CN114871571A - An integrated main and auxiliary beam splitting device for a blue-light laser welding robot - Google Patents

Abstract

The invention discloses an integrated main and auxiliary beam splitting device of a blue laser welding robot, and belongs to the technical field of blue semiconductor lasers. The system comprises a blue light beam collimation system, a blue light primary and secondary beam splitting system, a focusing system, a secondary light beam collimation beam linear conversion system, a linear blue light scanning position-finding light path and a CCD camera; the blue light beam is expanded and collimated by the blue light beam collimating system and then enters the blue light primary and secondary light beam splitting system to decompose the blue light beam to form a primary light beam for blue light welding and a secondary light beam for blue light scanning; the focusing system is used for converging the main light beam at a specified welding position for welding; the secondary beam collimated beam linear conversion system is used for converting the secondary beam into linear laser; the linear blue light scanning position-finding light path is used for carrying out image processing according to the reflected light of the linear laser received by the CCD camera to obtain the actual welding point position. The laser welding and laser welding seam detection device integrates two functions of laser welding and laser welding seam detection into one welding device, and does not need calibration by hands and eyes.

Description

An integrated main and auxiliary beam splitting device for a blue-light laser welding robot

technical field

The invention belongs to the technical field of blue light semiconductor lasers, and more particularly relates to an integrated main and auxiliary beam splitting device of a blue light laser welding robot.

Background technique

In recent years, there has been a great demand for laser processing of copper materials with high thermal conductivity and electrical conductivity. However, the laser absorption rate of copper materials for the infrared band is only 10% or less. On the other hand, as the wavelength continues to decrease As small as 500nm or less, the absorption rate of copper material to light increases sharply, which enables the use of blue light to weld high-reflection materials to achieve better processing results. In the fields of high precision requirements such as automotive power batteries, aerospace and electrical, blue semiconductor laser welding has strong advantages.

Laser welding is generally bound to joint robots. Compared with manual welding, welding robots can greatly reduce the production cost of welding labor and improve the automatic and intelligent operation of welding work. Welding robots can automatically complete a lot of work in a certain workspace. This type of welding action can automatically adjust the posture of each welder at will. In the process of robot welding, due to the deformation of the workpiece, the non-standard welding seam itself, and the deviation of the workpiece placement, the welding seam trajectory often does not match the robot preset trajectory. In order to change the era of welding programming education called blind welding, in the existing laser welding robot system, the Metropolis uses additional visual sensors and triangulation principle to obtain the real position of the welding seam, and then conduct real-time tracking of the robot's trajectory. Adjustment.

The working principle of laser vision is based on the principle of laser triangulation, which is a traditional and relatively mature method. The line laser is used as an external active light source in the whole system. When it is irradiated obliquely to the surface of the workpiece weld, a laser spot will be generated. The image of the laser spot enters the camera after reflection, and is imaged as an image point on the photosensitive detector of the camera. In the system, the welding torch and the laser vision sensor are rigidly fixed by the fixed bracket. When the weld surface of the detected workpiece moves up and down or left and right, the position of the image point will also change accordingly, so it can be determined according to the position change relationship of the image point. Calculate the height and horizontal position where the welding torch occurs.

At present, in the laser welding robot equipment, the laser welding head and the laser sensor are mostly two independent entities. In the actual production and application process, after the user installs the laser sensor on the laser welding head, it needs to be calibrated by hand and eye. Realize the conversion of the sensor scanning position to the actual position of the weld. If the user removes the laser welding head from the robot during equipment maintenance, it will cause a positional deviation between the sensor and the laser welding head, and the user needs to re-calibrate the hand-eye to ensure the accuracy of the scanning when the equipment is running.

SUMMARY OF THE INVENTION

In view of the defects of the prior art, the purpose of the present invention is to provide an integrated main and auxiliary beam splitting device of a blue-light laser welding robot, which integrates the laser welding head and the line scanning sensor required for blue-light welding, aiming to solve the problem in practical application. In the production welding process, frequent hand-eye calibration is required to ensure the accuracy of the equipment.

