Abstract

We present a rotated polarization directional coupler (RPDC) on a photonic chip. We demonstrate a double-track approach to modify the distribution of the refractive index between adjacent tracks and form a single waveguide with an arbitrary birefringent optical axis. We construct a RPDC with the two axis-rotated waveguides coupled in a strong regime. The obtained extinction ratios on average are about 16 dB and 20 dB for the corresponding orthogonal polarizations. We perform reconstruction of the Stokes vector to test the projection performance of our RPDC, and observe the average fidelities up to 98.1% and 96.0% for the perfectly initialized states in 0° and 45° RPDCs, respectively.

© 2018 Optical Society of America

Polarization manipulation of photons is a crucial approach in optical information processing, quantum communication, and quantum computing. Especially in free space, there are many mature bulk optical devices for the preparation, control, and projection of polarized states in very high precision. However, the scalability, robustness, cost, and insertion loss have activated developing integrated polarization devices for large-scale applications [16].

Femtosecond laser direct writing [715], taking advantage of precise control of highly localized material modifications through nonlinear absorption processes, has been an emerging technique for controlling birefringence [1621] and constructing on-chip polarization devices. Moreover, its single-step, mask-free, phase-stable, and cost-effective features have promised a powerful polarization-based technique for integrated photonics, such as polarization-dependent light attenuators [22], polarization-insensitive directional couplers [2326] (analogous to bulk optical beam splitters), polarization directional couplers [2729], birefringent retarders, wave plates [3032], and integrated source [33]. However, the orientation of these polarization devices is often fixed as horizontal or vertical with the standard configuration of femtosecond laser direct writing.

In this Letter, we present an on-chip rotated polarization directional coupler (RPDC) that can be constructed at arbitrary orientation by using a double-track approach. We adjust the relative radial and azimuthal positions of the adjacent tracks to artificially set the birefringent optical axis of a single-mode waveguide with a high transmittance. With the two axis-rotated waveguides, we can construct a RPDC along the same rotated axis, with which we are able to make the projection measurement on any pair on an orthogonal basis, rather than either horizontal or vertical.

As shown in Fig. 1, the waveguide (gray) is composed of two adjacent parallel tracks (dark red) with specific offsets (radial and azimuthal). Different from the method of defect tracks [31] written nearby, our track pairs are laid out quite close and a bit overlapped. Therefore, the two tracks here are able to transmit light simultaneously without energy dissipation into the defect track. An optimal radial separation can be found where the track pairs are able to guide light, behaving as a single-mode waveguide [34]. Moreover, a precise change of relative position in the track pairs may induce the anisotropic and inhomogeneous distribution of the refractive index, generating an artificial rotation in the birefringent optical axis.

 

Fig. 1. Illustration of the fabricating process for double-track waveguides and sketch of the coupling cross section of RPDCs. Each waveguide (gray) is composed of two adjacent tracks (dark red) with geometrically radial and azimuthal (θ) offsets. The two waveguide arms are located parallel to the fast axis (blue lines with arrows at both ends) or orthogonally to the slow axis (red lines with arrows at both ends) to enable evanescent couplings.

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We employ a regenerative amplifier based on Yb:KGW lasing medium, with a pulse duration of 290 fs, a repetition rate of 1 MHz, and a central wavelength of 513 nm, to conduct the fabrication. A 100× (0.70 NA) microscope objective is adopted to tightly focus the subsequent linearly polarized (along the writing direction) laser pulses into a depth of 170 μm underneath a commercially available fused silica substrate surface. We move the substrate and laser focal spot with three air-bearing translation stages, and the formed trajectory of the focal spot is the track. The writing speed in our experiment is about 1.268 mm/s with a resolution of 0.1 μm. We fix the writing laser at optimized pulse energy of 180 nJ and shape the beam by using a cylindrical lens [35] with a focal length of 70 cm.

The single-track is able to guide a single mode at 780 nm as well as the double-track waveguide. However, its birefringence is around the order of 105106, leaving the adjacent track to tenfold stress the birefringence [36]. Also, the single-track waveguide modes are generally larger compared to double-track waveguides, due to decreased index change at lower fluence [37].

