JP2014066615A - Electrooptical coefficient measuring method and electrooptical coefficient measuring apparatus - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an electrooptical coefficient measuring method and an electrooptical coefficient measuring apparatus being capable of highly accurate measurement of a large number of electrooptical coefficients with a simple configuration without coordinate conversion.SOLUTION: A crystal S to which an AC electric field is applied is arranged between a polarizer 20 and an analyser 22 arranged so that transmission axes are parallel or orthogonal to one another. Light with a narrow light emission wavelength band from a light source 10 is made incident into the crystal S through the polarizer 20 to detect and photoelectrically convert the light passing through the crystal S and the analyser 22. On the basis of a light detection signal thus detected, an electrooptical coefficient r(ris any one of r, r, r, r, r, r, rand r) is acquired. The crystal S is a rectangular parallelepiped crystal being composed of a uniaxial crystal or a biaxial crystal. Each normal line of six surfaces of the rectangular parallelepiped coincides with any one of main axes X, Xand Xof the crystal S. The direction of the AC electric field applied to the crystal S coincides with any one of the main axes.

Description

  The present invention relates to an electro-optic coefficient measuring method and an electro-optic coefficient measuring apparatus for measuring an electro-optic coefficient of a crystal.

Conventionally, in order to measure the electro-optic coefficients r ₄₁ , r ₄₂ , r ₄₃ , r ₅₁ , r ₅₂ , r ₆₁ , r ₆₂ , r ₆₃ of the crystal, it is necessary to perform appropriate coordinate transformation. For example, the principal axes X ₁ and X ₂ of the uniaxial crystal are rotated by 45 ° in the plane where both coordinate axes exist, new coordinate axes X ₁ ′ and X ₂ ′ are determined, and the electro-optic coefficient r ₄₁ is measured. A method is known (see Non-Patent Document 1).

Further, a method is known in which the electro-optic coefficient r ₅₁ of the point group 3m is measured using a complicated configuration in which the applied electric field direction and the incident light direction form an angle of 45 ° (see Non-Patent Document 2). In this method, an electric field is applied to the LiNbO ₃ crystal, and r ₅₁ = 28 pm / V is measured in a constant strain state. Since this value was measured in a frequency region where no dynamic phase change due to the inverse piezoelectric effect occurred, it was about 10% smaller than the value in the constant stress state. This value is well known as the r ₅₁ standard for LiNbO ₃ crystals at high frequencies. As another method for measuring the electro-optic coefficient r ₅₁ of the LiNbO ₃ crystal, methods shown in Non-Patent Documents 3 and 4 are known.

I. P. Kaminow and E. H. Turner, “Electro-optic light modulators” Appl. Opt. Vol.5, pp.1612-1628 (1966). E. H. Turner, “High-frequency electro-optic coefficients of lithium niobate”, Appl. Phys. Lett. Vol. 8, pp. 303-304 (1966). Kuniharu Takizawa, Akiko Ibayashi: "Measurement of sign and absolute value of electro-optic coefficient r51 and piezoelectric constant d15 of LiNbO3 crystal", Optics, & Photonics Japan 2009, 24aE5. Kuniharu Takizawa, Akiko Ibayashi: “The wavelength dependence of the electro-optic coefficient r51 of LiNbO3 crystals”, Optics, & Photonics Japan 2009, 24aE6.

の１軸性結晶には適しているが、２軸性結晶には不向きである。また、非特許文献３、４に示す方法は、ＬｉＮｂＯ_３結晶のようにｒ_５１以外の電気光学係数が符号も含めて精度よく測定されている場合に限り有効になる方法であり、殆どの結晶には不適である。 The conventional method has a common problem that it is specialized for a specific crystal and lacks versatility. For example, the method shown in Non-Patent Document 1 uses a point group to which KH ₂ PO ₄ and NH ₄ H ₂ PO ₄ belong.

It is suitable for the uniaxial crystal, but is not suitable for the biaxial crystal. The methods shown in Non-Patent Documents 3 and 4 are effective only when electro-optic coefficients other than r ₅₁ are accurately measured including the sign, such as LiNbO ₃ crystal. Not suitable for.

  Further, in the methods shown in Non-Patent Documents 2 to 4, since a value including a large error is measured due to the influence of the fringe effect of the electric field and the influence of various electro-optic coefficients other than the target electro-optic coefficient, high accuracy Not suitable for measurement.

  The present invention has been made in view of the above-described problems, and an object of the present invention is to provide an electro-optic that can measure a large number of electro-optic coefficients with a simple configuration without converting coordinates. A coefficient measuring method and an electro-optic coefficient measuring apparatus are provided.

