High speed non-mechanical two-dimensional KTN beam deflector enabled by space charge and temperature gradient deflection

Ju-Hung Chao; Wenbin Zhu; Chang-Jiang Chen; Adrian L. Campbell; Michael G. Henry; Shizhuo Yin; Robert C. Hoffman

doi:10.1364/OE.25.015481

1. Introduction

High-speed optical beam deflectors are widely used in various important applications such as low coherence optical tomography [1], laser displays [2], high speed beam scanners [3], switchable waveguides [4], and spectral multiplexed holograms [5]. In comparison to traditional mechanical beam deflectors, the beam deflector based on the electro-optic (EO) effect not only features a faster frequency response [6], but also is free from mechanical movement-induced noise. Additionally, the deflection angle can be precisely controlled by modulating the EO phase delay. Thus, different materials, such as KH₂PO₄ [7], PZT [8], and LiTaO₃ [9], have been explored to make EO beam deflectors. However, their performances are limited by a relatively small EO coefficient.

Among many different types of EO materials, potassium tantalate niobate (KTN) has the greatest potential due to its large EO coefficient when operated in the paraelectric phase [10]. A quadratic EO coefficient as large as 6.94 × 10⁻¹⁴ m²/V² has been reported [11], which was by far the largest EO Kerr effect found among all EO media.

By taking advantage of the large EO coefficient, KTN beam deflectors have a number of advantages, such as high-speed response, low driving voltage, and the capability to work at room temperature. These features have made KTN a popular choice of material in various applications, for instance, optical coherent tomography [12], high-speed spectrometer [13], wavelength tunable lasers [14], optical wave guides [15], and holographic memory systems [16].

In the recent development of KTN devices, a type of KTN beam deflector using the space-charge-controlled (SCC) mechanism was reported [17, 18], which is also currently commercially available. It is realized by injecting electrons into the KTN crystal and then performing EO modulation in the paraelectric phase. The injected electrons generate a non-uniform electric field, which could result in an electric field-dependent prism-shaped refractive index distribution via the quadratic EO effect. Such an electric field-dependent prism-shaped refractive index distribution could deflect the incoming light beam.

Despite all the great qualities of the KTN beam deflector, it does have a few weaknesses that often have to be dealt with. One of them is the field-induced phase transition [19–21], which is a rise of Curie temperature that occurs when the crystal is applied with a high biasing field. Resolution, which is essential for optical scanning devices, is another property that is in need of improvement. Two major factors that determine the resolution are the aperture size as well as the deflection angle. Although the KTN beam deflector enabled by the space-charge-modulation has a decent deflection angle, the aperture size is limited by the penetration depth of the injected charge. Furthermore, only one dimensional beam deflection has been achieved on a single KTN crystal.

A two-dimensional (2-D) beam deflection can be realized by lining up two deflectors appropriately. However, it will require more power to operate, and more complexity to achieve 2-D scanning. To realize 2-D deflection in a compact and cost-effective way, in this paper, we report the 2-D scanning based on the combination of two different physical scanning mechanisms. In addition to the space-charge-controlled beam deflection in the x direction, we harness another beam deflecting mechanism on the same KTN crystal by inducing a temperature gradient in the y direction. Since the relative permittivity of KTN near its Curie temperature is strongly dependent on the operating temperature in the paraelectric phase, a gradient in refractive index would form, in accordance with the induced temperature gradient inside the crystal, to bend the incoming laser beam in the y direction. As a result, a beam deflection along both the x and y direction can be achieved in a single KTN crystal. Note that, there have been previous works of employing temperature gradient for other applications (e.g., pyroliton [22]). However, in our paper, it is the first time to harness temperature gradient to achieve gradient permittivity distribution for the rapid beam deflections.

2. Physical mechanisms of 2-D scanning based on the combination of space-charge-controlled (SCC) KTN beam deflection and temperature gradient

2.1 1-D Space-charge-controlled KTN beam deflection

To understand the physical mechanisms of 2-D deflection based on the combination of space-charge controlled KTN beam deflection and temperature gradient, it will be helpful to briefly review the physical mechanism of 1-D deflection when only SCC is involved.

