The scanning speed of the PZT-actuated resonant fiber-optic scanner depends on the mechanical resonant frequency of its fiber cantilever, and the frequency of driving signal applied to the PZT must be tuned to the resonant frequency in order to obtain a maximum scanning field. In our previous designs [

11,

16], the resonant frequency

*ω* of the fiber-optic cantilever is given by

Eq. (1) [

17]:

where

*β* is a mode-related constant,

*E*,

*ρ*,

*R* and

*L* represent the Young’s modulus, mass density, radius and length of the fiber cantilever, respectively. For a single-mode fiber (SMF such as SMF-28) with a diameter 125 μm, the resonant frequency is ~1 kHz for a cantilever length

*L* = 10 mm. To reduce the resonant frequency to ~100 Hz, several solutions can be considered. The first method is to thin the radius of the fiber cantilever by 10 times through chemical etching. However, this approach might be problematic due to the resulted mechanical fragility, potential leakage of guided light and difficulty in controlling the etching uniformity. The second method is to use a longer cantilever, i.e. ~33-mm long for reducing the resonant frequency down to ~100 Hz. The endoscope made of such a long cantilever would have a long rigid length after packaging and become difficult to pass through the Y-shape entry port of a standard gastroscope. The third approach is to put additional weight to the cantilever and change its mechanical property, which is equivalent to changing the ratio of

*E* and

*ρ*. In this study, we first investigate how the resonance frequency can be changed with the combination of different cantilever length and additional weight added to the cantilever.

The modified structure of a resonant fiber-optic scanner is schematically shown in
. The distal portion of the SMF cantilever is covered with a stainless hypodermic tube of an ultrathin wall, which adds an additional weight to the cantilever and partially changes its mechanical property. The uncovered end of the SMF is fixed (i.e., to the PZT actuator) and the end covered by the hypodermic tube is kept free. This mechanical system can be modeled as a two-sectional stepped beam with one clamped end and one free end. Each section has its specific diameter, mass density and elasticity. There are innumerous vibration modes in this system. In this research, we are only interested in the fundamental transversal vibration mode which is used in the experiment.

The general transversal vibration solution of each section of the beam can be written as

Eq. (2) [

17]:

where

*y*_{i} is the vertical displacement of the

*i*th section (

*i*=1,2) of the cantilever and

Here in

Eq. (3)
*A* and

*I* represent the cross-sectional area and the second moment of inertia of the

*i*th section, respectively.

(

*i*=1,2;

*j*=1,2,3,4) are constants to be determined. By applying proper boundary conditions and connective conditions of continuity in displacements, displacement slopes, moments and shear forces on different sections, one can obtain the following equation if a nontrivial solution of

in

Eq. (2) exists:

where

*q*
_{i}=

*K*
_{i}
*L*
_{i} and

*L*
_{i} is the length of the

*i*th section of the cantilever.

Equation (4) is only a function of the resonant frequency

*ω*, and it can be solved numerically. There is more than one solution with each solution representing a different resonant frequency of transversal vibration modes. The fundamental resonant frequency of interest in our study is obtained as the one with the smallest absolute value.

shows the contour plot of the calculated resonant frequency of the hybrid cantilever as a function of the total SMF cantilever length and the ratio of the cantilever covered by a stainless hypodermic tube. The frequency was calculated based on a 305/406-μm inner/outer diameter stainless hypodermic tube, as used in experiments, added to the distal end of a 125-μm SMF cantilever.

In the previous design, only the cantilever length

*L* could be used to tune the resonant frequency according to

Eq. (1). The available scanning frequency range is thus limited due to the concern of the cantilever length and so forth the overall rigid length of the endoscope. With the modified cantilever structure as described above, the tunable frequency range is significantly extended without significantly increasing the cantilever length. As shown in , for a reasonable cantilever length below 20 mm, the resonant scanning frequency can be tuned between ~1 kHz to ~60 Hz by carefully selecting the combination of the total cantilever length and the metal tube length. The wide resonant frequency range provides the flexibility to match a wide range of FD-OCT systems with A-scan rates varying between ~30 kHz to ~500 kHz. also indicates that the slowest resonant scanning frequency can be achieved for a 20-mm fiber cantilever when ~50% of the cantilever is covered by a metal tube. In this paper, our target resonant frequency is around ~60 Hz in order to match a 40-kHz FDML-based SS-OCT system.

Due to the resonant nature of the scanning scheme, the scanning frequency is fixed once the probe is made. However, the above theoretical analysis provides a very good prediction of the probe configuration required for a given resonant frequency, which is pre-determined to match a given FD-OCT imaging speed. The resonance frequency can then be easily tuned/changed according to the design configuration during probe engineering.