A precise modeling method for multi-reflection cavity based on vector method

Abstract

A method for precise modeling of a multi-reflection cavity based on a vector method has high accuracy and strong universality, is suitable for modeling various types of reflective cavities, can solve the precision error caused by the traditional paraxial approximation method, and is of great significance for improving the accuracy of reflective cavity design and the improvement of reflective cavity measurement accuracy. It is characterized in that it includes a first front reflective mirror M1 and a second rear reflective mirror M2, which are arranged along the z-axis of a Cartesian rectangular coordinate system and face each other, wherein the first front reflective mirror M1 has a light hole in the middle, and the incident point A0 and the exit point in the light hole are both the coordinate origin O, the second rear reflective mirror M2 has a first reflection point A1 and a third reflection point A2, the relative distance between the first front reflective mirror M1 and the second rear reflective mirror M2 is d, and the second rear reflective mirror M2 includes the following parameters: δ, I, S, R, and N, wherein δ is the relative rotation angle, I is the laser incident vector, R is the laser reflection vector, and N is the mirror direction vector.

Description

Technical Field

[0001] The present invention relates to the technical field of optical multi-reflection cavities, and in particular to a method for accurately modeling a multi-reflection cavity based on a vector method. The multi-reflection cavity modeling method has high accuracy and strong universality, is applicable to the modeling of various types of reflective cavities, and can resolve the precision errors caused by traditional paraxial approximation methods. It is of great significance for improving the accuracy of reflective cavity design and the improvement of reflective cavity measurement accuracy. Background Art

[0002] Multi-reflection cavities can reflect incident light multiple times before emitting it, increasing the optical path length within a small volume. Currently, they are widely used in the fields of laser absorption spectroscopy and quantum precision measurement to improve measurement sensitivity and break through the minimum detection signal limit. However, the traditional paraxial approximation design method for multi-reflection cavities uses two paraxial approximation theories: all light rays are at a small angle to the optical axis, and the optical path length between reflectors is approximately a constant. This limits the accuracy of designing the cavity mirror parameters for high-reflection multi-reflection cavities and the position angle of the output light from small-volume reflective cavities. This can cause the designed cavity mirror spot distribution to be inconsistent with the actual one, and the light beam to escape prematurely. Therefore, accurate modeling of the reflective cavity is of great significance for improving the accuracy of reflective cavity design and the measurement accuracy of the reflective cavity. Summary of the Invention

[0003] In response to the defects or shortcomings of the existing technology, the present invention provides a multi-reflection cavity precise modeling method based on the vector method. The multi-reflection cavity modeling method has high accuracy and strong universality, is suitable for modeling various types of reflection cavities, and can solve the precision errors caused by traditional paraxial approximation methods. It is of great significance to improve the accuracy of reflection cavity design and the improvement of reflection cavity measurement accuracy.

[0004] The technical solutions of the present invention are as follows:

[0005] A method for accurately modeling a multi-reflection cavity based on a vector method, characterized by comprising the following steps:

[0006] Step 1: Arrange a first front reflecting mirror M1 and a second rear reflecting mirror M2 facing each other along the z-axis of a Cartesian rectangular coordinate system, wherein the first front reflecting mirror M1 has a light hole in the middle, and the incident point A0 and the exit point in the light hole are both the coordinate origin O, the second rear reflecting mirror M2 has a first reflection point A1 and a third reflection point A2, the relative distance between the first front reflecting mirror M1 and the second rear reflecting mirror M2 is d, and the second rear reflecting mirror M2 includes the following parameters: δ, I, S, R, and N, where δ is the relative rotation angle, I is the laser incident vector, R is the laser reflection vector, and N is the mirror direction vector;

[0007] Step 2: The number of laser reflections N between M1 and M2 is determined by d and δ. The laser enters the multi-reflection cavity formed by M1 and M2 from the light-through hole, and is emitted from the light-through hole after multiple reflections in the multi-reflection cavity.