In order to achieve the above purpose, the present invention provides an integrated main and auxiliary beam splitting device for a blue light laser welding robot, including a blue light beam collimation system, a blue light main and auxiliary beam splitting system, a focusing system, and a auxiliary beam collimation system placed along the optical path. Beamline conversion system, linear blue light scanning positioning optical path, CCD camera for observation. The blue light beam output by the blue light semiconductor laser is expanded and collimated by the blue light beam collimation system and then incident on the blue light main and sub-beam splitting system, and the blue light main and subsidiary beam splitting system is used for decomposing the blue light beam to form Main beam for blue light welding and secondary beam for blue light scanning;

The focusing system is used for converging the main beam at the designated welding position for welding;

The side beam collimation beam line conversion system is used to convert the side beam into a line laser, during the welding process, the line laser is located in front of the welding position, and can be received by the CCD camera after reflection;

The linear blue light scanning positioning optical path is used for image processing to obtain the actual welding spot position according to the reflected light of the linear laser received by the CCD camera.

Preferably, the beam collimation system includes two Kepler telescope structures formed by placing convex lenses along the optical path, wherein the focal length of the first convex lens is smaller than the focal length of the second convex lens. The blue semiconductor laser is coupled to the optical fiber output to the beam collimation system, and is expanded and collimated by a telescope system composed of two convex lenses.

Preferably, the blue light main and auxiliary beam splitting system includes a total reflection mirror and a beam splitter, wherein the beam splitter is preferably a reflector with a transmittance of 0.1%, so as to realize the decomposition of the blue light collimated beam and form a beam for blue light welding. main beam and secondary beam for blue light scanning. After the collimated beam is incident on the total reflection mirror, the beam direction is deflected by 90° and incident on the beam splitter with a transmittance of 0.1%, of which 99.9% of the blue light is reflected and incident to the beam focusing system, and 0.1% of the blue light is transmitted, and then passes through first. A reflector adjusts the beam direction, and then enters the secondary beam collimated beam linear conversion system.

Preferably, the main beam of the laser collimated beam passes through the focusing system, and the main beam is incident on the focusing mirror and converges at the designated welding position.

Preferably, the secondary beam collimating beam linear conversion system comprises a third convex lens, a fourth convex lens and two cylindrical lenses for beam shaping, which are placed along the optical path for condensing blue light, and the secondary beam first passes through the two convex lenses. A beam-condensing system is formed. After reducing the beam radius, it is incident on two cylindrical mirrors in turn, and finally converted into a line-shaped laser output. During the welding process, the linear laser is located just in front of the welding position, and can be received by the CCD camera at the same height as the focusing mirror after being reflected.

Preferably, the linear blue light scanning and positioning optical path includes two flat mirrors for reflecting the line laser, the CCD camera is located at the focusing lens, and the collimated laser is incident on the first flat mirror and the second after passing through the collimated beam linear conversion device. On the plane mirror, after two reflections, the linear beam irradiates the Cu material splicing weld, and makes it just in the forward direction of the welding. According to the principle of triangulation, the reflected light reflected by the Cu material will be incident on the CCD camera. , and then obtain the actual welding spot position through image processing.

As a further preference of the present invention, the central axis of each optical lens coincides with the central axis of the light beam.

As a further preference of the present invention, in the beam collimation system, the focal length of the first convex lens is smaller than that of the second convex lens, the two convex lenses are placed in parallel, and the separation distance is greater than the sum of the focal lengths of the two convex lenses, and the blue light output from the end face of the optical fiber has a certain The divergence angle of the light beam, the light beam converges at the focal position of the second convex lens after passing through the first convex lens. The distance from this point to the second convex lens is exactly equal to the focal length of the second convex lens. After the blue light beam passes through the second convex lens, the divergence angle pole can be obtained. Small parallel blue light beam output.

As a further preference of the present invention, all optical lenses are coated with an antireflection coating for blue light to reduce power loss caused by reflection or scattering on the lens surface, and a layer of special medium is coated on the light-emitting surface of the beam splitter, so that 99.9% of the blue light beam is plated with a special medium. % of the energy is reflected into the laser collimated beam focusing system and the remaining 0.1% is passed through the beamsplitter. The rest of the mirrors are coated with a film that enhances blue light reflection to reduce power loss;

As a further preference of the present invention, in the secondary beam collimation beam linear conversion system, the focal length of the third convex lens is greater than that of the fourth convex lens, the two convex lenses are placed in parallel, and the separation distance is exactly equal to the sum of the focal lengths of the two convex lenses, and the beam passes through the third convex lens. After the convex lens converges on the focal position of the third convex lens, since the distance from this point to the fourth convex lens is exactly equal to the focal length of the fourth convex lens, the blue light beam can obtain a parallel blue light beam output with a smaller spot size after passing through the fourth convex lens.

