Analyses of optical rays in KTN optical beam deflectors for device design

Tadayuki Imai; Masahiro Ueno; Yuzo Sasaki; Tadashi Sakamoto

doi:10.1364/AO.56.007277

1. INTRODUCTION

Optical beam deflectors, such as mirror galvanometers, are widely used in the industry. In addition to their use in various laser display systems, they are employed in laser markers and laser machining equipment. They are also used as key devices in surveying systems, including lidars and microscopes. While the most widely used and conventional device is the mirror galvanometer, there are also acousto-optic deflectors, devices based on microelectromechanical systems and electro-optic (EO) devices. EO deflectors utilize the EO effect of certain oxide crystal materials [1,2]. They offer outstanding speed compared with other devices, but their scanning angular ranges are small. A space-charge-controlled (SCC) optical beam deflector is classified as an EO device [3]. However, its deflection angle is much larger than those of conventional EO devices because of its cumulative deflecting nature and the huge EO effect of single crystals of potassium tantalate niobate ( ${KTa}_{1 - x} {Nb}_{x} O_{3}$ , KTN) [4,5]. The full deflection range exceeds 10 deg. Moreover, owing to the nature of the EO effect, the deflector works much faster than conventional galvanometer mirrors. A repetition rate of 350 kHz has been reported [6]. Recently, a repetition rate of 700 kHz was also reported with a modified device structure [7]. By combining the deflector with a grating and a semiconductor optical amplifier, we have developed high-speed wavelength-swept light sources for optical coherence tomography systems [8,9].

Figure 1 shows the SCC deflector. It consists of a KTN single crystal block with a pair of film electrodes. The space charge is formed by applying a voltage and injecting electrons into the crystal through the cathode. Then, the input optical beam is bent simply by applying a voltage. The deflection angle is proportional to the voltage. As easily supposed from Fig. 1, we can obtain a large deflection angle with a long KTN block. However, the figure also shows that the bottom electrode face does not allow the beam to exit the block when the deflection angle is too large. This limitation is more severe with a longer block. Therefore, it is not necessarily a good idea to extend the block length indefinitely.

Fig. 1. Illustration of the KTN SCC optical beam deflector.

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On the other hand, the thickness of the block $d$ is also an important parameter. If we regard the deflector as an imaging device, to obtain a good resolution, we require a large aperture and thus a large $d$ . However, higher voltages are required to obtain sufficiently large deflection angles for a thick device, and so, in practice, the thickness is limited by the available voltage sources. Moreover, in addition to the deflection function, an SCC deflector has a lens function, which is a by-product of the deflection function originating from the space charge [10]. As collimated beams are useful for many applications, the deflector is commonly used with glass lenses to compensate for the lens effect [11]. These compensating lenses change the deflection angle.

In this paper, we analyze curving optical rays in an SCC optical beam deflector and discuss deflection performance. First, we deal with the KTN block as an active cylindrical lens, analyze the optical rays based on the theory of the graded index lens, and discuss its deflection functions. Next, we describe how to use the block with collimated input and output beams by using compensating glass lenses. Finally, we report the conditions needed to avoid the vignetting caused by the electrode faces and thus obtain the optimum resolvable spot numbers as the most important parameter of the deflector. We discuss an optimization approach where a collimated or slightly converging beam rather than a diverging beam is input into the KTN block.

2. OPTICAL RAYS IN KTN CRYSTAL BLOCK

Here, we analyze optical rays in a KTN crystal block to construct the basis for our theory. When a KTN deflector is used, electrons are injected into the KTN crystal block to form a space charge field. Then, this space charge field forms a spatially distributed electric field and, as a result, a spatially distributed refractive index. Input light rays are bent by this index distribution, and, thus, the KTN block works as an optical beam deflector. In this paper, we assume that the charge density is uniform in the KTN block. Even with a uniform density, the deflection angle depends on the incident position of the ray. In fact, the KTN block bends light rays in the same manner as a graded index lens [10], although the light confinement is one dimensional like a cylindrical lens.

We discuss the optical rays with the diagram shown in Fig. 2. $d$ is the thickness of the KTN block and is typically about 1.0 mm. $L$ is the block length, which is typically 4.0 mm. However, we usually deposit high reflection coatings on the block to realize internal reflection so that the light passes through the block three or five times [9,11]. Therefore, the effective length will be 12–20 mm. Now, let us consider a ray that starts at point P, which is both outside the block and on the optical axis of the block, as shown in Fig. 2 (As described, a KTN block works as a cylindrical lens, not an ordinary spherical lens. Therefore, the structure shown in this figure has translational symmetry in the direction perpendicular to the paper, not axial symmetry.) The ray is directed towards the block and is incident on the block surface (input face) with an incident angle of $ϕ_{1}$ . $d_{c i}$ is the distance between P and the input face. $r_{1}$ is the distance between the optical axis and the point of incidence of the ray on the input face. The input ray proceeds through the block but is bent by the index distribution, as shown in the figure. Then, the ray reaches the output face at a distance $r_{2}$ from the axis, exits the block with an angle of $ϕ_{2}$ , and crosses the axis at point Q. The distance between the output face and Q is $d_{c o}$ . The index distribution of a graded index lens is given as follows:

n (r) = n_{0} (1 - \frac{A}{2} r^{2}) = n_{0} - \frac{1}{2} n_{0} A r^{2} .

