JP2005331440A - Optical phase distribution measurement method and system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide an optical phase distribution measurement method and an optical phase distribution measuring system for measuring optical phase distribution by identifying the optical phase distribution from information on optical intensity distribution which is easily measured, without using special measurement apparatuses. <P>SOLUTION: The optical phase distribution measurement method uses the optical phase distribution measurement system, comprising an optical system and an optical wave detection sensor having a plurality of different known optical characteristics. The method comprises an intensity distribution measurement step which measures the intensity distribution of light to be measured; an observation equation setting step which sets an observation equation of the optical phase distribution measurement system, on the basis of the intensity distribution obtained in the intensity distribution measurement step and the optical characteristics of the optical system; a phase distribution identification inverse problem setting step which sets a phase distribution identification inverse problem which identifies the phase distribution of the light to be measured from the observation equation; an optimization problem formulation step which formulates the set phase distribution identification inverse problem as a nonlinear optimization problem; and a phase distribution identification step which identifies the phase distribution of the light to be measured, by solving the optimization problem. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to an optical phase distribution measurement method and measurement system, and more particularly, to an optical phase distribution measurement method and measurement system that enable an optical phase distribution to be measured easily and economically.

  The optical phase distribution (hereinafter also referred to as light phase distribution) includes information such as the wavefront shape of the light source, the refractive index distribution of the transmissive object, and the surface shape of the reflective object. Optical phase distribution measurement is used for optical system design evaluation, quality control of optical elements such as semiconductor laser elements and lenses, non-contact nondestructive inspection, surface shape measurement of microscopes, etc., light wavefront shaping and astronomical observation It is required in many fields such as wavefront adaptive optics. The quantitative measurement of the optical phase distribution is very important in engineering (see Non-Patent Document 1, Non-Patent Document 2, and Non-Patent Document 3).

  Conventionally, as a method for measuring optical phase distribution using hardware (that is, a special measuring device), for example, a method for obtaining a relative phase distribution between measured light and reference light using an interferometer (hereinafter referred to as interference) Optical phase distribution measurement method using a meter (see Non-Patent Document 4 and Non-Patent Document 5) and a method of detecting a phase distribution by detecting a phase gradient distribution using a Shack-Hartman sensor (hereinafter, optical phase distribution using a Shack-Hartman sensor) Measurement method) (see Non-Patent Document 6 and Non-Patent Document 7).

  The conventional optical phase distribution measuring method will be described in detail as follows.

  First, the optical phase distribution measurement method using a conventional interferometer is to divide the light to be measured into two or more lights, overlap them after passing through different optical paths, and capture the interference fringes generated by the optical path difference. To obtain the phase distribution of light.

  That is, as shown in FIG. 53, in the conventional optical phase distribution measurement method using an interferometer, the light to be measured is divided into two, and how much the other wavefront differs from one wavefront (phase distribution). Check out. The reference light is called reference light. The reference light is extracted as uniform wavefront light by the reference lens and the spatial filter, and interferes with the light to be measured. If the optical path difference between the reference light and the light to be measured is an integral multiple of the wavelength, the interfered light becomes bright, and if the optical path difference is shifted from the integral multiple of the wavelength by half the wavelength, the interference light becomes dark. When the optical path difference is not constant depending on the location, bright and dark interference fringes are observed.

  In order to obtain the phase distribution from the interference fringe image, for example, it is necessary to analyze a plurality of interference fringe images having different optical path lengths as shown in FIG. A typical analysis method is a fringe scan method disclosed in Non-Patent Document 4, which scans a sinusoidal intensity distribution of interference fringes generated by interference and obtains a phase from a change in intensity.

  In the conventional optical phase distribution measuring method using the Shack-Hartmann sensor, a Shack-Hartmann wavefront sensor (SHWS) is composed of a large number of small lenses and a CCD. As shown in FIG. 55, the position where the light entering the small lens is focused is shifted by Δs on the focal plane due to the local phase gradient of the light. This shift is detected in all small lenses. The image formation pattern on the focal plane as shown in FIG. 56 is called a Hartmann pattern and is detected by the CCD.

