Wavefront sensing using a liquid-filled photonic ... - OSA Publishing

Wavefront sensing using a liquid-filled photonic crystal fiber Denise Valente,1,∗ Diego Rativa,2 and Brian Vohnsen1 1 AOI

2

Group, School of Physics, University College Dublin, Dublin 4, Ireland Polytechnic School of Pernambuco, University of Pernambuco, 50720-001, Recife-PE, Brazil *[email protected]

Abstract: A novel wavefront sensor based on a microstructural array of waveguides is proposed. The method is based on the sensitivity in light-coupling efficiency to the wavefront gradient present at the entrance aperture of each waveguide in an array, and hence the amount of incident light that couples is influenced by wavefront aberrations. The concept is illustrated with wavefront measurements that have been performed using a liquid-filled photonic crystal fiber (LF-PCF) working as a coherent fiber bundle. The pros and cons of the LF-PCF based sensor are discussed. © 2015 Optical Society of America OCIS codes: (010.7350) Wave-front sensing; (280.4788) Optical sensing and sensors; (060.5295) Photonic crystal fibers; (220.1010) Aberrations (global).

References and links 1. D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007). 2. R. N. Smartt and W. H. Steel, “Theory and application of point diffraction interferometers,” Jpn. J. Appl. Phys. 14(14-1), 351–356 (1975). 3. B. Vohnsen, S. Castillo, and D. Rativa, “Wavefront sensing with an axicon,” Opt. Lett. 36, 846–848 (2011). 4. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988). 5. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990). 6. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996). 7. T. R. Rimmele, “Solar adaptive optics,” Proc. SPIE 4007, 218–231 (2000). 8. R. K. Tyson, Principles of Adaptive Optics, 3rd ed. (Taylor & Francis Group, Inc., 2011). 9. J. Hartmann, “Bemerkungen u¨ ber den bau und die justierung von spektrographen,” Z. Instrumentenkd. 20, 47–58 (1900). 10. R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656–660 (1971). 11. B. Platt, R. Shack, “History and principles of Shack-Hartmann wave-front sensing,” J. Refract. Surg. 17, 573–577 (2001). 12. J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994). 13. B. Vohnsen and D. Rativa, “Ultrasmall spot size scanning laser ophthalmoscopy,” Biomed. Opt. Express 2, 1597– 1609 (2011) 14. J. W. Hardy, “Adaptive optics: a progress review,” Proc. SPIE 1542, 2-17 (1991). 15. D. Rativa, R. E. de Araujo, A. S. L. Gomes, and B. Vohnsen, “Hartmann-Shack wavefront sensing for nonlinear materials characterization,” Opt. Express 17, 22047–22053 (2009). 16. Qi Tian and Michael N. Huhns, “Algorithms for subpixel registration,” Comput. Vision Graph. Image Process. 35, 220–233 (1986). 17. V. Akondi, S. Castillo, and B. Vohnsen, “Digital pyramid wavefront sensor with tunable modulation,” Opt. Express 21, 18261–18272 (2013). 18. B. Vohnsen and D. Rativa, “Absence of an integrated Stiles-Crawford function for coherent light,” J. Vis. 11, 19 (2011).

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Received 11 Feb 2015; revised 29 Apr 2015; accepted 30 Apr 2015; published 8 May 2015 18 May 2015 | Vol. 23, No. 10 | DOI:10.1364/OE.23.013005 | OPTICS EXPRESS 13005

19. B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A 22, 2318–2328 (2005). 20. B. Vohnsen, “Photoreceptor waveguides and effective retinal image quality,” J. Opt. Soc. Am. A 14, 597–607 (2007). 21. B. Vohnsen, I. Iglesias, and P. Artal, “Coherent fibre-bundle wavefront sensor,” in Proceedings of 6th Adaptive Optics for Industry and Medicine, C. Dainty, ed. (Galway, Ireland, 2008), pp. 163–168. 22. D. Rativa and B.Vohnsen, “Simulating human photoreceptor optics using a liquid-filled photonic crystal fiber,” Biomed. Opt. Express 2, 543–551 (2011). 23. R. H. Rediker, B. G. Zollars, T. A. Lind, R. E. Hatch, and B. E. Burke, “Measurement of the wave front of a pulsed dye laser using an integrated-optics sensor with 200-nsec temporal resolution,” Opt. Lett. 14, 381–383 (1989). 24. J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. (Springer-Verlag, Inc., 1999).

