CN114779486A - Flat-top light spot regulation and control method - Google Patents
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Abstract
The invention discloses a flat-topped light spot regulating and controlling method, which comprises the following steps: after the laser light source outputs a light beam, the light beam passes through the single mode fiber and the collimating lens and enters the scatterer; the scatterer adjusts the received light beam; the adjusted light beam is incident into the multimode optical fiber through the focusing lens, and then the light beam is collected through the objective lens, the focusing lens and the CCD camera for observation; and observing the change of the incident angle and the change of the emergent light spot along with the rotation of the scatterer to finish the regulation and control of the flat-top light spot. The invention realizes an easy-to-operate light spot regulation and control method based on the angle modulation when coupling SMF to MMF; the mode distribution of the light beam is regulated and controlled along with different offset emission angles or emission modes, so that the required optimal output light field is achieved; meanwhile, the method is easy to operate and realize, has extremely low cost, and has feasible application value in the fields of laser technology and optical communication.
Description
Technical Field
The invention relates to the field of all-fiber light field regulation and control systems, in particular to a flat-top light spot regulation and control method.
Background
The light emitted by many common lasers is approximately gaussian, and the uneven energy distribution makes processing and communication difficult, and in recent years, solving this problem is a goal of researchers. For example, in the invention of various beam shapers, a flat-top light spot generated by the beam shaper is commonly used for laser processing, and can prevent over-exposure or under-exposure of a specific area, and typical applications of the beam shaper include: cutting, ablation, perforation, scoring, annealing, medical and aesthetic, microscopic and scientific, slide cytometry, etc. The formation of a uniform energy beam makes the application of laser light more advantageous. Or beam shaping by the lens group, effectively changes the profile of the beam. Then, the above-described method using spatial optics is not applicable to an optical fiber system.
In recent years, the demand for bit rates of Local Area Networks (LANs) has sharply increased, asynchronous transfer mode and Ethernet standards have been achieved at bit rates of 100Mb/s and 622Mb/s, respectively, and gigabit per second Ethernet standardization is currently being considered. Such high speed links are expected to be needed primarily for office and campus backbones. Since the main fiber base currently installed in buildings is multimode fiber (MMF), its modal bandwidth imposes an upper limit on the achievable transmission speed and link distance. For such high speed transmission modes, even though alternative approaches such as Single Mode Fiber (SMF) technology and ribbon fiber links exist, it is desirable for cost reasons to develop the link using an already installed base of multimode fiber (MMF). Therefore, overcoming the modal bandwidth problem of multimode optical fibers is an important technique. And in the application of laser, the requirement of large-scale output with uniform energy on the emission technology is also severe. The key problem is that the lasers widely used today for signal transmission have a gaussian shaped beam profile and the power distribution across the beam is not uniform. This causes a number of inconveniences, and beam shaping techniques are required to solve such problems.
Disclosure of Invention
Based on the technical problems in the background art, the invention provides a flat-top light spot regulation and control method which is suitable for an all-fiber light field regulation and control system, realizes the regulation and control of a conduction mode in a multimode fiber (MMF) based on the angle modulation when the MMF is coupled by a Single Mode Fiber (SMF), and finally forms light spot energy homogenization and beam shaping; the method is suitable for application scenes of laser energy homogenization, such as an indoor multi-user optical wireless communication system, can ensure that the receiving signal-to-noise ratio of each user is consistent, and can achieve the maximization of transmission capacity.
The technical scheme adopted by the invention is as follows:
a flat-top light spot regulation and control method is characterized by comprising the following steps:
(1) after the laser light source outputs a light beam, the light beam passes through the single mode fiber and the collimating lens and enters the scatterer, and the scatterer is used for changing the angle of the light beam incident into the multimode fiber;
(2) the scatterer adjusts the received light beam: the scatterer modifies the transmission ratio of the deflected light and the meridian light of the light beam, changes the angle of the light beam incident into the multimode fiber, further changes the phase constant of the internal optical waveguide of the multimode fiber, when the phase constant is changed, the corresponding excitation electric field intensity is solved in the Helmholtz equation, and when the excitation electric field intensity is superposed on the power coupling coefficient, the expression form of the power coupling coefficient is changed, so that different output light spots are obtained;
(3) the adjusted light beam is incident into the multimode optical fiber through the focusing lens, and then the light beam is collected through the objective lens, the focusing lens and the CCD camera for observation; and observing the change of the incident angle and the change of the emergent light spot along with the rotation of the scatterer to finish the regulation and control of the flat-top light spot.