As disclosed in Non-Patent Document 8, Δs and the phase gradient are in a proportional relationship. From this, it is possible to know local phase gradient information, and to obtain the total phase distribution by combining them. it can.
Kay. A. Nugen (KANugent), Day. Paganini, tea. E. Co-authored by TEGureyev, “A Phase Odyssey”, Physics Today, Vol. 54, No. 8, 2001, p27-32. El. A. LAThompson, “Adaptive Optics in Astronomy”, Physics Today, 47, 12, 1994, p24-31 Takashi Miyoshi, “Latest Trends in Nano-In-process Measurement Technology for Optical Applications”, Research on Machines, Vol. 54, No. 9, 2002, p928-933 "Optometer Guide", Optronics, 1998 Motohiro Takada, "Wavefront aberration measurement of laser light with an interferometer", Optical Alliance, 2nd edition, 2001, p19-22 W. H. WHSouthwell, “Wave front estimation from wavefront slope measurements”, J. Opt. Soc. Am., No. 70, No. 8, 1980, p998-1005 R. R.Irwan, Earl. Gee. Co-authored by RGLane, “Analysis of optimal centroid estimation applied to Shack-Hartmann sensing”, Applied Optics, 38, 32, 1999, p6737-6743 ILIT RAS, “Shack-Hartmann Wavefront Sensor for Laser Beam Analysis”, Internet <http://www.laser.ru/adopt/Science/sens/sensor .htm> Yadakai Toyohiko, "Light and Fourier Transform", Asakura Shoten, 1992 Takemura Yoshimura, "Basics of Optical Information Engineering", Corona, 2000 Yabe and Hachiman, "Nonlinear Programming", Asakura Shoten, 1999

  However, a special measuring device (that is, hardware such as an interferometer and a Shack-Hartman sensor) used in the conventional optical phase distribution measuring method described above is very expensive because it is composed of precision optical equipment. In addition, there is a problem that a space for installing such a special measuring device is required.

  The present invention has been made under the circumstances as described above, and an object of the present invention is to identify an optical phase distribution from information of a light intensity distribution that can be easily measured without using a special measuring apparatus. An object of the present invention is to provide an optical phase distribution measuring method and a measuring system capable of measuring an optical phase distribution.

  The present invention relates to an optical phase distribution measurement system, and the object of the present invention is to provide an optical system having a plurality of different optical characteristics and a light wave detection sensor, and to input light to be measured to each of the optical systems. And the output of the light to be measured is detected by the light wave detection sensor, and the intensity distribution of the detected light to be measured is measured as an image, or the light wave detection sensor is a CCD image sensor. Or the plurality of optical systems can be effectively achieved by comprising a diffractive system and a lens focusing system.

  The present invention also relates to an optical phase distribution measuring method, and the above object of the present invention is an optical phase distribution measuring method for measuring the phase distribution of light to be measured, using the optical phase distribution measuring system of the present invention. An observation equation of the optical phase distribution measurement system based on the intensity distribution obtained in the intensity distribution measurement step and the optical characteristics of each optical system, An observation equation setting step for setting the phase distribution identification inverse problem setting step for identifying the phase distribution identification inverse problem for identifying the phase distribution of the light to be measured from the observation equation, and the set phase distribution identification inverse problem nonlinearly An optimization problem formulation step that is formulated as an optimization problem of the above, and a phase distribution identification step that identifies the phase distribution of the measured light by solving the optimization problem Or, in the optimization problem, the design variable is a complex vector representing the light to be measured, and the objective function includes an optimization function set based on a priori information on the phase distribution of the light to be measured. Or the a priori information is information that the phase of the light under measurement will be distributed smoothly, or the objective function of the optimization problem is minimized. By using the Newton method, or representing a predetermined point adjacent to the design variable of the optimization problem as a second design variable, a small first size expressed by the second design variable is used. It is effectively achieved by using the estimated solution of the optimization problem of 2 as the initial estimated solution of the design variable of the optimization problem and solving the optimization problem.

  According to the optical phase distribution measurement method and the optical phase distribution measurement system according to the present invention, it is simple and economical without using an expensive optical instrument such as a special measurement device used in the conventional optical phase distribution measurement method. The optical phase distribution can be stably estimated from the information of the measurable light intensity distribution, and it has an excellent effect that the optical phase distribution can be measured easily and economically.

The best mode for carrying out the present invention will be described below with reference to the drawings.
<1> Mathematical Model First, a mathematical model of a light wave or optical phase distribution measurement system used in the present invention will be described.

