Referierte Publikationen

2021

Z. Sun, F. Tuitje, and C. Spielmann
A review of high‐resolution microscopic ghost imaging with a low‐dose pseudothermal light
J. Microsc., 284 :3 (July 2021)
B. Böning, and S. Fritzsche
Above-threshold ionization driven by Gaussian laser beams: beyond the electric dipole approximation
J. Phys. B, 54 :144002 (July 2021)
Abstract:
Strong-field atomic experiments have recently become sensitive to nondipole (magnetic) interactions. In particular, photoelectrons emitted in above-threshold ionization possess a nonzero momentum along the beam axis as a result of the Lorentz force. Here, we show how this longitudinal momentum can be theoretically calculated based on a nondipole strong-field approximation that accounts not only for the temporal but also the spatial dependence of the laser field in the photoelectron continuum. If the driving laser beam is approximated as a plane wave, the theoretical values differ from known experimental results by a constant offset. We demonstrate that this offset can successfully be removed if a realistic Gaussian beam profile is accounted for in the quantum description of ATI. We also discuss the influence of the size of the beam waist in the focus.
V. Schuster, V. Hilbert, R. Klas, C. Liu, M. Tschernajew, B. Bernhardt, J. Rothhardt, and J. Limpert
Agile spectral tuning of high order harmonics by interference of two driving pulses
Opt. Express, 29 :22117 (July 2021)
Abstract:
In this work, the experimental realization of a tunable high photon flux extreme ultraviolet light source is presented. This is enabled by high harmonic generation of two temporally delayed driving pulses with a wavelength of 1030 nm, resulting in a tuning range of 0.8 eV at the 19th harmonic at 22.8 eV. The implemented approach allows for fast tuning of the spectrum, is highly flexible and is scalable towards full spectral coverage at higher photon energies.
M. Alamoudi, M. Sattari, M. Balubaid, E. Eftekhari-Zadeh, E. Nazemi, O. Taylan, and E. Kalmoun
Application of Gamma Attenuation Technique and Artificial Intelligence to Detect Scale Thickness in Pipelines in Which Two-Phase Flows with Different Flow Regimes and Void Fractions Exist
Symmetry, 13 :1198 (July 2021)
Abstract:
Scale deposits can reduce equipment efficiency in the oil and petrochemical industry. The gamma attenuation technique can be used as a non-invasive effective tool for detecting scale deposits in petroleum pipelines. The goal of this study is to propose a dual-energy gamma attenuation method with radial basis function neural network (RBFNN) to determine scale thickness in petroleum pipelines in which two-phase flows with different symmetrical flow regimes and void fractions exist. The detection system consists of a dual-energy gamma source, with Ba-133 and Cs-137 radioisotopes and two 2.54-cm x 2.54-cm sodium iodide (NaI) detectors to record photons. The first detector related to transmitted photons, and the second one to scattered photons. The transmission detector recorded two signals, which were the counts under photopeak of Ba-133 and Cs-137 with the energy of 356 keV and 662 keV, respectively. The one signal recorded in the scattering detector, total counts, was applied to RBFNN as the inputs, and scale thickness was assigned as the output.
Y. Ma, S. Salman, C. Li, C. Mahnke, Y. Hua, S. Droste, J. Fellinger, A. Mayer, O. Heckl, C. Heyl, and I. Hartl
Compact, All-PM Fiber Integrated and Alignment-Free Ultrafast Yb:Fiber NALM Laser With Sub-Femtosecond Timing Jitter
J. Lightwave Technol., 39 :4431 (July 2021)
J. Gollwitzer, L. Bocklage, R. Röhlsberger, and G. Meier
Connecting Fano Interference and the Jaynes-Cummings Model in Cavity Magnonics
npj Quantum Inf, 7 :114 (July 2021)
Abstract:
We show that Fano interference can be realized in a macroscopic microwave cavity coupled to a spin ensemble at room temperature. Via a formalism developed from the linearized Jaynes-Cummings model of cavity electromagnonics, we show that generalized Fano interference emerges from the photon–magnon interaction at low cooperativity. In this regime, the reflectivity approximates the scattering cross-section derived from the Fano-Anderson model. Although asymmetric lineshapes in this system are often associated with the Fano formalism, we show that whilst Fano interference is actually present, an exact Fano form cannot be achieved from the linear Jaynes-Cummings model. In the Fano model an additional contribution arises, which is attributed to decoherence in other systems, and in this case is due to the resonant nature of the photonic mode. The formalism is experimentally verified and accounts for the asymmetric lineshapes arising from the interaction between magnon and photon channels. As the magnon–photon coupling strength is increased, these channels merge into hybridized magnon–photon modes and the generalized Fano interference picture breaks down. Our results are universally applicable to systems underlying the linearized Jaynes-Cummings Hamiltonian at low cooperativity and connect the microscopic parameters of the quantum optical model to generalized Fano lineshapes.