1.

Introduction

The measuring of wavefront aberrations is common to optical testing, active/adaptive optics, and image restoration [1]. Different approaches have been used to measure wavefront aberrations: using a wavefront reference as in the case of interferometry [2, 3], analyzing the irradiance variations due to local curvatures of the wavefront [4, 5], or measuring local wavefront gradients (i.e. microtilts) of the beam profile. The latter is the principle of different commercial wavefront sensors such as the pyramidal [6], cross-correlation [7], shearing interferometers [8] and the Hartmann-Shack wavefront sensor (H-S) [9, 10]. The H-S is the most versatile wavefront sensor, with numerous applications in astronomy [11], ophthalmology [12, 13], adaptive optics [14] and nonlinear optical analysis [15]. The H-S relies on an array of micro-lenses to create focal spots on a CCD. The position shifts of the focal spots are governed by the average tilt over each lenslet such that the grid of focal spots provide a measure of the micro-tilts across the entire pupil being analysed. These microtilts are integrated to reconstruct the total wavefront of the incident radiation, such that the total wavefront reconstruction using a H-S sensor is essentially limited by the sensor pitch (100µm − 200µm for most commercial H-S sensors). Furthermore, with a typical 6.3mm focal length and a CCD pixel period of 4.6µm this amounts to an angular resolution of δ θ ∼ 2.50 , although higher resolution can be obtained at a sub-pixel level [16]. The search for new wavefront sensing methods that can provide more resolution or higher dynamical range remains an attractive option such as e.g. the use of reconfigurable sensors using spatial light modulators [17]. Rather than using lenslets to capture wavefront tilts as with the H-S, an array of waveguides could potentially fulfil the same goal as the amount of light coupled to each would be dependent on local wavefront tilt. This is well known in visual optics where the retinal photoreceptors, with diameters ranging from 2µm to 8µm, are sensitive to the amplitude and phase gradient of the illumination field [18–21]. Here, this mechanism is transferred to a sensor design in which a bundle of waveguides sample an incoming wavefront. The main idea consist of making use of the sensitivity in coupled light power to the wavefront gradient present at the entrance face of each fibre in the array. As the fibre-based wavefront sensor is not based on an angular reading but rather on power, the sensitivity can be made arbitrarily large but would be power limited and sensitive to power fluctuations although the latter can be overcome by taking reference measurements. Here, the method has been implemented using a liquid-filled photonic crystal fiber (LF-PCF) with waveguide characteristics managed by temperature [22]. Tilt, defocus, astigmatism and coma aberrations are introduced using a deformable mirror conjugated to the entrance face of the LF-PCF, such that after propagation the guided light is imaged onto a CCD whereby the coupled light power from each individual waveguide can be determined.

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2.

Method

The fraction of light coupled is a function not just of characteristics like geometry and refractive indices but also of the wavefront and amplitude distribution of the light at the entrance of the fiber. The coupling efficiency to a waveguide with M modes is given by [19] 2 Z Z i2π ∗ Aexp φ r cos(θ ) Ψm rdrdθ = ∑ λ m=1 M

Ptotal

(1)

where A is the amplitude of the incident field with angle φ at the point (r, θ ) at the entrance of the waveguide and Ψm is the normalized field of the guided m-th mode. This equation shows that light transmission decreases with increasing angle of incidence. The analysis of power cannot distinguish between the phase gradient of the wavefront in two orthogonal directions and also it makes no distinction between positive and negative derivatives of phase. One way to solve this ambiguity has been proposed by Rediker et al. [23], where a couple of waveguides with different optical lengths has been implemented such that a phasedifference allows to discriminate a positive from a negative phase angle. An alternative solution is a fiber design with an inclined entrance face such that a bias angle (θ0 ) is introduced in each orthogonal direction (θ0,x , θ0,y ) whereby as shown in Fig. 1, the change in power can detect both the sign and relative values of the phase gradient in each orthogonal component.

(a)

(a)

(b)

(b)



+𝜃0,𝑦

Fig. 1. (a) Oblique wavefronts probed with a tilted single-mode fibre with positive +θ0,y bias angle; (b) Indicative changes in coupled light power obtained from Eq. 1.