(1) The present invention provides an electro-optic coefficient measuring method for measuring an electro-optic coefficient of a crystal,
A crystal to which an alternating electric field is applied is arranged between first and second polarizers arranged so that their transmission axes are parallel or orthogonal to each other, and light from a light source having a narrow emission wavelength band is converted into the first polarized light. The light is incident on the crystal through a child, the light transmitted through the crystal and the second polarizer is detected and photoelectrically converted, and the electro-optic coefficient r _mn (r _mn of the crystal is based on the detected light detection signal) , R ₄₁ , r ₄₂ , r ₄₃ , r ₅₁ , r ₅₂ , r ₆₁ , r ₆₂ , r ₆₃ )
The crystal is a cuboid crystal composed of a uniaxial crystal or a biaxial crystal, and the normals of the six faces of the cuboid coincide with any one of the principal axes X ₁ , X ₂ , and X ₃ of the crystal, respectively. Or an internal angle between the normal and the main axis is 0.1 ° or less,
The direction of the alternating electric field applied to the crystal coincides with any one of the main axes, or the internal angle between the direction of the alternating electric field and the main axis is 0.1 ° or less.

According to the present invention, a large number of electro-optic coefficients (r ₄₁ , r ₄₂ , r ₄₃ , r ₅₁ , r ₅₂ , r ₆₁ , r ₆₂ , r ₆₃ ) are accurately measured with a simple configuration without coordinate conversion. be able to. Since the crystal sample used for the measurement is a simple rectangular parallelepiped, it can be easily processed and can be measured with high accuracy. In addition, since a parallel Nicol or orthogonal Nicol optical system is configured, stable measurement is possible and excellent reproducibility is achieved.

(2) In the present invention,
The light from the light source is directed in a direction coinciding with any one of the principal axes so that the formula of the refractive index ellipsoid of the crystal to which the alternating electric field is applied is converted into an eigenvalue equation represented by the following formula: It may be propagated.

ただし、係数Ａ、Ｂは前記結晶の屈折率で表され、且つＡ≠Ｂであり、係数Ｃは前記電気光学係数ｒ_ｍｎ及び前記結晶に印加される電界で表され、ｉ及びｊは１〜３のいずれか１つの整数であり、且つ、ｉ≠ｊである。

Where the coefficients A and B are expressed by the refractive index of the crystal, and A ≠ B, the coefficient C is expressed by the electro-optic coefficient r _mn and the electric field applied to the crystal, and i and j are 1 to Any one of 3 and i ≠ j.

(3) In the present invention,
One of the main axes is parallel to the propagation direction of light from the light source, and the other two of the main axes form an angle of 45 ° or −45 ° with the transmission axis of the first polarizer. Also good.

(4) In the present invention,
From the light detection signal, measuring a dynamic phase difference theta (t) that varies with time t between the linearly polarized light that oscillates in the spindle X _j direction of the linearly polarized light and the crystal oscillating the main axis X _i direction of the crystal By doing so, the electro-optic coefficient r _mn of the crystal may be obtained.

  However, i and j are any one integer of 1 to 3, and i ≠ j.

  According to the present invention, by measuring the dynamic phase difference and obtaining the electro-optic coefficient, the influence of the phase change due to the expansion and contraction of the crystal length due to the inverse piezoelectric effect can be ignored, and a highly accurate measurement is possible. .

(5) In the present invention,
The refractive index n _i of crystals linearly polarized light oscillating the main axis X _i direction of the crystal feel _the crystals of the main axis X _j direction linearly polarized light vibrating in feel refractive index n _j of the _crystal, said light source having a center wavelength λ When the length of the crystal through which the light from the light propagates is L and the alternating electric field of the frequency f applied to the crystal is Esin (2πft), the absolute value of the dynamic phase difference θ (t) is given by May be represented.

（６）また本発明において、
前記光検出信号から直流成分Ｉ_０と２ｆ（ｆは、前記交流電界の周波数）の周波数成分の実効値Ｉ_ｒｍｓを抽出し、抽出した前記直流成分Ｉ_０と前記実効値Ｉ_ｒｍｓとに基づいて、前記結晶の電気光学係数ｒ_ｍｎを求めてもよい。

(6) In the present invention,
An effective value I _rms of a frequency component of DC components I ₀ and 2f (f is the frequency of the AC electric field) is extracted from the photodetection signal, and based on the extracted DC component I ₀ and the effective value I _rms. The electro-optic coefficient r _mn of the crystal may be obtained.

  According to the present invention, by measuring the electro-optic coefficient using a frequency component that is twice the frequency of the AC electric field applied to the crystal, other unnecessary electro-optic coefficients included in the amplitudes of frequencies other than the double frequency. Can be eliminated, and highly accurate measurement is possible.

(7) In the present invention,
The refractive index n _i of crystals linearly polarized light oscillating the main axis X _i direction of the crystal feel _the crystals of the main axis X _j direction linearly polarized light vibrating in feel refractive index n _j of the _crystal, said light source having a center wavelength λ When the length of the crystal through which the light from the light propagates is L, the electro-optic coefficient r _mn of the crystal may be obtained based on the following equation.

ただし、φは前記結晶の主軸Ｘ_ｉ方向に振動する直線偏光と前記結晶の主軸Ｘ_ｊ方向に振動する直線偏光との間の時間ｔに依存しない静的位相差であり、ｉ及びｊは１〜３のいずれか１つの整数であり、且つ、ｉ≠ｊである。

However, phi is a static phase difference is not dependent on the time t between the linearly polarized light oscillating the main axis X _j direction of the crystal and linearly polarized light that vibrates the main axis X _i direction of the crystal, i and j are 1 Any one of ˜3 and i ≠ j.

