Method for determining an interaction between an electromagnetic radiation and a material
The exemplary embodiments of the method, system, software arrangement and computer-accessible medium according to the present invention facilitates an analysis of interactions between nonlinear absorbing materials and an incident electromagnetic wave based on material properties and characteristics of the incident beam of the electromagnetic energy. Using the exemplary embodiments of the present invention, it is possible to determine laser beam propagation in a variety of multiphoton absorbing materials. Energy levels associated with such materials, which may be associated with various electron absorption and/or relaxation phenomena, may be added to and/or removed from the analysis. Incident laser beams can vary from continuous wave to attoseconds in duration and a numerical solution can be obtained that is radially and/or temporally dependent. Certain exemplary embodiments of the present invention can also be used to determine certain contributions of individual electronic energy levels within the materials to the total 15 absorption.
1. A method for analyzing interactions between electromagnetic radiation and an absorbing material having a plurality of energy states, comprising:
obtaining on a computer processor first information related to the plurality of energy states associated with the absorbing material, the first information comprising a plurality of absorption and/or relaxation blocks where the absorption blocks characterize forward absorption and /or reverse absorption of electrons or excitons from one energy state to another energy state in the material and the relaxation blocks represent electron, exciton and/or phonon relaxations that occur between energy states in the material;
obtaining on the computer processor second information related to electromagnetic radiation incident on the material; and
generating on the computer processor a third information based on the first and second information, where the third information is a solution to a rate equation for a population density vector function for the energy states of the material and a propagation equation for an electric field in the material.
2. The method according to claim 1 , wherein the electromagnetic radiation is an electromagnetic wave.
3. The method according to claim 1 , wherein the electromagnetic radiation is an electric field.
4. The method according to claim 1 , where the electromagnetic radiation comprises a laser pulse.
5. The method according to claim 1 , where the electromagnetic radiation comprises a plurality of laser pulses, and where the second information comprises at least one of a duration, an intensity or an electric field associated with each of the pulses.
6. The method according to claim 1 , where the electromagnetic radiation comprises a continuous electromagnetic wave.
7. The method of claim 1 , wherein the electromagnetic radiation is coherent.
8. The method of claim 1 , wherein the electromagnetic radiation is a fundamental mode of a cylindrically symmetric waveguide.
9. The method according to claim 1 , wherein the first information comprises at least one of an energy level diagram or an energy level string.
10. The method according to claim 1 , wherein the second information comprises at least one of a wavelength, a pulse width, an intensity or an electric field level.
11. A system for analyzing interactions between electromagnetic radiation and an absorbing material having a plurality of energy states comprising:
a non-transitory computer-accessible medium which includes thereon a set of instructions, wherein the set of instructions are configured to program a processing arrangement to:
receive first information related to the plurality of energy states associated with the absorbing material, the first information comprising a plurality of absorption and/or relaxation blocks where the absorption blocks characterize forward absorption and /or reverse absorption of electrons or excitons from one energy state to another energy state in the material and the relaxation blocks represent electron, exciton and/or phonon relaxations that occur between energy states in the material;
receive second information related to electromagnetic radiation incident on the material; and
generate on the processing arrangement third information based on the first and second information, where the third information is a solution to a rate equation for a population density vector function for the energy states of the material and a propagation equation for an electric field in the material.
12. A software arrangement embedded in a suitable non-transitory recording medium for analyzing interactions between electromagnetic radiation and an absorbing material having a plurality of energy states, comprising:
a first set of instructions which, when executed by a processing arrangement, is capable of receiving first information related to the plurality of energy states associated with the absorbing material the first information comprising a plurality of absorption and/or relaxation blocks where the absorption blocks characterize forward absorption and /or reverse absorption of electrons or excitons from one energy state to another energy state in the material and the relaxation blocks represent electron, exciton and/or phonon relaxations that occur between energy states in the material;
a second set of instructions which, when executed by the processing arrangement, is capable of receiving second information related to electromagnetic radiation incident on the material; and
a third set of instructions which, when executed by the processing arrangement, is capable of generating a third information based on the first and second information, where the third information is a solution to a rate equation for a population density vector function for the energy states of the material and a propagation equation for an electric field in the material.
13. The method of claim 1 wherein the rate and propagation equations are coupled and solved iteratively.
14. The method of claim 1 wherein the absorption blocks and/or relaxation blocks are used to populate matrices in the rate equation.
15. The method of claim 1 wherein the absorption blocks and/or the relaxation blocks are used to determine an electronic population for at least one of the plurality of energy levels or to determine a transmission or absorption of electromagnetic radiation in the material.
16. A method for characterizing a material having a plurality of energy states using a rate equation for a population density vector function comprising:
forming in a computer processor a plurality of absorption blocks and/or relaxation blocks where the absorption blocks characterize forward absorption and/or reverse absorption of electrons or excitons from one energy state to another energy state in the material and the relaxation blocks represent electron, exciton and/or phonon relaxations that occur between energy states in the material;
populating in the computer processor matrices of the rate equation with the absorption and/or relaxation blocks; and
solving the rate equation using the computer processor.
17. The method of claim 16 wherein the rate equation is solved iteratively using a Taylor series expansion.
18. The method of claim 17 wherein the Taylor series expansion is a second-order Taylor series expansion.
19. A method for analyzing an interaction between a beam of electromagnetic radiation and a material having a plurality of energy states, said method using a rate equation for a population density vector function for the material and a propagation equation for an electric field associated with the beam of electromagnetic radiation incident on the material, said method comprising:
forming in a computer processor a plurality of absorption blocks and/or relaxation blocks where the absorption blocks characterize forward absorption and/or reverse absorption of electrons or excitons from one energy state to another energy state in the material and the relaxation blocks represent electron, exciton and/or phonon relaxations that occur between energy states in the material;
populating in the computer processor matrices of the rate equation and vectors of the propagation equation with the absorption and/or relaxation blocks: and
solving the rate and propagation equations using the computer processor.
20. The method of claim 19 wherein the rate and propagation equations are coupled and solved iteratively.
21. A method for analyzing an interaction between electromagnetic radiation and a material having a plurality of energy levels comprising:
forming on a computer a plurality of absorption blocks and/or relaxation blocks that describe optical transitions between the plurality of energy levels of the material wherein each of the absorption blocks and the relaxation blocks includes at least one numerical parameter, and
forming on the computer a general numerical model that allows for determining the interaction between the electromagnetic radiation and the material using the absorption blocks and/or the relaxation blocks to determine an electronic population for at least one of the plurality of energy levels or to determine a transmission or absorption of the electromagnetic radiation in the material.