Abstract

Single mode beam delivery in the mid-infrared spectral range 5.1-10.5 μm employing flexible hollow glass waveguides of 15 cm and 50 cm lengths, with metallic/dielectric internal layers and a bore diameter of 200 μm were demonstrated. Three quantum cascade lasers were coupled with the hollow core fibers. For a fiber length of 15 cm, we measured losses down to 1.55 dB at 5.4 μm and 0.9 dB at 10.5 μm. The influence of the launch conditions in the fiber on the propagation losses and on the beam profile at the waveguide exit was analyzed. At 10.5 µm laser wavelength we found near perfect agreement between measured and theoretical losses, while at ~5 µm and ~6 µm wavelengths the losses were higher than expected. This discrepancy can be explained considering an additional scattering loss effect, which scales as 1/λ2 and is due to surface roughness of the metallic layer used to form the high-reflective internal layer structure of the hollow core waveguide.

© 2015 Optical Society of America

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References

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2014 (3)

J. Kriesel, G. M. Hagglund, N. Gat, V. Spagnolo, and P. Patimisco, “Spatial mode filtering of mid-infrared (mid-IR) laser beams with hollow core fiber optics,” Proc. SPIE 8993, 89930V (2014).

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

M. Siciliani de Cumis, S. Viciani, S. Borri, P. Patimisco, A. Sampaolo, G. Scamarcio, P. De Natale, F. D’Amato, and V. Spagnolo, “Widely-tunable mid-infrared fiber-coupled quartz-enhanced photoacoustic sensor for environmental monitoring,” Opt. Express 22(23), 28222–28231 (2014).
[Crossref] [PubMed]

2013 (2)

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Mid-infrared fiber-coupled QCL-QEPAS sensor,” Appl. Phys. B 112(1), 25–33 (2013).
[Crossref]

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

2012 (3)

2005 (1)

2004 (1)

Y. Matsuura and M. Miyagi, “Hollow optical fibers for ultraviolet and vacuum ultraviolet light,” IEEE J. Sel. Top. Quant. 10(6), 1430–1434 (2004).
[Crossref]

2000 (1)

J. A. Harrington, “A review of IR transmitting, hollow waveguides,” Fiber and Integrated Opt. 19(3), 211–227 (2000).
[Crossref]

1999 (1)

1998 (1)

R. Nubling and J. A. Harrington, “Launch conditions and mode coupling in hollow glass waveguides,” Opt. Eng. 37(9), 2454–2458 (1998).
[Crossref]

1996 (1)

1995 (1)

1989 (1)

1984 (1)

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Technol. 2(2), 116–126 (1984).
[Crossref]

1981 (1)

1980 (1)

C. Dragone, “Attenuation and radiation characteristics of the HE11 mode,” IEEE T. Microw. Theory 28(7), 704–710 (1980).
[Crossref]

1964 (1)

E. A. J. Marcantili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Abel, T.

Adam, J.-L.

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

Bernacki, B. E.

Bledt, C. M.

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

C. M. Bledt, J. A. Harrington, and J. M. Kriesel, “Loss and modal properties of Ag/AgI hollow glass waveguides,” Appl. Opt. 51(16), 3114–3119 (2012).
[Crossref] [PubMed]

Borri, S.

Brilland, L.

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

Caillaud, C.

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

Calvez, L.

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

D’Amato, F.

De Natale, P.

Dragone, C.

C. Dragone, “Attenuation and radiation characteristics of the HE11 mode,” IEEE T. Microw. Theory 28(7), 704–710 (1980).
[Crossref]

Gat, N.

J. Kriesel, G. M. Hagglund, N. Gat, V. Spagnolo, and P. Patimisco, “Spatial mode filtering of mid-infrared (mid-IR) laser beams with hollow core fiber optics,” Proc. SPIE 8993, 89930V (2014).

George, R.

Gibson, D. J.

Hagglund, G. M.

J. Kriesel, G. M. Hagglund, N. Gat, V. Spagnolo, and P. Patimisco, “Spatial mode filtering of mid-infrared (mid-IR) laser beams with hollow core fiber optics,” Proc. SPIE 8993, 89930V (2014).

Harrington, J. A.