T. Kaaden, V. Tympel, M. Kober, F. Schmidl, M. Rettenmayr, and S. Lippmann
Electric pulse heating device for the analysis of solid/solid phase transformations
Rev. Sci. Instrum., 92 :074703 (July 2021)
Abstract:
Ohmic pulse heating is applied to investigate diffusion and interface controlled solid-state phase transformations. The developed device uses extensive solid-state electronics providing a high current, low voltage approach that overcomes the limitations of existing setups, most notably the use of sample geometries that allow for the reliable measurement of local temperatures and their assignment to microstructures. Power for heating is supplied by a capacitor array with adjustable voltage, and the process is controlled by microcontrollers and a solid-state relay, which allows for controlled pulses that are adjustable in microseconds. Electric currents of up to 22 kA at 90 V can be realized by the setup. Electric data are monitored and collected during the experiments, and temperature data are captured using a high-resolution infrared camera at high frame rates (1200 fps). The capabilities of the setup are demonstrated by rapid heating (10(6) K/s) and subsequent cooling of a brass sample. Two distinct areas of the sample are analyzed in detail, showing similar heating, but different cooling curves with rates of 10(4) and 10(2) K/s. Local microstructure analysis shows that different phase transformation mechanisms were dominant, and thus, the setup fulfills its purpose.
P.-M. Hillenbrand, K. N. Lyashchenko, S. Hagmann, O. Andreev, D. Banaś, E. P. Benis, I. Bondarev, C. Brandau, E. De Filippo, O. Forstner, J. Glorius, R. E. Grisenti, A. Gumberidze, D. L. Guo, M. O. Herdrich, M. Lestinsky, Y. Litvinov, V. Pagano, N. Petridis, M. S. Sanjari, D. Schury, U. Spillmann, S. Trotsenko, M. Vockert, A. B. Voitkiv, G. Weber, and T. Stoehlker
Electron-loss-to-continuum cusp in collisions of U89+ with N-2 and Xe
Phys. Rev. A, 104 :012809 (July 2021)
Abstract:
We study the electron-loss-to-continuum (ELC) cusp experimentally and theoretically by comparing the ionization of U89+ projectiles in collisions with N-2 and Xe targets, at a beam energy of 75.91 MeV/u. The coincidence measurement between the singly ionized projectile and the energy of the emitted electron is used to compare the shape of the ELC cusp at weak and strong perturbations. A significant energy shift for the centroid of the electron cusp is observed for the heavy target of Xe as compared to the light target of N-2. Our results provide a stringent test for fully relativistic calculations of double-differential cross sections performed in the first-order approximation and in the continuum-distorted-wave approach.
J. Hofbrucker, B. Böning, A. V. Volotka, and S. Fritzsche
Elliptical dichroism in biharmonic ionization of atoms
Phys. Rev. A, 104 :013102 (July 2021)
Abstract:
In multiphoton ionization of atoms, elliptical dichroism may arise in the photoelectron angular distributions due to the interference of the possible ionization pathways. We here consider the interaction of atoms with an elliptically polarized biharmonic $(ømega + 2ømega)$ field which simultaneously allows one- and two-photon ionization of the atoms. The interference between these two ionization pathways introduces contributions to the elliptical dichroism in addition to the dichroism that arises from the two-photon ionization alone. We show that these additional dichroism contributions can lead to a stronger dichroism in comparison to the one arising from two-photon ionization only. We present a relativistic analysis of the corresponding photoelectron angular distributions and discuss individual contributions to the dichroic phenomena. Detailed computations have been performed for biharmonic ionization of neutral helium atoms.
