An optical gyroscope with adjustable detection sensitivity based on microcavity structure and angular velocity measurement method

Abstract

The invention discloses an optical gyroscope with adjustable detection sensitivity based on a microcavity structure and a method for measuring angular velocity. The optical gyroscope includes a straight waveguide, a microring resonant cavity, a rotating platform, a spatial periodic microelectrode, a coupling region, and an adjustable voltage source. The straight waveguide and the microring resonant cavity are in a coupled state to form a coupling region. The rotating platform is used to drive the microring resonant cavity to rotate. The incident light enters the microring resonant cavity through the coupling region via the straight waveguide. The spatial periodic microelectrode is arranged in a non-coupled arc region of the microring resonant cavity, acts on the microring resonant cavity, applies a periodically distributed electric field to modulate the electrode corresponding region, forms a periodic refractive index distribution, and generates an equivalent Bragg grating with controllable modulation depth. The adjustable voltage source is used to adjust the voltage of the spatial periodic microelectrode and adjust the refractive index of the electrode corresponding region. The invention can realize the measurement of angular velocity by measuring the difference in the resonant split peak transmittance of the transmission spectrum of the gyroscope.

Description

Technical Field

[0001] The invention relates to the technical field of optical gyroscopes, and in particular to an optical gyroscope with adjustable detection sensitivity based on a microcavity structure and a method for measuring angular velocity. Background Art

[0002] The resonant optical gyroscope mainly measures angular velocity by detecting the resonant frequency difference between the clockwise mode and the counterclockwise mode in the cavity caused by rotation. The key lies in the design of the sensitive unit, which mainly includes the whispering gallery mode resonant cavity, the optical waveguide ring resonant cavity, etc. This type of gyroscope has the advantages of high precision, light weight, and miniaturization, and is widely used in aviation, aerospace, autonomous driving and other fields.

[0003] At present, the optical gyroscope based on the whispering gallery mode resonator mainly measures the angular velocity by bidirectional input light field and measuring the resonant frequency drift. The sensitive unit design includes: single-ring coupled single waveguide structure and multi-ring optical waveguide resonant cavity structure. The single-ring structure requires the micro-ring to be finely made to obtain a high quality factor. The two incident light beams that meet the resonance conditions and transmit in opposite directions are coupled into the micro-ring to form a counterclockwise mode and a clockwise mode. The angular velocity can be calculated when the rotation causes the resonance peak to split. Compared with the single-ring resonator of the same area, the multi-ring optical waveguide resonant cavity structure can provide a more obvious resonance difference to obtain higher sensitivity, but the coupling between the rings needs to be precisely designed and controlled, and the coupling loss is higher.

[0004] Currently, the detection sensitivity of existing resonant optical gyroscopes is fixed, and the angular velocity is calculated by measuring the splitting of the resonance peak caused by rotation. If an optical gyroscope with adjustable detection sensitivity and angular velocity can be calculated by measuring transmittance, the required sensitivity can be flexibly adjusted according to actual conditions, which has important theoretical and practical significance. Summary of the invention

[0005] The object of the present invention is to provide an optical gyroscope with adjustable detection sensitivity based on a microcavity structure and a method for measuring angular velocity, which can measure the angular velocity by measuring the resonant split peak transmittance difference of the gyroscope transmission spectrum.

[0006] In one aspect of the present invention, the present invention proposes an optical gyroscope with adjustable detection sensitivity based on a microcavity structure. According to an embodiment of the present invention, the optical gyroscope includes a straight waveguide, a microring resonant cavity, a rotating platform, a spatial periodic microelectrode, a coupling region, and an adjustable voltage source;

[0007] The straight waveguide and the micro-ring resonant cavity are in a coupled state to form a coupling region;

[0008] The rotating platform is connected to the micro-ring resonant cavity, and the rotating platform is used to drive the micro-ring resonant cavity to rotate;

[0009] The incident light is coupled into the straight waveguide and coupled into the microring resonant cavity in the coupling region through the evanescent field on the surface of the straight waveguide. When the incident light meets the resonance condition, a counterclockwise transmission mode is formed, which is reflected in the transmission spectrum as a single transmission peak at the resonant wavelength.