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

C. M. Bledt, J. A. Harrington, and J. M. Kriesel, “Loss and modal properties of Ag/AgI hollow glass waveguides,” Appl. Opt. 51(16), 3114–3119 (2012).
[Crossref] [PubMed]

R. George and J. A. Harrington, “Infrared transmissive, hollow plastic waveguides with inner Ag-Agl coatings,” Appl. Opt. 44(30), 6449–6455 (2005).
[Crossref] [PubMed]

J. A. Harrington, “A review of IR transmitting, hollow waveguides,” Fiber and Integrated Opt. 19(3), 211–227 (2000).
[Crossref]

C. D. Rabii, D. J. Gibson, and J. A. Harrington, “Processing and characterization of silver films used to fabricate hollow glass waveguides,” Appl. Opt. 38(21), 4486–4493 (1999).
[Crossref] [PubMed]

R. Nubling and J. A. Harrington, “Launch conditions and mode coupling in hollow glass waveguides,” Opt. Eng. 37(9), 2454–2458 (1998).
[Crossref]

R. L. Kozodoy, A. T. Pagkalinawan, and J. A. Harrington, “Small-bore hollow waveguides for delivery of 3-µm laser radiation,” Appl. Opt. 35(7), 1077–1082 (1996).
[Crossref] [PubMed]

Y. Matsuura, T. Abel, and J. A. Harrington, “Optical properties of small-bore hollow glass waveguides,” Appl. Opt. 34(30), 6842–6847 (1995).
[Crossref] [PubMed]

Hongo, A.

Kawakami, S.

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Technol. 2(2), 116–126 (1984).
[Crossref]

Kozodoy, R. L.

Kriesel, J.

J. Kriesel, G. M. Hagglund, N. Gat, V. Spagnolo, and P. Patimisco, “Spatial mode filtering of mid-infrared (mid-IR) laser beams with hollow core fiber optics,” Proc. SPIE 8993, 89930V (2014).

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Mid-infrared fiber-coupled QCL-QEPAS sensor,” Appl. Phys. B 112(1), 25–33 (2013).
[Crossref]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Part-per-trillion level SF6 detection using a quartz enhanced photoacoustic spectroscopy-based sensor with single-mode fiber-coupled quantum cascade laser excitation,” Opt. Lett. 37(21), 4461–4463 (2012).
[Crossref] [PubMed]

Kriesel, J. M.

Marcantili, E. A. J.

E. A. J. Marcantili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Matsuura, Y.

Mechin, D.

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

Miyagi, M.

Y. Matsuura and M. Miyagi, “Hollow optical fibers for ultraviolet and vacuum ultraviolet light,” IEEE J. Sel. Top. Quant. 10(6), 1430–1434 (2004).
[Crossref]

Y. Matsuura, M. Saito, M. Miyagi, and A. Hongo, “Loss characteristics of circular hollow waveguides for incoherent infrared light,” J. Opt. Soc. Am. A 6(3), 423–427 (1989).
[Crossref]

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Technol. 2(2), 116–126 (1984).
[Crossref]

M. Miyagi, “Bending losses in hollow and dielectric tube leaky waveguides,” Appl. Opt. 20(7), 1221–1229 (1981).
[Crossref] [PubMed]

Nubling, R.

R. Nubling and J. A. Harrington, “Launch conditions and mode coupling in hollow glass waveguides,” Opt. Eng. 37(9), 2454–2458 (1998).
[Crossref]

Pagkalinawan, A. T.

Patimisco, P.

J. Kriesel, G. M. Hagglund, N. Gat, V. Spagnolo, and P. Patimisco, “Spatial mode filtering of mid-infrared (mid-IR) laser beams with hollow core fiber optics,” Proc. SPIE 8993, 89930V (2014).

M. Siciliani de Cumis, S. Viciani, S. Borri, P. Patimisco, A. Sampaolo, G. Scamarcio, P. De Natale, F. D’Amato, and V. Spagnolo, “Widely-tunable mid-infrared fiber-coupled quartz-enhanced photoacoustic sensor for environmental monitoring,” Opt. Express 22(23), 28222–28231 (2014).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Mid-infrared fiber-coupled QCL-QEPAS sensor,” Appl. Phys. B 112(1), 25–33 (2013).
[Crossref]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Part-per-trillion level SF6 detection using a quartz enhanced photoacoustic spectroscopy-based sensor with single-mode fiber-coupled quantum cascade laser excitation,” Opt. Lett. 37(21), 4461–4463 (2012).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

Rabii, C. D.

