Axial scanning employing tunable lenses: Fourier optics based system design

Katrin Philipp; Jürgen Czarske

doi:10.1364/OSAC.2.001318

1. Introduction

When designing optical systems, there is often the need to shift the measurement volume relative to a specimen under investigation. Conventionally, this is achieved by either moving the specimen itself or by moving optical elements such as microscope objectives along the optical axis. In the last few years, tunable lenses (TL) have evolved into a serious alternative for axial scanning in imaging systems, as they are compact and operate inertia-free. In particular, several microscopic techniques such as confocal [1–4], two-photon [5,6], structured illumination [7,8], light-sheet [9] as well as standard widefield microscopy [10,11] have benefited from using tunable lenses for axial scanning.

With few exceptions [12], there are two main approaches for the placement of tunable lenses in an optical system. In the first approach, the tunable lens is placed in front of an objective lens, with a distance of typically several millimeters up to a few centimeters between them [1,7]. In the second approach, a telecentric configuration is used [11]. Here, an additional relay-system is used, that consists of two lenses that are divided by the sum of their respective focal length (4f-system). The objective and tunable lenses are located on the front and back focal plane of this relay system, respectively.

The placement of the tunable lenses in the imaging system is commonly determined empirically or with the help of ray-tracing-based simulation packages. While it is known that the telecentric configuration is optimal in terms of minimal induced aberrations [11] and that there exists a trade-off between induced aberrations and maximum tuning range, a systematic and quantitative investigation of these effects is not yet available in the literature to the best of the authors knowledge. Additionally, the high diversity of commercial and particularly custom-built tunable lenses make a direct comparison between achieved tuning ranges and spatial resolutions difficult.

This article aims to provide a simple, yet powerful, purely analytic description of axial-scanning systems employing tunable lenses. We apply the proposed method on the most common configurations of tunable lens-based scanning systems to derive analytic expressions for their effective focal length and focal spot size. Further, this paper intents to be a hands-on guide for the experimentalist to find the optimal configuration for each particular target application.

2. Operator algebra

In order to describe the optical scanning systems, we use an operator algebra formalism based on the Fresnel approximation of the scalar diffraction theory introduced in [13] and borrow the notation proposed in [14]. This approach allows a high-level description of Fourier optics employing the Fresnel approximation. Without any loss of generality, we use operators for the one-dimensional case. The extension to two dimensions is straightforward, but much more difficult to read. For the full description of the optical setups covered in this article, we need the definitions of the operators $\mathcal {R}$ for a free-space propagation, $Q$ for a multiplication by a quadratic-phase exponential (as a representation of ideal, thin lenses) and $\mathcal {V}$ for the scaling by a scalar. The notation of an operator is such, that the argument of the operator follows in square brackets. The operand is enclosed by curly brackets. We define the operators as follows:

Scaling by a scalar is represented by the symbol $\mathcal {V}$ and defined by

(1)$$V[b]\{U(x)\} = |b|^{\frac{1}{2}} U(bx),$$

with $U(x)$ being the complex wave field with the spatial coordinate $x$. Multiplication by a quadratic-phase exponential is represented by the operator

(2)$$Q[c]\{U(x)\} = e^{i\frac{k}{2}cx^2}U(x)$$

with $k=2\pi /\lambda$. Note that $Q[-f^{-1}]$ represents a thin lens with focal length $f$. Free space propagation over a distance $d$ is described using the operator $\mathcal {R}[d]$ employing the Fresnel approximation of the scalar diffraction theory by

(3)$$ R[d]\{U(x_1)\} = \frac{1}{\sqrt{i\lambda d}} \int^\infty_{-\infty} U(x_1) e^{i\frac{k}{2d}(x_2-x_1)^2} \mathrm{d}x_1.$$

Here, $x_2$ denotes the coordinate after propagating the distance $d$. A list of useful relations between operators is extracted from [14]:

(4)$$\mathcal{Q}[c_1]\mathcal{Q}[c_2] = \mathcal{Q}[c_1+c_2]$$

(5)$$\mathcal{Q}[c]\mathcal{R}[d] = R[(d^{-1}+c)^{-1}]\mathcal{V}[1+cd]\mathcal{Q}[(c^{-1}+d)^{-1}]$$