Nonetheless, the method is sensitive to light intensity variations that cannot be separated from phase variations unless if highly multimode fibres non-sensitive to phase variations can be incorporated into the fiber bundle. 3.

Experimental setup

A microstructured array based on a photonic crystal fiber LMA-20 with 126 holes filled with castor oil by capillary forces has been implemented. The array presents a hexagonal packing arrangement, that achieve the highest-density lattice arrangement of circles based on their symmetry [24]. The LF-PCF is placed inside of a precision oven allowing accurate thermooptical control. Experimental details about the LF-PCF method are given in [22]. As represented in Fig. 2, a spatially-filtered HeNe laser beam (633nm, CW) is expanded to a diameter of 3mm and collimated. A Boston Micromachines 140-actuator deformable mirror (D-M) conjugated to the entrance face of the LF-PCF is used to generate well-controlled optical aberrations at the LF-PCF plane. Three telescopes consisting of the L1 - L6 achromatic lenses are used to reduce the illumination beam from a 3.0mm diameter at the D-M plane to approximately 150µm at the LF-PCF plane, approximately the same width as that of the array #234457 - $15.00 USD © 2015 OSA


(a) (a)

BS

(b)

(b)

Deformable mirror 𝑳𝟏

𝑳𝟐 BS 𝑴𝟏

Beam stopper

𝑳𝟕

Coupling efficiency (normalized)

of waveguides and each of which will be used to detect the local tilt φ of the wavefront. Finally, L7 is used to conjugate the D-M with the Hartmann-Shack. V=2.89 (LP01, LP11) 1 0.8 0.6 0.4 0.2 0 -10° -8° -6° -4° -2°

0° 2° 4°

Angle of incidence 

6° 8° 10°

𝑳𝟑

Spatial filter

HartmannShack

PCF

HeNe (633nm, cw)

𝑳𝟔

𝑳𝟒 𝒀𝒃𝒊𝒂𝒔 𝑴𝟐

𝑳𝟓

𝑿𝒃𝒊𝒂𝒔

Fig. 2. (a) Schematic of the optical setup (not to scale). D-M: deformable mirror, BS: Beam splitter, H-S: Hartmann-Shack wavefront sensor, M1 : fixed mirror, M2 : rotable mirror, PCF: photonic crystal fiber, and achromatic lenses are denoted by L1 to L7 ; (b) Theoretical power coupled fraction versus wavefront slope in degrees for a waveguide with radius a = 3.5µm obtained from Eq. 1 with two coupled modes (LP01 and LP11 ). The dashed line correspond to the bias angle with largest sensitivity.

Different tilted fiber conditions with bias angles in both spatial coordinates θ0,x and θ0,y , one at a time, are analyzed with a conjugated-plane tilted illumination by using a flat mirror (M2 ) in a gimbal mount conjugated to the coupling face of the LF-PCF. Then, the observed wavefront is given by: −θ0 x + ∑∞j=1 c j Z j if the θ0 bias angle is in x-direction W (x, y) = (2) −θ0 y + ∑∞j=1 c j Z j if the θ0 bias angle is in y-direction where Z j are the Zernike polynomials and c j are the Zernike coefficients, expressed in micrometers. Prior to the measurements, the LF-PCF is imaged by an upright microscope (Olympus BX61, 20X) with a CCD camera (Olympus SC30) [Fig. 3(a)] and the software package Fiji-ImageJ™ is used to detect the position of each waveguide [Figs. 3(b) and 3(c)]. This is used to define a mask function for Matlab™ analysis of the coupled light power by each oil column in the ensemble. As represented in Figs. 4–9, similar to the LF-PCF waveguide transversal section, an array of waveguides is represented using a MATLAB program where the light-coupled intensity values are given by colors that allow for an easy visualisation of individual intensity results. The spatial resolution of this array is ≈ 12.7µm, that corresponds to the distance between two adjacent waveguides. However, the packing density could be even higher using either smaller waveguide cores or reducing the spacing between the waveguides further while avoiding any significant overlap of evanescent waves. For the chosen LF-PCF the minimum waveguide spacing would be ≈ 11.5µm. #234457 - $15.00 USD © 2015 OSA


(a) (a)

(b) (b)

(c)

𝑑𝑖𝑛𝑡𝑒𝑟 = 12.7𝜇𝑚

𝑑𝑤𝑔 = 7.0𝜇𝑚

Fig. 3. (a) Transmission image of the LF-PCF illuminated by a white light, with a magnification of 20X; (b) The Color Threshold function of the Fiji-ImageJ software set the RGB thresholds and is used to detect objects of consistent colour values; (c) The Analyze Particles function was used to compute the precise position of each waveguide oil column in the array.