Renversez, G.

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

Saito, M.

Sampaolo, A.

Scamarcio, G.

M. Siciliani de Cumis, S. Viciani, S. Borri, P. Patimisco, A. Sampaolo, G. Scamarcio, P. De Natale, F. D’Amato, and V. Spagnolo, “Widely-tunable mid-infrared fiber-coupled quartz-enhanced photoacoustic sensor for environmental monitoring,” Opt. Express 22(23), 28222–28231 (2014).
[Crossref] [PubMed]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Mid-infrared fiber-coupled QCL-QEPAS sensor,” Appl. Phys. B 112(1), 25–33 (2013).
[Crossref]

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Part-per-trillion level SF6 detection using a quartz enhanced photoacoustic spectroscopy-based sensor with single-mode fiber-coupled quantum cascade laser excitation,” Opt. Lett. 37(21), 4461–4463 (2012).
[Crossref] [PubMed]

Schmeltzer, R. A.

E. A. J. Marcantili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Siciliani de Cumis, M.

Spagnolo, V.

J. Kriesel, G. M. Hagglund, N. Gat, V. Spagnolo, and P. Patimisco, “Spatial mode filtering of mid-infrared (mid-IR) laser beams with hollow core fiber optics,” Proc. SPIE 8993, 89930V (2014).

M. Siciliani de Cumis, S. Viciani, S. Borri, P. Patimisco, A. Sampaolo, G. Scamarcio, P. De Natale, F. D’Amato, and V. Spagnolo, “Widely-tunable mid-infrared fiber-coupled quartz-enhanced photoacoustic sensor for environmental monitoring,” Opt. Express 22(23), 28222–28231 (2014).
[Crossref] [PubMed]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Mid-infrared fiber-coupled QCL-QEPAS sensor,” Appl. Phys. B 112(1), 25–33 (2013).
[Crossref]

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Part-per-trillion level SF6 detection using a quartz enhanced photoacoustic spectroscopy-based sensor with single-mode fiber-coupled quantum cascade laser excitation,” Opt. Lett. 37(21), 4461–4463 (2012).
[Crossref] [PubMed]

Tredicucci, A.

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

Troles, J.

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

Viciani, S.

Vitiello, M. S.

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

Appl. Opt. (6)

Appl. Phys. B (2)

P. Patimisco, V. Spagnolo, M. S. Vitiello, A. Tredicucci, G. Scamarcio, C. M. Bledt, and J. A. Harrington, “Coupling external cavity mid-IR quantum cascade lasers with low loss hollow metallic/dielectric waveguides,” Appl. Phys. B 108(2), 255–260 (2012).
[Crossref]

V. Spagnolo, P. Patimisco, S. Borri, G. Scamarcio, B. E. Bernacki, and J. Kriesel, “Mid-infrared fiber-coupled QCL-QEPAS sensor,” Appl. Phys. B 112(1), 25–33 (2013).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcantili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Fiber and Integrated Opt. (1)

J. A. Harrington, “A review of IR transmitting, hollow waveguides,” Fiber and Integrated Opt. 19(3), 211–227 (2000).
[Crossref]

IEEE J. Sel. Top. Quant. (1)

Y. Matsuura and M. Miyagi, “Hollow optical fibers for ultraviolet and vacuum ultraviolet light,” IEEE J. Sel. Top. Quant. 10(6), 1430–1434 (2004).
[Crossref]

IEEE T. Microw. Theory (1)

C. Dragone, “Attenuation and radiation characteristics of the HE11 mode,” IEEE T. Microw. Theory 28(7), 704–710 (1980).
[Crossref]

J. Lightwave Technol. (1)

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Technol. 2(2), 116–126 (1984).
[Crossref]

J. Opt. Soc. Am. A (1)

Materials (1)

C. Caillaud, G. Renversez, L. Brilland, D. Mechin, L. Calvez, J.-L. Adam, and J. Troles, “Photonic bandgap propagation in all-solid chalcogenide microstructured optical fibers,” Materials 7(9), 6120–6129 (2014).
[Crossref]

Opt. Eng. (1)

R. Nubling and J. A. Harrington, “Launch conditions and mode coupling in hollow glass waveguides,” Opt. Eng. 37(9), 2454–2458 (1998).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

J. Kriesel, G. M. Hagglund, N. Gat, V. Spagnolo, and P. Patimisco, “Spatial mode filtering of mid-infrared (mid-IR) laser beams with hollow core fiber optics,” Proc. SPIE 8993, 89930V (2014).