(6)$$\mathcal{R}[d_1]\mathcal{R}[d_2]= R[d_1+d_2]$$

(7)$$\mathcal{V}[t_1]\mathcal{V}[t_2] = \mathcal{V}[t_1t_2]$$

(8)$$\mathcal{V}[t]\mathcal{R}[d] = R[d/t^2]\mathcal{V}[t]$$

(9)$$\mathcal{R}[d]\mathcal{V}[t] = \mathcal{V}[t]R[t^2d]$$

In the following, we assume ideal lenses and, thus, describe the influence of a lens with focal length $f$ on a propagating wave using the operator $\mathcal {Q}[-f^{-1}]$.

3. Axial scanning systems employing a tunable lens

In this section, axial scanning systems employing a tunable lens with an increasing degree of generality are discussed: the strict telecentric configuration in Section 3.1, the quasi-telecentric configuration in Section 3.2 and the general case in Section 3.3.

3.1 Telecentric regime

As it is the simplest configuration from a mathematical point of view, we start with the strictly telecentric configuration. The telecentric imaging system consists of a tunable lens (TL), a 4f-geometry and an objective lens (OL) as depicted in Fig. 1. Using the operator algebra introduced in Section 2, the system is described as

(10)$$S_{\mathrm{telecentric}} = \mathcal{Q}[-f_{\mathrm{OL}}^{-1}]\cdot S_{4f}\cdot \mathcal{Q}[-f_{\mathrm{TL}}^{-1}]$$

with the operator $S_{4f}$ denoting the 4f system, that is given by

(11)$$ S_{4f} = \mathcal{R}[f] \cdot \mathcal{Q}[-f^{-1}] \cdot \mathcal{R}[2f] \cdot \mathcal{Q}[-f^{-1}] \cdot \mathcal{R}[f] .$$

Applying Eq. (5) on the second and third factors of Eq. (11) yields

(12)$$ S_{4f} = \mathcal{R}[f] \cdot \mathcal{R}[-2f] \cdot \mathcal{V}[-1] \cdot \mathcal{Q}[f^{-1}] \cdot \mathcal{Q}[-f^{-1}] \cdot \mathcal{R}[f]$$

(13)$$ = R[-f] \cdot V[-1] \cdot R[f]$$

(14) $$= V[-1],$$

whereby Eq. (4), Eq. (8) and Eq. (6) were used for the simplifications. We confirm the common knowledge, that the output of a 4f-system is a reversed replica of its input.

Fig. 1. Telecentric setup in forward direction. The tunable and the objective lens are placed in the front and back focal plane of a 4f system, respectively. The whole telecentric system can be described using the effective focal length $f_{\mathrm {tc}}$.

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Using this result for the description of the telecentric system by inserting Eq. (14) into Eq. (10) yields

(15)$$S_{\mathrm{telecentric}} = \mathcal{V}[-1]\cdot \mathcal{Q}[-(f_{\mathrm{OL}}^{-1}+f_{\mathrm{TL}}^{-1})].$$

Note that the order of the tunable lens and the objective lens does not affect the behavior of the system according Eq. (15), since $\mathcal {Q}[-f_{\mathrm {OL}}^{-1}]\cdot \mathcal {Q}[-f_{\mathrm {TL}}^{-1}] = \mathcal {Q}[-f_{\mathrm {TL}}^{-1}]\cdot \mathcal {Q}[-f_{\mathrm {OL}}^{-1}]$ as a consequence of Eq. (4). Hence, the telecentric lens system is equivalent to a single lens with an effective focal length

(16)$$ f_{\mathrm{tc}}= \frac{f_{\mathrm{OL}}f_{\mathrm{TL}}}{f_{\mathrm{OL}} + f_{\mathrm{TL}}}$$

or, alternatively, an effective refractive power

(17)$$P_{\mathrm{tc}}= \frac{1}{f_{\mathrm{tc}}} = P_{\mathrm{TL}} + P_{\mathrm{OL}}.$$