(8) In the present invention,
The static phase difference φ may be adjusted so that the effective value I _rms of the 2f frequency component is maximized.

(9) The present invention provides an electro-optic coefficient measuring apparatus for measuring an electro-optic coefficient of a crystal,
First and second polarizers arranged such that their transmission axes are parallel or orthogonal;
A light irradiator that causes light from a light source having a narrow emission wavelength band to be incident on a crystal to which an alternating electric field has been applied via the first polarizer;
A light detection unit that detects and photoelectrically converts light transmitted through the crystal and the second polarizer;
An operation for calculating an electro-optic coefficient r _{mn of the} crystal based on a light detection signal from the light detection unit (m is an integer of any one of 4 to 6, and n is an integer of any of 1 to 3). An arithmetic processing unit that performs processing,
The crystal is a cuboid crystal composed of a uniaxial crystal or a biaxial crystal, and the normals of the six faces of the cuboid coincide with any one of the principal axes X ₁ , X ₂ , and X ₃ of the crystal, respectively. Or an internal angle between the normal and the main axis is 0.1 ° or less,
The direction of the alternating electric field applied to the crystal coincides with any one of the main axes, or the internal angle between the direction of the alternating electric field and the main axis is 0.1 ° or less.

According to the present invention, a large number of electro-optic coefficients (r ₄₁ , r ₄₂ , r ₄₃ , r ₅₁ , r ₅₂ , r ₆₁ , r ₆₂ , r ₆₃ ) are accurately measured with a simple configuration without coordinate conversion. be able to.

The figure which shows an example of a structure of the electro-optic coefficient measuring apparatus of this embodiment. The figure which shows the crystal point group to which the electro-optic coefficient measuring method of this embodiment is applied, and its electro-optic coefficient. The figure which shows the structure of the crystal | crystallization S used as a measuring object, an applied electric field direction, and a light propagation direction. It shows the appearance of the LiNbO ₃ crystal used in the measurement. The figure for demonstrating the measuring method of static phase difference (phi). And amplitude χ dynamic phase difference, the measurement results indicating the relationship between the amplitude V _a of the AC voltage applied to the LiNbO ₃ crystal.

  Hereinafter, this embodiment will be described. In addition, this embodiment demonstrated below does not unduly limit the content of this invention described in the claim. In addition, all the configurations described in the present embodiment are not necessarily essential configuration requirements of the present invention.

1. Measurement Principle and Configuration A measurement principle employed by the electro-optic coefficient measurement method and the electro-optic coefficient measurement apparatus according to this embodiment will be described. FIG. 1A is a diagram illustrating an example of the configuration of the electro-optic coefficient measuring apparatus according to the present embodiment.

  The electro-optic coefficient measuring apparatus shown in FIG. 1A includes a light source 10 (for example, a laser light source) having a narrow emission wavelength band, a polarizer 20 (first polarizer), and an analyzer 22 (second polarizer). And a photodetector 30 including a photoelectric sensor (photodetector), a low-pass filter 40, a lock-in amplifier 42, a signal generator 44, an amplifier 46, and an arithmetic device 50 (arithmetic processing unit).

In the example shown in FIG. 1A, the polarizer 20 and the analyzer 22 are arranged so that their transmission axis directions P ₁ and P ₂ are parallel to each other to constitute a parallel Nicol optical system. The crystal S to be measured is arranged between the analyzer 22 and the analyzer 22. Further, the main axes X ₁ and X _{3 of} the crystal S form an angle of 45 ° with the transmission axes P ₁ and P _{2 of} the polarizer 20 and the analyzer 22 (see FIG. 1B), and the main axis X of the crystal S ₂ are arranged in parallel with the propagation direction of the light from the light source 10. The orthogonal Nicol optical system may be configured by arranging the transmission axes P ₁ and P _{2 of} the polarizer 20 and the analyzer 22 so as to be orthogonal to each other.

  The light from the light source 10 passes through the polarizer 20 to become linearly polarized light and enters the crystal S. The light transmitted through the crystal S enters the photodetector 30 through the analyzer 22. The photodetector 30 receives the light transmitted through the analyzer 22, converts the intensity of the received light into a current or a voltage (photoelectric conversion), and outputs the light detection signal I to the low-pass filter 40 and the lock-in amplifier 42. .

The signal generator 44 outputs a control signal (drive signal) to the amplifier 46 to control the drive of the amplifier 46 and outputs the control signal to the lock-in amplifier 42 as a reference signal. The amplifier 46 amplifies the control signal output from the signal generator 44 and applies an alternating electric field to the crystal S. The lock-in amplifier 42 extracts a frequency component I ₁ that is twice the frequency of the reference signal output from the signal generator 44 from the light detection signal I output from the light detector 30 and outputs it to the arithmetic device 50. To do. The low-pass filter 40 extracts the DC component I ₀ from the light detection signal I output from the light detector 30 and outputs it to the arithmetic device 50.