Further, the method for regulating and controlling the flat-topped facula is characterized in that the multimode fiber is a step-index multimode fiber or a graded-index multimode fiber.
Further, the method for regulating and controlling the flat-topped light spot is characterized in that a main factor in the step-index multimode fiber is that a front-end scatterer rotates to generate a certain proportion of oblique light rays, when a mode of the certain proportion of oblique light rays is coupled into the step-index multimode fiber, the mode shows that the normalized cut-off frequency of the mode is different, so that a phase constant is changed, and the change of the phase constant brings the change of an excitation electric field, thereby influencing a final power coupling coefficient, and the method for regulating and controlling the mode field based on the step-index multimode fiber comprises the following steps:
(1) field solution for step-index multimode fibers
Refractive index n of core and cladding of step-index multimode fiber1And n2Is constant, longitudinal electric field EzAnd a longitudinal magnetic field HzAnd satisfying the same equation, and obtaining a two-dimensional wave equation under a cylindrical coordinate system by a Maxwell equation set:
solving equation (1) by adopting a separation variable method, and aligning radius r and angle
And (3) separating functions to obtain a radial equation and an angle field equation, thereby deducing longitudinal field solutions in the fiber core and the cladding of the optical fiber:
two new characteristic parameters U and W are introduced to replace kca and aca, by definition, characteristic quantity U
And W and k0N and beta have the following relations:
(2) electric field radially offset from gaussian beam
When an analytical expression of the power coupling coefficient is deduced, an expression of the incident radial deviation Gaussian beam amplitude is needed; one spot diameter is ρsThe amplitude of the gaussian beam can be expressed as a when the offset of the offset launch x is a
C is a constant, in cylindrical coordinates the formula is
(3) Normalized cut-off frequency
The two equations of equation set (4) are subtracted to define the normalized frequency of the fiber:
when the normalized parameter W of the used step-index optical fiber cladding is 0, the normalized radial phase constant U in the core is recorded as UcThe normalized frequency is denoted as Vc；
The normalized cut-off frequencies of the mode groups under different orders are different, when the mode group with higher order is excited, the root of the corresponding m-order Bessel function is gradually increased, and the corresponding normalized frequency V is at the momentcIncreased, normalized radial phase constant UcThe longitudinal propagation characteristic constant beta at the corresponding high-order mode group is changed along with the increase of the longitudinal propagation characteristic constant beta; the phase constant β of a conventional step-index fiber is solved by the helmholtz equation as follows:
the longitudinal characteristic propagation constant beta is changed and is also used as the formula (8) to obtain the corresponding excitation electric field strength formula eiThe variation of the power coupling coefficient can be obtained from the radial total electric field in the equations (5) and (6):
further, the method for regulating and controlling a flat-top light spot is characterized in that a main factor in the graded-index multimode fiber is a certain proportion of deflected light generated by rotation of a front-end scatterer, when the modal coupling of the deflected light enters the graded-index multimode fiber, a perturbation solution is generated in a transverse modal electric field corresponding to the mode under the weak derivative approximation, the perturbation solution is represented as a polarized modal electric field, and a final power coupling coefficient is changed under the excitation of the polarized electric field, and the method for regulating and controlling the mode field based on the graded-index multimode fiber comprises the following steps:
(1) the refractive index profile was studied first, and the gradient refractive index profile was expressed as
Where α ═ 2 is an infinite parabolic refractive index profile, and n iscoIs the maximum refractive index of the core, nclIs the refractive index of the cladding; the refractive index of a multimode fiber is modeled by an infinite parabolic profile, the normalized radius R is given by R ═ R/ρ, ρ is the core diameter;
n(R)＝nco(1-ΔR^2) (11)
profile parameter Δ of
Given, wherein nclIs the cladding refractive index at normalized radius R ═ 1; when n iscl≈ncoAt this time,. DELTA.ident (n)co-ncl)nco；
(2) Modal electric field of infinite parabolic refractive index multimode optical fiber
When the actual fiber and the ideal fiber both satisfy the weak guiding condition, the field solutions both satisfy the wave equation, that is
Under weak guidance approximation, a transverse modal electric field of MMF with infinite parabolic refractive index distribution is obtained by solving a scalar wave equation, and the spatial dependence of the transverse electric field is controlled by a scalar wave equation;
(3) Overlap integral of power coupling coefficients
Total power or total effective optical power p of bound light entering the coreeTotal radiation power p of the active part of the light sourcetThe ratio is defined as the coupling efficiency between the light source and the optical fiber, having
The beam having a Gaussian distribution is launched to a position radially offset from the center of the fiber, the relative power of the modes of the fiber coupled by the power coupling coefficient, the variation of the power coupling coefficient is also obtained from the total electric field in equations (5) and (6)
Wherein etIs the transverse field of the fiber mode, EincIs the electric field of the incident beam, AcoreIs the cross-sectional area of the fiber core; let the polarization electric field elm(given by the formula (15)) instead of the transverse electric field etThe power coupling coefficient becomes
The invention has the advantages that:
the invention realizes an easy-to-operate light spot regulation and control method based on the angle modulation when coupling SMF to MMF; the mode distribution of the light beam is regulated and controlled along with different offset emission angles or emission modes, so that the required optimal output light field is achieved; meanwhile, the method is easy to operate and realize, the cost is extremely low, and the method has feasible application value in the fields of laser technology and optical communication.