[0008] Step 3: Calculate R using the following relationship:

[0009] R=I-2(I·N)R

[0010] I=(i x ,i y ,i z )

[0011] R=(r x ,r y ,r z )

[0012] where i x 、i y 、i z are the normalized coordinate values ​​of I in the x-axis, y-axis, and z-axis directions, r x , r y , r z are the normalized coordinate values ​​of R in the x-axis, y-axis, and z-axis directions respectively;

[0013] Step 4: Take A1 on M2 as the second reflection incident point on M1, and the second reflection point on M1 is the incident point of A2 on M2. The coordinates and direction vectors of each reflection point are obtained by calculation. By analogy, the trajectory of each reflected light ray in the multi-reflection cavity mirror and the mirror spot position can be accurately calculated, thereby realizing accurate modeling of the multi-reflection cavity.

[0014] Both M1 and M2 are reflective mirrors that can establish space curve equations.

[0015] Step 4 includes the following relationships:

[0016]

[0017] Where D is the effective optical path of the laser propagating in the multi-reflection cavity, n is the serial number, x n 、y n 、z n They are the three-axis coordinates of the incident point, x n+1 、y n+1 、z n+1 They are the three-axis coordinates of the reflection point.

[0018] The second rear reflector surface M2 in step 1 also includes the following parameters: S, where S is an auxiliary vector:

[0019]

[0020] M1 and M2 are both reflecting surfaces of cylindrical mirrors. The surface equation of M1 is as follows:

[0021]

[0022] Where r is the radius of the reflector;

[0023] The surface equation of M2 is as follows:

[0024]

[0025] N on M2 takes the following expression:

[0026] N=(2cos(δ)(cos(δ)x+sin(δ)y),2sin(δ)(sin(δ)y+cos(δ)x),2(z-(dr)))

[0027] The mirror direction vector N' on M1 is expressed as follows:

[0028] N'=(2x,0,2(zr)).

[0029] The equation of the incident light L(x,y,z) of I in step 3 is as follows:

[0030]

[0031] where x n 、y n 、z n are the three-axis coordinates of the incident point.

[0032] The technical effects of the present invention are as follows: The present invention provides a multi-reflection cavity precise modeling method based on the vector method, which has high accuracy and strong universality, is suitable for modeling various types of reflective cavities, solves the precision error caused by the traditional paraxial approximation method, and can accurately calculate the position and effective optical path of light, which is of great significance for improving the accuracy of reflective cavity design and the improvement of reflective cavity measurement accuracy.

[0033] The advantages of the present invention compared with the prior art are:

[0034] (1) The present invention accurately models the light in the multi-reflection cavity through the vector method, which solves the accuracy error caused by the traditional paraxial approximation method and can accurately calculate the position of the light pattern and the effective optical path. It is of great significance to improve the accuracy of the reflective cavity design and the improvement of the reflective cavity measurement accuracy.

[0035] (2) The method of the present invention is applicable to the precise modeling of a multi-reflection cavity composed of two or more cylindrical mirrors, spherical mirrors, discrete mirrors and other types of reflectors, and has high applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 The present invention is a schematic diagram of a multi-reflection cavity reflection light path involved in a multi-reflection cavity precise modeling method based on a vector method.

[0037] Figure 2 It is a schematic diagram of the relationship between the incident laser light and the reflected laser light.

[0038] The reference numerals are explained as follows: M1-first front reflecting mirror; M2-second rear reflecting mirror; A0-incident point (i.e., coordinate origin O); A1-first reflection point; A2-third reflection point; d-relative distance; δ-relative rotation angle; A-mirror reflection point; I-laser incident vector; S-auxiliary vector; R-laser reflection vector; N-mirror direction vector; xyz-three axes of the Cartesian rectangular coordinate system (i.e., x-axis, y-axis, and z-axis). DETAILED DESCRIPTION

[0039] Below is the attached figure ( Figure 1-Figure 2 ) and Examples illustrate the present invention.