[0010] The spatial periodic microelectrode is arranged in the uncoupled arc area of ​​the microring resonant cavity, acts on the microring resonant cavity, applies a periodically distributed electric field to modulate the electrode corresponding area, forms a periodic refractive index distribution, and generates an equivalent Bragg grating with controllable modulation depth, and the equivalent grating can cause reflection; the method for generating mode splitting of a single transmission peak at the resonant wavelength of the transmission spectrum is as follows: the spatial periodic microelectrode generates refractive index modulation in the area corresponding to the electrode of the microring resonant cavity, equivalently forming a Bragg grating, and the counterclockwise transmission mode will generate reflection after satisfying the Bragg reflection condition in the refractive index modulation area to form a clockwise mode, and the clockwise mode and the counterclockwise mode are coupled with each other to generate transmission peak splitting.

[0011] The adjustable voltage source is used to adjust the voltage of the spatially periodic microelectrode and the refractive index of the region corresponding to the electrode.

[0012] In addition, the optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to the above embodiment of the present invention may also have the following additional technical features:

[0013] In some embodiments of the present invention, the material of the microring resonator is a low-loss electro-optical material in the near-infrared band.

[0014] In some embodiments of the present invention, the material of the microring resonator is lithium niobate.

[0015] In some embodiments of the present invention, the position of the spatially periodic microelectrode is fixed and does not contact the microring resonant cavity. The position of the formed equivalent Bragg grating does not change as the microring resonant cavity rotates. The refractive index modulation of the microring resonant cavity at the non-coupled arc is adjusted by changing the voltage to achieve gyroscope detection sensitivity control.

[0016] In another aspect of the present invention, the present invention proposes a sensitivity control method for an optical gyroscope with adjustable detection sensitivity based on a microcavity structure. According to an embodiment of the present invention, the refractive index of the microring resonant cavity at the non-coupled arc is adjusted by changing the voltage magnitude through an adjustable voltage source, the reflection intensity of the equivalent grating is changed, the coupling intensity of the two modes in the microring resonant cavity is affected, and the degree of splitting of the two modes of the transmission spectrum is affected; different degrees of mode splitting result in different transmittance differences caused by the same rotation speed, thereby realizing the control of gyroscope detection sensitivity.

[0017] In another aspect of the present invention, the present invention proposes a method for measuring angular velocity of an optical gyroscope with adjustable microcavity structure detection sensitivity. According to an embodiment of the present invention, the optical gyroscope with adjustable microcavity structure detection sensitivity is used for measurement, which specifically includes the following steps:

[0018] The incident light enters the microring resonant cavity through the coupling region via a straight waveguide. When the wavelength of the incident light meets the resonance condition, a counterclockwise transmission mode is formed, which is reflected in the transmission spectrum as a transmission peak at the resonance wavelength. The refractive index modulation of the corresponding area of ​​the microring resonant cavity is equivalent to forming a Bragg grating through spatial periodic microelectrodes. When the counterclockwise transmission mode meets the Bragg reflection condition in the refractive index modulation area, reflection will occur to form a clockwise mode. The two modes are mutually coupled, which is reflected in the transmission spectrum as mode splitting. When the rotating platform drives the microring resonant cavity to rotate, the transmittance of the two split modes changes in the transmission spectrum. By measuring the transmittance difference, the angular velocity of the gyroscope can be measured.

[0019] In addition, the angular velocity measurement method of the optical gyroscope with adjustable detection sensitivity based on the microcavity structure according to one of the above embodiments of the present invention may also have the following additional technical features:

[0020] In some embodiments of the present invention, the gyroscope rotation angular velocity calculation formula is as follows:

[0021]

[0022] Where ΔT is the transmittance difference of the microring resonator, T(ω) is the transmission spectrum of the microring resonator, ω0 is the resonant frequency of the microring resonator, and ω + ,ω - are the resonant frequencies of the microring resonator in the counterclockwise and clockwise modes without voltage modulation after rotation, and the coupling coefficient of the clockwise mode to the counterclockwise mode is κ + , the coupling coefficient of the counterclockwise mode to the clockwise mode is κ - ;

[0023] Sagnac frequency shift n is the refractive index of the microring resonator, R is the radius of the microring resonator, Ω is the rotation speed of the microring resonator, c is the speed of light in vacuum, and λ is the resonant wavelength of the microring resonator.