P. Wustelt, F. Oppermann, S. Mhatre, M. Kuebel, A. Sayler, M. Lein, S. Graefe, and G. Paulus
Laser-Driven Anharmonic Oscillator: Ground-State Dissociation of the Helium Hydride Molecular Ion by Midinfrared Pulses
Phys. Rev. Lett., 127 :043202 (July 2021)
Abstract:
The vibrational motion of molecules represents a fundamental example of an anharmonic oscillator. Using a prototype molecular system, HeH+, we demonstrate that appropriate laser pulses make it possible to drive the nuclear motion in the anharmonic potential of the electronic ground state, increasing its energy above the potential barrier and facilitating dissociation by purely vibrational excitation. We find excellent agreement between the frequency-dependent response of the helium hydride molecular cation to both classical and quantum mechanical simulations, thus removing any ambiguities through electronic excitation. Our results provide access to the rich dynamics of anharmonic quantum oscillator systems and pave the way to state-selective control schemes in ground-state chemistry by the adequate choice of the laser parameters.
S. Willing, K. Schlage, L. Bocklage, M. M. R. Moayed, T. Gurieva, G. Meier, and R. Röhlsberger
Novel Tunnel Magnetoresistive Sensor Functionalities via Oblique-Incidence Deposition
ACS Appl. Mater. Interfaces, 13 :32343 (July 2021)
Abstract:
Controlling the magnetic properties of ultrathin films remains one of the main challenges to the further development of tunnel magnetoresistive (TMR) device applications. The magnetic response in such devices is mainly governed by extending the primary TMR trilayer with the use of suitable contact materials. The transfer of magnetic anisotropy to ferromagnetic electrodes consisting of CoFeB layers results in a field-dependent TMR response, which is determined by the magnetic properties of the CoFeB as well as the contact materials. We flexibly apply oblique-incidence deposition (OID) to introduce arbitrary intrinsic in-plane anisotropy profiles into the magnetic layers. The OID-induced anisotropy shapes the magnetic response and eliminates the requirement of additional magnetic contact materials. Functional control is achieved via an adjustable shape anisotropy that is selectively tailored for the ultrathin CoFeB layers. This approach circumvents previous limitations on TMR devices and allows for the design of new sensing functionalities, which can be precisely customized to a specific application, even in the high field regime. The resulting sensors maintain the typical TMR signal strength as well as a superb thermal stability of the tunnel junction, revealing a striking advantage in functional TMR design using anisotropic interfacial roughness.
R. Hollinger, E. Haddad, M. Zapf, V. Shumakova, P. Herrmann, R. Roeder, I. Uschmann, U. Reisloehner, A. Pugžlys, A. Baltuska, F. Légaré, M. Zuerch, C. Ronning, C. Spielmann, and D. Kartashov
Role of free-carrier interaction in strong-field excitations in semiconductors
Phys. Rev. B, 104 :035203 (July 2021)
Abstract:
The interaction of laser pulses with condensed matter forms the basis of light-wave-driven electronics potentially enabling tera- and petahertz switching rate applications. Carrier control using near- and midinfrared pulses is appealing for integration into existing platforms. Toward this end, a fundamental understanding of the complexity of phenomena concerning sub-band-gap driven semiconductors such as high harmonic generation, carrier excitation due to multiphoton absorption, and interband tunneling as well as carrier-carrier interactions due to strong acceleration in infrared transients is important. Here, stimulated emission from polycrystalline ZnO thin films for pump wavelengths between 1.2 mu m (1 eV) and 10 mu m (0.12 eV) is observed. Contrary to the expected higher intensity threshold for longer wavelengths, the lowest threshold pump intensity for stimulated emission is obtained for the longest pump wavelength corroborating the importance of collisional excitation upon intraband electron acceleration.
F. Liu, Z. Chen, T. Morishita, K. Bartschat, B. Böning, and S. Fritzsche
Single-cycle versus multicycle nonsequential double ionization of argon
Phys. Rev. A, 104 :013105 (July 2021)
Abstract:
Using an improved quantitative rescattering model, we calculate the correlated two-electron momentum distributions (CMDs) for nonsequential double ionization of Ar exposed to intense laser pulses with a wavelength of 790 nm at a peak intensity of 1.0×10¹⁴ W/cm². We analyze the drastic variations in the CMDs that were observed by Kübel et al. [New J. Phys. 16, 033008 (2014)] in the transition from near-single-cycle to multicycle driving laser pulses. Our model reproduces their experimental data well. We also find that the transition from near-single-cycle to multicycle driving laser pulses depends strongly on the details of the pulse envelope. Special attention is paid to the mechanisms responsible for the cross-shaped structure observed experimentally with 4 fs pulses. Our analysis reveals that the cross-shaped structure in the carrier-envelope phase-averaged CMD for near-single-cycle pulses can be attributed to strong backward scattering of the recolliding electron as well as the narrow momentum distributions of the tunnel-ionized electrons compared to those for long pulses. This also explains why the cross-shaped distributions collapse to a rather structureless distribution when the pulse duration is increased to 8 fs.