As a further preference of the present invention, when the welding robot sets the tool TCP, the intersection of the main beam after the focusing lens focused by the main beam needs to be used as the tool point. When the focal length of the focusing lens is selected, the main beam is perpendicular to the Cu material welding seam. In the case of a plane, the intersection of the main beam and the line-shaped scanning spot reflected on the Cu material weld through the second plane mirror should be on the same plane.

As a further preference of the present invention, the focusing lens for focusing the main beam and the CCD camera for receiving the linear reflected light should be placed at the same height, and the receiving surface of the CCC camera should be parallel to the beam, which can minimize the spatter generated during welding. and the effect of interfering light on position tracking.

Through the above technical solutions conceived by the present invention, compared with the prior art, the following beneficial effects can be achieved:

1. The integrated main and auxiliary beam splitting device of the blue-light laser welding robot provided by the present invention concentrates the two functions of laser welding and laser welding seam detection on one welding equipment, and can obtain the scanning point and The position transmission matrix relationship at the actual welding focus, the user only needs to perform hand-eye calibration once when debugging the equipment.

2. In the present invention, after collimating the laser beam output by the blue semiconductor laser, the beam is split by a reflector with a transmittance of 0.1%, wherein 99.9% of the beam energy is reflected and focused by a focusing mirror as For the welding light of welding Cu material, after 0.1% of the beam energy is transmitted through the reflector, it is shaped into a linear light spot for the detection of the actual welding seam position.

3. For the welding of Cu materials, when the welding power of blue light increases, the influence of spatter and interference light at the welding position will become larger, but since the beam transmittance of the beam splitter mirror is 0.1%, when the output of blue light is When the power increases, the energy of the secondary beam used for scanning will also increase, so that the main source of the beam received by the CCD camera is the reflection of the linear scanning spot.

Description of drawings

FIG. 1 is a schematic diagram of an integrated main and auxiliary beam splitting device of a blue-light laser welding robot according to a specific embodiment of the present invention.

FIG. 2 is a diagram of a beam collimation system according to a specific embodiment of the present invention.

FIG. 3 is a schematic diagram of a main and sub-beam splitting principle of blue light according to a specific embodiment of the present invention.

FIG. 4 is a positional relationship diagram of the main and sub-beams of blue light according to a specific embodiment of the present invention.

FIG. 5 is a diagram of a linear conversion device for a collimated beam of a secondary beam according to a specific embodiment of the present invention.

FIG. 6 is a schematic diagram of a linear blue light scanning positioning optical path according to a specific embodiment of the present invention.

Detailed ways

In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, but not to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

In order to achieve the above purpose, the present invention provides an integrated main and auxiliary beam splitting device for a blue light laser welding robot, including a blue light collimation beam expanding system, a blue light beam splitting system, a collimated beam shaping system, a linear blue light beam placed along the optical path Scanning and positioning optical path, collimated beam convergence system, CCD camera for observation.

The invention provides an integrated main and auxiliary beam splitting device of a blue light laser welding robot, comprising a blue light beam collimation system, a blue light main and auxiliary beam splitting system, a focusing system, a side beam collimation beam line shape transformation system, which are placed along the optical path. Linear blue light scanning positioning optical path, CCD camera for observation. The blue light beam output by the blue light semiconductor laser is expanded and collimated by the blue light beam collimation system and then incident on the blue light main and sub-beam splitting system, and the blue light main and subsidiary beam splitting system is used for decomposing the blue light beam to form Main beam for blue light welding and secondary beam for blue light scanning;

The focusing system is used for converging the main beam at the designated welding position for welding;

The side beam collimation beam line conversion system is used to convert the side beam into a line laser, during the welding process, the line laser is located in front of the welding position, and can be received by the CCD camera after reflection;

The linear blue light scanning positioning optical path is used for image processing to obtain the actual welding spot position according to the reflected light of the linear laser received by the CCD camera.