The SCC beam deflection was first reported by Nakamura et al. [23]. It is believed that the external field induces a rise in the energy level of electrons at the electrodes such that the charges are able to drift across the Schottky barrier between the crystal and the electrode, and are driven towards the other electrode inside the crystal. With the charges being injected into the crystal, the electric field distribution inside the crystal is then altered. The refractive index inside the crystal is thus changed accordingly, causing the input beam to deflect. It was later found by Miyazu et al. [17] that the injected charges could actually be trapped due to the existence of the defect states inside the crystal. These charges could be stored in the crystal for a number of hours before being naturally released, if not pushed out by the excitation of other energy sources. Based on the assumption of uniformly distributed trapped charges of density N inside the crystal, Miyazu et al. derived a mathematical model describing the behavior of space-charge-preinjected KTN beam deflection. From Gauss’s law, the overall electric field distribution inside the crystal with an applied external field can be expressed by [17]

E (x) = - \frac{e N}{ε} (x - \frac{G}{2} + \frac{ε V}{e N G}),

where

e

is the electric charge,

ϵ

is the permittivity of the crystal, G is the gap between electrodes and V is the applied voltage. Therefore, the change in refractive index due to the electro-optic (EO) Kerr effect is given by [24]

\begin{matrix} Δ n (x) = - \frac{1}{2} n^{3} g_{11} ε^{2} E {(x)}^{2} \\ = - \frac{1}{2} n^{3} g_{11} e^{2} N^{2} {(x - \frac{G}{2} + \frac{ε V}{e N G})}^{2}, \end{matrix}

where n is the refractive index and

g_{11}

is the EO coefficient of the crystal in polar form when the polarization of the beam is parallel to the applied electric field. For a beam that has traveled the distance L in the crystal, the deflection angle can therefore be given by

\begin{matrix} θ (x) = L \frac{d}{d x} Δ n (x) \\ = - n^{3} g_{11} e^{2} N^{2} L (x - \frac{G}{2} + \frac{ε V}{e N G}) . \end{matrix}

It can be seen in Eq. (3) that the deflection angle $θ$ is proportional to the external voltage. Therefore, the scanning speed of the deflector can be as fast as the voltage source. Figure 1 shows the configuration of a space-charge-controlled KTN beam deflector.

Fig. 1 A sketch of a 1-D space-charge-controlled (SCC) KTN beam deflector.

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2.2 2-D beam deflection based on the combination of Temperature-gradient-controlled and SCC KTN beam deflection

To realize 2-D beam deflection, the input beam has to experience a gradient of refractive index in two dimensions within the KTN crystal. To achieve this goal, we assume that the space charge is uniformly injected into the KTN crystal by a pair of parallel conductive plates in the x direction with a charge density, $ρ$ , and there is a temperature gradient in the y direction by attaching a pair of temperature controlled plates on the bottom and top surfaces of KTN crystal, as illustrated in Fig. 2. The red and blue arrows represent the directions of the temperature gradient and electric field, respectively. The thick green line denotes the light propagation directions. The temperatures of bottom and top plates are kept at $T_{1}$ and $T_{2}$ , respectively. In this case, an inhomogeneous permittivity, $ε (y) = ε_{0} ε_{r} (y)$ , is induced in the y direction.

Fig. 2 A sketch of a 2-D temperature-gradient and SCC combined KTN beam deflector.

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For the purpose of simplicity, we assume that the temperature gradient is linear in the y direction and the distance between bottom and top plates is $H .$ Therefore, the temperature at an arbitrary location, y, can be derived as

T (y) = T_{1} + (T_{2} - T_{1}) \frac{y}{H},

where

T_{2} > T_{1}

. In the paraelectric phase, the relative permittivity of KTN follows the Curie-Wiess law [24]:

ε_{r} = \frac{C_{1}}{T - T_{c}},

where

C_{1}

is a material-specific Curie constant and

T_{c}

is the Curie temperature of KTN. Substituting Eq. (4) into Eq. (5), we obtain

ε_{r} (y) = \frac{C_{1}}{T_{1} + (T_{2} - T_{1}) \frac{y}{H} - T_{c}} .