The tuning range of an optical system is obtained using Eq. (16), resulting in

(18)$$\Delta f_{\mathrm{tc}}(f_1, f_2) = \frac{f_{\mathrm{OL}}^2\left|f_2-f_1\right|}{(f_{\mathrm{OL}}+f_1)(f_{\mathrm{OL}}+f_2)},$$

with the minimal and maximum refractive power of the adaptive lens $P_1 = 1/f_1$ and $P_2 = 1/f_2$, respectively, i. e. $P_1 \leq P_{\mathrm {TL}} \leq P_2$. Let us have a look at the tuning range of the telecentric system employing two common types of tunable lenses: The first type, positive-only tunable lenses operate in the positive focal length domain, i. e. $f_2 > f_1 > 0$. The second type are symmetric tunable lenses, i. e. $f:=f_1=-f_2$ or equivalently $-\frac {1}{f_1} \leq P_{\mathrm {TL}} \leq \frac {1}{f_1}$. The tuning range of these two cases are

(19)$$\Delta f_{\mathrm{TL, pos}} = \frac{f_{\mathrm{OL}}^2}{f_{\mathrm{OL}}+f_1} \mbox{ for } f_2 \to \infty$$

(20)$$\Delta f_{\mathrm{TL, sym}} = \frac{2f_1f_{\mathrm{OL}}^2}{f_{\mathrm{1}}^2-f_{\mathrm{OL}}^2} \mbox{ for } f_2 = -f_1.$$

Comparing Eqs. (19) and (20) yields that the tuning range employing symmetric lenses is higher than the one employing positive-only lenses as long as $f_1>\frac {f_{\mathrm {OL}}}{2}$, which is usually the case for practical imaging systems.

3.2 Quasi-telecentric regime

Let us now consider a modification of the telecentric design, where the tunable lens is shifted by a small distance $d$ away from the front focal plane of the first lens of the 4f-system, or, equivalently the objective lens is shifted away from the back focal plane of the second lens of the 4f system. As long as the assumptions $d \ll f_{\mathrm {OL}}$ and $d \ll f_{\mathrm {TL}}$ are valid, this system is referred to as quasi-telecentric. The operator describing this configuration reads

(21)$$\mathcal{S} = \mathcal{Q}[-f_{\mathrm{OL}}^{-1}]\cdot \mathcal{S}_{4f} \cdot \mathcal{R}[d]\cdot \mathcal{Q}[-f_{\mathrm{TL}}^{-1}]$$

(22)$$= \mathcal{V}[-1] \cdot \mathcal{Q}[-f_{\mathrm{OL}}^{-1}]\cdot \mathcal{R}[d]\cdot \mathcal{Q}[-f_{\mathrm{TL}}^{-1}]$$

Applying Eqs. (4), (5), (7) and (9) to Eq. (22), we get the simplified expression

(23)$$\mathcal{S} = \mathcal{V}\left[\frac{d-f_{\mathrm{OL}}}{f_{\mathrm{OL}}}\right] R\left[\frac{(f_{\mathrm{OL}}-d)d}{f_{\mathrm{OL}}}\right]Q\left[-\frac{1}{f_{\mathrm{qtc}}}\right]$$

with the effective focal length $f_{\mathrm {qtc}}$ and refractive power $P_{\mathrm {qtc}}$

(24)$$f_{\mathrm{qtc}} = \frac{f_{\mathrm{TL}}(f_{\mathrm{OL}}-d)}{f_{\mathrm{TL}}+f_{\mathrm{OL}}-d}, \quad P_{\mathrm{qtc}} = P_{\mathrm{TL}}+\frac{1}{f_{\mathrm{OL}}-d}.$$

The reversed quasi-telecentric configuration, i.e. when the positions of the objective and tunable lenses are swapped, is described equivalently by the operator