Here,

r

is the distance from the lens axis, which is now aligned with the optical axis of the KTN block.

n_{0}

and

A

are constants. It is known that the following equation holds for this configuration [12]:

[\begin{matrix} r_{2} \\ {\dot{r}}_{2} \end{matrix}] = [\begin{matrix} \cos (L \sqrt{A}) & \frac{1}{n_{0} \sqrt{A}} \sin (L \sqrt{A}) \\ - n_{0} \sqrt{A} \sin (L \sqrt{A}) & \cos (L \sqrt{A}) \end{matrix}] [\begin{matrix} r_{1} \\ {\dot{r}}_{1} \end{matrix}] .

The dots that appear over

r_{1}

and

r_{2}

indicate differentiation by

z

, which is a coordinate assigning a horizontal position. Then, the derivatives are related to the angles

ϕ_{1}

and

ϕ_{2}

as follows:

{\dot{r}}_{1} = \tan ϕ_{1} \approx ϕ_{1}, {\dot{r}}_{2} = - \tan ϕ_{2} \approx - ϕ_{2} .

Note that the effective length of the block is much greater than its thickness, and the approximation in Eq. (3) is valid with errors of less than 1%. We use this approximation throughout this paper. In addition, the following equation is obtained from Fig. 2:

r_{1} = d_{c i} \tan ϕ_{1} \approx d_{c i} ϕ_{1}, r_{2} = d_{c o} \tan ϕ_{1} \approx d_{c o} ϕ_{1} .

Substituting Eqs. (3) and (4) in Eq. (2) gives

(d_{c o} - d_{c} + f_{G}) (d_{c i} - d_{c} + f_{G}) = f_{G}^{2},

\frac{ϕ_{2}}{ϕ_{1}} = \frac{1}{f_{G}} (d_{c i} + f_{G} - d_{c}),

where

f_{G}

and

d_{c}

are defined by the following formulae:

f_{G} \equiv \frac{1}{n_{0} \sqrt{A} \sin (L \sqrt{A})},

d_{c} \equiv \frac{1}{n_{0} \sqrt{A}} \cot (\frac{L \sqrt{A}}{2}) .

Equation (5) holds regardless of

ϕ_{1}

. This means that every ray passing through point P also passes through point Q, and so lights diverging from P converge to Q. We can rewrite Eq. (5) as

\frac{1}{d_{c o} + d_{p}} + \frac{1}{d_{c i} + d_{p}} = \frac{1}{f_{G}},

which is the well-known thin lens formula. The parameter

d_{p}

is defined as

d_{p} \equiv 2 f_{G} - d_{c} .

d_{p}

specifies the principal planes of this lens, that is, one of the principal planes is located at

d_{p}

inward from the input face, and the other is located at

d_{p}

inward from the output face. Equation (9) also gives the meaning of

d_{c}

; if the ray in Fig. 2 is horizontally symmetric,

d_{c i} = d_{c o} = d_{c}

.

Fig. 2. Diagram of an optical ray that diverges at point P on the optical axis of a KTN block, enters the block, bends there, and exits the block.

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Next, we consider the case where point P is not on the lens axis. The height of P is $h_{P}$ from the axis, but the distance from the input face is $d_{c i}$ , which is the same as the previous case. It is easily confirmed that Eqs. (5)–(9) also hold for this case, and, regardless of the height of P, $h_{P}$ , rays diverging from P converge to a point on the plane that is a distance $d_{c o}$ from the output face. The height of the converging point Q is

h_{Q} = - \frac{d_{c o} + f_{G} - d_{c}}{f_{G}} h_{P} = - \frac{f_{G}}{d_{c i} + f_{G} - d_{c}} h_{P} .

Here, the lateral magnification is

\frac{h_{Q}}{h_{P}} = - \frac{d_{c o} + f_{G} - d_{c}}{f_{G}} = - \frac{f_{G}}{d_{c i} + f_{G} - d_{c}} .

Now, we consider a KTN block whose charge density has a uniform spatial distribution. Therefore, the charge density $ρ$ is a constant. The space charge induces an electric field in the block that is normal to the electrodes. The magnitude of the electric field depends on the position, and the refractive index is modulated by this field with the Kerr effect. The index is expressed in the following form [13]:

n = n_{0} - \frac{1}{2} n_{0}^{3} g_{eff} ρ^{2} {(x - σ)}^{2} .

n_{0}

is the original index, and

g_{eff}

is an effective Kerr coefficient [14].

x

is a coordinate that indicates the height from the block axis in Fig. 2.

σ

is a function of the applied voltage

V

. We usually write

σ

as

σ (V) = - \frac{ϵ V}{ρ d},

where

ϵ

is the permittivity. Comparing Eq. (12) with Eq. (1), we regard the KTN block as a cylindrical graded index lens with

\sqrt{A} = n_{0} \sqrt{g_{eff}} | ρ | r = x - σ .