To obtain the 2-D refractive index modulation, first, we need to calculate the static electric field by solving the Maxwell equations. In this inhomogeneous dielectric case, the Maxwell’s equations have the form [25]

\nabla \times E = 0,

\nabla \cdot D = ρ .

where

E = (E_{x}, E_{y}, E_{z})

and

D = ε (y) E

. Since the permittivity is homogeneous in the z direction and there is no electric field applied on the z direction, due to symmetry, we obtain that

E_{z} = 0

. Substituting this condition in to Eq. (7), we obtain three scalar equations as

\frac{\partial E_{x}}{\partial z} = 0,

\frac{\partial E_{y}}{\partial z} = 0,

\frac{\partial E_{x}}{\partial y} = \frac{\partial E_{y}}{\partial x} .

Equations (9)a) and (9b) indicate that $E_{x}$ and $E_{y}$ are the only functions of x and y, as given by $E_{x} (x, y)$ and $E_{y} (x, y)$ . In this case, Eq. (8), can be written as

\frac{\partial (ε (y) E_{x})}{\partial x} + \frac{\partial (ε (y) E_{y})}{\partial y} = ρ .

Since $ε (y)$ is not a function of x, Eq. (10) can be simplified into the form of

ε (y) \frac{\partial E_{x}}{\partial x} + \frac{d ε (y)}{d y} E_{y} + ε (y) \frac{\partial E_{y}}{\partial y} = ρ .

To obtain a partial differential equation containing only $E_{x}$ , we apply $\frac{\partial}{\partial x}$ operation on both sides of Eq. (11). Then it becomes

ε (y) \frac{\partial^{2} E_{x}}{\partial x^{2}} + \frac{d ε (y)}{d y} \frac{\partial E_{y}}{\partial x} + ε (y) \frac{\partial \frac{\partial E_{y}}{\partial x}}{\partial y} = 0.

Substituting Eq. (9)c) into Eq. (12), we obtain

ε (y) \frac{\partial^{2} E_{x}}{\partial x^{2}} + \frac{d ε (y)}{d y} \frac{\partial E_{x}}{\partial y} + ε (y) \frac{\partial^{2} E x}{\partial y^{2}} = 0,

which only contains

E_{x}

. Similarly, by applying

\frac{\partial}{\partial y}

operation on both sides of Eq. (11) and inserting Eqs. (9)c) and (11) into new equation, we obtain

\begin{matrix} ε (y) (\frac{\partial^{2} E_{y}}{\partial x^{2}} + \frac{\partial^{2} E_{y}}{\partial y^{2}}) + \frac{d ε (y)}{d y} \frac{\partial E_{y}}{\partial y} + \\ [\frac{d^{2} ε (y)}{d y^{2}} - \frac{1}{ε (y)} {(\frac{d ε (y)}{d y})}^{2}] E_{y} + \frac{1}{ε (y)} \frac{d ε (y)}{d y} ρ = 0, \end{matrix}

which only contains

E_{y}

. Furthermore, we assume following boundary conditions:

E_{y} (0, y) = 0,

E_{y} (G, y) = 0,

\int_{0}^{G} E_{x} d x = V,

where G and V are the distance and voltage difference between two electrodes, respectively. Theoretically speaking, by solving Eqs. (13) – (15) simultaneously,

E_{x}

and

E_{y}

can be obtained. We further assume that the polarization of the incoming light is along the x direction. In this case, the corresponding refractive index modulation can be derived as

Δ n (x, y) = - \frac{1}{2} n^{3} g_{11} ε_{0}^{2} {(ε_{r} (y) - 1)}^{2} E_{x}^{2} - \frac{1}{2} n^{3} g_{12} ε_{0}^{2} {(ε_{r} (y) - 1)}^{2} E_{y}^{2},

where

g_{11}

and

g_{12}

are the quadratic electro-optic coefficient in the polar form and

n_{0}

is the refractive without applying the electric field. The beam deflection in the x and y directions can be further derived as