<1-1> Mathematical Model of Light Wave In the present invention, monochromatic light having a single wavelength component, for example, light to be measured for phase distribution (that is, light to be measured) is typified by laser light, for example. In addition, it is assumed that the phases at different times are always temporally coherent light having a constant correlation.

<1-1-1> Complex number display of light wave Light is a kind of electromagnetic wave, and is a wave that propagates while an electric field and a magnetic field vibrate. Maxwell's equations relating the electric field E, the magnetic field H, the dielectric constant ε, and the magnetic permeability μ are composed of the following four equations such as Equation 1, Equation 2, Equation 3, and Equation 4.

これらの式より、以下のような波動方程式を得る。

From these equations, the following wave equation is obtained.

電界Ｅに関する波動方程式である上記数５の解の一つは、下記数７のように表される。

One of the solutions of Equation 5 above, which is the wave equation for the electric field E, is expressed as Equation 7 below.

ここで、ａは振幅で、ｅ_ｐは電界の振動方向を表す単位偏光ベクトルで、ωは角周波数で、ｔは時刻で、ｋは波数ベクトルで、ｒは位置ベクトルで、φは初期位相である。また、ｉは虚数を表す。なお、磁場Ｈについても同様であるので以下省略する。

Here, a is the amplitude, a unit polarization vector representing the vibration direction of e _p is the electric field, omega is the angular frequency, t is time, k is the wave vector, r is at position vector, phi in the initial phase is there. I represents an imaginary number. Since the same applies to the magnetic field H, the description is omitted below.

  Here, the components that do not depend on the phase time are collectively put together as the following formula 8, and the time oscillation term is expressed separately as shown in the following formula 9.

本発明では、光学における計算において、このように光波の時間振動項を分離し、複素数ｖで光波の強度ａと位相θを表現する。複素数ｖは複素振幅と呼ばれ、下記数１０のようになる。

In the present invention, in the calculation in optics, the time oscillation term of the light wave is thus separated, and the intensity a and the phase θ of the light wave are expressed by a complex number v. The complex number v is called a complex amplitude and is expressed by the following formula 10.

また、複素振幅ｖを実数部ｖ_ｒと虚数部ｖ_ｉで分離して表現した場合に、下記数１１、数１２、数１３のようになる。

Also, when expressed by separating the complex amplitude v with real numbers v _r and the imaginary part v _i, the following equation 11, equation 12 becomes as Equation 13.

本発明では、上述したように、光を複素数で表現するものとすることを前提とする。

＜１−１−２＞光波の離散表示
後述するように、本発明では、光位相分布測定系において、被測定光を検出するための光波検出センサーとして、ＣＣＤ撮像素子（以下、単にＣＣＤとも呼ばれる）を用いることが好ましい。光波検出センサー（本実施形態では、ＣＣＤ）により、検出されて画像として測定されるのは、被測定光の強度、すなわち、上述した複素振幅の絶対値である。測定された強度の空間分布は、二次元の離散配列で与えられるが、本発明では、光波を表す複素振幅の分布を取扱う場合に、図１に示されるような複素数の二次元配列を離散的に一次元に並べ換え、下記数１４で表される複素数ベクトルｖで表現する。

In the present invention, as described above, it is assumed that light is expressed by complex numbers.

<1-1-2> Discrete Display of Light Wave As will be described later, in the present invention, in the optical phase distribution measurement system, as a light wave detection sensor for detecting light to be measured, a CCD imaging device (hereinafter also simply referred to as a CCD). ) Is preferably used. What is detected and measured as an image by the light wave detection sensor (CCD in this embodiment) is the intensity of the light to be measured, that is, the absolute value of the complex amplitude described above. The spatial distribution of the measured intensity is given by a two-dimensional discrete array. In the present invention, when dealing with a distribution of complex amplitudes representing light waves, a complex two-dimensional array as shown in FIG. These are rearranged in a one-dimensional manner and expressed by a complex vector v represented by the following equation (14).

ここで、光位相分布測定系で測定されるのは、複素数ベクトルのそれぞれの成分の絶対値である。以降、複素数ベクトルｖのそれぞれ成分の絶対値を並べた実数ベクトルを下記数１５のように表記するものとする。

Here, what is measured by the optical phase distribution measurement system is the absolute value of each component of the complex vector. Hereinafter, a real vector in which the absolute values of the respective components of the complex vector v are arranged will be expressed as the following formula 15.