Here, we took the absolute value of

ρ

because the origin of the charge is the injected electrons, and

ρ

is negative. Therefore, in the deflector, the graded index lens moves by

σ

along the

x

direction when

V

is changed. We can regard this lens motion as the direct cause of the light beam deflection.

Here, we present some experimental data that proves the above model. Figure 3 shows an example of the spatial distributions of index modulation $Δ n$ in a KTN single crystal block. The size of the KTN block was $3.2 mm \times 4.0 mm (L) \times 1.2 mm (d)$ . The sample temperature was set at 41.3°C, which was 7.2°C above the phase transition temperature of the crystal [3]. At this temperature, the crystal had a cubic crystallographic structure, and all the faces were parallel to the cubic {100} face. The relative permittivity was 13,400. Titanium film electrodes were evaporated onto the two $3.2 mm \times 4.0 mm$ faces. We injected electrons via the electrode by applying a high voltage (100–400 V) and generated a space charge in the block. We then switched off the voltage and evaluated the $Δ n$ distributions by using the phase shift method [15]. We have previously reported the experimental procedures, including the electron injection and the evaluation of the $Δ n$ distribution [16]. In fact, we were able to evaluate only the birefringence $Δ n_{x} - Δ n_{y}$ of the crystal blocks with the optical setup described in Ref. [16], and it was not possible to measure $Δ n_{x}$ and $Δ n_{y}$ individually. Here, however, we used a Mach–Zehnder interferometer and extracted $Δ n_{x}$ , which was with the light polarization perpendicular to the electrode face. (The polarization is the same throughout this paper. We do not deal with perpendicular polarization, since the EO effect is much weaker for this polarization.) The wavelength was 685 nm. The dots in Fig. 3 show the measured data, and they are fitted well with a parabolic function. This indicates that the crystal was well charged, so that the charge density $ρ$ was uniform, and, as a result, Eq. (12) was accurately realized. Also, $ρ$ can be evaluated using this index curve as $80 C / m^{3}$ .

Fig. 3. Example spatial distribution of $Δ n$ in the KTN crystal block.

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We also investigated the lens function of the index distribution by observing the profiles of the beams output from the KTN block. We input a collimated beam into a KTN block and observed the output beam with a microscope, while changing the observation position. The wavelength and the $1 / e^{2}$ full width of the input beam were 1.31 and 527 μm, respectively. The width was obtained by Gaussian fitting in the same way as those for the output beams. The KTN block had the same structure as that in Fig. 3. However, there is an important difference in the effective length. Here, we partly covered the $3.2 mm \times 1.2 mm$ faces of the block with high reflection coatings, as described above. Thus, the light beam passed through the block five times, and the effective length $L$ was 20 mm. Figure 4 shows the change in the output beam width with propagation distance. The horizontal axis indicates the propagation distance measured from the end face of the KTN block. The parameter is the charge density, which we regulated by controlling the voltage used to charge the block. The figure shows that the beam converged to a point and then diverged in the same manner as with an ordinary cylindrical lens, and the width variations were well fitted to the well-known theoretical Gaussian beam profile [12]. Also, the point of convergence became nearer as the charge density increased.

Fig. 4. Change of the width of the beam’s output from a KTN block with the distance from the end face of the block. The parameter was the charge density $ρ$ .

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The distance of the point of convergence from the KTN block of each line in Fig. 4 corresponds to $d_{c o}$ in Fig. 2. We plotted $d_{c o}$ as a function of $ρ$ in Fig. 5. In regards to this experiment, $d_{c o} = f_{G} - d_{p}$ , because the input beam was collimated, and $d_{c i}$ was infinity. The solid line fitted to the $d_{c o}$ plots was calculated with Eqs. (7), (8), and (10). It can be seen that $A$ increased with $ρ$ , and the focal length decreased according to Eq. (7). We also plotted experimental data $d_{p}$ and the theoretical line in the figure. To estimate $d_{p}$ , we determined the position of the principal plane by extrapolating the plot line in Fig. 4 into the KTN block and found the position where the $1 / e^{2}$ full width crossed 527 μm, which is the width of the input beam. We used the same values for the parameters $n_{0}$ and $g_{eff}$ to draw the two fitting curves in Fig. 5. The fitting was fairly successful, thus validating the model described above. Now, we discuss how rays are deflected when we apply a voltage (Fig. 6). We put point P on the axis of the block and consider rays diverging from this point. When a voltage is applied and the lens axis shifts from the block axis, P is off-axis in relation to the lens. Then, the rays from P converge to a point that is also off-axis. The height of the converging point Q from the block axis is deduced by using the lateral magnification in Eq. (11) as follows:

x_{Q} = \frac{d_{c i} + d_{p}}{d_{c i} + f_{G} - d_{c}} σ = \frac{d_{c o} + d_{p}}{f_{G}} σ .

