loss at 1550 nm is estimated to be 1.57 dB/cm. The 2D intensity distribution images presented in Figs. 5(b) and 5(c) show that the guided modes are single-modes at both 785 nm and 1550 nm.
Fig. 5 Optical micrograph and guided modes of a waveguide written through the continuous annular ring-shaped focal field. (a) Cross-section of the depressed cladding. The guided region is indicated within the yellow dotted circle. (b), (c) 2D intensity distribution images of the guided modes at 785 nm and 1550 nm, respectively.
5. Conclusion
In conclusion, we have demonstrated the single-scan fabrication of depressed cladding waveguides using two types of specially designed longitudinal annular ring-shaped focal fields. These waveguides exhibit single guided mode with almost symmetrical Gaussian shape and relatively low propagation loss. The waveguide quality can be further promoted through improving the algorithm for phase mask design, using XYZ-stage with higher resolution as well as through optimizing the fabrication parameters such as incident laser polarization, pulse energy, pulse duration and scan speed. This technique has potential for the fabrication of complex 3D structures composed of many depressed cladding waveguides inside transparent materials.
Funding
National Basic Research Program of China under Grant No. 2013CB921904; National Natural Science Foundation of China (NSFC) under Grant Nos. 11474010, 61590933 and 11527901; Opened Fund of the State Key Laboratory on Integrated Optoelectronics No. IOSKL2015KF36.
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(21), 1729–1731 (1996). [CrossRef] [PubMed]
2. Y. L. Zhang, Q. D. Chen, H. Xia, and H. B. Sun, “Designable 3D nanofabrication by femtosecond laser direct writing,” Nano Today 5(5), 435–448 (2010). [CrossRef]
3. S. Juodkazis, V. Mizeikis, and H. Misawa, “Three-dimensional microfabrication of materials by femtosecond lasers for photonics applications,” J. Appl. Phys. 106(5), 051101 (2009). [CrossRef]
4. S. Maruo and J. T. Fourkas, “Recent progress in multiphoton microfabrication,” Laser Photonics Rev. 2(1–2), 100–111 (2008). [CrossRef]
5. L. Li, R. R. Gattass, E. Gershgoren, H. Hwang, and J. T. Fourkas, “Achieving λ/20 resolution by one-color initiation and deactivation of polymerization,” Science 324(5929), 910–913 (2009). [CrossRef] [PubMed]
6. Y. Liao, J. Song, E. Li, Y. Luo, Y. Shen, D. Chen, Y. Cheng, Z. Xu, K. Sugioka, and K. Midorikawa, “Rapid prototyping of three-dimensional microfluidic mixers in glass by femtosecond laser direct writing,” Lab Chip 12(4), 746–749 (2012). [CrossRef] [PubMed]
7. S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A Mater. Sci. Process. 77(1), 109–111 (2003). [CrossRef]
8. A. M. Streltsov and N. F. Borrelli, “Fabrication and analysis of a directional coupler written in glass by nanojoule femtosecond laser pulses,” Opt. Lett. 26(1), 42–43 (2001). [CrossRef] [PubMed]
9. A. M. Kowalevicz, V. Sharma, E. P. Ippen, J. G. Fujimoto, and K. Minoshima, “Three-dimensional photonic devices fabricated in glass by use of a femtosecond laser oscillator,” Opt. Lett. 30(9), 1060–1062 (2005). [CrossRef] [PubMed]
10. A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G. Vallone, and P. Mataloni, “Integrated photonic quantum gates for polarization qubits,” Nat. Commun. 2, 566 (2011). [CrossRef] [PubMed]
11. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008). [CrossRef]
12. K. Itoh, W. Watanabe, S. Nolte, and C. B. Schaffer, “Ultrafast processes for bulk modification of transparent materials,” MRS Bull. 31(8), 620–625 (2006). [CrossRef]
13. M. Ams, G. D. Marshall, P. Dekker, J. A. Piper, and M. J. Withford, “Ultrafast laser written active devices,” Laser Photonics Rev. 3(6), 535–544 (2009). [CrossRef]
14. W. Watanabe, Y. Li, and K. Itoh, “Ultrafast laser micro-processing of transparent material,” Opt. Laser Technol. 78, 52–61 (2016). [CrossRef]
15. J. W. Chan, T. R. Huser, S. H. Risbud, and D. M. Krol, “Modification of the fused silica glass network associated with waveguide fabrication using femtosecond laser pulses,” Appl. Phys., A Mater. Sci. Process. 76(3), 367–372 (2003). [CrossRef]
16. M. Ams, G. D. Marshall, P. Dekker, M. Dubov, V. K. Mezentsev, I. Bennion, and M. J. Withford, “Investigation of ultrafast laser-photonic material interactions: challenges for directly written glass photonics,” IEEE J. Sel. Top. Quantum Electron. 14(5), 1370–1381 (2008). [CrossRef]
17. A. Ferrer, V. Diez-Blanco, A. Ruiz, J. Siegel, and J. Solis, “Deep subsurface optical waveguides produced by direct writing with femtosecond laser pulses in fused silica and phosphate glass,” Appl. Surf. Sci. 254(4), 1121–1125 (2007). [CrossRef]
18. A. Zoubir, C. Lopez, M. Richardson, and K. Richardson, “Femtosecond laser fabrication of tubular waveguides in poly(methyl methacrylate),” Opt. Lett. 29(16), 1840–1842 (2004). [CrossRef] [PubMed]
19. S. Gross, M. Dubov, and M. J. Withford, “On the use of the Type I and II scheme for classifying ultrafast laser direct-write photonics,” Opt. Express 23(6), 7767–7770 (2015). [CrossRef] [PubMed]
20. C. Florea and K. A. Winick, “Fabrication and characterization of photonic devices directly written in glass using femtosecond laser pulses,” J. Lightwave Technol. 21(1), 246–253 (2003). [CrossRef]
21. L. Shah, A. Arai, S. Eaton, and P. Herman, “Waveguide writing in fused silica with a femtosecond fiber laser at 522 nm and 1 MHz repetition rate,” Opt. Express 13(6), 1999–2006 (2005). [CrossRef] [PubMed]
22. K. Yamada, W. Watanabe, T. Toma, K. Itoh, and J. Nishii, “In situ observation of photoinduced refractive-index changes in filaments formed in glasses by femtosecond laser pulses,” Opt. Lett. 26(1), 19–21 (2001). [CrossRef] [PubMed]
23. 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(7), 1559–1567 (2003). [CrossRef]
24. J. Burghoff, S. Nolte, and A. Tünnermann, “Origins of waveguiding in femtosecond laser-structured LiNbO3,” Appl. Phys., A Mater. Sci. Process. 89(1), 127–132 (2007). [CrossRef]
25. G. A. Torchia, A. Rodenas, A. Benayas, E. Cantelar, L. Roso, and D. Jaque, “Highly efficient laser action in femtosecond-written Nd: yttrium aluminum garnet ceramic waveguides,” Appl. Phys. Lett. 92(11), 111103 (2008). [CrossRef]
26. F. Chen and J. R. V. de Aldana, “Optical waveguides in crystalline dielectric materials produced by femtosecond- laser micromachining,” Laser Photonics Rev. 8(2), 251–275 (2014). [CrossRef]
27. D. Choudhury, J. R. Macdonald, and A. K. Kar, “Ultrafast laser inscription: perspectives on future integrated applications,” Laser Photonics Rev. 8(6), 827–846 (2014). [CrossRef]
28. D. Beckmann, D. Schnitzler, D. Schaefer, J. Gottmann, and I. Kelbassa, “Beam shaping of laser diode radiation by waveguides with arbitrary cladding geometry written with fs-laser radiation,” Opt. Express 19(25), 25418–25425 (2011). [CrossRef] [PubMed]