[0024] In some embodiments of the present invention, the calculation formula of the microring resonator transmission spectrum T(ω) is as follows:

[0025]

[0026] Where i is a unit imaginary number, μ cis the coupling rate between the microring resonator and the straight waveguide, ω is the frequency of the incident light field, ω + ,ω - are the resonant frequencies of the microring resonator in the counterclockwise and clockwise modes without voltage modulation after rotation, γ is the intrinsic loss of the microring resonator, and the coupling coefficient of the clockwise mode to the counterclockwise mode is κ + , the coupling coefficient of the counterclockwise mode to the clockwise mode is κ - .

[0027] In some embodiments of the present invention, the refractive index n of the microring resonator is calculated as follows:

[0028] ε is the dielectric constant and μ0 is the magnetic permeability.

[0029] In some embodiments of the present invention, the spatially periodic microelectrode modulation form is expressed as follows:

[0030]

[0031] in is the change in dielectric constant caused by voltage regulation, Δε m is the modulation depth amplitude, L is the voltage modulation related parameter, is the azimuth coordinate.

[0032] Compared with the prior art, the present invention has the following beneficial effects:

[0033] The present invention applies an electric field through a spatial periodic microelectrode to modulate the refractive index at the uncoupled arc of the microring resonant cavity, forming a split peak on the transmission spectrum; after applying rotation, the mode transmittance difference is detected to realize angular velocity measurement, and at the same time, the voltage is changed by an adjustable voltage source to adjust the refractive index modulation depth of the corresponding area of ​​the spatial periodic microelectrode, thereby realizing adjustable sensitivity of the gyroscope. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 It is a schematic structural diagram of an optical gyroscope with adjustable detection sensitivity based on a microcavity structure in Example 1 of the present invention;

[0035] Figure 2 is a graph showing the relationship between the splitting peak difference and the rotation speed under different voltage adjustment amplitudes provided in Example 2 of the present invention;

[0036] Figure 3 are transmission spectra measured in three different scenarios provided in Example 3 of the present invention;

[0037] Figure 4 It is a transmission spectrum diagram at different rotation speeds under fixed voltage regulation provided in Example 3 of the present invention;

[0038] In the figure, 1. straight waveguide, 2. microring resonant cavity, 3. rotating platform, 4. spatially periodic microelectrode, 5. coupling region, 6. adjustable voltage source. DETAILED DESCRIPTION

[0039] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0040] Example 1

[0041] like Figure 1 As shown, an optical gyroscope with adjustable detection sensitivity based on a microcavity structure includes a straight waveguide 1, a microring resonant cavity 2, a rotating platform 3, a spatially periodic microelectrode 4, a coupling region 5, and an adjustable voltage source 6.

[0042] The straight waveguide 1 is used for inputting and outputting light. The straight waveguide 1 forms a coupling region with the micro-ring resonant cavity 2. The micro-ring resonant cavity 2 and the straight waveguide 1 are in a coupled state. The coupling rate between the two is μ c The material of the micro-ring resonator 2 is a near-infrared band low-loss electro-optical material, such as lithium niobate. The radius of the micro-ring resonator 2 is R, and the refractive index is n. ε is the dielectric constant, μ0 is the magnetic permeability, at this time μ0 is 1, the resonant frequency of the microring resonator 2 is set to ω0, and the intrinsic loss of the microring resonator 2 is γ.

[0043] The rotating platform 3 is connected to the micro-ring resonant cavity 2 , and the rotating platform 3 is used to drive the micro-ring resonant cavity 2 to rotate.

[0044] The incident light is coupled into the straight waveguide 1, and is coupled into the microring resonant cavity 2 in the coupling region 4 through the evanescent field on the surface of the straight waveguide 1. When the incident light meets the resonance condition, a counterclockwise transmission mode is formed, which is reflected in the transmission spectrum as a single transmission peak at the resonance wavelength.