Thus, point Q is swept in proportion to the lens shift

σ

with a voltage change. The deflection angle can be derived with a ray that diverges from P on the block axis and proceeds along this axis. Therefore, the angle of incidence for this input ray is zero. If we extrapolate this input ray and the corresponding output ray with straight lines, the lines cross at a point on the principal plane of the lens. As described above and shown in Fig. 6, this point is on the block axis at a distance

d_{p}

from the output face. This is the beam deflection pivot point for this device. Therefore, with Fig. 6, the deflection angle

θ

is deduced as

θ \approx \tan θ = \frac{x_{Q}}{d_{p} + d_{c o}} = \frac{σ}{f_{G}} .

If we use Eq. (7),

θ = n_{0} \sqrt{A} σ \sin (L \sqrt{A}) .

This formula indicates that the deflection efficiency increases with the block length

L

and

\sqrt{A} \propto | ρ |

. However, it peaks at

L \sqrt{A} = π / 2

. Therefore, we assume

L \sqrt{A} < π / 2

throughout this paper. We are able to make a further approximation of this expression when

L \sqrt{A}

is sufficiently small compared with 1. With Eqs. (7), (13), and (14), we have

θ \approx σ n_{0} A L = - n_{0}^{3} g_{eff} ρ L ϵ \frac{V}{d},

which has the same shape as the reported formula [13].

Fig. 5. $d_{c o}$ and $d_{p}$ as functions of charge density. The solid lines are theoretical fittings.

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Fig. 6. Shift of graded index lens and light ray deflection.

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In the last part of this section, we discuss the resolvable spot number. We have assumed that the charge density $ρ$ is uniform in the KTN block. This assumption gives an odd function $σ (V)$ . Then, an AC voltage with zero bias gives a bidirectional deflection, and the full scanning range of point Q becomes

x_{Qfull} = 2 \frac{d_{c o} + d_{p}}{f_{G}} σ (\frac{V_{p p}}{2}),

where

V_{p p}

is the peak-to-peak value of the AC voltage. The resolvable spot number should be this scanning range divided by the full width of the spot

w_{c}

at Q. Here, we assume that

w_{c} = \frac{4 λ}{π Δ ϕ_{o}},

where

λ

is the wavelength of light, and

Δ ϕ_{o}

is the

1 / e^{2}

angular spread of the output beam converging to point Q [12]. Then, the resolvable spot number

N_{c}

becomes

N_{c} = \frac{π Δ ϕ_{o}}{2 λ} \frac{d_{c o} + d_{p}}{f_{G}} σ (\frac{V_{p p}}{2}) .

Figure 2 indicates that

Δ ϕ_{o}

is the range of

ϕ_{2}

. Thus, it can be converted to a range of

ϕ_{1}

, that is, the angular spread of the input beam

Δ ϕ_{i}

, by using Eq. (6). So, with the help of Eq. (9), we can rewrite

N_{c}

with

Δ ϕ_{i}

as follows:

N_{c} = \frac{π Δ ϕ_{i}}{2 λ} \frac{d_{c i} + d_{p}}{f_{G}} σ (\frac{V_{p p}}{2}) .

Therefore, the resolvable spot number is proportional to the angular spread

Δ ϕ_{o}

or

Δ ϕ_{i}

and, of course, the lens shift

σ

. The angular ranges limit

N_{c}

through the diffraction effect. On the other hand,

σ

represents the device activity. It appears that, as

σ

increases with voltage

V

,

V

places another limit on

N_{c}

. However,

σ

has a ceiling that is set by the block structure, and it is lower than that set by commonly used high voltage suppliers. We will discuss this limit in a later section. In fact, there is an additional limit caused by the crystal characteristics, such as the field-induced phase transition [17].

3. USING COLLIMATED BEAMS AS INPUT AND OUTPUT BEAMS

In the previous section, we discussed device characteristics with a diverging input beam and a converging output beam. However, collimated beams are usually much more convenient for many applications. In this section, we analyze device characteristics with collimated input and output beams. Obviously lenses can be used for converting a diverging or converging beam to a collimated beam and vice versa. As the KTN lens is a cylindrical graded index lens, we use cylindrical concave or cylindrical convex lenses for this beam conversion. Figure 7(a) shows beam conversion with a cylindrical concave lens. The optical axes of the concave lens and the KTN block are aligned. The focal length of the concave lens is $- f_{c i}$ , and $b$ is the distance between the rear principal plane of the concave lens and the input face of the KTN block.

Fig. 7. Conversion of input collimated beams to diverging beams (a) with concave lens and (b) with convex lens.

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We first discuss optical rays in the input and output beams. As the input beam is collimated, here, we consider an optical ray that is parallel to the optical axis and a distance of $x_{0 i}$ from the axis. This ray corresponds to that in Fig. 2 with

ϕ_{1} \approx \frac{x_{0 i}}{f_{c i}} .

Also,

d_{c i}

in Fig. 2 corresponds to the distance between the rear focal plane of the concave lens and the input face of the block, as shown in Fig. 7(a). The converging output beams in Fig. 2 can be collimated in a similar manner to that shown in Fig. 7(a). We place a concave lens with a focal length of

- f_{c o}

on the right side of the KTN output face. The output ray converging to Q in Fig. 2 is bent by this lens, and the ray becomes parallel to the axis if the front focal point of the lens coincides with point Q. Then, with the height of the bent ray

x_{0 o}

,

ϕ_{2}

in Fig. 2 is expressed as

ϕ_{2} \approx \frac{x_{0 o}}{f_{c o}} .

