Equivalent Fresnel Number Research Articles

Refined beam expansion method for unstable resonator modes

A refined and simplified version of the beam expansion method for computing the modes of unstable resonators is presented here.The principle of the beam expansion method is to express the field pattern on a basis of Hermite-Gaussian functions. Propagating each individual component around the resonator enables a transfer matrix to be constructed from which the eigenvalues and eigenvectors of the modes can be determined.Whereas the earlier proof-of-principle approach performed well for low Fresnel numbers, the refined technique gives good results for the challenging test case of magnification M=1.9 and equivalent Fresnel number Neq=49.4. The performance enhancement has been achieved by incorporating a number of adjustments that improve the accuracy of the calculation.

Excess noise of the misaligned off-axis confocal unstable resonators

The misalignment of the output mirror will cause serious decline of the output power of the positive branch off-axial confocal unstable resonators. In this paper, we try to explain this phenomenon by investigating the excess noise factor K for the misaligned resonator. Numerical result shows that the jump of the value of K due to the well-known mode crossing also occurs periodically against the degree of the misalignment (ε), the locations and the period of the jump are related the equivalent Fresnel number Neq. In addition, it shows the value of K, the loss, and the far-field divergence angle are small even for large downward misalignment, it implies that it may be able to gain laser output under the condition.

Thin disk laser with unstable resonator and reduced output coupler

In this paper, feasibility of using unstable resonator with reduced output coupling in a thin disk laser is studied theoretically. Unstable resonator is modeled by wave-optics using Collins integral and iterative method. An Yb:YAG crystal with 250 micron thickness is considered as a quasi-three level active medium and modeled by solving rate equations of energy levels populations. The amplification of laser beam in the active medium is calculated based on the Beer-Lambert law and Rigrod method. Using generalized beam parameters method, laser beam parameters like, width, divergence, M2 factor, output power as well as near and far-field beam profiles are calculated for unstable resonator. It is demonstrated that for thin disk laser (with single disk) in spite of the low thickness of the disk which leads to low gain factor, it is possible to use unstable resonator (with reduced output coupling) and achieve good output power with appropriate beam quality. Also, the behavior of output power and beam quality versus equivalent Fresnel number is investigated and optimized value of output coupling for maximum output power is achieved.

Error analysis of the virtual-source method

The virtual source method (VSM) is well known as a very efficient approach in generating the mode profile of unstable resonators (UR). However, as far as we know, the error of this method has rarely been examined in details. In this paper, we compare the formula in VSM with the uniform asymptotic solution of the Fresnel-Kirchhoff diffraction integral for a two-aperture model. It is shown that, the accuracy of edge wave functions is sensitive to both the equivalent Fresnel number (Neq) and the geometrical magnification (M) of UR, the error becomes Intolerable when they become very small. A criterion is presented to exam the validity of VSM for specific combination of Neq and M. The origin of the so-called “mask effect” is also discussed, it may be caused by treating all the geometric boundary of EWs emitted from different virtual source as the same.

Research on beam characteristics in a large-Fresnel-number unstable-waveguide hybrid resonator with parabolic mirrors.

Large-Fresnel-number unstable-waveguide hybrid resonators employing spherical resonator mirrors suffer from spherical aberration, which adversely affects beam quality and alignment sensitivity. In this paper, we present experimental and numerical wave-optics simulations of the beam characteristics of a negative-branch hybrid resonator having parabolic mirrors with a large equivalent Fresnel number in the unstable direction. These results are compared with a resonator using spherical mirrors. Using parabolic mirrors, the output beam has a smaller beam spot size and higher power density at the focal plane. We found that the power extraction efficiency is 3.5% higher when compared with a resonator using spherical mirrors as the cavity length was varied between -1 and 1 mm from the ideal confocal resonator. In addition, the power extraction efficiency is 5.6% higher for mirror tilt angles varied between -6 and 6 mrad. Furthermore, the output propagating field is similar to a converging wave for a spherical mirror resonator and the output beam direction deviates 3.5 mrad from the optical axis. The simulation results are in good agreement with the experimental results.

