Bessel–Gauss Beams of Arbitrary Integer Order: Propagation Profile, Coherence Properties, and Quality Factor
Abstract
:1. Introduction
2. Paraxial Wave Equation for Parabolic Media
2.1. Space of Solutions (Stationary and Guided LG Modes)
2.2. Subspaces with a Well-Defined Optical Angular Momentum
3. Guided Bessel–Gauss Modes as Generalized Coherent States
- Modified Bessel and Bessel profiles. For the functions consist of a modified Bessel function modulated by a Gaussian distribution of standard deviation . The latter accelerates the radial decay of the beam. Depending on the eigenvalue-phase , the function is interchanged by , and vice versa (the same occurs at specific points along the propagation axis, see details below).
- Contribution of the p-th harmonic. The most likely eigenvalue occurring in the superposition is determined by the expectation value . The helicity parameter ℓ is fixed, so the relevant information is encoded in the expectation value of the number operator . The straightforward calculation yields
3.1. Variances and Standard Deviations
3.2. Beam Quality
3.3. Propagation Properties
3.3.1. Behavior for Real Eigenvalues
- Initial configuration and periodicity. Considering as the point of departure, the initial configuration of the beam is recovered at the points , with , see Figure 5. At any of these points, the Bessel function is real-valued and exhibits a denumerable set of zeros. The latter defines a radial distribution of the phase plane where produces a phase shift . In turn, the polar variable sweeps times the interval in every one of the regions defined by the sign of .
- Self-focusing. A second class of interesting points distributed along the propagation axis is defined by the rule , with . The self-focus of the field is produced twice in each period , just at the points of the propagation axis, see Figure 5.
- Maximum spreading of the beam. At the points , with , the coherent state changes its profile from the Bessel function of the first kind to the modified Bessel function . The latter yields the maximum radial uncertainty of the beam. The intensity distribution is blurred on the transverse plane, and the phase distribution occurs without the radial distribution of the previous cases (the Bessel function has no zeros along the real axis of the complex u-plane if is an integer [43]). Figure 8 and Figure 9 show the transverse intensity and phase distribution for and , respectively.
- Vortices. At any other point of the propagation axis, the Bessel function appearing in (10) is complex-valued. Thus, contributes to the global phase of with a term that depends on in general. As a consequence, the phase distribution of the initial configuration is distorted such that it exhibits vortices. This is illustrated in Figure 10 for , the real and imaginary parts of change sign in different regions of the transverse plane. The phase distribution is, therefore, characterized by the quotient of such signs.
3.3.2. Behavior for Pure Imaginary Eigenvalues
4. Discussion
5. Conclusions
- ∘
- The maximum transverse-spreading of any BG mode is parameterized by . The shorter the value of , the better the collimation of the corresponding beam.
- ∘
- The quality of the BG modes is also parameterized by : the shorter the value of , the closer the BG modes are to the Gaussian profile.
- ∘
- The optical angular momentum spoils the beam quality: poor beam quality results for large , no matter how small is.
- ∘
- The fundamental LG mode is dominant in any superposition intended to build high-quality BG modes. The contribution of the remaining LG modes can be treated as a disturbance (noise) that deviates the BG mode from the ideal Gaussian profile.
- ∘
- The profile of the BG mode is for , and for .
- ∘
- No matter the value of , the profile of the BG modes changes periodically from to as the beam propagates along the z-axis.
- ∘
- The transverse-spreading of the BG modes is always finite and changes periodically from its maximum value to its minimum value at very specific points along the propagation axis.
- ∘
- At any other point of the propagation axis, the phase distribution of the BG modes exhibits vortices.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. The Irreducible Representation Space of the Lie Algebra
- (i)
- The ensemble is in one-to-one correspondence with the energy spectrum of the 2D quantum oscillator.
- (ii)
- The square integrability condition for quantum bound states corresponds to finite transverse optical power for localized optical beams [24].
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1/2 | 1 | 3/2 | 2 | |
3/2 | 2 | 5/2 | 3 | |
5/2 | 3 | 7/2 | 4 | |
7/2 | 4 | 9/2 | 5 |
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Cruz y Cruz, S.; Gress, Z.; Jiménez-Macías, P.; Rosas-Ortiz, O. Bessel–Gauss Beams of Arbitrary Integer Order: Propagation Profile, Coherence Properties, and Quality Factor. Photonics 2023, 10, 1162. https://doi.org/10.3390/photonics10101162
Cruz y Cruz S, Gress Z, Jiménez-Macías P, Rosas-Ortiz O. Bessel–Gauss Beams of Arbitrary Integer Order: Propagation Profile, Coherence Properties, and Quality Factor. Photonics. 2023; 10(10):1162. https://doi.org/10.3390/photonics10101162
Chicago/Turabian StyleCruz y Cruz, Sara, Zulema Gress, Pedro Jiménez-Macías, and Oscar Rosas-Ortiz. 2023. "Bessel–Gauss Beams of Arbitrary Integer Order: Propagation Profile, Coherence Properties, and Quality Factor" Photonics 10, no. 10: 1162. https://doi.org/10.3390/photonics10101162
APA StyleCruz y Cruz, S., Gress, Z., Jiménez-Macías, P., & Rosas-Ortiz, O. (2023). Bessel–Gauss Beams of Arbitrary Integer Order: Propagation Profile, Coherence Properties, and Quality Factor. Photonics, 10(10), 1162. https://doi.org/10.3390/photonics10101162