Drawings
Fig. 1 is a schematic structural diagram of a flat-top light spot regulating device in the invention.
Fig. 2(a) is a light field distribution diagram obtained when the included angle between the meridional ray and the oblique ray is 7 ° for the scatterer.
Fig. 2(b) is a light field distribution diagram obtained when the included angle between the meridional ray and the oblique ray is 12 ° in the scatterer.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments.
Example 1.
The structure of the flat-top light spot shaping device adopted in the embodiment of the flat-top light spot regulating and controlling method comprises the following steps: a front end regulating device and a rear end receiving device. The front-end regulating device comprises a laser source (a table laser diode source or other laser sources), a single-mode optical fiber, a collimating lens, a diffuser, a focusing lens and a multi-mode optical fiber. The rear end receiving device includes an objective lens, a focusing lens, and a CCD camera. As shown in fig. 1, a laser diode light source is connected to a collimating lens through a single mode fiber, and a scattering body is added between the collimating lens and the focusing lens to adjust the angle of the incident multimode fiber, so as to achieve mode broadening and energy homogenization. Multimode optical fibers around multiple circumferences are used to remove unwanted modes for subsequent viewing, and objective lenses, focusing lenses and CCD camera devices are used to receive the signals.
A flat-top light spot regulation and control method comprises the following steps:
(1) after the laser light source outputs a light beam, the light beam passes through the single mode fiber and the collimating lens and enters the scatterer, and the scatterer is used for changing the angle of the light beam incident into the multimode fiber;
(2) the scatterer adjusts the received light beam: the scatterer modifies the transmission ratio of the deflected light and the meridian light of the light beam, changes the angle of the light beam incident into the multimode fiber, further changes the phase constant of the internal optical waveguide of the multimode fiber, when the phase constant is changed, the corresponding excitation electric field intensity is solved in the Helmholtz equation, and when the excitation electric field intensity is superposed on the power coupling coefficient, the expression form of the power coupling coefficient is changed, so that different output light spots are obtained;
when the step-index multimode fiber is used, the normalized cut-off frequency of an internal deflection light mode of the multimode fiber is changed, so that a propagation characteristic constant, namely a phase constant beta is changed, when the phase constant is changed, the corresponding excitation electric field intensity is solved in a Helmholtz equation, and when the phase constant is superposed on a power coupling coefficient, the power coupling coefficient is changed, so that the coupling form of light in different modes in the fiber is changed; when a graded-index multimode fiber is used, the mode that appears as a deflected ray produces a perturbation solution under the electric field scalar wave equation under the weak derivative approximation, so that here the perturbation solution: under the action of the polarization electric field, the final power coupling coefficient is influenced, so that different output light spots are obtained;
(3) the adjusted light beam is incident into the multimode optical fiber through the focusing lens, and then the light beam is collected through the objective lens, the focusing lens and the CCD camera for observation; and observing the change of the incident angle and the change of the emergent facula along with the rotation of the scatterer to finish the regulation and control of the flat-top facula.