＜１−２＞光位相分布測定系（以下、単に測定系とも称する）の数理モデル
＜１−２−１＞光位相分布測定系
光位相分布測定系は、光学特性が既知である複数（少なくとも２つ以上）の異なる光学系と光波検出センサーとから構成される。光位相分布測定系の光波検出センサーとしては、ＣＣＤ撮像素子を用いるのが好ましいが、それに限定されることなく、他のセンサーを用いても良い。なお、本実施形態において、光波検出センサーとしては、ＣＣＤを用いる。

<1-2> Mathematical model of optical phase distribution measurement system (hereinafter also simply referred to as measurement system) <1-2-1> Optical phase distribution measurement system The optical phase distribution measurement system includes a plurality of (at least 2 or more) different optical systems and light wave detection sensors. As the light wave detection sensor of the optical phase distribution measurement system, it is preferable to use a CCD imaging device, but the present invention is not limited to this, and other sensors may be used. In this embodiment, a CCD is used as the light wave detection sensor.

  In the present invention, using such an optical phase distribution measurement system, an intensity distribution image of the light to be measured is measured by a light wave detection sensor (CCD), and an intensity distribution image of the obtained light to be measured and an optical system Based on information such as optical characteristics, the phase distribution of the light to be measured can be measured by identifying the phase distribution of the light to be measured using an optical phase distribution measurement method.

  As described above, in the present invention, monochromatic light having a single wavelength component is used as an optical phase distribution measurement target (that is, measured light). For example, a wavelength other than the wavelength component focused on the measured light is selected. In the case where a light wave is included, the present invention can be applied by placing a filter in front of the light wave detection sensor (CCD) and extracting only the wavelength component of interest.

  Hereinafter, the measurement principle and mathematical model of the optical phase distribution measurement system will be described with reference to FIG. As shown in FIG. 2, the optical phase distribution measuring system is composed of k different optical systems whose optical characteristics are known and a light wave detection sensor (CCD).

  In the optical phase distribution measurement system, first, light to be measured, that is, light whose phase distribution is to be measured is input to an optical system with known optical characteristics, and the intensity and phase are modulated. Then, the output light wave is detected by a light wave detection sensor (CCD), and the intensity distribution of the detection light is measured as an image. These procedures are performed on the same light to be measured using a plurality of optical systems (here, k optical systems, that is, optical system 1, optical system 2,..., Optical system k).

  When the measurement principle of the optical phase distribution measurement system is mathematically expressed, it indicates that the complex vector g representing the light to be measured is converted by the conversion matrix H by the optical system to become a complex vector f representing the detection light. That is, it can be expressed as the following formula 16.

ここで、光波検出センサー（ＣＣＤ）により検出されるのは、検出光を表す複素数ベクトルｆのそれぞれの成分の絶対値を並べた実数ベクトルである。

Here, what is detected by the light wave detection sensor (CCD) is a real vector in which the absolute values of the respective components of the complex vector f representing the detected light are arranged.

  In the optical phase distribution measurement method of the present invention, the absolute value vector | f | of the detection light is used as an observation amount, and the complex number vector g representing the measurement light is identified, thereby enabling measurement of the phase distribution of the measurement light. Yes.

In the optical phase distribution measurement system of the present invention, the type and number of optical systems used are arbitrary. By applying an appropriate optical system, it is possible to overcome the ill-conditioned property of the observation equation of the optical phase distribution measurement system and obtain the only solution.

<1-2-2> Mathematical Model of Optical Wave Conversion by Optical System Any optical system that can be applied to the optical phase distribution measurement system of the present invention may be used as long as the optical characteristics are known. Examples of the optical system that can be easily used include an optical system such as a lens, a lens that is out of focus, a lens having a different pupil function, a diffraction grating, and diffraction in the air (that is, direct incidence without using an optical system).

Below, the example of three optical systems is given and the mathematical model of the conversion of the light wave by an optical system is demonstrated. For details, refer to Non-Patent Document 9, Non-Patent Document 10, and the like.

