How to Choose the Cam Curve Form of Zoom Lenses?

In the final stage of the zoom lens design, the cam curve needs to be designed. In order to balance the uniformity of the imaging magnification change and the cam rotation angle, a suitable cam curve form needs to be selected, and a certain functional relationship between the imaging magnification M and the cam rotation angle θ must be established.

According to the actual boundary conditions and function continuity to be met by the cam curve, several common functional relations between M and θ are established, and the design examples are simulated by Matlab.

The results show that when the established power function relationship between the imaging magnification M and the cam rotation angle θ and the pressure rise angle does not exceed the allowable value, the variable magnification change balance is the best, the magnification change curve is smooth, and there is no cam inflection point. This method can facilitate designers to change the cam curve shape according to different design requirements and find the best zoom cam curve.

In the zoom system, the cam is the component that controls the movement of the lens group and enables the system to change the focal length while keeping the image plane stable. The smooth, flexible, and fast zoom operation is an important content of the zoom performance, which is directly related to the roughness of the cam profile and the pressure angle of the cam curve.

If the cam curve has a larger inclination angle in a certain section, it appears steeper. In the process of rotating the cam, there will be a heavier hand feeling. The gear that drives the cam to rotate is worn out, and the edge of the cam groove may even be squeezed and deformed, which affects the imaging quality; when it is more serious, the cam will be stuck and cannot be used at all.

In the mechanically compensated zoom lens, by selecting the movement relationship between the movement of the zoom group and the cam angle, it can be satisfied that the imaging magnification changes at a uniform speed or at a uniform acceleration during the zooming process. This is very promising whether it is for television, film photography, or zoom lenses used in military tracking.

But at the same time, the pressure angle corresponding to the cam curve at both ends of the long and short focal lengths will be very large. When pursuing the miniaturization of the zoom lens, the rotation torque of the cam will often exceed the tolerable value. Therefore, it is necessary to choose the cam curve form and find a balance between the uniformity of the imaging magnification change and the cam rotation angle.

This paper establishes the power function relationship between the imaging magnification M and the cam rotation angle θ. Combined with the zoom system equation, the Matlab simulation interface is written to simulate the cam curve form. The results show that the established function M-θ is more linear than the traditional curve fitting and θ-x Method, the cam curve is smoother, the pressure rise angle is smaller overall (<45°), and the magnification change is more balanced.

The basic theory of the zoom system

The final stage of zoom lens design, that is, after the optical mechanism parameters (radius, interval, glass material) of each lens element is determined, it is also necessary to calculate the numerical relationship between the displacement of the zoom group and the compensation group to process the cam track. Below we discuss the two forms of the cam curve equation.

Form 1: Obtain the compensation group movement amount y from the variable magnification group movement amount x, and then obtain the variable magnification cam curve, namely

x→y→m₂* ，m₃* →M (1)

Form 2: Obtain the magnifications m2* and m3* of the variable magnification group and the compensation group from the variable magnification M required by the system, and obtain the movement x, y of the variable magnification group and the compensation group, and then obtain the cam curve, namely

M→m₂*，m₃*→x,y (2)

It can be seen from the above two forms that in order to establish the cam equation and then process the cam track, it is necessary to establish the functional relationship between the cam angle and one of the above variables.

Design of zoom cam curve

In order to balance the uniformity of the imaging magnification change and the cam angle, it is necessary to select a suitable cam curve form. This requires us to establish a certain functional relationship between the imaging magnification M and the cam rotation angle θ, which will be discussed below.

1. Formal discussion

First, specify the following symbols.

f₁,f₂,f₃,f₄： They are the focal lengths of the front fixed group, zoom group, compensation group, and rear fixed group;

m₂,m₃： Respectively the lateral magnification of the zoom group and the compensation group;

m₂₃=m₂m₃: The lateral magnification of the variable magnification part composed of the variable magnification group and the compensation group;

m₂₃₀：The horizontal magnification of the variable magnification part at the starting point (θ=0) at the focal position. This example is the shortest focal length position;

L： The maximum movement of the zoom group;

θ：The rotation angle of the cam;

α：The maximum rotation angle of the cam;

R： Drum radius:

M: Zoom magnification;

Max： Maximum zoom magnification;

x,y: Respectively represent the movement amount of the zoom group and the compensation group;

Among them, k_xand k_yare the slopes of the cam curve zoom group and the compensation group respectively, that is, the tangent value of their pressure rise angle.

When constructing the functional relationship between θ and M, two boundary conditions need to be paid attention to:

Boundary condition one： When θ=0°, M=1

Boundary condition two： When θ=α, M=Mmax

(1) θ has a linear relationship with M

It can make the magnification change produce a uniform effect, this relation can be expressed as

The above formula can be used to differentiate θ to obtain the rate of change of magnification:

(1) There is a linear relationship between θ and M

(a) First construct the power function relationship between θ and M:

M: Zoom magnification;

M_max： Maximum zoom magnification;

x,y: Respectively represent the movement amount of the zoom group and the compensation group;

Among them, k_x and k_y are the slopes of the cam curve zoom group and the compensation group respectively, that is, the tangent value of their pressure rise angle.