Figure 2(a) shows the induced rotation of the birefringent optical axis (α) dependent on the azimuthal orientation angles (θ) of the track pairs from 0° to 180° per 5° at a fixed optimized radial separation of 2 μm. We deduce the orientation of the birefringent optical axis of the waveguide by placing two crossed polarizers (Pol and Pol) before and after the chip [Fig. 2(e)] [31,32,38]. The minimum transmission reveals the orientation of the birefringent optical axis, and the maximum is always 45° away from the minimum. Although not linearly proportional, the fast and slow axes are confirmed as orthogonal and tunable in the plane marked in Fig. 2(b) by controlling the orientation angles (θ), behaving as a stable and controllable transmission of α-polarized or the corresponding orthogonal polarized light through the chip. The end-view microscope images (corresponding to an average area of 5.6μm×5.8μm) of waveguide cross sections composed of the track pairs can be found in Fig. 2(c), for the real pictures of changing applied writing parameters of control orientation (θ) and radial separation. Moreover, we find that the near-field profiles of the guided modes [Fig. 2(d)] (corresponding to an average area of 6.8μm×6.9μm width) at 780 nm wavelength are preserved well with a comparably small influence induced by differential θ. We also observe a high average total transmission rate up to 60% through 19.64 mm waveguide propagation with a 60× objective in free space, and over 50% transmission coupled into a single-mode fiber.

 

Fig. 2. (a) Experimental results and best-fit model of the artificial birefringence optical axis reorientation α as a function of the azimuthal offset control orientation θ. (b) Schematic of rotation plane of the two rotated orthogonal optical axes. (c) Morphology of the waveguide cross sections presented by microscope end-view of track pairs fabricated with 100× objective, 2 μm radial offset in fused silica substrate. (d) Corresponding near-field output modes of waveguides with different azimuthal offsets. (e) Setup for birefringent optical axis confirmation. POL, polarizer; QWP, quarter-wave plate.

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In order to achieve a polarization directional coupler, one needs to utilize the difference in coupling coefficients between the input light polarized parallel and perpendicular to the birefringent optical axis. It has been proved that the coupling constants for H- and V-polarized light vary from each other, caused not only by projection magnitude, but also due to anisotropy under different spatial dispositions of waveguide arms with fixed horizontal and vertical optical axes [39]. Also, a link among the geometry of the waveguides, polarization of light, and coupling constants has been revealed [40]. The wanted difference of evanescent coupling may become weak and complex if the two waveguides are kept in the same plane while their birefringent optical axes are rotated [Fig. 3(a)]. This enlightens us to form a 3D tilted coupler with feasible experimental modified configuration to produce a compact and balanced RPDC with rotated optical axes [Fig. 3(b)]. Once the birefringent optical axis is confirmed, the evanescent light coupling of the two waveguides should still be subjected to coupling mode theory [41], in which the field amplitude evolutions a1(z) and a1(z) in two waveguide arms follow the equation

{da1dz=iβ1(z)a1+K12a2,da2dz=K21a1iβ2(z)a2.
In the ideal case, we take the wavenumber dependent on the propagation coordinate β1=β2, and the coupling coefficient K12=K21, considering the modifications of the mode’s effective refractive indices are identical. It is not difficult to retrieve the shortest coupling length reaching a polarization directional coupler remains π/2(KSKF), where KS and KF are the two coupling coefficients for the two orthogonal polarization states, supposing KS>KF.

 

Fig. 3. (a) Configuration of two waveguides located in a plane of same depth. (b) Configuration of two waveguides located with a rotation of α from the plane. (c) Polarization analysis of the 45° rotated parallel coupling region with different linearly input states (H, V, D, A Pol.). (d) Normalized transmission power of anti-diagonal (135°, A Pol.) and diagonal (45°, D Pol.) polarized light through a 45° RPDC. (e) Extinction performance of the 45° RPDC.