(25)$$\mathcal{S}_{\mathrm{rev}} = \mathcal{V}\left[\frac{d-f_{\mathrm{TL}}}{f_{\mathrm{TL}}}\right] R\left[\frac{(f_{\mathrm{TL}}-d)d}{f_{\mathrm{TL}}}\right]Q\left[-\frac{1}{f_{\mathrm{qtc, rev}}}\right]$$

with

(26)$$f_{\mathrm{qtc, rev}} = \frac{(f_{\mathrm{TL}}-d)f_{\mathrm{OL}}}{f_{\mathrm{TL}}-d+f_{\mathrm{OL}}},\quad P_{\mathrm{qtc, rev}} = \frac{1}{f_{\mathrm{TL}}-d}+P_{\mathrm{OL}}.$$

Note that the system behavior in the quasi-telecentric configuration depends on the propagation direction through the system, as apparent from comparing Eqs. (23) and (25). The effective refractive power $P_{\mathrm {qtc}}$ depends linearly from $P_{\mathrm {TL}}$, whereby the distance $d$ results in an constant offset of $P_{\mathrm {qtc}}$, as shown in Fig. 2a. In particular, the effective refractive power does not equal the refractive power of the objective lens at $P_{\mathrm {TL}} = 0$ and $d\neq 0$ due to the imperfect mapping between tunable and objective lens by the 4f-system.

Fig. 2. Effective refractive power $P_{\mathrm {qtc}}$ of the a) quasi-telecentric and b) reversed quasi-telecentric configuration employing the weak approximation according Eqs. (28) and (29), respectively.

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In contrast, in the reversed quasi-telecentric configuration, an incoming wave propagates first through the objective lens which causes no imperfect imaging of the 4f-system at $P_{\mathrm {TL}} = 0$. Yet, imperfect imaging occurs at all $P_{\mathrm {TL}} \neq 0$ and the relation between the refractive powers of the optical system and of the tunable lens is nonlinear, see Fig. 2b. Note that the tuning range of the effective refractive power is larger in the forward quasi-telecentric configuration than in the reversed one.

For the quasi-telecentric approximation, the assumptions $d\ll \left |f_{\mathrm {TL}}\right |$ and $d\ll \left |f_{\mathrm {OL}}\right |$ are used to simplify Eq. (23) and Eq. (25), respectively. Note, that $d\ll \left |f_{\mathrm {OL}}\right |$ is a (much) stronger approximation as $d\ll \left |f_{\mathrm {TL}}\right |$ for most practical applications, especially in microscopy. Applying $\frac {(f_{\mathrm {TL}}-d)}{f_{\mathrm {TL}}} \approx 1$ and $\frac {(f_{\mathrm {OL}}-d)}{f_{\mathrm {OL}}} \approx 1$ to the arguments of the $\mathcal {R}$ and $\mathcal {Q}$-terms yields the strong quasi-telecentric approximation

(27)$$\mathcal{S}_{\mathrm{qtc, strong}} = \mathcal{S}_{\mathrm{qtc, rev, strong}} = V[-1] \cdot \mathcal{R}\left[d\right] \cdot Q\left[-\frac{1}{f_{\mathrm{tc}}}\right]$$

While the strongly approximated expression in Eq. (27) offers a simple description of the quasi-telecentric setup, it does not contain the refractive power change due to the shift of the distance $d$ and it also does not incorporate the dependency of the system behavior on the propagation direction through the quasi-telecentric system.

To provide a more accurate description particularly for larger $d$ and to take the propagation direction dependency into account, we introduce a weaker quasi-telecentric approximation: As the arguments of $\mathcal {R}$ and $\mathcal {Q}$ occur in the argument of an exponential function and the argument of $\mathcal {V}$ occurs only in a linear or square-rooted manner, the approximation in the former case has a much stronger effect. Hence, the (weak) quasi-telecentric approximations for forward and backward propagation are given by

(28)$$\mathcal{S}_{\mathrm{qtc}}= V[-1] \cdot \mathcal{R}\left[\Delta s_{\mathrm{qtc}}\right] \cdot Q\left[-P_{\mathrm{qtc}}\right],$$

(29)$$\mathcal{S}_{\mathrm{qtc, rev}} = V[-1] \cdot \mathcal{R}\left[\Delta s_{\mathrm{qtc, rev}}\right] \cdot Q\left[-P_{\mathrm{qtc, rev}}\right]$$

with the focal position shifts $\Delta s$ and $\Delta s_{\mathrm {qtc, rev}}$ compared to the telecentric design that are defined by