Further, the multimode fiber is a step-index multimode fiber or a graded-index multimode fiber.
Further, a main factor in the step-index multimode fiber is that a front-end scatterer rotates to generate a certain proportion of oblique light, when a mode of the certain proportion of oblique light is coupled into the step-index multimode fiber, the mode shows that the normalized cut-off frequency of the mode is different, so that a phase constant is changed, and the change of the phase constant brings the change of an excitation electric field, thereby influencing a final power coupling coefficient, and the mode field regulation and control method based on the step-index multimode fiber comprises the following steps:
(1) field solution for step-index multimode fibers
Refractive index n of core and cladding of step-index multimode fiber1And n2Is constant, longitudinal electric field EzAnd a longitudinal magnetic field HzAnd satisfying the same equation, and obtaining a two-dimensional wave equation under a cylindrical coordinate system by a Maxwell equation set:
solving equation (1) by adopting a separation variable method, and aligning radius r and angle
And (3) separating functions to obtain a radial equation and an angle field equation, thereby deducing a longitudinal field solution in the fiber core and the cladding of the optical fiber:
two new characteristic parameters U and W are introduced to replace kca and aca, by definition, characteristic variables U and W and k0N and beta have the following relations:
(2) electric field radially offset from gaussian beam
When an analytic expression of the power coupling coefficient is deduced, an expression of the amplitude of an incident radial offset Gaussian beam is needed; one spot diameter is ρsThe amplitude of the gaussian beam can be expressed as a when the offset of the offset launch x is a
C is a constant having the formula
(3) Normalized cut-off frequency
The two equations of equation set (4) are subtracted to define the normalized frequency of the fiber:
when the normalized parameter W of the step-index fiber cladding is 0, the normalized radial phase constant U in the core is recorded as UcThe normalized frequency at this time is denoted as Vc；
The normalized cut-off frequencies of the mode groups under different orders are different, and when the mode group with higher order is excited, the root of the corresponding m-order Bessel function is gradually changedGradually increase, at which time the corresponding normalized frequency VcIncreased, normalized radial phase constant UcThe longitudinal propagation characteristic constant beta at the corresponding high-order mode group is changed along with the increase of the longitudinal propagation characteristic constant beta; the phase constant β of a conventional step-index fiber is solved by the helmholtz equation as follows:
the longitudinal characteristic propagation constant beta is changed and is also used as the formula (8) to obtain the corresponding excitation electric field strength formula eiThe variation of the power coupling coefficient can be obtained from the radial total electric field in the equations (5) and (6):
furthermore, the main cause in the graded-index multimode fiber is a certain proportion of oblique light generated by rotation of a front-end scatterer, when the modal coupling of the oblique light enters the graded-index multimode fiber, a transverse modal electric field corresponding to the mode can generate a perturbation solution under the weak derivative approximation, the perturbation solution is represented as a polarized modal electric field, and under the excitation of the polarized electric field, the final power coupling coefficient can be changed, and the mode field regulation and control method based on the graded-index multimode fiber comprises the following steps:
(1) the refractive index distribution was studied first, and the gradient refractive index distribution was represented by
Where α ═ 2 is an infinite parabolic refractive index profile, and n iscoIs the maximum refractive index of the core, nclIs the refractive index of the cladding; the refractive index of the multimode fiber is modeled by an infinite parabolic profile, the normalized radius R is given by R ═ R/rho, and rho is the diameter of the fiber core;
n(R)＝nco(1-AR^2) (11) profile parameter Δ consists of
Given, wherein nclIs the cladding index at normalized radius R1; when n iscl≈ncoWhen Δ ≡ (n)co-ncl)nco；
(2) Modal electric field of infinite parabolic refractive index multimode optical fiber
When the actual optical fiber and the ideal optical fiber both satisfy the weak guiding condition, the field solutions thereof both satisfy the wave equation, that is
Under weak guidance approximation, a transverse modal electric field of MMF with infinite parabolic refractive index distribution is obtained by solving a scalar wave equation, and the spatial dependence of the transverse electric field is controlled by a scalar wave equation;
(3) Overlap integral of power coupling coefficients
Total power or total effective optical power p of bound light entering the coreeTotal radiation power p of the active part of the light sourcetThe ratio is defined as the coupling efficiency between the light source and the optical fiber, having
The light beam with Gaussian distribution is emitted to a position radially offset from the center of the fiber, the relative power of the power coupled into the fiber mode is given by the power coupling coefficient, and the variation of the power coupling coefficient can be obtained according to the total electric field in the formulas (5) and (6)
Wherein etIs the transverse field of the fiber mode, EincIs the electric field of the incident beam, AcoreIs the cross-sectional area of the fiber core; let the polarization electric field elm(given by the formula (15)) instead of the transverse electric field etThe power coupling coefficient becomes
The working principle of the invention is as follows:
a first part: after the light source outputs light, the light enters the scatterer after passing through the single mode fiber and the collimating lens. The function of the diffuser is to change the angle of incidence into the multimode fiber, thereby modifying the gaussian distribution (all meridional rays) and the output to a sweet-centered circle (all skew rays) by modifying the ratio of the skew rays and the meridional rays propagating rays (the mixture of meridional rays and skew rays).