29. G. Salamu, F. Jipa, M. Zamfirescu, and N. Pavel, “Watt-level output power operation from diode-laser pumped circular buried depressed-cladding waveguides inscribed in Nd: YAG by direct femtosecond-laser writing,” IEEE Photonics J. 8(1), 9 (2016). [CrossRef]
30. J. W. Chan, T. R. Huser, S. H. Risbud, J. S. Hayden, and D. M. Krol, “Waveguide fabrication in phosphate glasses using femtosecond laser pulses,” Appl. Phys. Lett. 82(15), 2371–2373 (2003). [CrossRef]
31. A. G. Okhrimchuk, A. V. Shestakov, I. Khrushchev, and J. Mitchell, “Depressed cladding, buried waveguide laser formed in a YAG:Nd3+ crystal by femtosecond laser writing,” Opt. Lett. 30(17), 2248–2250 (2005). [CrossRef] [PubMed]
32. D. G. Lancaster, S. Gross, H. Ebendorff-Heidepriem, K. Kuan, T. M. Monro, M. Ams, A. Fuerbach, and M. J. Withford, “Fifty percent internal slope efficiency femtosecond direct-written Tm3+:ZBLAN waveguide laser,” Opt. Lett. 36(9), 1587–1589 (2011). [CrossRef] [PubMed]
33. A. Okhrimchuk, V. Mezentsev, A. Shestakov, and I. Bennion, “Low loss depressed cladding waveguide inscribed in YAG:Nd single crystal by femtosecond laser pulses,” Opt. Express 20(4), 3832–3843 (2012). [CrossRef] [PubMed]
34. H. Liu, Y. Jia, J. R. Vázquez de Aldana, D. Jaque, and F. Chen, “Femtosecond laser inscribed cladding waveguides in Nd:YAG ceramics: fabrication, fluorescence imaging and laser performance,” Opt. Express 20(17), 18620–18629 (2012). [CrossRef] [PubMed]
35. Y. Liao, J. Qi, P. Wang, W. Chu, Z. Wang, L. Qiao, and Y. Cheng, “Transverse writing of three-dimensional tubular optical waveguides in glass with a slit-shaped femtosecond laser beam,” Sci. Rep. 6(28790), 28790 (2016). [CrossRef] [PubMed]
36. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]
37. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33(2), 209–218 (1994). [CrossRef] [PubMed]
38. B. Y. Gu and G. Z. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1(6), 517–522 (1981).
39. S. H. Yan, “Research on the weighted Yang-Gu algorithm,” Acta Photonica Sin. 36(3), 530 (2007).
40. X. Yu, K. Q. Chen, and Y. Zhang, “Optimization design of diffractive phase elements for beam shaping,” Appl. Opt. 50(31), 5938–5943 (2011). [CrossRef] [PubMed]
41. C. Guo, C. Wei, J. B. Tan, K. Chen, S. T. Liu, Q. Wu, and Z. J. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2017). [CrossRef]
42. S. Gross, M. Ams, G. Palmer, C. T. Miese, R. J. Williams, G. D. Marshall, A. Fuerbach, D. G. Lancaster, H. Ebendorff-Heidepriem, and M. J. Withford, “Ultrafast laser inscription in soft glasses: a comparative study of athermal and thermal processing regimes for guided wave optics,” Int. J. Appl. Glass Sci. 3(4), 332–348 (2012). [CrossRef]
43. G. Milewski, D. Engström, and J. Bengtsson, “Diffractive optical elements designed for highly precise far-field generation in the presence of artifacts typical for pixelated spatial light modulators,” Appl. Opt. 46(1), 95–105 (2007). [CrossRef] [PubMed]
44. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006). [CrossRef] [PubMed]
45. H. Lin, B. Jia, and M. Gu, “Generation of an axially super-resolved quasi-spherical focal spot using an amplitude-modulated radially polarized beam,” Opt. Lett. 36(13), 2471–2473 (2011). [CrossRef] [PubMed]