[0045] The spatial periodic microelectrode 4 is arranged in the non-coupled arc region of the microring resonant cavity 2, and acts on the microring resonant cavity 2, applying a periodically distributed electric field to modulate the corresponding region of the electrode, forming a periodic refractive index distribution, and generating an equivalent Bragg grating with controllable modulation depth, which can cause reflection; the spatial periodic microelectrode 4 is fixed in position and does not contact the microring resonant cavity 2, and the position of the formed equivalent Bragg grating will not change as the microring resonant cavity 2 rotates. The modulation form of the spatial periodic microelectrode 4 can be expressed as in is the change in dielectric constant caused by voltage regulation, Δε mTo modulate the depth amplitude, changing the voltage will affect Δε m value, L is the voltage modulation related parameter, is the azimuth coordinate.

[0046] The method of splitting the mode of a single transmission peak at the resonant wavelength of the transmission spectrum is as follows: the spatial periodic microelectrode generates refractive index modulation in the region corresponding to the electrode of the microring resonator 2, equivalently forming a Bragg grating, and the counterclockwise transmission mode will generate reflection after satisfying the Bragg reflection condition in the refractive index modulation region, forming a clockwise mode, and the clockwise mode and the counterclockwise mode are coupled to each other, resulting in transmission peak splitting, that is, symmetrical split resonance peaks on the transmission spectrum of the microring resonator 2. The resonant frequencies of the counterclockwise mode and the clockwise mode after rotation without voltage modulation are ω respectively. + ,ω - , where the coupling coefficient of the clockwise mode to the counterclockwise mode is κ + , the coupling coefficient of the counterclockwise mode to the clockwise mode is κ - , whose expression is k is the order of the resonance frequency mode. The frequency of the incident light field is ω. After applying counterclockwise rotation to the microring resonator 2, the difference between the resonance splitting peaks changes, and the transmission spectrum of the microring resonator 2 is expressed as

[0047]

[0048] The adjustable voltage source 6 is used to adjust the voltage of the spatial periodic microelectrode 4 and the refractive index of the region corresponding to the electrode.

[0049] Example 2

[0050] In Example 1, the sensitivity control method of the optical gyroscope with adjustable detection sensitivity based on the microcavity structure adjusts the refractive index modulation of the microring resonant cavity at the uncoupled arc by changing the voltage through an adjustable voltage source, thereby realizing the detection sensitivity control of the gyroscope.

[0051] The coupling between the micro-ring resonator 2 and the straight waveguide 1 is adjusted by adjusting the adjustable voltage source 6 to change the voltage and adjust Δε m The value changes the refractive index, thus affecting the mutual coupling coefficient κ of the counterclockwise mode and the clockwise mode. + , κ - , which affects the degree of mode splitting of the transmission spectrum and the transmittance difference ΔT of the microring resonator 2, and its expression is:

[0052]

[0053] The Sagnac frequency shift n is the refractive index, R is the radius of the microring resonator 2, ω0 is the resonant frequency of the microring resonator 2, Ω is the rotation speed, c and λ are the speed of light in vacuum and the resonant wavelength of the microring resonator 2, and Approx. After obtaining the transmittance difference from the transmission spectrum, the rotation speed Ω can be solved by the above formula.

[0054] Adjust the adjustable voltage source to change the voltage, affect the equivalent grating reflection intensity, and optimize κ in the transmittance difference ΔT ± Parameters and the degree of resonance peak splitting change the sensitivity of the measurement.

[0055] The adjustable voltage source 6 controls the spatial periodic microelectrode 4 to apply different voltages to the adjustment area, and measures the transmission spectrum of the gyroscope at different rotation speeds. Figure 2 The graphs are the relationship between transmittance difference and rotation speed under different voltage adjustment. Curve 1 is the relationship between rotation speed and transmittance difference without voltage adjustment, and curve 2 is the relationship between rotation speed and transmittance difference with voltage adjustment of 7.45x10 -4 The relationship between the rotation speed and the transmittance difference under the condition of V. Curve 3 is the voltage of 1.43x10 -3 Relationship between rotation speed and transmittance difference under the condition of V. It can be seen that under different voltage adjustment, the response of the gyroscope shown in the present invention to the rotation speed is different, which proves that the sensor of the present invention can adjust the sensitivity of the gyroscope by adjusting the voltage.