(30)$$\Delta s_{\mathrm{qtc}} = \frac{(f_{\mathrm{OL}}-d)d}{f_{\mathrm{OL}}} \mbox{ and } \Delta s_{\mathrm{qtc, rev}} = \frac{(f_{\mathrm{TL}}-d)d}{f_{\mathrm{TL}}}.$$

Eqs. (28) and (29) offer the intuitive interpretation, that the system behavior is composted by a synthetic lens with refractive power $P_{\mathrm {qtc}}$ ($P_{\mathrm {qtc, rev}}$) followed by a propagation of a distance $\Delta s_{\mathrm {qtc}}$ ($\Delta s_{\mathrm {qtc, rev}}$) to compensate the lens shift $d$ and the shift of the focal position by the tunable lens. Therefore, the effective focal lengths for the quasi-telecentric configuration equal

(31)$$\mathrm{EFL}_{\mathrm{qtc}} = f_{\mathrm{qtc}}-\Delta s_{\mathrm{qtc}}$$

(32)$$\mathrm{EFL}_{\mathrm{qtc, rev}} = f_{\mathrm{qtc, rev}}-\Delta s_{\mathrm{qtc, rev}}.$$

Unsurprisingly, the EFLs in the forward and backwards configuration are different, whereas deviations between configurations with different distance $d$ increase with decreasing refractive power of the tunable lens as shown in Fig. 3. As a result, tunable lenses that support negative refractive powers are beneficial in terms of tuning range for axial scanning systems as shown in Fig. 4.

Fig. 3. Effective focal lengths for axial scanning systems employing a tunable lens in a) forward and b) backwards quasi-telecentric configuration.

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Fig. 4. Tuning range for a) positive and b) symmetric tunable lenses in forward (solid lines) and backwards quasi-telecentric configuration (dashed lines).

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Another consequence is that the tuning range in forward direction strongly depends on distance $d$, whereas a negative distance $d$ (i.e. deceasing the distance between objective and tunable lens) leads to an increased tuning range and a positive $d$ yields a decreased tuning range, see Fig. 5. In contrast, the tuning range in backward direction is only slightly influenced by changing distance $d$, in particular in the practically relevant range $P_{\mathrm {TL}} = (-0.2\dots 0.2)\cdot P_{\mathrm {OL}}$.

Fig. 5. Tuning range depending on distance $d$ for a) positive and b) symmetric tunable lenses in forward (solid lines) and backwards quasi-telecentric configuration (dashed lines).

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3.3 General description

The general description includes the quasi-telecentric configuration introduced in Section 3.2 without restrictions for the distance $d$ beyond $d<f_{\mathrm {TL}}$ and $d<f_{\mathrm {OL}}$. The general description is also valid for a system, in which the tunable and the objective lens are placed directly behind each other with the distance $d$ between them. The absence of the 4f system in the compact configuration results in a sign change in the $\mathcal {V}[\cdot ]$-term in Eq. (23). For simplicity and without loss of generality we drop the sign change in the following considerations. Hence, the general system and its reversed version are represented by the operators

(33)$$\mathcal{S} = \mathcal{V}\left[\frac{f_{\mathrm{OL}}-d}{f_{\mathrm{OL}}}\right] \mathcal{R}\left[\frac{(f_{\mathrm{OL}}-d)d}{f_{\mathrm{OL}}}\right]Q\left[-\frac{1}{f_{\mathrm{qtc}}}\right],$$

(34)$$\mathcal{S}_{\mathrm{rev}} = \mathcal{V}\left[\frac{f_{\mathrm{TL}}-d}{f_{\mathrm{TL}}}\right] \mathcal{R}\left[\frac{(f_{\mathrm{TL}}-d)d}{f_{\mathrm{TL}}}\right]Q\left[-\frac{1}{f_{\mathrm{qtc, rev}}}\right].$$

In contrast to the previous Sections, the $\mathcal {V}$-terms in Eqs. (33) and (34) are also taken into account for the general description. As the $\mathcal {V}$ operator does not represent an (idealized) physical process such as refraction by a lens or free-space propagation, a physical interpretation is less straight-forward.