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The coupling ratio (field amplitudes) with the configuration shown in Fig. 3(b) can be calculated as

{T=sin(KZ),R=1T2.
To validate this theory, we fabricate the second double-track waveguide arm 7 μm beside (parallel to) the fast axis to enable the evanescent light coupling shown in Fig. 3(b). In order to exclude potential additional birefringent stress that the second waveguide arm may cause to rotate the optical axis, we made polarization analysis of the 45° rotated birefringent optical axis of the parallel straight coupling region by two crossed polarizers, as is mentioned above. As shown in Fig. 3(c), the linearly input H-polarized light is measured as V-polarized output light, etc. The input D/A-polarized light traveling along the optical axis retains visibility as high as 98%–99%. Transmission power of the 45° RPDC is characterized by different coupling lengths, as shown in Fig. 3(d). We calculate the ratio with beam diameter definition of full-width, half max. From our experiment, the coupling lengths reaching the 0° and 45° RPDCs are equal, which well verifies the theory of our RPDC. The two waveguide arms are split adiabatically through an “S curve,” as shown in Fig. 4(a). It should be noticed that evanescent coupling may also occur during the bending region, and a phase term should be added in the argument of the sine function. Therefore, the parallel coupling length reaching a RPDC could be shortened from 28.5 mm to 23 mm. Experimentally, we observe average extinction ratios up to 16 dB and 20 dB [Fig. 3(e)] for the diagonal (D) and anti-diagonal (A) polarizations, respectively. Here we define extinction ratio as
{ERT=|10logTFTS|(dB),ERR=|10logRFRS|(dB),
where pi=1,2 is the output power of different waveguide direction coupler output arms, respectively.

 

Fig. 4. (a) Schematic of the experimental setup for reconstruction of the Stokes vector. POL, polarizer; HWP, half-wave plate; QWP, quarter-wave plate. (b) Measured fidelities of different initialized states with 0° and 45° RPDCs. (c) Reconstructed density matrix obtained with 0° RPDC. (d) Reconstructed density matrix obtained with 45° RPDC. Blue parts represent real components and gray parts imaginary components.

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To further test the quality of the fabricated RPDC, we perform reconstruction of the Stokes vector by using a simple linear tomography [42,43] technique, shown in Fig. 4(a). In our experiment, we initialize high-purity polarization states with a high-extinction polarizer and wave plates at the state preparation part. In the state measurement part, we set the quarter-wave plate (QWP) and half-wave plate (HWP) to map the states to horizontal/vertical bases. Unlike the standard bulk optical implementation, the projection measurements are carried out by a RPDC rather than a polarization beam splitter. The additional HWP rotates the frame to our RPDC. In this way, we follow the standard procedure to perform state tomography while our RPDC acts as a polarization beam splitter to make projection measurements. The density matrix for the polarization degrees of light can be related to the Stokes parameters by the formula

ρ^=12i=03SiS0σ^i,
where σ^i=0 is the identity operator, and σ^i=1,2,3 are the Pauli operators:
{S0=P|0+P|1,S1=P12(|0+|1)P12(|0|1),S2=P12(|0+i|1)P12(|0i|1),S3=P|0P|1,
where P|ϕ represents the probability to measure the state ϕ.

Ideally, the fidelities measured with a perfect RPDC should be 100%. Here, we observe the average fidelities up to 98.1% and 96.0% in 0° and 45° RPDCs, respectively, where we rigorously perform a maximum likelihood estimation to keep the density matrix physical. The detailed reconstructed density matrices are shown in Figs. 4(c) and 4(d) with error bars omitted, as they are too small to be visible. The projection performance of RPDCs with not much difference identifies the ability to make general polarization projections on a photonic chip.

In conclusion, a new double-track approach is presented here to realize waveguides with rotated birefringent optical axes and to construct an on-chip RPDC with the two axis-rotated waveguides coupled in a strong regime. We test the quality of the on-chip RPDC by performing reconstruction of the Stokes vector with perfectly initialized polarization states. The features of straightforward fabrication, straightforward polarization projection, and high transmittance make this on-chip RPDC a good candidate for large-scale polarization-based optical communication and quantum information processing.

