Coma Aberration of a Nonlinear Aspherical Micro-Lens Created by Kerr medium
Main Article Content
Abstract
A nonlinear aspherical micro-lens (NAML) can be dynamically formed inside a Kerr medium under Gaussian beam (GB) illumination, providing an intensity-dependent optical surface that enables adaptive wavefront control. While previous studies have clarified the formation mechanism and spherical aberration of the NAML, the behavior of off-axis coma has not yet been fully investigated. In this work, we derive the complete third-order coma expression of the NAML by combining the nonlinear refractive-index distribution with the Seidel aberration formulation. The dependence of coma on the average laser power, nonlinear-layer thickness, and incident ray angle is examined through numerical simulations. The results show that the aspheric coma component dominates and scales strongly with , leading to substantial wavefront distortion at large power, thickness, or field angle. A practical operating window is identified in which the coma remains within a few wavelengths. These findings establish the NAML as a power-tunable micro-optical element capable of partial off-axis aberration control.
Keywords:
Nonlinear optics; Aspherical micro lens; Conic coefficient; Aberration optimization.
References
[1] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, John Wiley & Sons, 2019.
[2] H. Q. Quy, V. N. Sau, N. T. T. Tam and N. V. Hoa, Nonlinear Coupler for Optical Fiber Mach–Zehnder Interferometers, Commun. Phys. 20, 2010, pp. 45–50
[3] R. Julius, A.–B. M. A. Ibrahim, P. K. Choudhury, A. N. Alias and M. S. Abd Halim, Quantum Features of The Nonlinear Coupler with Competing Nonlinearity, Sci. Rep. 12, 2022, pp. 8245.
[4] Q. Han, S. J. Elston, W. Kamal, L. Xue and S. M. Morris, A Nonlinear Model of Flexoelectric Liquid-Crystal Diffraction Gratings, Opt. Laser Technol. 180, 2025, pp. 111502.
[5] B. Wang and L. Pu, Design and Fabrication of Nonlinear Grating by Holographic Technology, CCIS 244, ICICA 2011, pp. 614–620.
[6] N. T. Q. Anh et al., Multiple Electromagnetically Induced Grating in The 85rb Five-Level Atomic Medium, J. Opt. Soc. Am. B 41, 2024, pp. 976–983.
[7] H. Q. Quy, M. V. Luu, T. D. Thanh, B. X. Kien, N. M. Thang, H. D. Quang, Optical Bistability of Partial Reflection–Coated Thin Film of Oil Red O, Appl. Opt. 59, 2020, pp. 5664–5669.
[8] Y.–P. Cai and R.–G. Wan, Bistable Reflection and Beam Shifts with Excitation of Surface Plasmons in a Saturable Absorbing Medium, Opt. Express 30, 2022, pp. 20725–20736.
[9] A. Kowalewska-Kudłaszyk, W. Leoński, T. D. Nguyen, V. C. Long, K. F. Renk, Kicked Nonlinear Quantum Scissors and Entanglement Generation, Phys. Scr. T160, 2014, pp. 014023.
[10] A. Kowalewska-Kudłaszyk and W. Leoński, Nonlinear Coupler Operating on Werner-Like States—Entanglement Creation Enhancement and Preservation, J. Opt. Soc. Am. B 31, 2014, pp. 1290–1297
[11] H. Q. Quy, T. T. Doan, T. Q. Tuan, N. M. Thang, Enhancement of Optical Trapping Efficiency by Nonlinear Optical Tweezers, Opt. Commun. 427, 2018, pp. 341–347.
[12] H. Q. Quy, T. T. Doan, T. Q. Tuan, N. M. Thang, Nonlinear Optical Tweezers for Longitudinal Control of Dielectric Particles, Opt. Commun. 421, 2018, pp. 94–98.
[13] H. Q. Quy et al., Nonlinear Microscope Objective Using a Thin Layer of Organic Dye for Optical Tweezers, Eur. Phys. J. D 74, 2020, pp. 52.