The computing device 50 (computer) calculates the electrical power of the crystal S based on the DC component I ₀ output from the low-pass filter 40 and the frequency component I ₁ (effective value of the frequency component I ₁ ) output from the lock-in amplifier 42. An arithmetic process for calculating an optical coefficient is performed.

FIG. 2 is a diagram showing a crystal point group to which the electro-optic coefficient measurement method of the present embodiment is applied and its electro-optic coefficient. In the measurement method of the present embodiment, electro-optic coefficients r ₄₁ , r ₄₂ , r ₄₃ , r ₅₁ , r ₅₂ , r ₆₁ , r ₆₂ , r ₆₃ indicated by circles in the crystal point group shown in FIG. You can select and measure its absolute value.

Hereinafter, the measurement principle of this embodiment will be described by taking as an example the case of measuring the electro-optic coefficient r ₅₁ of a uniaxial crystal belonging to the crystal point group 3m.

FIG. 3 is a diagram showing a configuration of the crystal S to be measured. As shown in FIG. 3, the crystal S is a cuboid crystal, and the normals of the six faces of the cuboid are the principal axes X ₁ , X ₂ , X _{3 of the} crystal S (the principal axis X of the refractive index ellipsoid of the crystal S). ₁ , X ₂ , X ₃ ). The direction of the alternating electric field applied to the crystal S is consistent with the main axis X ₁ of the crystal S, the propagation of light incident on the crystal S (emitted from the light source 10, the linearly polarized light transmitted through the polarizer 20) direction coincides with the main axis X ₂ of the crystal S.

When added to X ₁ axis direction of the crystal S belonging to the electric field E to the point cloud 3m, induced by electro-optical effect (EO effect) by the electric field E, the refractive index change component of the formula of the refractive index ellipsoid of the crystal S ( (Dynamic phase) is expressed by the following equation.

式（１）より、結晶Ｓの屈折率回転楕円体は次式で与えられる。

From the formula (1), the refractive index spheroid of the crystal S is given by the following formula.

ただし、ｎ_ｏは、結晶Ｓの常光線屈折率であり、ｎ_ｅは、結晶Ｓの異常光線屈折率である。

However, n _o is the ordinary refractive index of the crystal S, n _e is the extraordinary ray refraction index of the crystal S.

Here, as shown in FIG. 3, when the propagating light along the X ₂ axis of the crystal S, the formula (2) is rewritten into the following equation.

ここで、

とおくと、式（３）の固有値方程式の解ｋ_１、ｋ_２は、

となる。電気光学係数は１００ｐｍ／Ｖ以下であるから、式（４）より、

となり、この結晶方位では電気光学効果は無視されるため、光デバイスや光計測への応用はあり得なかった。ここが盲点となって、これまでこの方位による電気光学係数の測定は行われていなかった。しかし、電気光学係数を測定することを目的とする場合には、電気光学係数が係数Ｃにだけ含まれるという条件のもとでは、電気光学効果による微弱な位相変化を検出することにより電気光学係数を求めることができる。式（３）は、この条件を満たしているため、電気光学係数ｒ_５１を求めることができる。

here,

Then, the solutions k ₁ and k ₂ of the eigenvalue equation of Equation (3) are

It becomes. Since the electro-optic coefficient is 100 pm / V or less, from equation (4),

Thus, since the electro-optic effect is ignored in this crystal orientation, it could not be applied to an optical device or optical measurement. This is a blind spot, and the electro-optic coefficient has not been measured in this direction so far. However, when the purpose is to measure the electro-optic coefficient, the electro-optic coefficient is detected by detecting a weak phase change due to the electro-optic effect under the condition that the electro-optic coefficient is included only in the coefficient C. Can be requested. Since Equation (3) satisfies this condition, the electro-optic coefficient r ₅₁ can be obtained.

Using equation (4) to (7), the refractive index _{n X1} of _{the X 1} axis direction of the crystal, the refractive index _{n X3} of X ₃ axis direction of the crystal is given by the following equation.

この結晶Ｓを図１の偏光子２０と検光子２２の間に配置したとき、結晶Ｓの主軸Ｘ_ｉ（主軸Ｘ_１）方向に振動する直線偏光と主軸Ｘ_ｊ（主軸Ｘ_３）方向に振動する直線偏光の間の時間tとともに変化する動的位相差をθ（ｔ）とし、時間tに依存しない静的位相差をφとすると、光検出器３０から出力される光検出信号Iは、次の式で与えられる。

When this crystal S is disposed between the polarizer 20 and the analyzer 22 in FIG. 1, the linearly polarized light that vibrates in the direction of the principal axis X _i (principal axis X ₁ ) of the crystal S and the vibration in the direction of the principal axis X _j (principal axis X ₃ ). Assuming that the dynamic phase difference between the linearly polarized light changing with time t is θ (t) and the static phase difference independent of time t is φ, the photodetection signal I output from the photodetector 30 is It is given by the following formula.