W. Eschen, S. Wang, C. Liu, R. Klas, M. Steinert, S. Yulin, H. Meissner, M. Bussmann, T. Pertsch, J. Limpert, and J. Rothhardt
Towards attosecond imaging at the nanoscale using broadband holography-assisted coherent imaging in the extreme ultraviolet
Communications Physics, 4 :154 (July 2021)
Abstract:
The inherently broad bandwidth of attosecond pulses conflicts with the coherence requirements of lensless imaging. Here, broadband holography-assisted coherent imaging is demonstrated with a resolution of less than 35 nm. Nanoscale coherent imaging has emerged as an indispensable modality, allowing to surpass the resolution limit given by classical imaging optics. At the same time, attosecond science has experienced enormous progress and has revealed the ultrafast dynamics in complex materials. Combining attosecond temporal resolution of pump-probe experiments with nanometer spatial resolution would allow studying ultrafast dynamics on the smallest spatio-temporal scales but has not been demonstrated yet. To date, the large bandwidth of attosecond pulses poses a major challenge to high-resolution coherent imaging. Here, we present broadband holography-enhanced coherent imaging, which enables the combination of high-resolution coherent imaging with a large spectral bandwidth. By implementing our method at a high harmonic source, we demonstrate a spatial resolution of 34 nm in combination with a spectral bandwidth of 5.5 eV at a central photon energy of 92 eV. The method is single-shot capable and retrieves the spectrum from the measured diffraction pattern.
V. Schuster, C. Liu, R. Klas, P. Dominguez, J. Rothhardt, J. Limpert, and B. Bernhardt
Ultraviolet dual comb spectroscopy: a roadmap
Opt. Express, 29 :21859 (July 2021)
Abstract:
Dual Comb Spectroscopy proved its versatile capabilities in molecular fingerprinting in different spectral regions, but not yet in the ultraviolet (UV). Unlocking this spectral window would expand fingerprinting to the electronic energy structure of matter. This will access the prime triggers of photochemical reactions with unprecedented spectral resolution. In this research article, we discuss the milestones marking the way to the first UV dual comb spectrometer. We present experimental and simulated studies towards UV dual comb spectroscopy, directly applied to planned absorption measurements of formaldehyde (centered at 343 nm, 3.6 eV) and argon (80 nm, 16 eV). This will enable an unparalleled relative resolution of up to 10-9 - with a table-top UV source surpassing any synchrotron-linked spectrometer by at least two and any grating-based UV spectrometer by up to six orders of magnitude.
E. Mosman, and F. Karbstein
Vacuum birefringence and diffraction at an x-ray free-electron laser: From analytical estimates to optimal parameters
Phys. Rev. D, 104 :013006 (July 2021)
Abstract:
We study vacuum birefringence and x-ray photon scattering in the head-on collision of x-ray free electron and high-intensity laser pulses. Resorting to analytical approximations for the numbers of attainable signal photons, we analyze the behavior of the phenomenon under the variation of various experimental key-parameters and provide new analytical scalings. Our optimized approximations allow for quantitatively accurate results on the one-percent level. We in particular demonstrate that an appropriate choice of the x-ray focus and pulse duration can significantly improve the signal for given laser parameters, using the experimental parameters to be available at the Helmholtz International Beamline for Extreme Fields at the European XFEL as example. Our results are essential for the identification of the optimal choice of parameters in a discovery experiment of vacuum birefringence at the high-intensity frontier.