4.

Results and discussion

The measurements were realized with the oil and fiber at 67◦C. At this temperature the characteristic waveguide number is V ≈ 2.89 which corresponds to multimode waveguiding (coupled modes LP01 and LP11 ). This dual mode guiding regimen was chosen since it has a smaller susceptibility to diameter fluctuations of the waveguides than a purely single mode would have. The bias angles were adjusted to θ0 = 4◦ , in the linear region of the angulardependent coupling regime where its sensitivity is largest, as shown in Fig. 2(b). The local tilt at the entrance of each waveguide are typically small so Lambert’s cosine law and very minor variations in the Fresnel coefficients were ignored for simplicity. The coupling to the LF-PCF of the light at the entry face with well-controlled monochromatic optical aberrations was investigated and the results are presented below. The image of the LFPCF for plane wavefront (null-aberrations) was used as reference. 4.1. Tilt, Z2 Z1−1 Initially, the coupled power as a function of tilt only was analyzed from c2 = −1.0µm up to c2 = 1.0µm. Higher tilt values are limited by the deformable mirror stroke. As shown in Fig. 4(a), the coupling efficiency is larger when the local slope of the wavefront gradient (φx ) and the bias angle θ0,x have the same sign. The deviation between the theoretically expected coupled light and experimentally measured in each waveguide is schematically represented in Fig. 4(b). The deviation was calculated as δ = 0.03 (3% with respect to the scale) for c2 = −1µm and δ = 0.14 (14% with respect to the scale) for c2 = 1µm. A possible reason for the discrepancy between the accuracy of positive and negative tilts could be surpassing of the paraxial limit for c2 = 1µm. For this case with a positive bias angle and positive tilt, the resultant angle after mirror M2 is 5.5◦ , corresponding to an incident ray displaced 1.9cm from the optical axis of lenses L5 and L6 which could plausibly generate additional aberrations.

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(a)

(a)

Theoretically

(b)

𝑐2 = 0𝜇𝑚

𝑐2 = 1𝜇𝑚

𝑐2 = −1𝜇𝑚

Experimentally 𝑐2 = −1𝜇𝑚

𝑐2 = 0𝜇𝑚

𝛿 = 0.03

𝑐2 = 1𝜇𝑚 𝛿 = 0.25

𝜽𝟎,𝒙 = 𝟒𝒐

𝜽𝟎,𝒙 = 𝟒𝒐

Ref.

(b)

(a)

(b)

𝑐2 = −1𝜇𝑚 𝛿 = 0.03

Deviation 𝑐2 = 0𝜇𝑚

𝑐2 = 1𝜇𝑚 𝛿 = 0.25

𝜽𝟎,𝒙 = 𝟒𝒐 Ref.

Fig. 4. (a) Intensity of the coupled light for tilt aberrations from c2 = −1µm to c2 = 1µm. The plane wavefront (null aberrations) was used as reference. LEFT: Simulation, assuming two guided modes (LP01 and LP11) and identical waveguides; RIGHT: Experimental results with the LF-PCF. Deviations (δ ) of measured signals are also indicated; (b) Deviation between the theoretically expected coupled light and experimentally measured in each waveguide.

4.2.