Funding

National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11374211, 11690033, 61734005); Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405, 17JC1400403); Shanghai Municipal Education Commission (SMEC) (16SG09, 2017-01-07-00-02-E00049).

Acknowledgment

The authors thank Roberto Osellame and Jian-Wei Pan for helpful discussions. X.-M. J. acknowledges support from the National Young 1000 Talents Plan.

REFERENCES

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References

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  1. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, Science 320, 646 (2008).
    [Crossref]
  2. S. Tanzilli, A. Martin, F. Kaiser, M. P. De Micheli, O. Alibart, and D. B. Ostrowsky, Laser Photon. Rev. 6, 115 (2012).
    [Crossref]
  3. J. O’Brien, B. Patton, M. Sasaki, and J. Vučković, New J. Phys. 15, 035016 (2013).
    [Crossref]
  4. B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, Nat. Photonics 8, 770 (2014).
    [Crossref]
  5. P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, Nat. Commun. 8, 13984 (2017).
    [Crossref]
  6. F. Flamini, N. Spagnolo, and F. Sciarrino, arXiv:1803.02790 (2018).
  7. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, Opt. Lett. 21, 1729 (1996).
    [Crossref]
  8. K. Minoshima, A. M. Kowalevicz, E. P. Ippen, and J. G. Fujimoto, Opt. Express 10, 645 (2002).
    [Crossref]
  9. A. M. Kowalevicz, V. Sharma, E. P. Ippen, J. G. Fujimoto, and K. Minoshima, Opt. Lett. 30, 1060 (2005).
    [Crossref]
  10. S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Y. Arai, Opt. Express 13, 4708 (2005).
    [Crossref]
  11. R. Osellame, N. Chiodo, V. Maselli, A. Yin, M. Zavelani-Rossi, G. Cerullo, P. Laporta, L. Aiello, S. D. Nicola, P. Ferraro, A. Finizio, and G. Pierattini, Opt. Express 13, 612 (2005).
    [Crossref]
  12. S. M. Eaton, H. Zhang, M. L. Ng, J. Li, W.-J. Chen, S. Ho, and P. R. Herman, Opt. Express 16, 9443 (2008).
    [Crossref]
  13. S. M. Eaton, M. L. Ng, R. Osellame, and P. R. Herman, J. Non-Cryst. Solids 357, 2387 (2011).
    [Crossref]
  14. R. Osellame, G. Cerullo, and R. Ramponi, Femtosecond Laser Micromachining: Photonic and Microfluidic Devices in Transparent Materials (Springer, 2012), Vol. 123.
  15. A. Arriola, S. Gross, N. Jovanovic, N. Charles, P. G. Tuthill, S. M. Olaizola, A. Fuerbach, and M. J. Withford, Opt. Express 21, 2978 (2013).
    [Crossref]
  16. F. Kapron, N. Borrelli, and D. Keck, IEEE J. Quantum Electron. 8, 222 (1972).
    [Crossref]
  17. E. Bricchi, B. G. Klappauf, and P. G. Kazansky, Opt. Lett. 29, 119 (2004).
    [Crossref]
  18. V. R. Bhardwaj, P. B. Corkum, D. M. Rayner, C. Hnatovsky, E. Simova, and R. S. Taylor, Opt. Lett. 29, 1312 (2004).
    [Crossref]
  19. L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, Opt. Express 20, 24103 (2012).
    [Crossref]
  20. F. Chen and J. R. V. Aldana, Laser Photon. Rev. 8, 251 (2014).
    [Crossref]
  21. B. McMillen and Y. Bellouard, Opt. Express 23, 86 (2015).
    [Crossref]
  22. F. Zhang, Y. Yu, C. Cheng, Y. Dai, and J. Qiu, Opt. Lett. 38, 2212 (2013).
    [Crossref]
  23. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, Phys. Rev. Lett. 108, 010502 (2012).
    [Crossref]
  24. I. Pitsios, F. Samara, G. Corrielli, A. Crespi, and R. Osellame, Sci. Rep. 7, 11342 (2017).
    [Crossref]
  25. G. Corrielli, S. Atzeni, S. Piacentini, I. Pitsios, A. Crespi, and R. Osellame, Opt. Express 26, 15101 (2018).
    [Crossref]
  26. C.-Y. Wang, J. Gao, Z.-Q. Jiao, L.-F. Qiao, R.-J. Ren, Z. Feng, Y. Chen, Z.-Q. Yan, Y. Wang, H. Tang, and X.-M. Jin, arXiv:1808.08586 (2018).
  27. A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G. Vallone, and P. Mataloni, Nat. Commun. 2, 566 (2011).
    [Crossref]
  28. L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, Opt. Express 19, 11992 (2011).
    [Crossref]
  29. I. V. Dyakonov, M. Y. Saygin, I. V. Kondratyev, A. A. Kalinkin, S. S. Straupe, and S. P. Kulik, Opt. Lett. 42, 4231 (2017).
    [Crossref]
  30. L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, Opt. Express 19, 18294 (2011).
    [Crossref]
  31. R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, Sci. Rep. 4, 4118 (2014).
    [Crossref]
  32. G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, Nat. Commun. 5, 4249 (2014).
    [Crossref]
  33. S. Atzeni, A. S. Rab, G. Corrielli, E. Polino, M. Valeri, P. Mataloni, N. Spagnolo, A. Crespi, F. Sciarrino, and R. Osellame, Optica 5, 311 (2018).
    [Crossref]
  34. N. Jovanovic, S. Gross, C. Miese, A. Fuerbach, J. Lawrence, and M. Withford, in Advanced Photonics & Renewable Energy (Optical Society of America, 2010), p. JThA28.
  35. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. D. Silvestri, and G. Cerullo, J. Opt. Soc. Am. B 20, 1559 (2003).
    [Crossref]
  36. L. A. Fernandes, J. R. Grenier, P. V. S. Marques, J. S. Aitchison, and P. R. Herman, J. Lightwave Technol. 31, 3563 (2013).
    [Crossref]
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G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, Nat. Commun. 5, 4249 (2014).
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P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, Nat. Commun. 8, 13984 (2017).
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P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, Nat. Commun. 8, 13984 (2017).
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A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G. Vallone, and P. Mataloni, Nat. Commun. 2, 566 (2011).
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C.-Y. Wang, J. Gao, Z.-Q. Jiao, L.-F. Qiao, R.-J. Ren, Z. Feng, Y. Chen, Z.-Q. Yan, Y. Wang, H. Tang, and X.-M. Jin, arXiv:1808.08586 (2018).