[0040] Figure 1 The present invention is a schematic diagram of a multi-reflection cavity reflection light path involved in a multi-reflection cavity precise modeling method based on a vector method. Figure 2 This is a schematic diagram of the relationship between the incident laser light and the reflected laser light. Figures 1 to 2 As shown, a method for accurately modeling a multi-reflection cavity based on a vector method includes the following steps: Step 1, arranging a first front reflector M1 and a second rear reflector M2 that are opposite to each other along the z-axis of a Cartesian rectangular coordinate system, wherein the middle portion of the first front reflector M1 has a light-through hole, the incident point A0 and the exit point in the light-through hole are both the coordinate origin O, the second rear reflector M2 has a first reflection point A1 and a third reflection point A2, the relative distance between the first front reflector M1 and the second rear reflector M2 is d, and the second rear reflector M2 includes the following parameters: δ, I, S, R, and N, where δ is the relative rotation angle, I is the laser incident vector, R is the laser reflection vector, and N is the mirror direction vector; Step 2, determining the number N of reflections of the laser between M1 and M2 by d and δ, the laser is incident from the light-through hole into the multi-reflection cavity formed by M1 and M2, and exits from the light-through hole after multiple reflections in the multi-reflection cavity;

[0041] Step 3: Calculate R using the following relationship:

[0042] R=I-2(I·N)N

[0043] I=(i x ,i y ,i z )

[0044] R=(r x ,r y ,r z )

[0045] where i x 、i y 、i z are the normalized coordinate values ​​of I in the x-axis, y-axis, and z-axis directions, r x , r y , r z are the normalized coordinate values ​​of R in the x-axis, y-axis, and z-axis directions respectively;

[0046] Step 4: Take A1 on M2 as the second reflection incident point on M1, and the second reflection point on M1 is the incident point of A2 on M2. The coordinates and direction vectors of each reflection point are obtained by calculation. By analogy, the trajectory of each reflected light ray in the multi-reflection cavity mirror and the mirror spot position can be accurately calculated, thereby realizing accurate modeling of the multi-reflection cavity.

[0047] Both M1 and M2 are reflective mirrors that can establish space curve equations.

[0048] Step 4 includes the following relationships:

[0049]

[0050] Where D is the effective optical path of the laser propagating in the multi-reflection cavity, n is the serial number, x n 、y n 、z n They are the three-axis coordinates of the incident point, x n+1 、y n+1 、z n+1 They are the three-axis coordinates of the reflection point.

[0051] The second rear reflector surface M2 in step 1 also includes the following parameters: S, where S is an auxiliary vector:

[0052]

[0053] M1 and M2 are both reflecting surfaces of cylindrical mirrors. The surface equation of M1 is as follows:

[0054]

[0055] Where r is the radius of the reflector;

[0056] The surface equation of M2 is as follows:

[0057]

[0058] N on M2 takes the following expression:

[0059] N=(2cos(δ)(cos(δ)x+sin(δ)y),2sin(δ)(sin(δ)u+cos(δ)a),2(z-(dr)))

[0060] The mirror direction vector N' on M1 is expressed as follows:

[0061] N'=(2x,0,2(zr)).

[0062] The equation of the incident light L(x,y,z) of I in step 3 is as follows:

[0063]

[0064] where x n 、y n 、z n are the three-axis coordinates of the incident point.

[0065] The present invention relates to the field of optical multi-reflection cavities, and specifically to a multi-reflection cavity modeling method based on a vector method. The multi-reflection cavity modeling method has high accuracy and strong universality, is applicable to the modeling of various types of reflective cavities, solves the precision errors caused by traditional paraxial approximation methods, can accurately calculate the position and effective optical path of light, and is of great significance for improving the accuracy of reflective cavity design and the improvement of reflective cavity measurement accuracy.