ただし、Ｉ_ｉは、入射光の強度に対応する光検出信号である。動的位相差θ（ｔ）と静的位相差φは、それぞれ次式で与えられる。

Here, I _i is a light detection signal corresponding to the intensity of incident light. The dynamic phase difference θ (t) and the static phase difference φ are given by the following equations, respectively.

ただし、Ｌは、光伝搬方向の結晶長（光源１０からの光が伝搬する結晶の長さ）であり、λは、光源１０の中心波長であり、Ｅｓｉｎ（２πｆｔ）は、結晶Ｓに印加される周波数ｆの交流電界である。

Where L is the crystal length in the light propagation direction (the length of the crystal through which light from the light source 10 propagates), λ is the center wavelength of the light source 10, and Esin (2πft) is applied to the crystal S. AC electric field of frequency f.

とすると、式（１０）は、次式に書き改められる。 Here, the amplitude χ of the dynamic phase difference θ (t) is

Then, the expression (10) is rewritten into the following expression.

χ＜＜１であるから、

とおけるため、式（１４）は、次式となる。

Since χ << 1,

Therefore, the equation (14) becomes the following equation.

ただし、Ｊ_ｈ（χ）は、ｈを次数としχを変数とする第１種ベッセル関数である。

However, J _h (χ) is a first type Bessel function in which h is an order and χ is a variable.

Here, when the light detection signal I is guided to the lock-in amplifier 42 and the h = 0 component, that is, the cos (4πft) component I ₁ is extracted, I ₁ is given by the following equation.

Ｊ_１（χ／２）成分が、変調周波数２πｆの２倍の周波数成分Ｉ_１から抽出されることは、本測定方法の特徴の１つであり、高感度測定の鍵となる技術である。

The extraction of the J ₁ (χ / 2) component from the frequency component I ₁ that is twice the modulation frequency 2πf is one of the features of this measurement method and is a key technique for high-sensitivity measurement.

Here, in the LiNbO ₃ crystal of X ₁ axis cut X ₂ axis propagation, χ / 2 << 1, so the effective value I _rms of the equation (17) is expressed by the following equation.

次に、式（１６）の直流成分Ｉ_０（ローパスフィルタ４０で抽出される直流成分）は、次式で与えられる。

Next, the direct current component I ₀ (the direct current component extracted by the low-pass filter 40) of Expression (16) is given by the following expression.

式（１８）、（１９）より、

となる。式（１３）と式（２０）から、次式を得る。

From equations (18) and (19),

It becomes. From the equations (13) and (20), the following equation is obtained.

式（２１）を整理すると、次式となる。

When formula (21) is arranged, the following formula is obtained.

式（２２）は、光検出信号の直流成分Ｉ_０と、印加電界の周波数の２倍の周波数成分Ｉ_ｒｍｓ及び静的位相差φを測定すれば、点群３ｍ結晶の電気光学係数ｒ_５１を、他の電気光学係数や圧電定数を含むことなく、単独で測定できることを表している。

Equation (22) is obtained by measuring the DC component I _{0 of the} photodetection signal, the frequency component I _rms twice the frequency of the applied electric field, and the static phase difference φ to obtain the electro-optic coefficient r ₅₁ of the point group 3m crystal. This means that it can be measured independently without including other electro-optic coefficients and piezoelectric constants.

The analysis method described above is not limited to the electro-optic coefficient r ₅₁ of the point group 3m crystal, but also the electro-optic coefficients r ₄₁ , r ₄₂ , r ₄₃ , r ₅₁ , r ₅₂ , r ₆₁ , of the crystal point group shown in FIG. It can be extended to the measurement of absolute values of r ₆₂ and r ₆₃ .

In order to measure each electro-optic coefficient indicated by a circle in the crystal point group shown in FIG. 2 in the same manner as the electro-optic coefficient r ₅₁ of the crystal of the point group 3m, the following three conditions must be satisfied: .

First, the 6-plane normal of the uniaxial crystal or biaxial crystal processed into a rectangular parallelepiped coincides with any one of the principal axes X ₁ , X ₂ , X ₃ of the refractive index ellipsoid of the crystal, and In the extremely simple configuration in which the direction of the AC electric field applied to the crystal coincides with any one of the main axes, the formula of the refractive index ellipsoid of the crystal in the presence of the applied AC electric field is given by the following formula (23): The first condition is that light having a narrow wavelength band is propagated in a direction coinciding with any one of the principal axes X ₁ , X ₂ , and X ₃ of the refractive index ellipsoid so as to be converted into the eigenvalue equation represented. is there.

ただし、結晶の主軸の添え字であるｉ及びｊは１〜３のいずれか１つの整数であり、且つ、ｉ≠ｊである。

However, i and j which are subscripts of the main axis of the crystal are integers of any one of 1 to 3, and i ≠ j.

  Even if the normals of the six surfaces do not completely coincide with the main axis, one or a plurality of electro-optic coefficients other than the object to be measured are mixed in the phase difference if the angle between them is 0.1 ° or less. However, it is possible to ignore the above and measure the electro-optic coefficient of the measurement object with high accuracy. Similarly, even if the direction of the AC electric field is not completely coincident with the main axis, it is possible to measure the electro-optic coefficient of the measurement object with high accuracy if the angle formed by the two is 0.1 ° or less. .