<A> Mathematical Model of Light Wave Conversion by Diffraction System As shown in FIG. 3, the complex amplitude distribution g (x, y) of the light wave is localized in the xy plane and propagated by the distance R in the z-axis direction. The complex amplitude distribution f (X, Y) in the later XY plane is assumed. At this time, considering the case where the distance R is sufficiently larger than the wavelength and the diffracted wave is present in the vicinity of the z-axis, a Fresnel diffraction expression representing the relationship before and after propagation of the light wave is obtained as shown in Equation 17 below.

ここで、ｋは波数であり、波長をλとすると、ｋ＝２π／λとなる。下記数１８とすると、下記数１９で表すたたみ込み積分の定義より、上記数１７は、下記数２０のようにたたみ込み積分で表すことができる。

Here, k is the wave number, and k = 2π / λ where λ is the wavelength. Assuming the following equation 18, the above equation 17 can be expressed by a convolution integral as shown in the following equation 20 from the definition of the convolution integral represented by the following equation 19.

上記数２０を離散化し、行列、ベクトル表示で書くと、下記数２１のようになる。

When the above equation 20 is discretized and written in matrix or vector display, the following equation 21 is obtained.

ここで、簡単のために、１次元で考え、具体的に書き出してみる。上記数２１において、下記数２２、数２３とすると、フレネル回折の変換マトリックスは、下記数２４のようになる。

Here, for the sake of simplicity, consider one-dimensionally and write it out specifically. In the above formula 21, if the following formula 22 and formula 23 are used, the conversion matrix of Fresnel diffraction becomes the following formula 24.

ここで、下記数２５が成立する。

Here, the following formula 25 holds.

ｈ(Ｘ,Ｙ)は、点光源入力に対する光学システムのインパルス応答を表し、点像応答関数（ＰＳＦ:point spread function）と呼ばれる。ＰＳＦは入力関数に依存せずシステム固有の性質を示す。光波の回折現象も上記数１８のＰＳＦを持つ光学システムと見なせる。この他の光学システムにおいても、ＰＳＦが前もって分かれば、入力ｇに対するシステムの出力ｆを求めることができる。

＜Ｂ＞結像レンズ系による光波の変換の数理モデル
図４に示すように、ｘｙ面内に光波の複素振幅分布ｇ(ｘ,ｙ)が局在しており、距離ｌだけ伝搬後、レンズを透過し、再び距離ｒだけ伝搬した後のＸＹ面内の複素振幅分布をｆ(Ｘ,Ｙ)とする。

h (X, Y) represents an impulse response of the optical system with respect to a point light source input, and is called a point spread function (PSF). The PSF does not depend on the input function and exhibits system-specific properties. The light wave diffraction phenomenon can also be regarded as an optical system having the PSF of the above formula (18). Also in this other optical system, if the PSF is known in advance, the output f of the system with respect to the input g can be obtained.

<B> Mathematical Model of Light Wave Conversion by Imaging Lens System As shown in FIG. 4, the complex amplitude distribution g (x, y) of the light wave is localized in the xy plane, and after propagating by the distance l, the lens Let f (X, Y) be the complex amplitude distribution in the XY plane after passing through and propagating again by the distance r.

The light wave g propagates by a distance l until just before the lens and is Fresnel diffracted. The light wave u ⁻ (x, y) immediately before the lens is represented by the following Expression 26.

この光波が焦点距離ｆのレンズによる位相変調および瞳関数ｐ(ｘ,ｙ)による振幅変調を受けると、レンズ直後の光波ｕ^＋(ｘ,ｙ)は、下記数２７のようになる。

When this light wave is subjected to phase modulation by the lens having the focal length f and amplitude modulation by the pupil function p (x, y), the light wave u ⁺ (x, y) immediately after the lens is expressed by the following equation (27).

ここで、下記数２８で表す瞳関数は、レンズの開口を表す関数である。

Here, the pupil function represented by the following formula 28 is a function representing the aperture of the lens.

この光波が観測面まで、距離ｒだけ伝搬し、再びフレネル回折する。観測される出力光波ｆ(ｘ,ｙ)は、下記数２９のようになる。

This light wave propagates to the observation surface by a distance r and again undergoes Fresnel diffraction. The observed output light wave f (x, y) is as shown in Equation 29 below.

以上の三つの関係から、観測面での光波を求めることができる。観測位置は結像条件を満足し、距離ｒ,ｌおよび焦点距離ｆについてレンズの公式、結像倍率Ｍ、瞳関数のフーリエ変換形をそれぞれ下記数３０、数３１、数３２とする。

From the above three relations, the light wave on the observation surface can be obtained. The observation position satisfies the imaging conditions, and the lens formula, imaging magnification M, and Fourier transform form of the pupil function for the distances r, l and the focal length f are the following equations 30, 31, and 32, respectively.