高功率激光系统中缺陷引起的近场调制分析

In high power laser device, near-field beam quality is evaluated through the intensity distribution. However, the measured distribution is a steady state, which can not reflect the evolution features of intensity modulation caused by noise disturbances. Some area encountering serious intensity modulation may be ignored in transmission process. Near-field transmission characteristics of the beam disturbed by noise disturbance are studied. Meanwhile, deeper understanding of the limitations of the measured near-field intensity distribution is obtained. To simplify the analysis of the near-field intensity distribution at different distance under the influence of single local defect, the characteristics of near-field intensity modulation is represented by equivalent Fresnel number. The results show that once a weak modulated information appears in the distribution of measured nearfield intensity, there may be a more serious modulation area before the measurement position. A highest modulation can be up to nine times for a phase type defects with pi delay corresponding to incident intensity.

A modified virtual source method for off-axis unstable resonators with large equivalent Fresnel number

Physical significance of the ‘edge wave’ in the asymptotic solution of the diffraction by a single slit is expounded in detail. Basing on this concept, we modify the virtual source method to the calculation of the lowest eigenmode mode profile in both the positive and negative branch strip off-axis unstable resonators. The advantages of the method are that it is conceptually simple and requires only minimal computer time specifically for the resonators with large equivalent Fresnel number.

Excess noise in strip off-axis confocal unstable resonators

The lowest-loss mode and higher-order modes in strip off-axis confocal unstable resonators are calculated by the eigenvector method. The moduli of all pairs of the adjacent order eigenvalues exhibit crossing behavior. For the lowest- and next-lowest-loss modes, crossings occur when the equivalent Fresnel number N eq is close to s + 0.08 (s positive integer), and the lowest-loss mode profile calculated by the Fox–Li iteration is no longer self-reproducing, but the superposition of both eigenmodes with unfixed complex weights varies as the number of iterations. Numerical calculations of excess noise factors K in strip off-axis confocal unstable resonators are described in detail. The value of K shows an irregular resonance switching against N eq. The hybrid model presented by van Eijkelenborg et al. is modified in this paper, and shows a good quantitative agreement with nonorthogonality theory.

Off-axial elliptical cosine-Gaussian beams and their propagation properties

In this paper, a new kind of light beam called off-axial elliptical cosine-Gaussian beam (ECosGBs) is defined by using the tensor method. An analytical propagation expression for the ECosGBs passing through axially nonsymmetrical optical systems is derived by using vector integration. The intensity distributions of ECosGBs on the input plane, on the output plane with the equivalent Fresnel number being equal to 0.1 and on the focal plane are respectively illustrated for the propagation properties. The results indicate that an ECosGB is eventually transformed into an elliptical cosh-Gaussian beam. In other words, ECosGBs and cosh-Gaussian beams act in a reciprocal manner after propagation.

Three-dimensional field structure in open unstable cavities Part II: Active cavity results

The three-dimensional field distribution of the diffractive cavity mode structure in an active, open, unstable laser resonator is presented as a function of the equivalent Fresnel number of the cavity. The active cavity mode structures are compared to that of the corresponding passive cavity so that the effects of a spatially extended, homogeneously broadened, saturable gain medium on the cavity field structure may be ascertained. The qualitative structure of this intracavity field distribution, including the central intensity core (or oscillator filament), is explained in terms of the Fresnel zone structure defined over the cavity feedback aperture.

Three-dimensional field structure in open unstable resonators Part I: Passive cavity results

The three-dimensional field distribution of the diffractive cavity mode structure in a passive, open, unstable resonator is presented as a function of the equivalent Fresnel number of the cavity. The qualitative structure of this intracavity field distribution, including the central intensity core (or oscillator filament), is characterized in terms of the Fresnel zone structure that is defined over the cavity feedback aperture. Previous related research is reviewed.