A second part: the ratio of the incident noon ray and the oblique ray in the multimode optical fiber is adjusted by adjusting the scatterer so as to achieve a required angle. The change in the incident angle brings about thereby the influence on the phase constant of the internal optical waveguide of the optical fiber. The phase propagation constant is a parameter which has an important influence on the emergent light spot in the optical waveguide, when the parameter is changed, the corresponding excitation electric field intensity can be solved in a Helmholtz equation, and when the parameter is superposed on the power coupling coefficient, the representation form of the power coupling coefficient is changed, so that different output light spots are achieved.
And a third part: the adjusted light beam is incident into the multimode optical fiber through the focusing lens, and then the light beam is collected through the objective lens, the focusing lens and the CCD camera for observation. The change of the incident angle and the change of the emergent facula along with the rotation of the scatterer are observed, thereby achieving the required purpose.
As shown in fig. 2(a), when the included angle between the noon light and the oblique light of the scatterer is 7 °, the light field distribution diagram obtained by the scatterer is stable and uniform in light intensity within a certain radius, the light spot is a flat-top light spot, and the intensity distribution approximately satisfies the high-order gaussian distribution, and satisfies the following formula:
wherein the numerical values of the parameters approximately satisfy the condition that A is 1 and x0＝519,ω＝385,β＝8。
As shown in fig. 2(b), when the eccentric incident angle is further increased to 12 °, a light field with a hollow-circle-shaped intensity distribution is obtained, and the intensity distribution approximately satisfies a parabolic function relationship, as shown in equation (21).
y＝A(x-x0)2+β (21)
Wherein the numerical values of the parameters approximately satisfy that A is 1.2 multiplied by 10-5,x0＝500,β＝0.4。
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered as the technical solutions and the inventive concepts of the present invention within the technical scope of the present invention.
Claims (4)
1. A flat-top light spot regulation and control method is characterized by comprising the following steps:
(1) after the laser light source outputs a light beam, the light beam passes through the single mode fiber and the collimating lens and enters the scatterer, and the scatterer is used for changing the angle of the light beam incident into the multimode fiber;
(2) the scatterer adjusts the received light beam: the scatterer modifies the transmission ratio of the deflected light and the meridian light of the light beam, changes the angle of the light beam incident into the multimode fiber, further changes the phase constant of the internal optical waveguide of the multimode fiber, when the phase constant is changed, the corresponding excitation electric field intensity is solved in the Helmholtz equation, and when the excitation electric field intensity is superposed on the power coupling coefficient, the expression form of the power coupling coefficient is changed, so that different output light spots are obtained;
(3) the adjusted light beam is incident into the multimode optical fiber through the focusing lens, and then the light beam is collected through the objective lens, the focusing lens and the CCD camera for observation; and observing the change of the incident angle and the change of the emergent light spot along with the rotation of the scatterer to finish the regulation and control of the flat-top light spot.
2. The method according to claim 1, wherein the multimode fiber is a step-index multimode fiber or a graded-index multimode fiber.