A. Gumberidze, D. Thorn, A. Surzhykov, C. Fontes, D. Banaś, H. Beyer, W. Chen, R. Grisenti, S. Hagmann, R. Hess, P.-M. Hillenbrand, P. Indelicato, C. Kozhuharov, M. Lestinsky, R. Märtin, N. Petridis, R. Popov, R. Schuch, U. Spillmann, S. Tashenov, S. Trotsenko, A. Warczak, G. Weber, W. Wen, D. Winters, N. Winters, Z. Yin, and T. Stöhlker
Angular Distribution of Characteristic Radiation Following the Excitation of He-Like Uranium in Relativistic Collisions
Atoms, 9 :20 (June 2021)
Abstract:
In this paper, we present an experimental and theoretical study of excitation processes for the heaviest stable helium-like ion, that is, He-like uranium occurring in relativistic collisions with hydrogen and argon targets. In particular, we concentrate on angular distributions of the characteristic K alpha radiation following the K -> L excitation of He-like uranium. We pay special attention to the magnetic sub-level population of the excited 1s2l(j) states, which is directly related to the angular distribution of the characteristic K alpha radiation. We show that the experimental data can be well described by calculations taking into account the excitation by the target nucleus as well as by the target electrons. Moreover, we demonstrate for the first time an important influence of the electron-impact excitation process on the angular distributions of the K alpha radiation produced by excitation of He-like uranium in collisions with different targets.
S. Salman, Y. Ma, K. Gürel, S. Schilt, C. Li, P. Pfäfflein, C. Mahnke, J. Fellinger, S. Droste, A. Mayer, O. Heckl, T. Südmeyer, C. Heyl, and I. Hartl
Comparison of two low-noise CEO frequency stabilization methods for an all-PM Yb:fiber NALM oscillator
OSA Continuum, 4 :1889 (June 2021)
H. Gies, and A. Salek
Curvature bound from gravitational catalysis in thermal backgrounds
Phys. Rev. D, 103 :125027 (June 2021)
Abstract:
We investigate the phenomenon of gravitational catalysis, i.e., curvature-induced chiral symmetry breaking and fermion mass generation, at finite temperature. Using a scale-dependent analysis, we derive a thermal bound on the curvature of local patches of spacetime. This bound quantifies regions in parameter space that remain unaffected by gravitational catalysis and thus are compatible with the existence of light fermions as observed in nature. While finite temperature generically relaxes the curvature bound, we observe a comparatively strong dependence of the phenomenon on the details of the curvature. Our bound can be applied to scenarios of quantum gravity, as any realistic candidate has to accommodate a sufficient number of light fermions. We argue that our bound therefore represents a test for quantum-gravity scenarios: A suitably averaged spacetime in the (trans-)Planckian regime that satisfies our curvature bound does not induce correspondingly large Planckian fermion masses by gravitational catalysis. The temperature dependence derived in this work facilitates to follow the fate of gravitational catalysis during the thermal history of the (quantum) Universe. In an application to the asymptotic-safety scenario of quantum gravity, our bound translates into a temperature-dependent upper bound on the number of fermion flavors.
Y. Ma, D. Seipt, K. Krushelnick, and A. G. R. Thomas
Generation of straight and curved hollow plasma channels by laser-generated nonlinear wakefields and studies of ultra-intense laser pulse guiding
Phys. Plasmas, 28 :063104 (June 2021)
Abstract:
We report on a study of highly controllable, quasi-static hollow plasma channels generated by ion motion following a laser–plasma interaction via numerical simulations. These channels are generated by ion motion in a nonlinear wakefield, which has an asymmetry between focusing and defocusing periods. Such plasma channels, driven by a fs duration, 100 TW level laser pulse, have the capability to guide more powerful laser pulses. Moreover, we show that curved plasma channels can be generated in a transverse density gradient plasma and all-optical guiding of a second, significantly more powerful laser pulse in such a curved plasma channel. This mechanism may be useful for creating plasma optics for multi-stage TeV laser plasma accelerators and compact synchrotron radiation sources.
B. Baghdasaryan, F. Steinlechner, and S. Fritzsche
Justifying the thin-crystal approximation in spontaneous parametric down-conversion for collinear phase matching
Phys. Rev. A, 103 :063508 (June 2021)
Abstract:
Spatially engineered photons from spontaneous parametric down-conversion (SPDC) are a valuable tool for studying and applying photonic entanglement. An advantage of SPDC is that simple expressions for the two-photon state can be obtained using justified approximations. In particular, the thin-crystal approximation has often been invoked in the engineering of high-dimensional entangled states. Knowledge of the conditions under which the thin-crystal approximation remains valid is essential for the realization of experimental setups. We provide a quantitative guideline on the validity of the thin-crystal approximation in calculating the two-photon spatial state. In particular, we show that the applicability of this regime is related to the focusing parameter (w) over barp = w(p)/root lambda(p) L, where w(p) and lambda(p) are the beam waist and wavelength of the pump beam, respectively, and L is the length of the nonlinear crystal. Additionally, the validity of the thin-crystal regime is investigated concerning the size of a subspace in the Laguerre Gaussian basis, into which the two-photon state can be projected in a given experiment.