\begin{matrix} θ_{x} = L \frac{\partial Δ n (x, y)}{\partial x} \\ = - L n^{3} g_{11} ε_{0}^{2} {(ε_{r} (y) - 1)}^{2} E_{x} \frac{\partial E_{x}}{\partial x} - L n^{3} g_{12} ε_{0}^{2} {(ε_{r} (y) - 1)}^{2} E_{y} \frac{\partial E_{y}}{\partial x}, \end{matrix}

\begin{matrix} θ_{y} = L \frac{\partial Δ n (x, y)}{\partial y} \\ = - L n^{3} g_{11} ε_{0}^{2} (ε_{r} (y) - 1) \frac{d ε_{r} (y)}{d y} E_{x}^{2} - L n^{3} g_{11} ε_{0}^{2} {(ε_{r} (y) - 1)}^{2} E_{x} \frac{\partial E_{x}}{\partial y} \\ = - L n^{3} g_{12} ε_{0}^{2} (ε_{r} (y) - 1) \frac{d ε_{r} (y)}{d y} E_{y}^{2} - L n^{3} g_{12} ε_{0}^{2} {(ε_{r} (y) - 1)}^{2} E_{y} \frac{\partial E_{y}}{\partial y}, \end{matrix}

where

L

is the propagation length of the light beam along the z direction.

Unfortunately, there is no obvious closed-form analytic solutions for Eqs. (13) and (14) when $ε (y)$ is not a constant and $ρ \neq 0$ . Thus, we conducted a numerical approach to solve Eqs. (13) and (14). We began by establishing the temperature gradient for this computation. To ensure that the result of the numerical computation is consistent with the experimental result, the following realistic experimental parameters were used in this calculation. The dimensions of the KTN crystal was 3 mm x 3 mm x 3 mm. The Curie temperature of KTN crystal was $T_{c} = 22 {}^{o}C$ . The bottom (y $= 0 mm$ ) and top (y $= 3 mm$ ) temperatures were set at $T_{1} = 23 {}^{o}C$ and $T_{2} = 38 {}^{o}C$ , respectively. Here we assumed $C_{1} = 110000$ in Eq. (6). Figure 3 shows the calculated permittivity distribution as a function of position in y-axis under above conditions. Note that, to avoid the complexity of electric fields near the edges (i.e., $y = 0$ and $y = H$ ) as well as the singularity when $T \to T_{c}$ , based on Eq. (6), only the $ε_{r} (y)$ within the range of 0.5mm-2.5 mm was drawn. The calculated numbers were on the same order as the experimental results reported by Refs. [17,19,23] for the KTN crystals with a similar Curie temperature.

Fig. 3 The calculated relative permittivity, $ε_{r} (y)$ , as a function of location y with following parameters: $T_{1} = 23 {}^{o}C$ , $T_{2} = 38 {}^{o}C$ , $T_{c} = 23 {}^{o}C$ , $C_{1} = 110, 000$ , and $H = 3 m m .$

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To investigate the influence of temperature gradient on the electric field distribution inside the KTN crystal, we compared the electric field distribution in KTN with and without the temperature gradient. In this computation, we assumed that the external applied voltage, $V = 800 V$ , $ε_{r} = 13000$ and the uniformly injected charge density, $ρ = - 3 C / m^{3}$ , which again were both realistic parameters and also within the same order of magnitude as the previously reported experimental results [17,19,23]. Figure 4(a) shows the electric field distribution without the temperature gradient (i.e., $T_{1} = T_{2}$ ). It is obvious that the electric field is only along the x direction. Then, based on Eqs. (13) – (15), we computed the electric field with a temperature gradient (i.e., $T_{1} = 23 {}^{o}C a n d T_{2} = 38 {}^{o}C$ ), as shown in Fig. 4(b). In this case, $E_{y}$ was no longer zero. However, it was too small compared to $E_{x}$ to be observed. Hence, we enlarged the $E_{y}$ in Fig. 4(b) by 200 times to show that the E-field distribution was in fact in both x and y directions.

Fig. 4 The calculated electric field under an external biasing field (a) without temperature gradient, in which case, $E_{y} = 0,$ and (b) with a temperature gradient of $T_{1} = 23 {}^{o}C a n d T_{2} = 38 {}^{o}C$ , in which case, $E_{y} \neq 0.$ The colors bar show the magnitude of electric field in V/m.