We also obtain the following with Eq. (6):

\frac{x_{0 o}}{f_{c o}} = \frac{d_{c i} + f_{G} - d_{c}}{f_{G}} \frac{x_{0 i}}{f_{c i}}, \frac{x_{0 i}}{f_{c i}} = \frac{d_{c o} + f_{G} - d_{c}}{f_{G}} \frac{x_{0 o}}{f_{c o}} .

Next, we derive the deflection angle expression for beams collimated by the concave lens with the help of Fig. 8. Point Q in this figure is the converging point with an applied voltage but without the collimating concave lens. The concave lens is placed so that its front focal plane contains Q. In such a situation, the following equation holds, which is drawn from the geometrical optics theory,

x_{Q} = f_{c o} θ_{c} .

Here,

θ_{c}

is the deflection angle after the beam has passed through the concave lens, and

x_{c}

is the height of Q. By using Eq. (15), we obtain

θ_{c} = \frac{1}{f_{c o}} \frac{d_{c i} + d_{p}}{d_{c i} + f_{G} - d_{c}} σ = \frac{d_{p} + d_{c o}}{f_{c o}} \frac{σ}{f_{G}} .

Compare this formula with Eq. (16), which was given for a KTN deflector without a collimating lens. If the input side and the output side are symmetrical, that is,

d_{c i} = d_{c o} = d_{c}

, Eq. (24) can be simplified to

θ_{c} = \frac{2}{f_{c o}} σ .

It is noteworthy that parameters

A

and

L

are not included in this formula, and that, for the symmetrical configuration, the deflection angle does not depend on the EO coefficient

g_{eff}

or the length of the KTN block.

Fig. 8. Changes in deflection angle and pivot when a compensating concave lens is installed.

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The concave lens also moves the original deflection pivot $O_{O}$ to $O_{O}^{'}$ . With the thin lens formula [which is similar to Eq. (9), as described], the distance between the rear principal plane $H^{'}$ of the lens and the new pivot $O_{O}^{'}$ is deduced as

c = f_{c o} (1 - \frac{f_{c o}}{d_{c o} + d_{p}}) .

Then, we evaluate the resolvable spot number with the collimating lenses. An optical beam with a finite width is diffracted with propagation, and the beam width becomes wider with the propagation distance in the region beyond the Rayleigh length. Then, it has a constant angular spread $Δ θ$ , and the full deflection angle $θ_{c w}$ divided by $Δ θ$ becomes a measure of the resolvable spot number. $Δ θ$ is evaluated with the same theory as that used for Eq. (18) and is given by the following formula [12]:

Δ θ = \frac{4 λ}{π w_{o 0}},

where

w_{o 0}

is the full beam width at the concave lens. The beam width is measured at

1 / e^{2}

intensity. As in the previous section, we can express the full deflection angle

θ_{c w}

as follows by using Eq. (24):

θ_{c w} = 2 \frac{d_{p} + d_{c o}}{f_{c o}} \frac{1}{f_{G}} σ (\frac{V_{p p}}{2}) .

Therefore, we assume that the resolvable spot number is evaluated with

N_{c} = \frac{θ_{c w}}{Δ θ} = \frac{π w_{o 0}}{2 λ} \frac{d_{p} + d_{c o}}{f_{c o}} \frac{1}{f_{G}} σ (\frac{V_{p p}}{2}) .

Here,

w_{o 0}

can be converted to

w_{i 0}

, which is the width of the input collimated beam, in a similar manner to the conversion of

x_{0 o}

to

x_{0 i}

with Eq. (23). Thus, with the help of Eq. (5), we obtain

N_{c} = \frac{π w_{i 0}}{2 λ} \frac{d_{p} + d_{c i}}{f_{c i}} \frac{1}{f_{G}} σ (\frac{V_{p p}}{2}),

where

N_{c}

is expressed with the input parameters but without the output parameters. It is a natural result that Eqs. (28) and (29) are equivalent to Eqs. (19) and (20), considering

\frac{w_{i 0}}{f_{c i}} = Δ ϕ_{i}, \frac{w_{o 0}}{f_{c o}} = Δ ϕ_{o} .

Hitherto, we have analyzed the characteristics of a KTN deflector with a pair of compensating concave lenses. However, it is possible to replace the compensating lenses with convex lenses, as shown in Fig. 7(b). We can replace one or both of the lenses with a convex lens without degrading the deflector performance. The difference is that an image is inverted when it passes through a convex lens. As a result, the beam deflection direction becomes opposite when the output side lens is replaced (Fig. 9). Therefore, with notations $f_{c i}$ and $f_{c o}$ for the focal lengths of the convex lenses, Eqs. (23), (28), and (29) also hold true for this lens type, whereas the signs of Eqs. (24) and (25) change. In addition, the pivot moves to a position further than the rear focal point of the convex lens, as indicated by Fig. 9. The distance $c$ in Fig. 9 is

c = \frac{h}{x_{Q}} f_{c o} = f_{c o} (1 + \frac{f_{c o}}{d_{c o} + d_{p}}) .