[14] B. X. Kien, D. V. Chau and H. Q. Quy, Aspherical Behaviors of The Kerr Medium Irradiated by a Gaussian Beam, J. Nonlinear Opt. Phys. Mater. Vol. 35, No. 01, 2026, pp. 2550004
[15] D. T. Moore, Gradient-Index Optics: A Review, Appl. Opt. 19, 1980, pp. 1035–1038.
[16] A. Hasnaoui, M. Fromager and K. Ait-Ameur, On the Validity of the Parabolic Approximation in Kerr Lensing Effect, Optik 193, 2019, pp. 162986.
[17] S. Leghmize et al., On The Different Ways for Defining the Effective Focal Length of a Kerr Lens, Laser Phys. 27, 2017, pp. 106201.
[18] A. Sattin et al., Aberration Correction in Long Grin Lens–Based Microendoscopes for Extended Field-of-View Two-Photon Imaging, eLife, 2024.
[19] W. M. Lee and S. H. Yun, Adaptive Aberration Correction of Grin Lenses for Confocal Endomicroscopy, Opt. Lett. 36, 2011, pp. 4608–4610.
[20] F. Bortoletto et al., Multiphoton Fluorescence Microscopy with Grin Objective Aberration Correction by Low-Order Adaptive Optics, PLoS ONE 6, 2011.
[21] R. G. González-Acuña and H. A. Chaparro-Romo, General Formula for Bi-Aspheric Singlet Lens Design Free of Spherical Aberration, Appl. Opt. 57, 2018, pp. 9341–9345.
[22] H. Qin, Aberration Correction of a Single Aspheric Lens with Particle Swarm Algorithm, Opt. Commun. 285, 2012, pp. 2996–3000.
[23] A. Miks and P. Pokorný, Spherical Aberration of an Optical System and Its Influence on Depth of Focus, Appl. Opt. 56, 2017, pp. 5099–5107.
[24] S. Vazquez-Montiel and O. García-Lievanos, Spherical Aberration Correction Using Aspheric Surfaces, Rev. Mex. Fis. 59, 2013, pp. 273–281.
[25] A. Hasnaoui et al., Structuring a Laser Beam Subject to Optical Kerr Effect for Improving Its Focusing Properties, Appl. Phys. B, Vol 127(5), 2021, pp.75.
[26] K. F. Renk, Basics of Laser Physics, Springer, 2017.
[27] A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, CRC Press, 2016.
[28] R. Schuhmann, Description of Aspheric Surface, Adv. Opt. Technol. 8, 2019, pp. 267–278.
[29] J. Sasian, Introduction to Lens Design, Cambridge Univ. Press, 2019.
[30] J. A. Strother, Reduction of Spherical and Chromatic Aberration in Axial-Scanning Optical Systems with Tunable Lenses, Biomed. Opt. Express 12, 2021, pp. 3530–3552.
[31] G. Forbes, Shape Specification for Axially Symmetric Optical Surfaces, Opt. Express 15, 2007, pp. 5218–5226.
[32] R. R. Krishnamurthy and R. Alkondan, Nonlinear Characterization of Mercurochrome Dye for Optical Limiting, Opt. Appl. 40, 2010, pp. 187–196.
[33] H. A. Badran et al., Laser-Induced Optical Nonlinearities in Orange G Dye: Polyacrylamide Gel, Can. J. Phys. 89, 2011, pp. 1219–1224.
[34] S. Jeyaram and T. Geethakrishnan, Third-Order Nonlinear Optical Properties of Acid Green 25 Dye by Z-Scan Method, Opt. Laser Technol. 89, 2017, pp. 2570–2574.
[35] L. T. Nguyen et al., Numerical Methods for Analyzing Z-Scan Data, J. Nonlinear Opt. Phys. Mater. 23, 2014, pp. 1450020.
[36] K. Lee et al., Non-Resonant Power-Efficient Directional Nd:Yag Ceramic Laser Using a Scattering Cavity, Nat. Commun. 12, 2021, pp. 8.
[37] K. M. Abedin et al., Power Stability of Different Lasers and Its Effect on Phase-Stepping Shearography, Results Opt. 12, 2023, pp. 100490.