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C.-Y. Wang, J. Gao, Z.-Q. Jiao, L.-F. Qiao, R.-J. Ren, Z. Feng, Y. Chen, Z.-Q. Yan, Y. Wang, H. Tang, and X.-M. Jin, arXiv:1808.08586 (2018).

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P. Yang, G. R. Burns, J. Guo, T. S. Luk, and G. A. Vawter, J. Appl. Phys. 95, 5280 (2004).
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F. Chen and J. R. V. Aldana, Laser Photon. Rev. 8, 251 (2014).
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Nat. Commun. (3)

P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, Nat. Commun. 8, 13984 (2017).
[Crossref]

A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G. Vallone, and P. Mataloni, Nat. Commun. 2, 566 (2011).
[Crossref]

G. Corrielli, A. Crespi, R. Geremia, R. Ramponi, L. Sansoni, A. Santinelli, P. Mataloni, F. Sciarrino, and R. Osellame, Nat. Commun. 5, 4249 (2014).
[Crossref]

Nat. Photonics (1)

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, Nat. Photonics 8, 770 (2014).
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New J. Phys. (1)

J. O’Brien, B. Patton, M. Sasaki, and J. Vučković, New J. Phys. 15, 035016 (2013).
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Opt. Express (12)

Opt. Lett. (6)

Optica (1)

Phys. Rev. A (2)

S. Rojas-Rojas, L. Morales-Inostroza, U. Naether, G. B. Xavier, S. Nolte, A. Szameit, R. A. Vicencio, G. Lima, and A. Delgado, Phys. Rev. A 90, 063823 (2014).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 (2001).
[Crossref]

Phys. Rev. Lett. (1)

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, Phys. Rev. Lett. 108, 010502 (2012).
[Crossref]

Sci. Rep. (2)

I. Pitsios, F. Samara, G. Corrielli, A. Crespi, and R. Osellame, Sci. Rep. 7, 11342 (2017).
[Crossref]

R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, Sci. Rep. 4, 4118 (2014).
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Science (1)

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, Science 320, 646 (2008).
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Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399 (1851).

