# Extended the depth of field and zoom microscope with varifocal lens

Schematic illustration of the proposed microscope is shown in Fig. 1a, which consists of four major components: an EDOF and zoom microscope objective, an image sensor, a lighting system and an image processing system. As the essential element, the varifocal lens is specially designed and fabricated to help the system achieve not only continuous optical zoom with a large zoom ratio, but axial scanning with constant magnification. Combined with the proposed image fusion algorithm, the proposed microscope can realize two functions: EDOF and optical zoom. For the EDOF function, the proposed microscope can realize EDOF with constant magnification and high resolution. By varying the curvature of the varifocal lenses, a series of 2D images with the same magnification are obtained, and then processed by the proposed image algorithm rapidly to generate the EDOF image. as shown in Fig. 1b. The optical zoom function is shown in Fig. 1c. The combination of the varifocal lenses is optimized to achieve different magnifications without any mechanical movement. Thus, the proposed microscope can obtain the EDOF images at any magnifications in real time without mechanical movement, which is not available for existing microscopes.

As the key part of the proposed microscope, the EDOF and zoom microscope objective consists of several glass lenses and four varifocal lenses. Simplified schematic diagram of the objective microscope is shown in Fig. 2a. Four varifocal lenses provide four variables to realize optical axial scanning, zooming, and keeping constant magnification with high resolution. In this way mechanical displacement of the sample during scanning and zooming can be avoided. For conventional optical sectioning techniques, the magnification of the acquired image varies with the scanning depth, and it is difficult to correct the problem of inconstant magnification by using only one focal length as the variable. However, the introduction of varifocal lenses can increase the degree of freedom of the objective microscope, which can be used to correct the problem of inconsistency when extending the DOF and realize optical zoom.

In order to make full use of the zoom capability of the varifocal lens, we adopt a splicing design of the front and rear groups in the optical path. The image plane of the front group coincides with the object plane of the back group, and the image space NA of the front group matches the object space NA of the rear group. In this way, we can use the front group to increase the NA of the objective, while the rear group is used to expand the zoom ratio. In addition, both the front and rear groups have zoom capability, so they can work together to achieve continuous optical zoom by controlling the curvature of the four varifocal lenses. In addition, the change of the magnification (M _ {{ text {Front}}} ) of the front group is caused by the change of the object distance, and the change of the magnification (M _ {{{ text {Rear}}}} ) of the rear group is caused by the change of the image distance. The optical distance of the front and rear groups remains unchanged, so the equation of zoom magnification can be obtained as follows,

$$frac {{1 – M_ {Front} ^ {2}}} {{M_ {Front} ^ {2}}} f_ {Front} ^ { prime} dM_ {Front} + frac {- 1 – M _ {{{ text {Re}} ar}} ^ {2}}} {{M _ {{{ text {Re}} ar}} ^ {2}}} f _ {{{ text {Re}} ar}} ^ { prime} dM _ {{{ text {Re}} ar}} = 0,$$

(1)

where (M _ {{{ text {Front}}}} ), (M _ {{{ text {Rear}}}} ) are magnifications of the front group and rear group of the proposed objective, respectively. Besides, (f _ {{{ text {Front}}}} ^ { prime} ), (f _ {{{ text {Rear}}}} ^ { prime} ) are the focal length of the objective, the front group and rear group, respectively.

The transverse magnification of the objective microscope ( (M _ {{{ text {Ob}}}} )) can be given by the below equation.

$$M _ {{{ text {Ob}}}} = M _ {{{ text {Front}}}} * M _ {{{ text {Rear}}}},$$

(2)

Considering the characteristics of the front and rear group in the light path, we use the two varifocal lenses in the front group as the focusing group to extend EDOF of microscope and keep constant magnification. As shown in Fig. 2b, By adjusting the curvature of the two varifocal lenses, microscope objective focus quickly in a certain depth, extending the DOF of microscope from (z_ {0} ) durable (z_ {1} ). Since there is only one surface with variable curvature in the varifocal lens, we can simplify it to a simple plano-convex lens for ease of calculation. To keep the magnification of the proposed objective constant, the focal length of two varifocal lenses are required to satisfy Eq. (3).

 frac {{f_ {P11} ^ { prime}}} {{f_ {P21} ^ { prime}}} = frac {{z_ {1} – f_ {G} ^ { prime}} } {{z_ {0} – f_ {G} ^ { prime}}} * frac {{f_ {P12} ^ { prime}}} {{f_ {P22} ^ { prime}}}, 

(3)

where (f {^ { prime}} _ {P11} ), (f {^ { prime}} _ {P21} ), (f {^ { prime}} _ {P12} ) and (f {^ { prime}} _ {P22} ) are the corresponding focal lengths of varifocal lens 1 and lens 2 when the focusing distances of objective are z0 and z1, respectively. (f {^ { prime}} _ {G} ) is the focal length of glass lens group.

According to the theoretical analysis above, the proposed objective can realize continuous optical zoom and extending the DOF with constant magnification based on specially designed parameters.