Specifically, the beam collimation system includes a Kepler telescope structure composed of two convex lenses placed along the optical path, wherein the focal length of the first convex lens is smaller than the focal length of the second convex lens. The blue semiconductor laser is coupled to the optical fiber output to the beam collimation system, and is expanded and collimated by a telescope system composed of two convex lenses.

Specifically, the blue light main and auxiliary beam splitting system includes a total reflection mirror and a beam splitter with a transmittance of 0.1%, which realizes the decomposition of the blue light collimated beam and forms the main beam for blue light welding and blue light scanning. the secondary beam. After the collimated beam is incident on the total reflection mirror, the beam direction is deflected by 90° and incident on the beam splitter with a transmittance of 0.1%, of which 99.9% of the blue light is reflected and incident to the beam focusing system, and 0.1% of the blue light is transmitted, and then passes through first. A reflector adjusts the beam direction, and then enters the secondary beam collimation beam linear conversion system.

Specifically, the main beam of the laser collimated beam passes through the focusing system, and the main beam is incident on the focusing mirror and converges at the designated welding position.

Specifically, the secondary beam collimating beam linear conversion system includes a third convex lens, a fourth convex lens and two cylindrical lenses for beam shaping, which are placed along the optical path for blue light convergence. The secondary beam first passes through the two convex lenses. A beam-condensing system is formed. After reducing the beam radius, it is incident on two cylindrical mirrors in turn, and finally converted into a line-shaped laser output. During the welding process, the linear laser is located just in front of the welding position, and can be received by the CCD camera at the same height as the focusing mirror after being reflected.

Specifically, the linear blue light scanning and positioning optical path includes two flat mirrors for reflecting the line laser, the CCD camera is located at the focusing lens, and the collimated laser is incident on the first flat mirror and the second after passing through the collimated beam linear conversion device. On the plane mirror, after two reflections, the linear beam irradiates the Cu material splicing weld, and makes it just in the forward direction of the welding. According to the principle of triangulation, the reflected light reflected by the Cu material will be incident on the CCD camera. , and then obtain the actual welding spot position through image processing.

Specifically, the central axis of each optical lens coincides with the central axis of the light beam.

Specifically, in the beam collimation system, the focal length of the first convex lens is smaller than that of the second convex lens, the two convex lenses are placed in parallel, and the separation distance is greater than the sum of the focal lengths of the two convex lenses, and the blue light output from the end face of the optical fiber has a certain beam divergence angle, After the light beam passes through the first convex lens, it converges at the focal position of the second convex lens. The distance from this point to the second convex lens is exactly equal to the focal length of the second convex lens. After the blue light beam passes through the second convex lens, parallel blue light with a very small divergence angle can be obtained. beam output.

Specifically, all optical lenses are coated with an anti-reflection coating for blue light to reduce the power loss caused by reflection or scattering on the surface of the lens, and a special medium is coated on the light-emitting surface of the beam splitter, so that 99.9% of the energy in the blue light beam is absorbed Reflected into the laser collimated beam focusing system, the remaining 0.1% of the energy passes through the beamsplitter. The rest of the mirrors are coated with a film that enhances blue light reflection to reduce power loss;

Specifically, in the secondary beam collimation beam linear conversion system, the focal length of the third convex lens is greater than that of the fourth convex lens, the two convex lenses are placed in parallel, and the separation distance is exactly equal to the sum of the focal lengths of the two convex lenses, and the beam passes through the third convex lens. At the focal position of the third convex lens, since the distance from this point to the fourth convex lens is exactly equal to the focal length of the fourth convex lens, after the blue light beam passes through the fourth convex lens, a parallel blue light beam output with a smaller spot size can be obtained.

Specifically, when the welding robot sets the tool TCP, the intersection of the main beam after the focusing lens focused by the main beam needs to be used as the tool point. When the focal length of the focusing lens is selected, when the main beam is perpendicular to the plane of the Cu material weld , the intersection of the main beam and the linear scanning spot reflected on the Cu material weld through the second plane mirror should be on the same plane.

Specifically, the focusing lens for focusing the main beam and the CCD camera for receiving the linear reflected light should be placed at the same height, and the receiving surface of the CCD camera should be parallel to the beam, which can minimize the spatter and disturbing light generated during welding. The impact of location tracking.