Here, again,

c

is measured from the rear principal plane of the convex lens. Figures 7–9 show that configurations using convex lenses are bulky compared with those with concave lenses. However, device tunability is improved. As seen in Fig. 7(a),

d_{c i} (d_{c o})

must be longer than

f_{c i} (f_{c o})

when using concave lenses. It is preferable to inject a greater number of electrons into a KTN block in order to obtain good device performance, such as

N_{c}

, which will be discussed later. Then,

d_{c}

becomes short, as do

d_{c i}

and

d_{c o}

. This fact limits our choice of concave lenses. In regards to convex lenses, there is no such restriction, which makes it easier for us to optimize the performance.

Fig. 9. Changes in deflection angle and pivot when a compensating convex lens is installed.

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4. RAY ANALYSES FOR OBTAINING GOOD RESOLVING PERFORMANCE

For an optical beam deflector, the resolvable spot number $N_{c}$ is one of the most important specifications. We have proposed expressions for $N_{c}$ , such as Eq. (20). According to the equation, $N_{c}$ is proportional to the angular spread of the input beam $Δ ϕ_{i}$ and the shift of the KTN index lens $σ$ . As $σ$ increases with the applied voltage, $N_{c}$ also increases with the voltage, which is simply as a result of the increase in the deflection angle. However, with an increasing deflection angle, some rays collide with the wall of the KTN block and are stopped. On the other hand, such ray stopping tends to occur with a small deflection angle when $Δ ϕ_{i}$ is large. Namely, the block size restricts $N_{c}$ . In this section, we propose methods to avoid the ray being stopped by the wall and to attain good resolving performance.

Equation (2) is the relation between the input and output rays of a KTN block. Here, we replace $L$ in this equation with $z$ whose origin we set at the input face. Then, Eq. (2) with Eqs. (3), (4), and (14) define the shapes of rays in the block. We define

{\dot{x}}_{1} = ϕ_{1}, x_{1} = d_{c i} ϕ_{1}

and, then, obtain the ray shape as

x = \cos (z \sqrt{A}) d_{c i} ϕ_{1} + \frac{1}{n_{0} \sqrt{A}} \sin (z \sqrt{A}) ϕ_{1} - {\cos (z \sqrt{A}) - 1} σ .

At the output face of the KTN block, the ray passes a point with

z = L, x = x_{L} \equiv (\frac{d_{c i} + d_{p}}{f_{G}} d_{c} - d_{c i}) ϕ_{1} + \frac{d_{p}}{f_{G}} σ .

Also, Eq. (32) has extrema and the first extremum (maximum) is at

z = z_{p} \equiv \frac{1}{\sqrt{A}} \arctan (\frac{1}{n_{0} \sqrt{A}} \frac{ϕ_{1}}{d_{c i} ϕ_{1} - σ}), x = x_{p} \equiv \sqrt{{(d_{c i} ϕ_{1} - σ)}^{2} + d_{p} d_{c} ϕ_{1}^{2}} + σ .

See Fig. 2 and the extremum (peak of the ray) in the figure. If

z_{p} < L

, the extremum is inside the block, and

x_{p} < d / 2

is the condition if the ray is to avoid colliding with the wall. If

z_{p} > L

, the extremum is outside the block. Then, in the block,

x

becomes the maximum at the output face, and

x_{L} < d / 2

is the condition for avoiding the ray collision. Thus,

(z_{p} < L) \frac{d}{2} > \sqrt{{(d_{c i} ϕ_{1} - σ)}^{2} + d_{p} d_{c} ϕ_{1}^{2}} + σ (z_{p} > L) \frac{d}{2} > (\frac{d_{c i} + d_{p}}{f_{G}} d_{c} - d_{c i}) ϕ_{1} + \frac{d_{p}}{f_{G}} σ,

are the conditions for avoiding the ray collision. It is possible to transform this condition to a more convenient one. With parameters defined by

ϕ_{1 b} \equiv \frac{d}{2 (d_{p} + d_{c i})}, σ_{b} \equiv \frac{d}{2} (1 - \frac{f_{G}}{d_{p} + d_{c i}}) = \frac{d}{2} - f_{G} ϕ_{1 b},

the condition is

(σ < σ_{b}) ϕ_{1} < \frac{\sqrt{\frac{d}{2} (\frac{d}{2} - 2 σ) (d_{c i}^{2} + d_{p} d_{c}) + d_{c i}^{2} σ^{2}} + d_{c i} σ}{d_{c i}^{2} + d_{p} d_{c}},

(σ > σ_{b}) ϕ_{1} < \frac{1}{2} \frac{f_{G} d - 2 d_{p} σ}{d_{p} d_{c} + (d_{c} - f_{G}) d_{c i}} .