  Next, the second condition is that the coefficients A and B of the eigenvalue equation of Expression (23) are expressed by the refractive index of the crystal to be measured, and A ≠ B.

  Furthermore, the third condition is that the electro-optic coefficient to be measured and the applied electric field are included in the coefficient C of the eigenvalue equation of Expression (23).

2. Measurement Result The electro-optic coefficient r ₅₁ of the LiNbO ₃ crystal was measured using the measurement system of FIG. In this measurement, a He—Ne laser (wavelength λ = 632.8 nm, output = 7 mW) was used as the light source 10.

The non-doped congruent LiNbO ₃ crystal used in this measurement is shown in FIG. The crystal is processed into a rectangular parallelepiped, and the laser light input / output surface and the two surfaces on which the electrodes are deposited are optically polished. Normal input and output surfaces and the X ₂ axis, and the normal is X ₁ axis of two planes which electrodes are deposited, further normal of the remaining two faces are an X ₃ axis and processed in parallel respectively . As a result, the internal angle between each normal line and the corresponding crystal principal axis was 1/1000 radians or less.

P ₁ in FIG. 4 is the direction of light transmission axis of the polarizer 20, the direction P ₂ of the light transmission axis of the analyzer 22 is parallel to the P _1. That is, a parallel Nicol optical system is configured. Further, the main shaft X ₁ axis and X ₃ axis of the crystal is arranged at an angle of transmission axis 45 ° polarizer 20.

Prior to this measurement, a static phase difference φ was obtained. A method for measuring the static phase difference φ will be described with reference to FIG. First, as shown in FIG. 5A, the laser light from the light source 10 is directly incident on the photodetector 30, and the output I _a = I _i of the low-pass filter 40 is measured. However, the laser beam is linearly polarized, and the vibration direction of the electric field is made to coincide with the transmission axes (P ₁ and P ₂ directions in FIG. 4) of the polarizer 20 and the analyzer 22.

Next, as shown in FIG. 5B, the LiNbO ₃ crystal (crystal S) of FIG. 4 is inserted, and the output I _b of the low-pass filter 40 is measured. From the measurement results of I _a and I _b , the transmittance α of the LiNbO ₃ crystal is given by the following equation.

次に、図５（Ｃ）に示すように、ＬｉＮｂＯ_３結晶を取り除き、偏光子２０と検光子２２を挿入して、ローパスフィルタ４０の出力Ｉ_ｃを測定する。偏光子２０及び検光子２２の透過軸Ｐ_１、Ｐ_２は、図４に示すようにＬｉＮｂＯ_３結晶の主軸Ｘ_１、Ｘ_３と同一平面にあり、主軸Ｘ_３と４５°の角度をなす。また、上述したように、偏光子２０及び検光子２２の透過軸Ｐ_１、Ｐ_２とレーザ光の電界の振動方向は完全に一致している。図５（Ｃ）に示す構成におけるローパスフィルタ４０の出力Ｉ_ｃは、Ｉ_ｃ＝βＩ_ｉで表される。従って、偏光子２０と検光子２２の透過率βは、次式で与えられる。

Next, as shown in FIG. 5C, the LiNbO ₃ crystal is removed, the polarizer 20 and the analyzer 22 are inserted, and the output I _c of the low-pass filter 40 is measured. The transmission axes P ₁ and P ₂ of the polarizer 20 and the analyzer 22 are in the same plane as the main axes X ₁ and X ₃ of the LiNbO ₃ crystal as shown in FIG. 4, and form an angle of 45 ° with the main axis X ₃ . Further, as described above, the transmission axes P ₁ and P ₂ of the polarizer 20 and the analyzer 22 and the vibration direction of the electric field of the laser light are completely coincident. The output I _c of the low-pass filter 40 in the configuration shown in FIG. 5C is represented by I _c = βI _i . Accordingly, the transmittance β of the polarizer 20 and the analyzer 22 is given by the following equation.

最後に、図５（Ｄ）に示すように、ＬｉＮｂＯ_３結晶を挿入し、ＬｉＮｂＯ_３結晶に電界Ｅを加えた状態で、ローパスフィルタ４０の出力Ｉ_ｄと、ロックインアンプ４２の出力Ｉ_ｒｍｓを測定する。Ｉ_ｄ、Ｉ_ｒｍｓは、それぞれ次式で与えられる。

Finally, as shown in FIG. 5D, with the LiNbO ₃ crystal inserted and the electric field E applied to the LiNbO ₃ crystal, the output I _d of the low-pass filter 40 and the output I _rms of the lock-in amplifier 42 are taking measurement. I _d and I _rms are given by the following equations, respectively.

式（２４）〜（２６）より、次式を得る。

From the equations (24) to (26), the following equation is obtained.