Mode-crossing interpretation of resonances in the transverse excess noise factor

It is well known that the excess noise factor K of an unstable-cavity laser shows very pronounced peaks at values of the equivalent Fresnel number for which the losses of the two lowest-loss modes become equal. This makes it natural to explore the validity of a two-mode description of K. We report that these resonances of K, as far as their shape and position is concerned, are indeed revealed by a two-mode formalism. The magnitude of the resonances, however, cannot be explained within a two-mode context.

Resonance of Quantum Noise in an Unstable Cavity Laser.

We have measured the quantum-limited lpinewidth of a hard-edged unstable cavity gas laser. Our results confirm the predicted resonant behavior of the quantum-noise strength as a function of equivalent Fresnel number. This behavior is due to the nonorthogonality of the transverse eigenmodes.

Optimization of coherent lidar performance with graded-reflectance transmitter resonator optics in the low equivalent Fresnel number regime

Using a diffractive eigenmode treatment to model the laser output we show that graded-reflectance resonator optics offer significant efficiency benefits over conventional hard-edge coupled unstable resonators in the context of coherent detection lidar applications. Extending previous research pertinent to the high equivalent Fresnel number regime, we have modeled the optimum performance of a notional super-Gaussian coupled cavity as a function of the key resonator parameters in the low equivalent Fresnel number (<3) regime. The findings from this study are applicable to the design of coherent lidar transmitters operated within this regime.

Diffractive analysis of unstable optical resonators with super-Gaussian mirrors

The azimuthally symmetric diffractive transverse eigenmodes in an unstable optical resonator with a super-Gaussian reflectivity output mirror are computed in the low-Fresnel-number region to quantify the limits of validity of the geometrical-optics solution. It is found that for low cavity magnifications (M ≤ 2) and with mirrors of super-Gaussian orders n ≤ 8, the diffractive eigenvalues deviate significantly from the geometrical-optics value for equivalent Fresnel numbers Neq 2. These new results indicate that for the geometrical-optics approximation to be valid, resonators with M ≤ 2 and n ≤ 8 should be designed with Neq ≥ 2. This will ensure that the cavity exhibits good azimuthally symmetric transverse-mode discrimination and a super-Gaussian fundamental mode.

Enhancement of modal feedback in unstable resonators using mirrors with a phase step

Numerical calculations using the Prony method have shown that the feedback coefficient in unstable laser resonators can be increased by factors of as much as 5 over the geometrical value using a mirror with a phase step at its center. A phase shift of close to pi over an area of an equivalent Fresnel number of 0.5 leads to minimum output losses. Experiments with a TEA CO(2) laser confirm the prediction. The results can be attributed to a cancellation of the output wave by destructive interference, which confines the laser beam around the resonator axis.

Mode properties and characteristics of negative branch unstable ring resonators

Laboratory-scale negative-branch unstable ring resonators were designed to simulate large Fresnel number, large-mode cross section positive-branch high-power ring lasers. These laboratory-scale lasers can be built having small-mode cross sections, long lengths, and large Fresnel numbers by incorporating imaging systems within the resonators. Large variations of Fresnel numbers can also be obtained by simple positioning of the output aperture. To verify that the diffractive modes of the high-power lasers are accu rately si mulated, ba re cavity mode discri mi nation properties of these ring resonator designs were calculated by various 3-D diffractive models. Modeling was completed for resonators with equivalent Fresnel numbers up to 1 1 and with equivalent Fresnel numbers of 10 and above using a fast Fourier transform (FFT) and a virtualsource code, respectively. Comparison of the FFT and the virtual source code results, at moderate Fresnel numbers, indicated good agreement between the different modeling techniques. Good mode discrimination was observed at half-integral equivalent Fresnel numbers, whereas poor mode discrimination was observed at integral equivalent Fresnel numbers. These results show that less computational intensive codes, such as the virtual source code, can quickly and cheaply model a resonator having a wide range of Fresnel numbers. These results also show that the equivalent Fresnel number of a near-imaging negative-branch ring resonator is a reasonably valid parameter for determining resonator diffractive phase and irradiance profiles even for resonators having large Fresnel numbers. The modeling results show that the modes of positive- and negative-branch resonators, having the same Fresnel number, behave the same to within a complex conjugate.