これらの関係式を用いて整理し、定数項を無視すれば、入力光と出力光の関係は、下記数３３のようなたたみ込み積分で書ける。詳細は非特許文献１０を参照する。

If these relational expressions are used for arrangement and the constant term is ignored, the relationship between the input light and the output light can be written by the convolution integral as shown in the following Expression 33. Refer to Non-Patent Document 10 for details.

ここで、レンズの直径をＤとし、１次元で扱うと、関数Ｐは矩形関数のフーリエ変換で表される。すなわち、図５に示されるような結像光学系の点像応答関数ＰＳＦは、下記数３４で表すことが分かる。

Here, assuming that the diameter of the lens is D and handled in one dimension, the function P is represented by Fourier transform of a rectangular function. That is, it can be seen that the point image response function PSF of the imaging optical system as shown in FIG.

具体的な変換マトリックスの形は、下記数３５とすれば、数２４と同様に結像光学系の変換マトリックスを書くことができる。

If the shape of the specific transformation matrix is the following Equation 35, the transformation matrix of the imaging optical system can be written in the same manner as Equation 24.

上記数３５式は、Ｘ＝ｘ近傍で値を持つので、結像光学系の変換行列は、帯行列になることがわかる。

Since Equation 35 has a value in the vicinity of X = x, it can be seen that the conversion matrix of the imaging optical system is a band matrix.

When the lens diameter D → ∞, the PSF is h (x) = δ (x), and a complete image with no spread of the point image and no blur is output to the observation surface. That is, the conversion matrix is a unit matrix.

<C> Mathematical Model of Light Wave Conversion by Diffraction Grating As shown in FIG. 6, the complex amplitude distribution g (x, y) of the light wave is localized in the xy plane, and after propagating by the distance l, Let f (X, Y) be the complex amplitude distribution in the XY plane after passing through and propagating again by the distance r.

The input light wave g propagates by a distance l until just before the diffraction grating and is Fresnel diffracted. The light wave u ⁻ (x, y) immediately before the diffraction grating is represented by the following Expression 36.

次に、回折格子の複素透過率をｔ(ｘ,ｙ)とすると、回折格子を透過した直後の光波ｕ^＋(ｘ,ｙ)は、下記数３７のようになる。

Next, assuming that the complex transmittance of the diffraction grating is t (x, y), the light wave u ⁺ (x, y) immediately after passing through the diffraction grating is expressed by the following Expression 37.

この光波が観測面まで、距離ｒだけ伝搬し、再びフレネル回折する。観測される出力光波ｆ(ｘ,ｙ)は、下記数３８のようになる。

This light wave propagates to the observation surface by a distance r and again undergoes Fresnel diffraction. The observed output light wave f (x, y) is as shown in Equation 38 below.

以上の処理をまとめると、下記数３９のようになる。

Summarizing the above processing, the following equation 39 is obtained.

上記数３９を離散化し、行列ベクトル表示で書くと、下記数４０、数４１のようになる。

When the above equation 39 is discretized and written in matrix vector display, the following equations 40 and 41 are obtained.

ここで、１次元で考え、具体的に書き出してみる。行列[ｈ_ｒ],[ｈ_ｌ]は、数２４と同様である。また、行列[ｔ]は、対角行列となり、下記数４２のようになる。

Here, think in one dimension and write it out specifically. The matrices [h _r ] and [h _l ] are the same as in Expression 24. Further, the matrix [t] is a diagonal matrix and is represented by the following formula 42.

＜２＞光位相分布測定方法（以下、位相分布同定手法とも呼ばれる）
以下では、位相分布同定逆問題を設定し、具体的な位相分布同定手法について説明する。

＜２−１＞位相分布同定逆問題の設定
＜２−１−１＞光位相分布測定系の観測方程式
入力光（つまり、被測定光）ｇをｎ個に離散化し、ｇ∈Ｃ^ｎ、異なるｋ個の光学系を通過後の光をそれぞれｍ個に離散化しｆ_１,ｆ_２,…,ｆ_ｋ∈Ｃ^ｍ、これらの光学系による変換マトリックスをそれぞれ[Ｈ_１],[Ｈ_２],…,[Ｈ_ｋ]∈Ｃ^ｍ×ｎとする。また、光波検出センサーであるＣＣＤにより測定された強度分布を

とすると、光位相分布測定系の観測方程式は、下記数４３、数４４のように書くことができる。

<2> Optical phase distribution measurement method (hereinafter also referred to as phase distribution identification method)
In the following, a phase distribution identification inverse problem is set and a specific phase distribution identification method will be described.