[38] P. D. Quy et al., Calculation and Design of Thermal Object Lens Using an Aspherical Lens for Handheld Observation Devices, J. Military Sci. Technol. 76, 2022, pp. 127–136.
[39] S. Sparrold and A. Lansing, Spherical Aberration Compensation Plates, Opt. Comp. Photonic Int., 2011.
[40] D. C. Compertore et al., Development of a Calibration Standard for Spherical Aberration, Proc. SPIE, 2013.
[41] D. Malacara, Handbook of Lens Design, Marcel Dekker, 1994.
[2] H. Q. Quy, V. N. Sau, N. T. T. Tam and N. V. Hoa, Nonlinear Coupler for Optical Fiber Mach–Zehnder Interferometers, Commun. Phys. 20, 2010, pp. 45–50
[3] R. Julius, A.–B. M. A. Ibrahim, P. K. Choudhury, A. N. Alias and M. S. Abd Halim, Quantum Features of The Nonlinear Coupler with Competing Nonlinearity, Sci. Rep. 12, 2022, pp. 8245.
[4] Q. Han, S. J. Elston, W. Kamal, L. Xue and S. M. Morris, A Nonlinear Model of Flexoelectric Liquid-Crystal Diffraction Gratings, Opt. Laser Technol. 180, 2025, pp. 111502.
[5] B. Wang and L. Pu, Design and Fabrication of Nonlinear Grating by Holographic Technology, CCIS 244, ICICA 2011, pp. 614–620.
[6] N. T. Q. Anh et al., Multiple Electromagnetically Induced Grating in The 85rb Five-Level Atomic Medium, J. Opt. Soc. Am. B 41, 2024, pp. 976–983.
[7] H. Q. Quy, M. V. Luu, T. D. Thanh, B. X. Kien, N. M. Thang, H. D. Quang, Optical Bistability of Partial Reflection–Coated Thin Film of Oil Red O, Appl. Opt. 59, 2020, pp. 5664–5669.
[8] Y.–P. Cai and R.–G. Wan, Bistable Reflection and Beam Shifts with Excitation of Surface Plasmons in a Saturable Absorbing Medium, Opt. Express 30, 2022, pp. 20725–20736.
[9] A. Kowalewska-Kudłaszyk, W. Leoński, T. D. Nguyen, V. C. Long, K. F. Renk, Kicked Nonlinear Quantum Scissors and Entanglement Generation, Phys. Scr. T160, 2014, pp. 014023.
[10] A. Kowalewska-Kudłaszyk and W. Leoński, Nonlinear Coupler Operating on Werner-Like States—Entanglement Creation Enhancement and Preservation, J. Opt. Soc. Am. B 31, 2014, pp. 1290–1297
[11] H. Q. Quy, T. T. Doan, T. Q. Tuan, N. M. Thang, Enhancement of Optical Trapping Efficiency by Nonlinear Optical Tweezers, Opt. Commun. 427, 2018, pp. 341–347.
[12] H. Q. Quy, T. T. Doan, T. Q. Tuan, N. M. Thang, Nonlinear Optical Tweezers for Longitudinal Control of Dielectric Particles, Opt. Commun. 421, 2018, pp. 94–98.
[13] H. Q. Quy et al., Nonlinear Microscope Objective Using a Thin Layer of Organic Dye for Optical Tweezers, Eur. Phys. J. D 74, 2020, pp. 52.
[14] B. X. Kien, D. V. Chau and H. Q. Quy, Aspherical Behaviors of The Kerr Medium Irradiated by a Gaussian Beam, J. Nonlinear Opt. Phys. Mater. Vol. 35, No. 01, 2026, pp. 2550004
[15] D. T. Moore, Gradient-Index Optics: A Review, Appl. Opt. 19, 1980, pp. 1035–1038.
[16] A. Hasnaoui, M. Fromager and K. Ait-Ameur, On the Validity of the Parabolic Approximation in Kerr Lensing Effect, Optik 193, 2019, pp. 162986.