Modeling of multimode operation of a laser with an unstable resonator

A description is given of a new method for calculating the parameters of nonlinear multimode lasing and for obtaining a stability criterion of the solution representing multimode emission. The uniqueness of the stable solution is discussed. The results are given of investigations of the parameters of multimode radiation in a laser with an inertial saturable medium. The conditions for the appearance of multimode lasing are determined and the energy characteristics of multimode radiation are found as a function of the small-signal gain of the active medium, equivalent Fresnel number, and resonator magnification. It is shown that a change from single-mode to two-mode lasing occurs quasiperiodically when the equivalent Fresnel number is changed by unity and the small-signal gain is slightly higher than the threshold value. The optimal conditions for ensuring high brightness of multimode radiation are identified. The proposed numerical method can be used in calculations of the parameters of multimode operation of optically coupled lasers with unstable resonators, as well as lasers with three-mirror and stable resonators.

Transverse mode properties of beam-rotated unstable resonators for free electron lasers

A novel class of ring unstable resonators (UR90) with 90-deg rotation of the beam on each round-trip has been proposed as a method of avoiding disadvantages of obscuration in the output beam. We herein consider exten- sion of this idea to include an internally focused beam, so that the portion of the beam near the focus can fit within the narrow gain region of a free electron laser. Modeling of transverse mode properties of empty cavities with or without an internal focus proceeds in the same general way. We have calculated the transverse mode in- tensity and phase distributions and the power retention fraction PR as a function of equivalent Fresnel number Feq over various ranges of F^. There is quasi-oscillatory behavior similar to that found in conventional unstable resonators, with poor mode loss separation occurring near an interger TV plus 7/8; hence, designs should avoid such values of Fpn. REE electron lasers (FEL's) operated as high-power oscillators present special problems regarding resonator design. The optical beam must be small in transverse extent in the (wiggler) gain region, but the optical flux on resonator mirrors must not be allowed to damage them. The flux can be reduced to acceptable levels in at least two ways. Very long paths from gain region to mirrors can be used to spread the beam by diffraction, but the great distances involved are undesirable. Alternatively, grazing incidence beam ex- panders/contractors can be used. With the latter, one has the choice of operation in either the stable or the unstable domain. Stable resonators can provide the advantage of an unobscured output beam if a practical means is available to separate out a fixed fraction of the total circulating beam for output, but this poses practical problems (materials damage, etc.) at high flux levels. Operation in the unstable regime avoids the latter dif- ficulty by using a scraper output mirror that outputs all of the outer portion of the beam, i.e., the scraper need not be a par- tially transmitting element. The special resonator problems of FEL's are, of course, related to the general question of choice of most favorable resonator design, e.g., stable or unstable; positive or negative branch; conventional or, perhaps, novel designs for special applications, etc. The excellent transverse mode separation properties of (both positive and negative branches) unstable resonators are well known and such resonators are very widely used. Because of the internal focus of the negative branch type and the possibility of breakdown in gaseous or solid gain media, the positive branch class is most widely used. For FEL's the breakdown problem is absent because gain takes place in a vacuum, and negative branch resonators are acceptable.

Losses and mode structure of unstable resonators with spherical mirrors

Losses and near and far field intensity distributions of unstable resonators with equivalent Fresnel numbers in the range up to three have been investigated both experimentally and theoretically. The losses were determined absolutely using a new method for loss measurement. Intensity distributions of lowest loss modes were recorded by means of an image-processing system and compared to numerical solutions of Kirchhoff ’s integral equation of the empty cavity. The experiments were performed with a pulsed Nd-YAG system.