Other (4)

F. Flamini, N. Spagnolo, and F. Sciarrino, arXiv:1803.02790 (2018).

R. Osellame, G. Cerullo, and R. Ramponi, Femtosecond Laser Micromachining: Photonic and Microfluidic Devices in Transparent Materials (Springer, 2012), Vol. 123.

N. Jovanovic, S. Gross, C. Miese, A. Fuerbach, J. Lawrence, and M. Withford, in Advanced Photonics & Renewable Energy (Optical Society of America, 2010), p. JThA28.

C.-Y. Wang, J. Gao, Z.-Q. Jiao, L.-F. Qiao, R.-J. Ren, Z. Feng, Y. Chen, Z.-Q. Yan, Y. Wang, H. Tang, and X.-M. Jin, arXiv:1808.08586 (2018).

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Figures (4)

Fig. 1.
Fig. 1. Illustration of the fabricating process for double-track waveguides and sketch of the coupling cross section of RPDCs. Each waveguide (gray) is composed of two adjacent tracks (dark red) with geometrically radial and azimuthal ( θ ) offsets. The two waveguide arms are located parallel to the fast axis (blue lines with arrows at both ends) or orthogonally to the slow axis (red lines with arrows at both ends) to enable evanescent couplings.
Fig. 2.
Fig. 2. (a) Experimental results and best-fit model of the artificial birefringence optical axis reorientation α as a function of the azimuthal offset control orientation θ . (b) Schematic of rotation plane of the two rotated orthogonal optical axes. (c) Morphology of the waveguide cross sections presented by microscope end-view of track pairs fabricated with 100 × objective, 2 μm radial offset in fused silica substrate. (d) Corresponding near-field output modes of waveguides with different azimuthal offsets. (e) Setup for birefringent optical axis confirmation. POL, polarizer; QWP, quarter-wave plate.
Fig. 3.
Fig. 3. (a) Configuration of two waveguides located in a plane of same depth. (b) Configuration of two waveguides located with a rotation of α from the plane. (c) Polarization analysis of the 45° rotated parallel coupling region with different linearly input states (H, V, D, A Pol.). (d) Normalized transmission power of anti-diagonal (135°, A Pol.) and diagonal (45°, D Pol.) polarized light through a 45° RPDC. (e) Extinction performance of the 45° RPDC.
Fig. 4.
Fig. 4. (a) Schematic of the experimental setup for reconstruction of the Stokes vector. POL, polarizer; HWP, half-wave plate; QWP, quarter-wave plate. (b) Measured fidelities of different initialized states with 0° and 45° RPDCs. (c) Reconstructed density matrix obtained with 0° RPDC. (d) Reconstructed density matrix obtained with 45° RPDC. Blue parts represent real components and gray parts imaginary components.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

{ d a 1 d z = i β 1 ( z ) a 1 + K 12 a 2 , d a 2 d z = K 21 a 1 i β 2 ( z ) a 2 .
{ T = sin ( K Z ) , R = 1 T 2 .
{ ER T = | 10 log T F T S | ( dB ) , ER R = | 10 log R F R S | ( dB ) ,
ρ ^ = 1 2 i = 0 3 S i S 0 σ ^ i ,
{ S 0 = P | 0 + P | 1 , S 1 = P 1 2 ( | 0 + | 1 ) P 1 2 ( | 0 | 1 ) , S 2 = P 1 2 ( | 0 + i | 1 ) P 1 2 ( | 0 i | 1 ) , S 3 = P | 0 P | 1 ,

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