R. Soguel, A. Volotka, D. Glazov, and S. Fritzsche
Many-Electron QED with Redefined Vacuum Approach
Symmetry, 13 :1014 (June 2021)
Abstract:
The redefined vacuum approach, which is frequently employed in the many-body perturbation theory, proved to be a powerful tool for formula derivation. Here, we elaborate this approach within the bound-state QED perturbation theory. In addition to general formulation, we consider the particular example of a single particle (electron or vacancy) excitation with respect to the redefined vacuum. Starting with simple one-electron QED diagrams, we deduce first- and second-order many-electron contributions: screened self-energy, screened vacuum polarization, one-photon exchange, and two-photon exchange. The redefined vacuum approach provides a straightforward and streamlined derivation and facilitates its application to any electronic configuration. Moreover, based on the gauge invariance of the one-electron diagrams, we can identify various gauge-invariant subsets within derived many-electron QED contributions.
M. Mueller, J. Buldt, H. Stark, C. Grebing, and J. Limpert
Multipass cell for high-power few-cycle compression
Opt. Lett., 46 :2678 (June 2021)
Abstract:
A multipass cell for nonlinear compression to few-cycle pulse duration is introduced composing dielectrically enhanced silver mirrors on silicon substrates. Spectral broadening with 388 W output average power and 776 mu J pulse energy is obtained at 82% cell transmission. A high output beam quality (M-2 < 1.2) and a high spatio-spectral homogeneity (97.5%), as well as the compressibility of the output pulses to 6.9 fs duration, are demonstrated. A finite element analysis reveals scalability of this cell to 2 kW average output power.
Y. Ma, D. Seipt, A. E. Hussein, S. Hakimi, N. F. Beier, S. B. Hansen, J. Hinojosa, A. Maksimchuk, J. Nees, K. Krushelnick, A. G. R. Thomas, and F. Dollar
The effects of laser polarization and wavelength on injection dynamics of a laser wakefield accelerator
Phys. Plasmas, 28 :063101 (June 2021)
Abstract:
Here, we investigate the effects of laser polarization and wavelength on electron injection dynamics in a laser wakefield accelerator. During the ionization process, electrons gain residual momentum and kinetic energy via above threshold ionization, which has a strong dependence on laser polarization. A circularly polarized laser pulse results in a much higher residual momentum and kinetic energy gain for the ionized electrons compared with the linearly polarized case. This residual momentum results in particle injection because of the sensitivity of particle trapping to the initial conditions and enhanced the total injected beam charge in both experiments and particle-in-cell simulations. Due to the strong correlation of above threshold ionization with laser wavelength, in this work we extended the investigation to long wavelength (up to 20 μm) drive pulses using particle-in-cell simulations. Owing to the gain in kinetic energy, it may be expected that the charge trapped would consistently increase for circular polarization with increasing laser wavelength, but this was not observed. Instead, there are oscillations with wavelength in the relative trapped charge between linear and circular polarization cases, which arise because of ionization and heating effects on the plasma. Our studies highlight the complex interplay between several different physical effects, including injection regimes—above threshold ionization assisted injection, wave-breaking injection by carrier-envelope-phase effects and ionization injection—ionization gradient induced laser pulse evolution, and thermal modifications to the wake structure that need considering when extrapolating laser wakefield acceleration to different wavelength regimes.
G. von Gersdorff, S. Panahiyan, and W. Chen
Unification of topological invariants in Dirac models
Phys. Rev. B, 103 :245146 (June 2021)
Abstract:
Topological phases of materials are characterized by topological invariants that are conventionally calculated by different means according to the dimension and symmetry class of the system. For topological materials described by Dirac models, we introduce a wrapping number as a unified approach to obtain the topological invariants in arbitrary dimensions and symmetry classes. Given a unit vector that parametrizes the momentum dependence of the Dirac model, the wrapping number describes the degree of the map from the Brillouin zone torus to the sphere formed by the unit vector that we call the Dirac sphere. This method is gauge-invariant and originates from the intrinsic features of the Dirac model and moreover places all known topological invariants, such as the Chern number, winding number, Pfaffian, etc, on equal footing.