[0056] Example 3

[0057] In the angular velocity measurement method of the optical gyroscope with adjustable detection sensitivity based on the microcavity structure in Example 1, the microring resonant cavity 2 is arranged on the lithium niobate chip, and the radius is in the order of millimeters. The measurement method specifically comprises the following steps:

[0058] The incident light enters the microring resonant cavity through the coupling zone via a straight waveguide. When the wavelength of the incident light meets the resonance condition, a counterclockwise transmission mode is formed, which is reflected in the transmission spectrum as a transmission peak at the resonance wavelength. The refractive index modulation of the corresponding area of ​​the microring resonant cavity is equivalent to forming a Bragg grating through spatial periodic microelectrodes. When the counterclockwise transmission mode meets the Bragg reflection condition in the refractive index modulation area, reflection will occur to form a clockwise mode. The two modes are mutually coupled, which is reflected in the transmission spectrum as mode splitting. When the rotating platform drives the microring resonant cavity to rotate, the transmittance of the two split modes changes in the transmission spectrum. By measuring the transmittance difference ΔT, the angular velocity of the gyroscope can be measured.

[0059] The transmittance difference ΔT of micro-ring resonator 2 is expressed as:

[0060]

[0061] The Sagnac frequency shift n is the refractive index, R is the radius of the microring resonator 2, ω0 is the resonant frequency of the microring resonator 2, Ω is the rotation speed, c and λ are the speed of light in vacuum and the resonant wavelength of the microring resonator 2, and Approx. After obtaining the transmittance difference from the transmission spectrum, the rotation speed Ω can be solved by the above formula.

[0062] The adjustable voltage source is adjusted to change the voltage, which affects the equivalent grating reflection intensity, optimizes the κ± parameter and the resonance peak splitting degree in the transmittance difference ΔT, and changes the measurement sensitivity.

[0063] Three scenarios were set up: no voltage adjustment and rotation, voltage adjustment without rotation, and voltage adjustment and rotation at the same time, and the gyroscope transmission spectrum was measured. Figure 3 The transmission spectra of the gyroscope in three scenarios are shown in Figure 1. Curve 1 is the transmission spectrum without voltage adjustment and rotation, Curve 2 is the transmission spectrum with voltage adjustment only, and Curve 3 is the transmission spectrum with voltage adjustment and rotation at the same time. It can be seen that after voltage adjustment only, the resonance peak is obviously split, and the difference ΔT of the split peak generated after voltage adjustment and rotation is calculated. The rotation speed can be obtained as 701.76 rad / s.

[0064] After applying voltage adjustment, the micro-ring resonant cavity 2 is provided with different rotation speeds through the rotating platform 3 to measure the gyroscope transmission spectrum. Figure 4 The transmission spectra of the gyroscope at different rotation speeds, where the rotation speed of curve 1 is 0 rad / s, the rotation speed of curve 2 is 1403.52 rad / s, and the rotation speed of curve 3 is 2807.04 rad / s. It can be seen that after applying different rotation speeds, the difference ΔT between the two split peaks increases with the increase of the rotation speed, proving the feasibility of the gyroscope of the present invention to measure angular velocity.

[0065] The above contents are merely examples and explanations of the structure of the present invention. The technicians in this technical field may make various modifications or additions to the specific embodiments described or replace them in a similar manner. As long as they do not deviate from the structure of the present invention or exceed the scope defined by the claims, they should all fall within the protection scope of the present invention.

Claims

1. An optical gyroscope with adjustable detection sensitivity based on a microcavity structure, characterized in that: It includes a straight waveguide, a micro-ring resonator, a rotating platform, a spatially periodic micro-electrode, a coupling region, and an adjustable voltage source; The straight waveguide and the micro-ring resonant cavity are in a coupled state to form a coupling region; The rotating platform is connected to the micro-ring resonant cavity, and the rotating platform is used to drive the micro-ring resonant cavity to rotate; The coupling region is used to couple the incident light from the straight waveguide into the micro-ring resonant cavity; The spatial periodic microelectrode is arranged in the non-coupled arc region of the microring resonant cavity, acts on the microring resonant cavity, applies a periodically distributed electric field to modulate the region corresponding to the electrode, forms a periodic refractive index distribution, and generates an equivalent Bragg grating with controllable modulation depth, and the equivalent Bragg grating can cause reflection; The adjustable voltage source is used to adjust the voltage of the spatially periodic microelectrodes and the refractive index of the region corresponding to the electrodes; The spatial periodic microelectrode is fixed in position and does not contact the microring resonant cavity.