Another form of this condition is

(ϕ_{1} > ϕ_{1 b}) 2 σ < \frac{\frac{d^{2}}{4} - (d_{c i}^{2} + d_{p} d_{c}) ϕ_{1}^{2}}{\frac{d}{2} - d_{c i} ϕ_{1}},

(ϕ_{1} < ϕ_{1 b}) 2 σ < \frac{f_{G} d - 2 ϕ_{1} {d_{p} d_{c} + (d_{c} - f_{G}) d_{c i}}}{d_{p}} .

Equations (36) and (38) are for

z_{p} < L

, that is, the problem is collision inside the block. Equations (37) and (39) are for

z_{p} > L

, that is, the problem is collision at the output face.

We drew these curves in Fig. 10 for several values of $| ρ |$ . Here, we set $d_{c i} = d_{c o} = d_{c}$ , namely, we employed a symmetrical setup. The filled circles in the figure show the boundary points defined with Eq. (35). We used Eq. (36) on the right side of the circle and Eq. (37) on the left side for each curve. The area below a curve shows the permitted values of $σ$ and $ϕ_{1}$ with the corresponding charge density. The figure indicates that, although a considerably large $σ$ is permitted at $ϕ_{1} = 0$ , the lens shift $σ$ becomes restricted by increasing $ϕ_{1}$ . This is because, $if ϕ_{1}$ is large, the ray approaches closer to a wall of the KTN block and collides with the wall with a small lens shift. If $ϕ_{1}$ is too large, the ray is stopped in the block regardless the size of $σ$ . This $ϕ_{1}$ limit is extended by increasing $| ρ |$ , which bends a ray moving towards a wall so that it comes back towards the optical axis of the block. Another noticeable point in regard to the curves in Fig. 10 is that $σ$ can be larger with a smaller $| ρ |$ and a small $ϕ_{1}$ . This means that we can obtain large deflection angles with this condition. However, we must note here that Fig. 10 indicates conditions for avoiding the ray stopping in the block. The approximate deflection angle of Eq. (17) is proportional to $ρ$ . Therefore, to obtain large deflection angles with a small $| ρ |$ , we need to apply high voltages sufficient to supplement the small $| ρ |$ .

Fig. 10. Lens shift $σ$ as a function of the incident angle of an input ray $ϕ_{1}$ , which indicates limitations in regards to these parameters for avoiding ray collision. The curves are drawn for different charge densities. The area below each curve is the permitted region.

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Then, we derive an expression for the maximum resolvable spot number $N_{c}$ , restricted by the block size. For a diverging input beam with an angular spread $Δ ϕ_{i}$ , the maximum value for $ϕ_{1}$ is $Δ ϕ_{i} / 2$ . Then, the $σ$ limit is derived with Eqs. (38) and (39), and the $N_{c}$ limit is deduced with Eq. (20):

(Δ ϕ_{i} > 2 ϕ_{1 b}) N_{c \lim} = \frac{π}{8 λ} \frac{d_{c i} + d_{p}}{f_{G}} Δ ϕ_{i} \frac{d^{2} - (d_{c i}^{2} + d_{p} d_{c}) Δ ϕ_{i}^{2}}{d - d_{c i} Δ ϕ_{i}},

(Δ ϕ_{i} < 2 ϕ_{1 b}) N_{c \lim} = \frac{π}{4 λ} \frac{d_{c i} + d_{p}}{f_{G}} Δ ϕ_{i} \times \frac{f_{G} d - Δ ϕ_{i} {d_{p} d_{c} + (d_{c} - f_{G}) d_{c i}}}{d_{p}} .

This is the maximum resolvable spot number of a KTN optical beam deflector. We plotted

N_{c \lim}

as a function of

Δ ϕ_{i}

in Fig. 11 for a symmetrical setup (

d_{c i} = d_{c o} = d_{c}

) and for different charge densities. Other parameters are also shown in the figure. Here again, the circles show boundaries. We used Eq. (41) on the left side of the circles and Eq. (40) on the right side of the circles. In this figure,

N_{c \lim}

is zero at

Δ ϕ_{i} = 0

because

Δ ϕ_{o}

is also zero, and the output beam cannot be confined, namely

w_{c} = \infty

in Eq. (18).

N_{c \lim}

increases with

Δ ϕ_{i}

, reaches its maximum value, and then returns to zero. We can improve

N_{c \lim}

by increasing

Δ ϕ_{i}

because the output beam confinement is improved. However, if we increase

Δ ϕ_{i}

too much, the margin for the lens shift

σ

(or deflection angle) becomes narrow, and

N_{c \lim}

is degraded (see Fig. 10). Therefore, it is possible to improve the resolvable spot number by adjusting the system arrangement via

Δ ϕ_{i}

. The maximum

N_{c \lim}

obtained by this adjustment increases with

| ρ |

. It appears that the improvement is not very great. However, here again, we must note that the deflection angle is proportional to

ρ

. The maximum

N_{c \lim}

values can be realized with lower voltages when

| ρ |

is high.

Fig. 11. Maximum resolvable spot number of a KTN optical beam deflector as a function of $Δ ϕ_{i}$ with a symmetrical configuration.