The following is a detailed description in conjunction with optional embodiments:

In this embodiment, as shown in FIG. 1 , the application scenario of the integrated main and auxiliary beam splitting device of the blue laser welding robot is the welding seam detection and scanning welding of Cu material splicing welds, and the integrated main and auxiliary beam splitting of the blue laser welding robot The device is packaged in a welding device 5 which is fixed on the flange 4 of the industrial six-axis robot 2 . First set the welding trajectory of the industrial six-axis robot 2. After the movement starts, the industrial six-axis robot 2 sends a signal to the blue light semiconductor laser 1. The blue light semiconductor laser 1 generates a blue light beam, which is transmitted to the welding device 5 through the optical fiber 6. After the integrated main and auxiliary beam splitting device of the blue-light laser welding robot, the main beam intersection 22 focused by the convex lens 21 moves horizontally to the right along the splicing seam 3 of the Cu material for splicing welding, and at the same time, the auxiliary beam is shaped to form a linear scanning spot. 18 is located in the forward direction of the robot. The reflected light information collected by the CCD camera 19 is processed by the image processing system 20 , and then the coordinate information of the actual weld is sent to the industrial six-axis robot 2 .

In this embodiment, as shown in Fig. 2, the beam collimation system consists of two convex lenses 7 and 8. The focal length of the convex lens 7 is f1, and the focal length of the convex lens 8 is f2. The two convex lenses are placed in parallel with a distance slightly larger than f1+f2 . When the blue light generated by the blue laser is output through the fiber, it has a certain blue light divergence angle. After the convergence of the convex lens 7, it converges at the focal point of the convex lens 8, and then passes through the convex lens 8 to obtain a collimated blue laser output.

In this embodiment, as shown in FIG. 3 , the collimated light beam passing through the convex lens 8 passes through a reflecting mirror 9 to realize the change of the propagation direction of the light beam, and then enters the reflecting mirror 10 which is made of special material. The reflectivity is 99.9%, and the blue light is split at the mirror 10 and decomposed into reflected light (main beam) containing 99.9% energy and transmitted light (sub-beam) containing 0.1% energy, and the transmitted light by the mirror 10 contains 0.1 The collimated blue light beam with % energy, after being reflected by the mirror 11, is incident into the linear transformation system of the collimated light beam.

In this embodiment, as shown in FIG. 4 , the placement position of the convex lens 21 is at the same height as the CCD camera 19 , and the main beam intersection 22 after being focused by the convex lens 21 just converges on the Cu material splicing weld 3 , and the position satisfies the Coplanar with the linear scanning spot 18 and behind the linear scanning spot 18 in the welding direction.

In this embodiment, as shown in FIG. 5 . The secondary beam collimated beam linear transformation system is composed of two convex lenses 12 , 13 and two cylindrical mirrors 14 , 15 . The focal lengths of the convex lens 12 and the convex lens 13 are f3 and f4, respectively, and f3 is greater than f4. The two convex lenses are placed in parallel, and the separation distance is exactly equal to the sum of the focal lengths of the two convex lenses. The parallel beam passes through the convex lens 12 and converges at the focal position of the convex lens 12. , since the distance from this point to the convex lens 13 is exactly equal to the focal length of the convex lens 13 , after the blue light beam passes through the convex lens 13 , a parallel blue light beam output with a smaller spot size can be obtained. The condensed blue light parallel light will be quickly focused in the lens after contacting the first surface of the cylindrical mirror 14, resulting in a very large divergence angle of the light beam, resulting in a linear effect on the image plane, and then passing through the cylindrical surface The mirror 15 further shapes the diverging beam, and finally obtains a line-shaped laser suitable for laser scanning.