Wavefront aberrations

To examine the angular variation present in the wavefront it was necessary to introduce sequentially an X-bias andexperiments were realized angle and a Y-bias angle. Simulations with defocus Z4 Z20 [Fig. 5], astigmatism Z5 Z2−2 and Z6 Z22 [Figs. 6 and 7] and coma Z7 Z3−1 and Z8 Z31 [Figs. 8 and 9]. The deviation (δ ) between the theoretically expected coupled light and experimentally measured are indicated in absolute values in each image. In the case of defocus, as show in Fig. 5, the intensity values of guided-light analysed for both spatial coordinates show rotational symmetry, characteristic of the local slope. The deviation is on average δ = 0.18 (10% with respect to the scale). Figures 6 and 7 show results for astigmatism and reveal antisymmetry according to the X and Y axes as well as the local slope of the function, outlined by a saddle point. Average deviations are respectively δ = 0.17 (9% with respect to the scale) for Z5 and δ = 0.23 (17% with respect to the scale) for Z6 . Finally, the coma aberration is characterized by a wavefront with peak and valley laterally positioned in Y-direction for Z7 and in X-direction for Z8 . This results in antisymmetric coupled power to positive and negative Zernike coefficients when the bias angle is in the peak-to-valley axis and complementary coupled power when the bias angle is orthogonal to the peak-to-valley axis [Figs. 8 and 9]. Average deviations were calculated as δ = 0.22 (14% with respect to the

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scale) for Z7 and δ = 0.21 (16% with respect to the scale) for Z8 . Experimental results are in overall good accordance with the theoretical expectations confirming the potential of the approach to measure monochromatic aberrations. The average accuracy of the system was calculated as 87%. Possible factors that could deteriorate the accuracy are variations in guiding characteristics of each waveguide in the bundle and the fact that local tilts in the aberrated wavefronts are expected to vary from 0◦ to 16◦ while the coupling efficiency presents a linear dependency on angle only in the range of 2.5◦ to 5.5◦ . A LF-PCF with smaller holes could minimize this limitation due to a wider acceptance angle which would improve the accuracy. This reduction in diameter of the guides would also result in a higher spatial resolution of the wavefront sensor thereby allowing the measurement of wavefront aberrations of higher orders. Other limitations of the system are the need of a non-aberrated wavefront as a reference and the dependence of the coupling efficiency on wavelength, restricting the measurement to monochromatic light. However, this last problem could be overcome with the association of the system with a spectrometer, permiting not just wavefront analysis in broadband light but spectroscopy at the same time.

Fig. 5. Intensity for defocus aberrations from c4 = −1µm to c4 = 1µm . The plane wavefront (null aberrations) was used as reference. LEFT: Simulation, assuming two guided modes (LP01 and LP11); RIGHT: Experimental results. Deviations (δ ) of measured signals are also indicated.

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Fig. 6. Intensity for astigmatism aberrations from c5 = −1µm to c5 = 1µm. The plane wavefront (null aberrations) was used as reference. LEFT: Simulation, assuming two guided modes (LP01 and LP11); RIGHT: Experimental results. Deviations (δ ) of measured signals are also indicated.

Fig. 7. Intensity for astigmatism aberrations from c6 = −1µm to c6 = 1µm. The plane wavefront (null aberrations) was used as reference. LEFT: Simulation, assuming two guided modes (LP01 and LP11); RIGHT: Experimental results. Deviations (δ ) of measured signals are also indicated.

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Fig. 8. Intensity for coma aberrations from c7 = −1µm to c7 = 1µm. The plane wavefront (null aberrations) was used as reference. LEFT: Simulation, assuming two guided modes (LP01 and LP11); RIGHT: Experimental results. Deviations (δ ) of measured signals are also indicated.

Fig. 9. Intensity for coma aberrations from c8 = −1µm to c8 = 1µm. The plane wavefront (null aberrations) was used as reference. LEFT: Simulation, assuming two guided modes (LP01 and LP11); RIGHT: Experimental results. Deviations (δ ) of measured signals are also indicated.

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5.

Conclusion

A wavefront sensor based on the angular sensitivity of light coupling efficiency to an optical fiber has been experimentally implemented using a LF-PCF. Experimental results show fair agreement with simulations, showing the potential of this device. The LF-PCF is an innovative technique for sensing wavefronts with potential to overcome some intrinsic problems of the Hartmann-Shack such as the wavefront sensing in diminutive spaces. The fabrication of PCF fibers with solid micro-structures with refractive index that permits the guiding of light in each waveguide, in a similar way as realized with the LF-PCF, could also facilitate wavefront sensing in liquid environments. While the bend radius of the fiber is kept larger than the critical radius of curvature will birefringence and leakage have a negligible effect on the results. Wavefront reconstruction algorithms and new designs of the sensor will be implemented in the future. Acknowledgments This research was supported by the Science Without Borders program - CAPES foundation Brazil and the Science Foundation Ireland grant 08/IN.1/B2053.

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