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We then computed the refractive index distribution by inserting the electric field distribution obtained in Fig. 4(a) and 4(b) into Eq. (16). The results are shown in Fig. 5(a) and 5(b), respectively. It can be seen in Fig. 5(a) that there was a refractive index modulation in only the x direction, thus only the beam deflection in the x direction could occur in this case. In Fig. 5(b), on the other hand, with the induced temperature gradient of $T_{1} = 23 {}^{o}C a n d T_{2} = 38 {}^{o}C$ , we could clearly see a refractive index distribution along both x and y directions, which enabled the two-dimensional beam deflection.

Fig. 5 The calculated refractive index distribution (a) without the temperature gradient (i.e., $T_{1} = T_{2})$ and (b) with a temperature gradient of $T_{1} = 23 {}^{o}C a n d T_{2} = 38 {}^{o}C$ .

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Finally, by further substituting the calculated refractive index modulation into Eqs. (17)(a) and 17(b), the beam deflection as a function of external electric field could be obtained. The computed beam deflections were also compared with the experimental results, as discussed in detail in the following section.

3. Experiment results and discussion

To validate the proposed 2D beam deflection based on the combination of SCC beam deflection and temperature gradient enabled beam deflection, we conducted the following experiments. In the experiment, a cubic shaped KTN crystal with a side length of 3 mm was coated with a pair of Ti-Au electrodes on the surfaces along the x direction, as illustrated in Fig. 2. The Curie temperature of this KTN crystal was $T_{c} = 22 {}^{o}C$ . Two Peltier thermo-electric modules, both connected to a temperature controller, were attached on the surfaces along y direction. The temperatures of the bottom and top Peltier thermo-electric modules were set at $T_{1} = 23 {}^{o}C a n d T_{2} = 38 {}^{o}C$ , respectively. A horizontally polarized diode-pumped solid-state laser beam with an output wavelength of 532 nm travelled in the z direction through the center of x-y surface. A DC biasing field was applied on the crystal in the x direction via the electrodes. The output beam was projected on a graph sheet, as illustrated in Fig. 6.

Fig. 6 The experimental setup used to measure the deflection angle.

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In the experiment, the deflection angles were measured at different applied voltages, from 100 V to 800 V with a step of 100 V. The measured data points formed a 2-D trace on the projection surface and was analyzed separately in both x and y directions, as shown in Fig. 7(a)-7(b). The circularly shaped dots in Fig. 7(a) show the experimentally measured deflection angles in the x direction. As a comparison, the computed deflection angles based on Eq. (17)(a) with the same parameters (i.e., $T_{1} = 23 {}^{o}C a n d T_{2} = 38 {}^{o}C$ ), are depicted as the solid line in Fig. 7(a). In Fig. 7(b), the triangularly shaped dots represent the measured deflection angles in y direction with respect to different applied voltages. To quantify the difference between experimental and theoretical results, we computed the average relative difference (i.e., the average deflection difference/deflection range) between the theoretical and experimental results of deflection angles in x and y directions, which were 2.9% and 7.2%, respectively. This confirmed that the theoretical and experimental results agreed well. Furthermore, as shown in Fig. 7(b), deflection shows a slight linear trend at the low voltages (e.g., <200V) and a parabolic increase at the high voltages (e.g., >200 V). In other words, the quadratic term (i.e., the first term of Eq. (17)(b)) dominates in that equation as the applied field increases. Note that, the measured deflection angle was smaller than the one reported by Ref. 19, which was due to the lower injected charge density of our KTN sample. As mentioned in Ref 26, there were large variations in injected charge densities at a given external electric field and penetration depth for different KTN samples, even with similar compositions due to the uncontrollable perturbations (e.g., temperature fluctuations) in the growing process. For example, as reported in Ref 26, with the same applied electric field (300 V/mm) and a penetration depth 0.5 mm, the sample A had an injected charge density $ρ = - 70 C ⁄ m^{3}$ whereas sample B had an injected charge density $ρ = - 5 C ⁄ m^{3}$ . The value of Sample B was comparable to the injected charge density used in our paper $ρ = - 3 C ⁄ m^{3}$ at an applied field of 266.7 V/mm. In other words, although the value of injected charge density in our paper was small, it was still within the range of previously reported values.