2. The optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to claim 1, characterized in that: The material of the micro-ring resonant cavity is a low-loss electro-optical material in the near-infrared band.

3. The optical gyroscope with adjustable detection sensitivity based on microcavity structure according to claim 1, characterized in that: The material of the micro-ring resonant cavity is lithium niobate.

4. A sensitivity control method for an optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to any one of claims 1 to 3, characterized in that: By changing the voltage through an adjustable voltage source, the refractive index of the microring resonant cavity in the corresponding area of ​​the electrode is adjusted, the reflection intensity of the equivalent grating is changed, the coupling intensity of the clockwise mode and the counterclockwise mode in the microring resonant cavity is affected, and the degree of splitting of the two modes of the transmission spectrum is affected to achieve gyroscope detection sensitivity control.

5. A method for measuring angular velocity of an optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to any one of claims 1 to 3, characterized in that: The following steps are involved: The incident light enters the microring resonant cavity through the coupling zone via a straight waveguide. When the wavelength of the incident light meets the resonance condition, a counterclockwise transmission mode is formed, which is reflected in the transmission spectrum as a transmission peak at the resonance wavelength. The spatial periodic microelectrodes produce refractive index modulation in the corresponding area of ​​the microring resonant cavity electrode, which is equivalent to forming a Bragg grating. When the counterclockwise transmission mode meets the Bragg reflection condition in the refractive index modulation area, it will be reflected to form a clockwise mode. The two modes are mutually coupled, which is reflected in the transmission spectrum as mode splitting. When the rotating platform drives the microring resonant cavity to rotate, the transmittance of the two split modes changes in the transmission spectrum. By measuring the transmittance difference, the angular velocity of the gyroscope can be measured.

6. The angular velocity measurement method of an optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to claim 5, characterized in that: The gyroscope rotation angular velocity calculation formula is as follows: Among them, ΔT is the transmittance difference of the microring resonant cavity, T(ω) is the transmission spectrum of the microring resonant cavity, ω0 is the resonant frequency of the microring resonant cavity, ω+, ω - are the resonant frequencies of the microring resonator in the counterclockwise and clockwise modes without voltage modulation after rotation, and the coupling coefficient of the clockwise mode to the counterclockwise mode is κ + , the coupling coefficient of the counterclockwise mode to the clockwise mode is κ - ; The Sagnac frequency shift n is the refractive index of the microring resonator, R is the radius of the microring resonator, Ω is the rotation speed of the microring resonator, c is the speed of light in vacuum, and λ is the resonant wavelength of the microring resonator.

7. The angular velocity measurement method of an optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to claim 6, characterized in that: The calculation formula of the transmission spectrum T(ω) of the microring resonator is as follows: Where i is a unit imaginary number, μ c is the coupling rate between the microring resonator and the straight waveguide, ω is the frequency of the incident light field, ω + ,ω - are the resonant frequencies of the microring resonator in the counterclockwise and clockwise modes without voltage modulation after rotation, γ is the intrinsic loss of the microring resonator, and the coupling coefficient of the clockwise mode to the counterclockwise mode is κ + , the coupling coefficient of the counterclockwise mode to the clockwise mode is κ - .

8. The angular velocity measurement method of an optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to claim 6, characterized in that: The refractive index n of the microring resonator is calculated as follows: ε is the dielectric constant and μ0 is the magnetic permeability.

9. The angular velocity measurement method of an optical gyroscope with adjustable detection sensitivity based on a microcavity structure according to claim 5, characterized in that: The spatial periodic microelectrode modulation form is expressed as follows: in is the change in dielectric constant caused by voltage regulation, Δε m is the modulation depth amplitude, L is the voltage modulation related parameter, is the azimuth coordinate.

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