式（２８）は、図４に示す構成のローパスフィルタ４０の出力Ｉ_ａ、Ｉ_ｂ、Ｉ_ｃ、Ｉ_ｄを
測定すれば、静的位相差φを定めることができることを示している。ただし、目的の電気光学係数ｒ_５１は、式（２７）の動的位相差χに含まれているため、図５（Ｄ）に示す構成のローパスフィルタ４０の出力Ｉ_ｒｍｓはできるだけ大きいことが望ましい。そこで、ＬｉＮｂＯ_３結晶をＸ_３軸に移動させて、Ｉ_ｒｍｓが最大になるように調整したあとで、ローパスフィルタ４０の出力Ｉ_ａ、Ｉ_ｂ、Ｉ_ｃ、Ｉ_ｄを測定することが望ましい。

Equation (28) indicates that the static phase difference φ can be determined by measuring the outputs I _a , I _b , I _c , and I _d of the low-pass filter 40 configured as shown in FIG. However, since the target electro-optic coefficient r ₅₁ is included in the dynamic phase difference χ in Expression (27), it is desirable that the output I _rms of the low-pass filter 40 configured as shown in FIG. . Therefore, it is desirable to measure the outputs I _a , I _b , I _c , and I _d of the low-pass filter 40 after moving the LiNbO ₃ crystal to the X ₃ axis and adjusting it so that I _rms becomes maximum.

A 1 kHz AC voltage (amplitude V _a = 150 to 895 V) is applied to the LiNbO ₃ crystal, _a 2 kHz component I _rms (effective value of a frequency component twice the modulation frequency) is extracted by the lock-in amplifier 42, and the low-pass filter 40 The DC component I _d (I ₀ ) was extracted with An example of the measurement result is shown in Table 1. Table 2 shows the static phase difference φ and the dynamic phase difference amplitude χ obtained from these values.

図６は、動的位相差の振幅χと、ＬｉＮｂＯ_３結晶に加えた交流電圧の振幅Ｖ_ａの関係を示す測定結果である。図６の黒塗り点は測定値を示す。また図６の測定値を結ぶ実線は最小二乗法から求めた実験式であり、次式で与えられる。

FIG. 6 is a measurement result showing _a relationship between the amplitude χ of the dynamic phase difference and the amplitude Va of the AC voltage applied to the LiNbO ₃ crystal. Black dots in FIG. 6 indicate measured values. The solid line connecting the measured values in FIG. 6 is an empirical formula obtained from the least square method and is given by the following formula.

図６及び式（２９）は、式（２１）を裏付けている。

FIG. 6 and equation (29) support equation (21).

と表される。以上の実験を１０回繰り返し、式（２９）を含む１０個の実験式を求めた。各実験式にＶ_ａ＝６００Ｖを代入して式（３０）により求めた電気光学係数ｒ_５１の実測値は、３２．０５±０．０１ｐｍ／Ｖであった。標準偏差値は十分小さく、精度の高い測定が行われたことを示している。 From equation (21), the electro-optic coefficient r ₅₁ is

It is expressed. The above experiment was repeated 10 times, and 10 empirical formulas including the formula (29) were obtained. The actual measurement value of the electro-optic coefficient r ₅₁ calculated by the equation (30) by substituting V _a = 600 V into each empirical equation was 32.05 ± 0.01 pm / V. The standard deviation value is sufficiently small, indicating that a highly accurate measurement was performed.

As described above, according to the measurement method of the present embodiment, a large number of electro-optic coefficients shown in FIG. 2 can be measured without coordinate conversion. In addition, very sensitive measurement is possible. In experiments demonstrating the present invention, a dynamic phase difference of 10 ⁻³ radians was measured as a change of 180 μV. The lock-in amplifier can detect even a nanovolt level, and even a small estimate can detect a dynamic phase difference that is two to three orders of magnitude smaller.

  Since the crystal sample used for measurement is a very simple rectangular parallelepiped, it can be easily processed and can be measured with high accuracy. When the normal line of the rectangular parallelepiped and one of the main axes of the crystal are substantially parallel, if the angle formed by the normal line and the main axis is 0.1 ° or less, the influence of other electro-optic coefficients is eliminated and the target electric The optical coefficient can be measured with high accuracy. In addition, since a parallel Nicol optical system (or a crossed Nicol optical system) is configured, very stable measurement is possible and excellent reproducibility is achieved.

  In addition, in order to measure the target electro-optic coefficient by extracting a frequency component twice the frequency of the alternating electric field applied to the crystal, other unnecessary electro-optic coefficients included in the amplitude of the frequency other than the double frequency The impact can be avoided. Further, since the dynamic phase difference is measured, the influence of the phase change due to the expansion and contraction of the crystal length due to the inverse piezoelectric effect can be ignored.

  The technical scope of the present invention is not limited to the above-described embodiment, and appropriate modifications can be made without departing from the spirit of the present invention.