Fig. 7 The deflection angles (a) in x direction at different applied voltages with the circularly shaped dots being the experimentally measured results and the solid line denoting the calculated deflection angles based on Eqs. (17)(a), and 17(b) in y direction at different applied voltages, with the triangularly shaped dots being the experimentally measured results and the solid line denoting the calculated deflection angles based on Eq. (17)(b).

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As aforementioned, in our approach, the deflection in x direction was caused by the space charge injection induced gradient electric field whereas the deflection in y direction was caused by the temperature gradient induced gradient electric field as well as the gradient permittivity distribution. To show the two-dimensional (2-D) deflection capability, we measured the deflections as a function of voltage in x and y directions at different temperature gradients, as shown in Fig. 8. One could see that the deflection angles could cover the entire x-y plane by adjusting the applied voltage and the magnitude of temperature gradient. In other words, the beam can potentially be deflected to any location in x-y plane by selecting the proper combination of applied voltage and the magnitude of the temperature gradient.

Fig. 8 The deflection locations in x-y plane measured at different voltages 100V-800V with a step of 100V and under different temperature gradients. The square, diamond, and circular dots denote the measured deflection locations corresponding to the temperature gradients 23-28 °C, 23-33 °C, and 23-38 °C, respectively.

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Finally, to demonstrate the capability of high-speed beam deflection, we also measured the deflection speed by using the experimental setup, as illustrated in Fig. 9. An iris was placed in between the beam deflector and a high-speed photodetector (FEMTO Messtechnik GmbH HCA-S-200M-Si) which was connected to an oscilloscope. When inducing a fast-changing voltage signal, the beam was deflected and then blocked by the iris. Hence, the photodetector could measure the change in light intensity as a function of time, which indicated the deflection speed. In this experiment, a fast-changing voltage signal with a rise time around 1 ns was applied to the KTN crystal. Figure 10 shows the drop in detected light intensity as a function of time. Despite some slight noises from the circuit, we can still see a clear response time of 77 ns (from 90% to 10%). The capacitance of the KTN crystal in this experiment was 0.54 nF, and the impedance of the circuit was 68 ohm, thus the rise time of the RC circuit was 81 ns, which is close to the measured response time. Therefore, the speed of this device was mainly limited by the RC time constant of the circuit in this experiment. This result confirm that a very fast deflection speed can be achieved.

Fig. 9 The experimental setup used to measure the deflection speed

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Fig. 10 The experimentally measured time response of the beam deflection.

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4. Conclusion

In this report, two-dimensional KTN beam deflection was successfully achieved by combining two deflecting mechanisms: the space-charge-controlled beam deflection and the temperature-gradient-controlled beam deflection, along the x and y direction respectively. By applying an external biasing field along the x direction and inducing a temperature gradient in the y direction on a single KTN crystal, a deflection in both x and y direction was observed. The effect of space-charge and temperature-gradient mechanisms were quantitatively analyzed and also experimentally confirmed. The maximum difference between experimental results and theoretical analyses was around 7%. Furthermore, high-speed (~80 ns) deflection capability was also experimentally demonstrated. Such a compact non-mechanical high speed beam deflector can be very useful for a variety of applications, including high-speed 3D laser printing, high-resolution high-speed scanning imaging, and free space reconfigurable laser communications.

Note

The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.

Funding

US Army Research Laboratory under Cooperative Agreement Number W911NF-14-2-0067.

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High speed non-mechanical two-dimensional KTN beam deflector enabled by space charge and temperature gradient deflection

Abstract

1. Introduction

2. Physical mechanisms of 2-D scanning based on the combination of space-charge-controlled (SCC) KTN beam deflection and temperature gradient

2.1 1-D Space-charge-controlled KTN beam deflection

2.2 2-D beam deflection based on the combination of Temperature-gradient-controlled and SCC KTN beam deflection

3. Experiment results and discussion

4. Conclusion

Note

Funding

References and links

Cited By

Figures (10)

Equations (22)

Optics Express