Sensors (Basel) (1)

P. Patimisco, V. Spagnolo, M. S. Vitiello, G. Scamarcio, C. M. Bledt, J. A. Harrington, and J. A. Harrington, “Low-loss hollow waveguide fibers for mid-infrared quantum cascade laser sensing applications,” Sensors (Basel) 13(1), 1329–1340 (2013).
[Crossref] [PubMed]

Other (4)

https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=7062#SingleMode .

J. A. Harrington, Infrared Fibers and Their Applications (SPIE, 2004).

B. Lendl and B. Mizaikoff, Optical Fibers for Mid-infrared Spectrometry, Handbook of Vibrational Spectroscopy Vol. 2 (John Wiley & Sons Ltd, 2002).

R. S. Quimby, Photonics and Lasers: An Introduction (John Wiley & Sons Ltd, 2006).

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Figures (7)

Fig. 1
Fig. 1

A schematic of the experimental setup. The laser beam is focused into the HCW entrance using a coupling lens. The beam profile at the waveguide exit is acquired with an infrared pyrocamera. QCL –Quantum Cascade Laser; HCW – Hollow-Core Waveguide.

Fig. 2
Fig. 2

Far field spatial intensity distribution of λa-QCL (a) and λc-QCL (b). The beam profile was measured directly by shining the QCL output on to the detector, positioned at ~2.5 cm from the QCL.

Fig. 3
Fig. 3

Far field spatial intensity distribution of λa-QCL upon exiting at 15 cm-long [(a), (b) and (c)] and 50 cm-long [(d), (e) and (f)] HCW, employing coupling lenses with focal lengths: f = 25 mm [(a), (d)], f = 50 mm [(b), (e)] and f = 76 mm [(c), (f)]. The beam profiles have been obtained with the experimental scheme illustrated in Fig. 1. The distance between the fiber output and the pyrocamera has been fixed to 2.5 cm.

Fig. 4
Fig. 4

Total losses (dots) calculated from the ratio between input/output power values of λa-QCL [(a) and (b)] and λb-QCL [(c) and (d)] at the 15 cm-long [(a) and (c)] and 50 cm-long [(b) and (d)] hollow waveguides exit/entrance. The solid lines are theoretical losses α1m calculated using Eq. (2) and (3).

Fig. 5
Fig. 5

(a) Far field spatial intensity distribution of λc-QCL upon exiting a 15 cm long HCW by using the coupling lens with f = 25 mm. (b) Experimental losses (dots) as a function of 2ω0/d ratio. The solid line represents the theoretical losses trend calculated by using Eqs. (2) and (3) for λc.

Fig. 6
Fig. 6

Differences between theoretical and experimental losses measured for λa, λb and λc for a 50 cm-long HWC plotted as a function of 1/λ2.

Fig. 7
Fig. 7

(a) Bending losses (dots) for 50cm-long HCW measured by using λa-QCL and the coupling lens with f = 50 mm. Solid line is the best linear fit of the data below the critical radius Rc = 1.15 m. (b) Mode profile at the HCW exit bent at a radius of curvature of 0.111 m. (c) Mode profile at the HCW exit bent at a radius of curvature of 0.302 m. The corresponding data points are marked by arrows.

Tables (2)

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Table 1 Ratios 2ω0/d calculated for three coupling lenses for both the λa-QCL (beam radius of 1.18 mm) and the λb-QCL (beam radius of 1.33 mm).

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Table 2 Ratios 2ω0/d calculated for three different coupling lenses and λc-QCL.

Equations (4)

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ω 2 (z)= ω 0 2 [ 1+ ( z 0 R ) 2 ]
L p (dB)=10Lo g 10 ( m η 1m e 2 α 1m L )
α 1m = ( u 1m 2π ) 2 λ 2 a 3 ( n n 2 k 2 ){ 1 2 [ 1+ n d 2 ( n d 2 1 ) 1/2 ] 2 }
θ= u 11 λ 2πa

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