10 light source, 20 polarizer (first polarizer), 22 analyzer (second polarizer), 30 photodetector, 40 low-pass filter, 42 lock-in amplifier, 44 signal generator, 46 amplifier, 50 arithmetic processing (Arithmetic processing unit), S crystal

Claims (9)

In an electro-optic coefficient measuring method for measuring an electro-optic coefficient of a crystal,
A crystal to which an alternating electric field is applied is arranged between first and second polarizers arranged so that their transmission axes are parallel or orthogonal to each other, and light from a light source having a narrow emission wavelength band is converted into the first polarized light. The light is incident on the crystal through a child, the light transmitted through the crystal and the second polarizer is detected and photoelectrically converted, and the electro-optic coefficient r _mn (r _mn of the crystal is based on the detected light detection signal) , R ₄₁ , r ₄₂ , r ₄₃ , r ₅₁ , r ₅₂ , r ₆₁ , r ₆₂ , r ₆₃ )
The crystal is a cuboid crystal composed of a uniaxial crystal or a biaxial crystal, and the normals of the six faces of the cuboid coincide with any one of the principal axes X ₁ , X ₂ , and X ₃ of the crystal, respectively. Or an internal angle between the normal and the main axis is 0.1 ° or less,
An electro-optic characterized in that a direction of an alternating electric field applied to the crystal coincides with any one of the main axes, or an internal angle between the direction of the alternating electric field and the main axes is 0.1 ° or less. Coefficient measurement method. In claim 1,
The light from the light source is directed in a direction coinciding with any one of the principal axes so that the formula of the refractive index ellipsoid of the crystal to which the alternating electric field is applied is converted into an eigenvalue equation represented by the following formula: An electro-optic coefficient measuring method characterized by propagating.

Where the coefficients A and B are expressed by the refractive index of the crystal, and A ≠ B, the coefficient C is expressed by the electro-optic coefficient r _mn and the electric field applied to the crystal, and i and j are 1 to Any one of 3 and i ≠ j. In claim 1 or 2,
Any one of the principal axes is parallel to the propagation direction of light from the light source, and the remaining two of the principal axes form an angle of 45 ° or −45 ° with the transmission axis of the first polarizer. An electro-optic coefficient measuring method. In any one of Claims 1 thru | or 3,
From the light detection signal, measuring a dynamic phase difference theta (t) that varies with time t between the linearly polarized light that oscillates in the spindle X _j direction of the linearly polarized light and the crystal oscillating the main axis X _i direction of the crystal To obtain an electro-optic coefficient r _mn of the crystal.
However, i and j are any one integer of 1 to 3, and i ≠ j. In claim 4,
The refractive index n _i of crystals linearly polarized light oscillating the main axis X _i direction of the crystal feel _the crystals of the main axis X _j direction linearly polarized light vibrating in feel refractive index n _j of the _crystal, said light source having a center wavelength λ When the length of the crystal through which the light from the light propagates is L and the alternating electric field of the frequency f applied to the crystal is Esin (2πft), the absolute value of the dynamic phase difference θ (t) is given by An electro-optic coefficient measuring method, characterized in that:

In any one of Claims 1 thru | or 5,
An effective value I _rms of a frequency component of DC components I ₀ and 2f (f is the frequency of the AC electric field) is extracted from the photodetection signal, and based on the extracted DC component I ₀ and the effective value I _rms. An electro-optic coefficient measurement method characterized by obtaining an electro-optic coefficient r _mn of the crystal. In claim 6,
The refractive index n _i of crystals linearly polarized light oscillating the main axis X _i direction of the crystal feel _the crystals of the main axis X _j direction linearly polarized light vibrating in feel refractive index n _j of the _crystal, said light source having a center wavelength λ A method of measuring an electro-optic coefficient, wherein the length of a crystal through which light from the light propagates is L, and the electro-optic coefficient r _mn of the crystal is obtained based on the following equation.

However, phi is a static phase difference is not dependent on the time t between the linearly polarized light oscillating the main axis X _j direction of the crystal and linearly polarized light that vibrates the main axis X _i direction of the crystal, i and j are 1 Any one of ˜3 and i ≠ j. In claim 7,
The electro-optic coefficient measuring method, wherein the static phase difference φ is adjusted so that the effective value I _rms of the frequency component of 2f is maximized. In an electro-optic coefficient measuring device for measuring the electro-optic coefficient of a crystal,
First and second polarizers arranged such that their transmission axes are parallel or orthogonal;
A light irradiator that causes light from a light source having a narrow emission wavelength band to be incident on a crystal to which an alternating electric field has been applied via the first polarizer;
A light detection unit that detects and photoelectrically converts light transmitted through the crystal and the second polarizer;
An operation for calculating an electro-optic coefficient r _{mn of the} crystal based on a light detection signal from the light detection unit (m is an integer of any one of 4 to 6, and n is an integer of any of 1 to 3). An arithmetic processing unit that performs processing,
The crystal is a cuboid crystal composed of a uniaxial crystal or a biaxial crystal, and the normals of the six faces of the cuboid coincide with any one of the principal axes X ₁ , X ₂ , and X ₃ of the crystal, respectively. Or an internal angle between the normal and the main axis is 0.1 ° or less,
An electro-optic characterized in that a direction of an alternating electric field applied to the crystal coincides with any one of the main axes, or an internal angle between the direction of the alternating electric field and the main axes is 0.1 ° or less. Coefficient measurement device.

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