3. The method according to claim 2, wherein a main factor in the step-index multimode fiber is represented by a certain proportion of deflected light generated by rotation of a front scatterer, and when a mode of the certain proportion of deflected light is coupled into the step-index multimode fiber, the mode is represented by a difference of normalized cutoff frequencies of the mode, so that a phase constant is changed, and the change of the phase constant brings a change of an excitation electric field, thereby affecting a final power coupling coefficient, and the method for adjusting and controlling the mode field based on the step-index multimode fiber comprises:
(1) field solution for step-index multimode fibers
Refractive index n of core and cladding of step-index multimode fiber1And n2Is constant, longitudinal electric field EzAnd a longitudinal magnetic field HzAnd satisfying the same equation, and obtaining a two-dimensional wave equation under a cylindrical coordinate system by a Maxwell equation system:
solving equation (1) by adopting a separation variable method, and aligning radius r and angle
And (3) separating functions to obtain a radial equation and an angle field equation, thereby deducing a longitudinal field solution in the fiber core and the cladding of the optical fiber:
two new characteristic parameters U and W are introduced to replace kca and aca, by definition, characteristic variables U and W and k0N and beta have the following relations:
(2) electric field radially deflecting Gaussian beam
When an analytic expression of the power coupling coefficient is deduced, an expression of the amplitude of an incident radial offset Gaussian beam is needed; one spot diameter is ρsThe amplitude of the Gaussian beam can be tabulated when the offset of the offset launch x is aShown as
C is a constant having the formula
(3) Normalized cut-off frequency
The two equations of equation set (4) are subtracted to define the normalized frequency of the fiber:
when the normalized parameter W of the step-index fiber cladding is 0, the normalized radial phase constant U in the core is recorded as UcThe normalized frequency is denoted as Vc；
The normalized cut-off frequencies of the mode groups under different orders are different, when the mode group with higher order is excited, the root of the corresponding m-order Bessel function is gradually increased, and the corresponding normalized frequency V is at the momentcIncreased, normalized radial phase constant UcThe longitudinal propagation characteristic constant beta at the corresponding high-order mode group is changed along with the increase of the longitudinal propagation characteristic constant beta; the phase constant β of a conventional step-index fiber is solved by the helmholtz equation as follows:
the longitudinal characteristic propagation constant beta is changed and is also used as the formula (8) to obtain the corresponding excitation electric field strength formula eiThe variation of the power coupling coefficient can be obtained from the radial total electric field in the equations (5) and (6):
4. the method according to claim 2, wherein the main factor of the graded-index multimode fiber is a certain proportion of deflected light generated by rotation of a front scatterer, when the mode of the deflected light is coupled into the graded-index multimode fiber, a perturbation solution is generated in a transverse mode electric field corresponding to the mode under a weak derivative approximation, the perturbation solution is represented as a polarized mode electric field, and a final power coupling coefficient is changed under excitation of the polarized electric field, and the method for regulating and controlling the mode field based on the graded-index multimode fiber comprises the following steps:
(1) the refractive index distribution was studied first, and the gradient refractive index distribution was represented by
Where α -2 is an infinite parabolic refractive index profile and n iscoIs the maximum refractive index of the core, nclIs the refractive index of the cladding; the refractive index of a multimode fiber is modeled by an infinite parabolic profile, the normalized radius R is given by R ═ R/ρ, ρ is the core diameter;
n(R)＝nco(1-ΔR^2) (11)
profile parameter Δ is given by
Given, wherein nclIs the cladding index at normalized radius R1; when n iscl≈ncoAt this time,. DELTA.ident (n)co-cnl)nco；
(2) Modal electric field of infinite parabolic refractive index multimode optical fiber
When the actual optical fiber and the ideal optical fiber both satisfy the weak guiding condition, the field solutions thereof both satisfy the wave equation, that is
Under weak guidance approximation, a transverse modal electric field of MMF with infinite parabolic refractive index distribution is obtained by solving a scalar wave equation, and the spatial dependence of the transverse electric field is controlled by a scalar wave equation;
(3) Overlap integral of power coupling coefficients
Total power or total effective optical power p of bound light entering the coreeTotal radiation power p of the active part of the light sourcetThe ratio is defined as the coupling efficiency between the light source and the optical fiber, having
The beam having a Gaussian distribution is launched to a position radially offset from the center of the fiber, the relative power of the modes of the fiber coupled by the power coupling coefficient, the variation of the power coupling coefficient is also obtained from the total electric field in equations (5) and (6)
Wherein etIs the transverse field of the fiber mode, EincIs the electric field of the incident beam, AcoreIs the cross-sectional area of the fiber core; let the polarization electric field elm(given by the formula (15)) instead of the transverse electric field etThe power coupling coefficient becomes
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