In order to investigate the effect of the $\mathcal {V}\left [\frac {f_{\mathrm {TL}}-d}{f_{\mathrm {TL}}}\right ]$-term on a focal spot, we assume a Gaussian beam that propagates through the axial scanning system. Without any loss of generality, we describe the forward direction through the system and later generalize the results to the reversed configuration. We assume, that the $\mathcal {R}$- and $Q$-term in Eq. (33) have formed an Gaussian beam that is described by

(35)$$U_{\mathrm{in}}(x, z) = U_0 \frac{w_0}{w(z)} \exp \! \left( \! \frac{-x^2}{w(z)^2}\right ) \exp\left(-i \! \left(kz +k \frac{x^2}{2R(z)} - \psi(z) \! \right) \! \! \right)$$

where $z$ is the position on the optical axis, with $z=0$ representing the focal plane position. The initial amplitude is $U_0=U_{\mathrm {in}}(0,0)$, the radius of curvature $R(z)=z\left [1+\left (\frac {z_{\mathrm {R}}}{z}\right )^2\right ]$, the beam width $w(z)=w_0 \sqrt {1+\left (\frac {z_{\mathrm {R}}}{z}\right )^2}$ and the Gouy phase $\psi (z) = \arctan \left ( \frac {z}{z_{\mathrm {R}}} \right )$ with the beam waist $w_0$ and the Rayleigh length $z_{\mathrm {R}}=\frac {\pi w_0^2}{\lambda }$. At the focal plane, this Gaussian beam is represented by

(36)$$U_{\mathrm{in}}(x, 0) = U_0 \exp\left(-\frac{x^2}{w_0^2}\right)$$

As the focal plane is located at the distance $\mathrm {EFL}_{\mathrm {qtc}}$ after the principle plane of the objective lens, the field passed to the $\mathcal {V}$ operator equals

(37)$$U_{\mathrm{out}}(x) = \left(\mathcal{R}\left[\mathrm{EFL}\right] \cdot \mathcal{V}\left[\frac{f_{\mathrm{OL}}-d}{f_{\mathrm{OL}}}\right]\right) \left\{ U_{\mathrm{in}}(x, z=-\mathrm{EFL}_{\mathrm{qtc}}) \right\}$$

(38)$$ = \mathcal{V}\left[\frac{f_{\mathrm{OL}}-d}{f_{\mathrm{OL}}}\right] U_{\mathrm{in}}\left(x, z_{\mathrm{focus}}\right)$$

with the focus position $z_{\mathrm {focus}}$ on the optical axis.

(39)$$z_{\mathrm{focus}} = \left(1-\frac{d}{f_{\mathrm{OL}}}\right)^2\mathrm{EFL}-\mathrm{EFL}_{\mathrm{qtc}}$$

As $z_{\mathrm {focus}} = 0$ by definition, it follows that

(40)$$U_{\mathrm{out}}(x) = \mathcal{V}\left[\frac{d-f_{\mathrm{OL}}}{f_{\mathrm{OL}}}\right] U_{\mathrm{in}}(x,0)$$

(41)$$ = U_{\mathrm{in},0} \left|\frac{d-f_{\mathrm{OL}}}{f_{\mathrm{OL}}}\right|^{1/2} \exp\left\{-\left(\frac{d-f_{\mathrm{OL}}}{f_{\mathrm{OL}}}\right)^2\frac{x^2}{w_0^2}\right\}$$