When constructing the functional relationship between θ and M, two boundary conditions need to be paid attention to:

Boundary condition one： When θ=0°, M=1

Boundary condition two： When θ=α, M=Mmax

(1) θ has a linear relationship with M

It can make the magnification change produce a uniform effect, this relation can be expressed as

The above formula can be used to differentiate θ to obtain the rate of change of magnification:

(1) There is a linear relationship between θ and M

(a) First construct the power function relationship between θ and M:

In the formula, M, M_max, α, θ are defined as above, n is the cam curve coefficient, which can also be regarded as an adjustment coefficient, and n≠0 is specified. (5) Differentiating the formula to θ can be obtained:

(b) Construct the exponential function relationship between θ and M

The formula (8) can be obtained by differentiating θ:

(c) Construct the logarithmic function relationship between θ and M

After differentiation, there is

(d) Construct the Gaussian function relationship between θ and M

(e) The above four functional relationships are used in the case of n≠0. For the case of n=0, we stipulate:

That is, there is a linear relationship between θ and x, which is currently the most commonly used relationship. Differentiate equation (16) and the cam equation to obtain the rate of change of the magnification as

The above results are discussed below.

(1) When θ has a linear relationship with M, it can be seen from the formula (4) that the system variable times and uniform speed are very desirable. However, due to the non-linear relationship between the movement amount x of the zoom group and M, it will inevitably lead to the non-uniformity of the movement speed of the lens group and the imbalance of the pressure angle.

(2) When θ and M are in a non-linear relationship, and θ and M are in a backup function relationship, if n is constant, the variable magnification and the rotation angle will change in a backup function relationship. It can be seen from the formula (7) that when the cam curve coefficient n approaches 0, the variable magnification and the rotation angle change exponentially, that is, the magnification increases faster and faster during the cam rotation.

When n=1 and the conditions of the above formula appear, the rate of change of magnification is a fixed value, that is, during the entire rotation of the cam from the end of the shortest focal length to the end of the longest focal length, the change of magnification is uniform, which becomes the situation discussed in formula (4).

When m=1/2 and the conditions of the above formula appear, so the magnification change is uniformly accelerated, which is also what we expect.

(3) The relationship between θ and M is exponential, logarithmic, and Gaussian. When n is constant, the rate of change of magnification is inversely proportional to θ, that is, when the cam gradually rotates from the shortest focal length to the longest focal length, the rate of change of magnification becomes slower and slower.

When n changes, the rate of change of magnification is nonlinear with the θ curve. When the cam curve coefficient n approaches 0, the rate of change of the magnification is a constant, that is, the change of the magnification is linear.

Generally speaking, the principle of selecting the cam curve is to select the curve form with the best variable magnification change and smooth magnification change curve when the pressure rise angle does not exceed the allowable value.

Through the above discussion and Matlab experimental simulation, we understand that when the relationship between θ and M is a backup function, by selecting the appropriate adjustment coefficient n, it can be achieved that when the pressure angle is close to the allowable value, the variable magnification change has the best balance and the magnification change. The curve is smooth and there is no cam inflection point.

However, none of the other functions can achieve better results. Figure 1 is a cam simulation diagram designed using three-dimensional drawing software based on the previous analysis results. The following describes the results of our simulation experiments.

Fig. 1 Simulation figure of cam

2. Analysis and simulation

Combine the two forms of cam curve solving methods given above, and the functional relationship between θ and M, to compile the Matlab GUI interface.

The program input parameters are:

Cam curve coefficient n, optical system's short focal length f₀, variable magnification group focal length f₂, compensation group focal length f₃, lead (movement distance of the variable magnification group from short focal length to long focal length) wl, objects in the variable magnification group at short focal length, the distance is l₂₀, the image distance of the compensation group is ll₃₀ at short focus, the maximum cam rotation angle α, the rotation angle interval mu in angle units, and the diameter of the cam drum dd.

The program output parameters are:

Rotation angle, movement of the zoom group, movement of the compensation group, focal length, pressure rise angle of the zoom group, pressure rise angle of the compensation group, and zoom ratio data and graphics.

Through this program interface, you can easily see the relationship between the zoom ratio and the angle of rotation, observe whether the pressure rise angle exceeds the allowable value, and change the adjustment coefficient to quickly change the above relationship to find the best cam curve required.

Take the 20^X continuous zoom positive compensation system designed by the research group as an example to select the cam curve.

The fixed parameters are:

Variable parameter: adjustment coefficient n

This program is editable for all the above parameters. In order to facilitate the discussion, here we mainly consider the influence of the adjustment coefficient n on the result.

Fig. 2 Matlab simulation interface

(1) When n=0, there is a linear relationship between θ and x, as shown in Figure 3.

(2) When n = 0. 047, the power function graph is shown in Figure 4.

(3) When n= 0. 000 1, the graph of the exponential function is shown in Figure 5.

(4) When n= 2.32, the logarithmic function graph is shown in Figure 6

(5) When n = 0.1, the graph of the Gaussian function is shown in Figure 7.

The experiment proves that the previous discussion on the relationship of various functions is consistent with the trend of the graph curve.