[17] S. Leghmize et al., On The Different Ways for Defining the Effective Focal Length of a Kerr Lens, Laser Phys. 27, 2017, pp. 106201.
[18] A. Sattin et al., Aberration Correction in Long Grin Lens–Based Microendoscopes for Extended Field-of-View Two-Photon Imaging, eLife, 2024.
[19] W. M. Lee and S. H. Yun, Adaptive Aberration Correction of Grin Lenses for Confocal Endomicroscopy, Opt. Lett. 36, 2011, pp. 4608–4610.
[20] F. Bortoletto et al., Multiphoton Fluorescence Microscopy with Grin Objective Aberration Correction by Low-Order Adaptive Optics, PLoS ONE 6, 2011.
[21] R. G. González-Acuña and H. A. Chaparro-Romo, General Formula for Bi-Aspheric Singlet Lens Design Free of Spherical Aberration, Appl. Opt. 57, 2018, pp. 9341–9345.
[22] H. Qin, Aberration Correction of a Single Aspheric Lens with Particle Swarm Algorithm, Opt. Commun. 285, 2012, pp. 2996–3000.
[23] A. Miks and P. Pokorný, Spherical Aberration of an Optical System and Its Influence on Depth of Focus, Appl. Opt. 56, 2017, pp. 5099–5107.
[24] S. Vazquez-Montiel and O. García-Lievanos, Spherical Aberration Correction Using Aspheric Surfaces, Rev. Mex. Fis. 59, 2013, pp. 273–281.
[25] A. Hasnaoui et al., Structuring a Laser Beam Subject to Optical Kerr Effect for Improving Its Focusing Properties, Appl. Phys. B, Vol 127(5), 2021, pp.75.
[26] K. F. Renk, Basics of Laser Physics, Springer, 2017.
[27] A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, CRC Press, 2016.
[28] R. Schuhmann, Description of Aspheric Surface, Adv. Opt. Technol. 8, 2019, pp. 267–278.
[29] J. Sasian, Introduction to Lens Design, Cambridge Univ. Press, 2019.
[30] J. A. Strother, Reduction of Spherical and Chromatic Aberration in Axial-Scanning Optical Systems with Tunable Lenses, Biomed. Opt. Express 12, 2021, pp. 3530–3552.
[31] G. Forbes, Shape Specification for Axially Symmetric Optical Surfaces, Opt. Express 15, 2007, pp. 5218–5226.
[32] R. R. Krishnamurthy and R. Alkondan, Nonlinear Characterization of Mercurochrome Dye for Optical Limiting, Opt. Appl. 40, 2010, pp. 187–196.
[33] H. A. Badran et al., Laser-Induced Optical Nonlinearities in Orange G Dye: Polyacrylamide Gel, Can. J. Phys. 89, 2011, pp. 1219–1224.
[34] S. Jeyaram and T. Geethakrishnan, Third-Order Nonlinear Optical Properties of Acid Green 25 Dye by Z-Scan Method, Opt. Laser Technol. 89, 2017, pp. 2570–2574.
[35] L. T. Nguyen et al., Numerical Methods for Analyzing Z-Scan Data, J. Nonlinear Opt. Phys. Mater. 23, 2014, pp. 1450020.
[36] K. Lee et al., Non-Resonant Power-Efficient Directional Nd:Yag Ceramic Laser Using a Scattering Cavity, Nat. Commun. 12, 2021, pp. 8.
[37] K. M. Abedin et al., Power Stability of Different Lasers and Its Effect on Phase-Stepping Shearography, Results Opt. 12, 2023, pp. 100490.
[38] P. D. Quy et al., Calculation and Design of Thermal Object Lens Using an Aspherical Lens for Handheld Observation Devices, J. Military Sci. Technol. 76, 2022, pp. 127–136.
[39] S. Sparrold and A. Lansing, Spherical Aberration Compensation Plates, Opt. Comp. Photonic Int., 2011.
[40] D. C. Compertore et al., Development of a Calibration Standard for Spherical Aberration, Proc. SPIE, 2013.
[41] D. Malacara, Handbook of Lens Design, Marcel Dekker, 1994.