[0066] refer to Figure 1 、 Figure 2 As shown, a method for accurately modeling a multi-reflection cavity based on a vector method is shown. A multi-reflection cavity is composed of a front reflector M1 and a rear reflector M2. There is a certain relative distance d and relative rotation angle δ between the front reflector M1 and the rear reflector M2. The relative distance d and the relative rotation angle δ determine the number of reflections of the laser between the front reflector M1 and the rear reflector M2. A light hole is opened on the front reflector M1. The laser is incident into the multi-reflection cavity through the light hole and is emitted from the light hole after multiple reflections in the multi-reflection cavity.

[0067] The laser reflection vector R is calculated by the relationship between the laser incident vector I and the mirror direction vector N at the reflection point A1:

[0068] R=I-2(I·N)n

[0069] Among them, the laser incident vector I=(i x ,i y ,i z ), laser reflection vector R=(r x ,r y ,r z ), i x 、i y 、i zare the normalized coordinate values ​​of the laser incident vector, r x , r y , r z are the normalized coordinate values ​​of the laser reflection vector.

[0070] The laser incident vector I passes through the reflection point A1 and is emitted in the direction of the laser reflection vector R. According to the laser reflection law, the reflection angle is equal to the incident angle. Therefore, the mirror direction vector N is twice the auxiliary vector S. Based on this, we can obtain:

[0071] R=I+2S

[0072] In addition, the auxiliary vector S can be expressed by the projection relationship between the laser incident vector I and the mirror direction vector N:

[0073]

[0074] Among them, N is recorded as a normalized vector form:

[0075]

[0076] The mirror surface equations of the front reflector M1 and the rear reflector M2 are determined by the geometric properties of the mirror surfaces. Here, the case where both the front reflector M1 and the rear reflector M2 are cylindrical mirrors is used as an example for explanation. According to the geometric properties of the front reflector M1 being a cylindrical mirror, the surface equation of the front reflector M1 can be written as:

[0077]

[0078] Where r is the mirror radius,

[0079] In the Cartesian coordinate system, based on the relative position relationship between the rear reflector M2 and the front reflector M1, the surface equation of the rear reflector M2 can be written as:

[0080]

[0081] Among them, δ is the relative rotation angle, d is the relative distance, and r is the mirror radius.

[0082] The mirror direction vector N at the reflection point A1 is calculated by taking partial derivatives of the spatial surface equation obtained according to the mirror geometry in the x, y, and z directions. According to the surface equation of the front reflector M1, the mirror direction vector N at any point on the front reflector M1 can be obtained as:

[0083] N=(2x,0,2(zr))

[0084] Similarly, according to the surface equation of the rear reflector M2, the mirror direction vector N of any point on the rear reflector M2 can be obtained as:

[0085] N=(2cos(δ)(cos(δ)x+sin(δ)y),2sin(δ)(sin(δ)y+cos(δ)x),2(z-(dr)))

[0086] According to the coordinates of the incident point (x n ,y n ,z n ) and the laser incident direction vector I to obtain the incident light equation:

[0087]

[0088] The coordinates of the reflection point (x n+1 ,y n+1 ,z n+1 ).

[0089] According to the coordinates of the incident point and the reflection point of each reflection, the effective optical path D of the light propagating in the multi-reflection cavity mirror can be accurately calculated:

[0090]

[0091] Where N is the number of reflections.

[0092] The light in the multi-reflection cavity mirror is incident from the incident point A0 with known incident coordinates and incident direction vector, intersects with the rear reflecting mirror M2 at the reflection point A1, and is reflected by the rear reflecting mirror M2 to the front reflecting mirror M1. The reflection point A1 is the incident point of the next reflection, and its coordinates and direction vector are obtained by calculation. By analogy, the trajectory of each reflected light in the multi-reflection cavity mirror and the mirror spot position can be accurately calculated, thereby realizing accurate modeling of the multi-reflection cavity.