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In fact, we can improve the resolvable spot number beyond those shown in Fig. 11. We calculated the $N_{c \lim}$ of a symmetrical configuration to draw the figure. The improvement can be realized by breaking this symmetry. To obtain a simplified perspective, we fix the beam width at the input face of the KTN block. When the full width of the beam there is $a_{i}$ ,

a_{i} = d_{c i} Δ ϕ_{i} .

We are able to rewrite Eqs. (40) and (41) with Eq. (42) as

(Δ ϕ {}_{i}< \frac{d - a_{i}}{d_{p}}) N_{c \lim} = \frac{π}{λ} \frac{(a_{i} + d_{p} Δ ϕ_{i}) (d^{2} - a_{i}^{2} - d_{p} d_{c} Δ ϕ_{i}^{2})}{8 f_{G} (d - a_{i})},

(Δ ϕ {}_{i}> \frac{d - a_{i}}{d_{p}}) N_{c \lim} = \frac{π}{4 λ} \frac{1}{d_{p}} (a_{i} + d_{p} Δ ϕ_{i}) \times {d + a_{i} - \frac{d_{c}}{f_{G}} (a_{i} + d_{p} Δ ϕ_{i})}

or

(d_{c i} < \frac{a_{i} d_{p}}{d - a_{i}}) N_{c \lim} = \frac{π}{8 λ} \frac{a_{i}^{3}}{f_{G} (d - a_{i})} (1 + \frac{d_{p}}{d_{c i}}) (\frac{d^{2} - a_{i}^{2}}{a_{i}^{2}} - \frac{d_{p} d_{c}}{d_{c i}^{2}}),

(d_{c i} > \frac{a_{i} d_{p}}{d - a_{i}}) N_{c \lim} = \frac{π}{4 λ} \frac{a_{i}^{2}}{d_{p}} (1 + \frac{d_{p}}{d_{c i}}) {1 + \frac{d}{a_{i}} - \frac{d_{c}}{f_{G}} (1 + \frac{d_{p}}{d_{c i}})} .

Figure 12 shows the resolvable spot number

N_{c \lim}

drawn with Eqs. (43) and (44). The vertical and horizontal axes are the same as those in Fig. 11, but the horizontal axis extends to the negative region in Fig. 12. This is because the maximum

N_{c \lim}

is obtained at a negative value of

Δ ϕ_{i}

with a certain condition. Clearly, every curve in Fig. 12 has its peak in the negative region. The peak of Eq. (44) appears at

Δ ϕ_{i} = \frac{f_{G} d - (2 d_{c} - f_{G}) a_{i}}{2 d_{c} d_{p}} = \frac{n_{0} \sqrt{A}}{2 \sin (L \sqrt{A})} [d - {1 + 2 \cos (L \sqrt{A})} a_{i}],

and, there, the peak value of

N_{c \lim}

is

\frac{π}{16 λ} \frac{f_{G}}{d_{c} d_{p}} {(d + a_{i})}^{2} = \frac{π}{16 λ} \frac{n_{0} \sqrt{A}}{\sin (L \sqrt{A})} {(d + a_{i})}^{2} .

Therefore, a large

a_{i}

is preferable for obtaining a large

N_{c}

. Then, Eq. (47) indicates that

Δ ϕ_{i}

tends to become negative.

Δ ϕ_{i} = 0

means that the input beam is collimated with the beam width

a_{i}

, and a negative

Δ ϕ_{i}

means that the input beam is converging rather than diverging. Therefore, in this framework, a collimated or a converging beam is a better choice as the input than a diverging beam if we are to obtain good spot resolving performance. This can be ascribed to the fact that rays hardly ever collide with the block wall with a converging beam. However, the converging point Q in Fig. 2 comes closer to the output face of the block in such a situation. Therefore, it would be difficult to choose a cylindrical concave lens for collimating the output beam, and a convex lens would be more suitable.

Fig. 12. Maximum resolvable spot number of a KTN optical beam deflector as a function of $Δ ϕ_{i}$ with a fixed beam width at the input face of the KTN block. The beam width $a_{i}$ is 0.8 mm.

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5. CONCLUSIONS

We analyzed curving optical rays in a KTN block with a spatial distribution of a refractive index to maximize deflector performance. We analyzed the rays with the conventional ray matrix of a graded index lens. The modeling was experimentally validated by the focusing properties of the KTN block. With this model, we constructed a method for improving the deflection performance of KTN optical deflectors. We concentrated particularly on the resolvable spot number, which is the most important figure for an optical beam deflector. In terms of this figure of merit, a converging beam or a collimated beam is better as the input beam than a diverging beam.

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Analyses of optical rays in KTN optical beam deflectors for device design

Abstract

1. INTRODUCTION

2. OPTICAL RAYS IN KTN CRYSTAL BLOCK

3. USING COLLIMATED BEAMS AS INPUT AND OUTPUT BEAMS

4. RAY ANALYSES FOR OBTAINING GOOD RESOLVING PERFORMANCE

5. CONCLUSIONS

REFERENCES

Cited By

Figures (12)

Equations (55)

Applied Optics