The corresponding intensity equals

(42)$$I_{\mathrm{out}}(x) = I_{\mathrm{out},0}\exp\left\{-2\left(1-\frac{d}{f_{\mathrm{OL}}}\right)^2\frac{x^2}{w_0^2}\right\}$$

with $I_{\mathrm {in}}(x)= \frac {\left |E_{\mathrm {in}}(x)\right |^2}{2\eta }$, $I_{\mathrm {out}}(x)=\left |1-\frac {d}{f_{\mathrm {OL}}}\right | \cdot I_{\mathrm {in}}(x)$ and $I_{\mathrm {out},0}=I_{\mathrm {out}}(0,0)$, where the constant $\eta$ is the characteristic impedance of the medium in which the beam is propagating ($\eta = \eta _0 \approx {377}\, {\Omega }$ for free space). Equaling Eq. (42) with $I_{\mathrm {focus},0}/e^2$ and solving the resulting equation for $x$ yields the output beam waist

(43)$$w_{0, \textrm{out}} = \frac{w_0}{1-\frac{d}{f_{\mathrm{OL}}}}$$

The output beam waist in the reversed quasi-telecentric configuration can be derived analogously to

(44)$$w_{0, \textrm{out, rev}} = \frac{w_0}{1-\frac{d}{f_{\mathrm{TL}}}}.$$

Note, however, that $w_0$ in Eqs. (43) and (44) depends on the numerical aperture of the system and consequently the focal length of the tunable lens, i. e. $w_0 = w_0(f_{\mathrm {TL}})$.

To take into account the effect of the varying numerical aperture $\textrm {NA}$ of the system, we define the global beam waist $w_{0, global}=w_0(P_{\mathrm {TL}}=0, d=0)$. As $w_0 \propto 1/\textrm {NA} \propto f_{\mathrm {qtc}}$, it follows that

(45)$$w_{0, \textrm{out}} = \frac{w_{0, \mathrm{global}}}{1-\frac{d}{f_{\mathrm{OL}}}}\frac{f_{\mathrm{qtc}}}{f_{\mathrm{OL}}} = w_{0, \textrm{out, rev}} = \frac{w_{0, \mathrm{global}}}{1-\frac{d}{f_{\mathrm{TL}}}}\frac{f_{\mathrm{qtc, rev}}}{f_{\mathrm{OL}}}$$

It is particularly interesting that the effective beam waist in units of the global initial beam waist $w_{0, \mathrm {global}}$ is independent from the propagation direction through the system as shown in Fig. 6.

Fig. 6. Beam waists in units of (a) local and (b) global beam waist, Solid lines correspond to forward and dashed lines to backward propagation through the system. b) The effective beam waist in units of the global initial beam waist $w_{0, \mathrm {global}}$ is independent from the propagation direction through the system.

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4. Conclusion

In summary, we provided a purely analytical Fourier optics description of some common lens systems employing tunable lenses. While our approach does neither fully include lens-induced aberrations nor thick lenses or lens systems such as microscope objectives, its analytical formulation allows a deep understanding on the underlying principle and might serve as a suitable tool for designing optical imaging systems employing tunable lenses. The proposed approach allows the extinct observation of system-induced aberrations without the need to take the imperfections of single optical elements into account.

The application of our approach on three common configurations for axial scanning systems yields the following practical considerations:

• In the strict telecentric configuration, no system-induced aberrations occur as long as ideal lenses are assumed. However, the spot size is affected by the change of numerical aperture due to the focal length tuning.
• The tuning range can be optimized when using a quasi-telecentric system with a negative position shift $d$ in forward configuration.
• The loss of spatial resolution in the quasi-telecentric in contrast to the strictly telecentric configuration can be traded off in exchange for an increased tuning range.
• In the strict telecentric regime, the effect of the optical system on a propagation wave is independent from the propagation direction through the system.
• In contrast, the focus shift depends on the propagation direction in the quasi- and non-telecentric configuration. This is especially important when designing systems in which the axial scanning system is passed twice as e.g. in reflection confocal microscopy as this induces an additional defocus aberration.

Funding

Deutsche Forschungsgemeinschaft (DFG) (CZ 55/32-1).

Acknowledgments

The authors thank Dr. Nektarios Koukourakis for helpful discussions.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Axial scanning employing tunable lenses: Fourier optics based system design

Abstract

1. Introduction

2. Operator algebra

3. Axial scanning systems employing a tunable lens

3.1 Telecentric regime

3.2 Quasi-telecentric regime

3.3 General description

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (6)

Equations (45)

OSA Continuum