[0093] The front reflector M1 and the rear reflector M2 can be any reflectors that can establish a spatial curve equation, and the multi-reflection cavity can be composed of more than two reflectors. The light trajectory of the laser propagating between each reflector can also be accurately calculated, thereby accurately designing the parameters of the multi-reflection cavity.

[0094] Any content not described in detail in this specification is prior art known to those skilled in the art. It should be noted that the above description is intended to help those skilled in the art understand the present invention, but does not limit the scope of protection of the present invention. Any equivalent substitution, modification, improvement, and / or simplification of the above description that does not depart from the essence of the present invention shall fall within the scope of protection of the present invention.

Claims

1. A method for accurate modeling of a multi-reflection cavity based on a vector method, characterized in that: The following steps are involved: Step 1: Arrange a first front reflecting mirror M1 and a second rear reflecting mirror M2 facing each other along the z-axis of a Cartesian rectangular coordinate system, wherein the first front reflecting mirror M1 has a light hole in the middle, and the incident point A0 and the exit point in the light hole are both the coordinate origin O, the second rear reflecting mirror M2 has a first reflection point A1 and a third reflection point A2, the relative distance between the first front reflecting mirror M1 and the second rear reflecting mirror M2 is d, and the second rear reflecting mirror M2 includes the following parameters: δ, I, S, R, and N, where δ is the relative rotation angle, I is the laser incident vector, R is the laser reflection vector, and N is the mirror direction vector; Step 2: The number of laser reflections N between M1 and M2 is determined by d and δ. The laser enters the multi-reflection cavity formed by M1 and M2 from the light-through hole, and is emitted from the light-through hole after multiple reflections in the multi-reflection cavity. Step 3: Calculate R using the following relationship: R=I-2(I·N)N I=(i x ,i y ,i z ) R＝(r x ,r y ,r z ) where i x 、i y 、i z are the normalized coordinate values ​​of I in the x-axis, y-axis, and z-axis directions, r x , r y , r z are the normalized coordinate values ​​of R in the x-axis, y-axis, and z-axis directions respectively; Step 4: Take A1 on M2 as the second reflection incident point on M1, and the second reflection point on M1 is the incident point of A2 on M2. The coordinates and direction vectors of each reflection point are obtained by calculation. By analogy, the trajectory of each reflected light ray in the multi-reflection cavity mirror and the mirror spot position can be accurately calculated, thereby realizing accurate modeling of the multi-reflection cavity.

2. The multi-reflection cavity accurate modeling method based on the vector method according to claim 1 is characterized in that: Both M1 and M2 are reflective mirrors that can establish space curve equations.

3. The multi-reflection cavity accurate modeling method based on the vector method according to claim 1 is characterized in that: Step 4 includes the following relationships: Where D is the effective optical path of the laser propagating in the multi-reflection cavity, n is the serial number, x n 、y n 、z n They are the three-axis coordinates of the incident point, x n+1 、y n+1 、z n+1 They are the three-axis coordinates of the reflection point.

4. The multi-reflection cavity accurate modeling method based on the vector method according to claim 1 is characterized in that: The second rear reflector surface M2 in step 1 also includes the following parameters: S, where S is an auxiliary vector:

5. The multi-reflection cavity precise modeling method based on the vector method according to claim 1 is characterized in that: M1 and M2 are both reflecting surfaces of cylindrical mirrors. The surface equation of M1 is as follows: Where r is the radius of the reflector; The surface equation of M2 is as follows: N on M2 takes the following expression: N=(2cos(δ)(cos(δ)x+sin(δ)y),2sin(δ)(sin(δ)y+cos(δ)x),2(z-(dr))) The mirror direction vector N' on M1 is expressed as follows: N'=(2x,0,2(zr)).

6. The multi-reflection cavity precise modeling method based on the vector method according to claim 1 is characterized in that: The equation of the incident light L(x,y,z) of I in step 3 is as follows: where x n 、y n 、z n are the three-axis coordinates of the incident point.

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