In this embodiment, as shown in FIG. 6 , the linear blue light scanning positioning system is composed of two reflective mirrors 16 and 17 at a certain angle, and the linear blue light after passing through the cylindrical mirror 15 continuously passes through the mirrors 16 and 17 After reflection, it is obliquely incident on the Cu board, and the linear scanning light spot 18 is in a vertical relationship with the Cu material splicing weld 3 in the horizontal plane, and is received by the CCD camera 19 after being reflected on the surface of the Cu material. In the principle of triangulation, the point light spot is imaged in On the CCD line array, and there is a unique correspondence between the imaging position and the depth position of the light spot, by measuring the center position of the real image formed on the CCD line array, the depth coordinates of the light spot at the moment can be obtained by the calculation method of geometric optics, so as to obtain For the depth parameter at this point on the measured surface, since the light spot used in scanning is a linear light spot, a set of data of the measured surface topography can be obtained by measuring several sampling points in the vertical direction of the weld. Since the blue light at the welding seam will pass through the welding seam and cannot be reflected back to the CCD camera 19 normally, the scanned image will have an obvious break point after processing. Since the relative position of the linear scanning beam and the welding focal point is fixed, the position transmission matrix between them is also fixed. 20. Calculate the position of the weld in the robot coordinate system according to the known position transmission matrix, and transmit it to the industrial six-axis robot 2 .

Those skilled in the art can easily understand that the above are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the spirit and principles of the present invention, etc., All should be included within the protection scope of the present invention.

Claims (8)

1. an integrated main and auxiliary beam splitter of a blue light laser welding robot, is characterized in that, comprises the blue light beam collimation system, the blue light main and auxiliary beam splitting system, the focusing system, the auxiliary beam collimation beam linear transformation that is placed along the optical path system, a linear blue light scanning positioning optical path, and a CCD camera for observation; the blue light beam output by the blue light semiconductor laser is beam expanded and collimated by the blue light beam collimation system and then incident on the blue light main and auxiliary beam splitting system. The blue light main and auxiliary beam splitting system is used to decompose the blue light beam to form the main light beam for blue light welding and the auxiliary light beam for blue light scanning; The focusing system is used for converging the main beam at the designated welding position for welding; The side beam collimation beam line conversion system is used to convert the side beam into a line laser, during the welding process, the line laser is located in front of the welding position, and can be received by the CCD camera after reflection; The linear blue light scanning positioning optical path is used for image processing to obtain the actual welding spot position according to the reflected light of the linear laser received by the CCD camera. 2 . The integrated main and auxiliary beam splitting device according to claim 1 , wherein the blue light beam collimation system comprises a first convex lens and a second convex lens placed along the optical path, forming a Kepler telescope structure, and compromising the blue light. 3 . The beam is expanded and collimated. 3. The integrated main and auxiliary beam splitting device according to claim 2, wherein the blue light main and auxiliary beam splitting system comprises a total reflection mirror and a beam splitter, and the collimated blue light beam is incident on the total reflection After the mirror, the beam direction is deflected by 90° and enters the beam splitter, the reflected light is the main beam, and the transmitted light is the secondary beam. 4. The integrated main and auxiliary beam splitting device according to claim 1 or 3, characterized in that, the auxiliary beam collimation beam linear conversion system comprises a third convex lens and a fourth convex lens for condensing blue light placed along the optical path. A convex lens, a first cylindrical lens and a second cylindrical lens for beam shaping, the sub-beams pass through the third convex lens and the fourth convex lens in turn to reduce the beam radius, and then enter the first cylindrical lens and the second cylindrical surface in turn mirror, which is converted into a line-shaped laser output. 5 . The integrated main and auxiliary beam splitting device according to claim 1 , wherein the linear blue light scanning and positioning optical path comprises a first plane mirror and a second plane mirror for reflecting the linear laser. 5 . After two reflections, it is irradiated on the splicing weld to form a linear scanning light spot. The reflected light reflected by the material will be incident on the CCD camera, and the actual welding spot position will be obtained through image processing. 6 . The integrated main and auxiliary beam splitting device according to claim 5 , wherein the focusing system is a focusing lens, the intersection point of the main beam converged by the focusing lens is a tool point, and the intersection point of the main beam and the linear The scanning spots are in the same plane. 7 . The integrated main and auxiliary beam splitting device according to claim 2 , wherein the focal length of the first convex lens is smaller than the focal length of the second convex lens, and the separation distance is greater than the sum of the focal lengths of the two convex lenses. 8 . 8 . The integrated main and auxiliary beam splitting device according to claim 4 , wherein the focal length of the third convex lens is greater than the focal length of the fourth convex lens, and the separation distance is equal to the sum of the focal lengths of the two convex lenses. 9 .

Images