<2-1> Setting of phase distribution identification inverse problem <2-1-1> Observation equation of optical phase distribution measurement system Discretizes input light (that is, light to be measured) g into n pieces, and g∈C ^{n is} different. The light after passing through the k optical systems is discretized into m pieces, respectively, f ₁ , f ₂ ,..., f _k ∈ C ^m , and conversion matrices obtained by these optical systems are [H ₁ ], [H ₂ ], ..., [H _k ] εC ^{m × n} . In addition, the intensity distribution measured by the light wave detection sensor CCD

Then, the observation equation of the optical phase distribution measurement system can be written as the following equations 43 and 44.

上記数４３は、光学系による被測定光ｇから検出光ｆへの変換式であり、上記数４４は、複素数ベクトルｆの絶対値ベクトル|ｆ|が測定された強度

として与えられる、つまり、測定量であることを表す。ここで、既知量は、光学系の光学特性を表す複素数マトリックス[Ｈ]、および測定強度を表す絶対値ベクトル

であり、未知量は、被測定光を表す複素数ベクトルｇと検出光を表す複素数ベクトルｆである。

The above equation (43) is a conversion formula from the light to be measured g to the detection light (f) by the optical system, and the above equation (44) is the intensity at which the absolute value vector | f |

It is given as, that is, it represents a measured quantity. Here, the known amount is a complex number matrix [H] representing the optical characteristics of the optical system, and an absolute value vector representing the measured intensity.

The unknown amounts are a complex vector g representing the measured light and a complex vector f representing the detected light.

を観測量とし、光位相分布測定系の観測方程式を満足する被測定光の複素数ベクトルｇを同定する位相分布同定逆問題を解くこととなる。

＜２−１−２＞最適化問題
上記数４３と数４４で表す、光位相分布測定系の観測方程式を満たす入力光（つまり、被測定光）ｇを求めるのが、本発明の光位相分布測定方法の目的となるが、これらの方程式系は、非線形方程式のため、直接解を求めるのは困難である。 The optical phase distribution measuring method of the present invention aims to obtain the phase distribution of the light to be measured, so that the complex amplitude of the light to be measured is identified. That is, multiple measured intensities

And the phase distribution identification inverse problem for identifying the complex vector g of the light to be measured that satisfies the observation equation of the optical phase distribution measurement system.

<2-1-2> Optimization Problem The optical phase distribution of the present invention is to obtain the input light (that is, the light to be measured) g that satisfies the observation equation of the optical phase distribution measurement system expressed by the above equations 43 and 44. Although it is the purpose of the measurement method, these equation systems are nonlinear equations, and it is difficult to obtain a direct solution.

  Therefore, in the optical phase distribution measurement method of the present invention, the inverse phase distribution identification problem for obtaining the input light (that is, the light to be measured) g is formulated as a nonlinear optimization problem (hereinafter also simply referred to as an optimization problem). . In the formulated optimization problem, the design variable is a complex vector g representing the light to be measured, and a nonlinear objective function (that is, an evaluation function) as shown in the following equation 45 is set.

この位相分布同定逆問題では、上記数４５を０とする、つまり、最小とする複素数ベクトルｇが推定解となる。

In this phase distribution identification inverse problem, the complex vector g having the above-described formula 45 of 0, that is, the minimum is the estimated solution.

  However, the observation equation of the optical phase distribution measurement system does not strictly hold due to an error in the measurement value of the optical phase distribution measurement system. This can make the phase distribution identification inverse problem ill-conditioned. Therefore, in order to overcome the ill-conditioning property of the phase distribution identification inverse problem, the a priori information on the measured light is used to optimize the phase distribution identification inverse problem. That is, the following formula 46 obtained by adding an appropriate term to the above formula 45 is used as the objective function.

ここで、ｗは適切化の重み係数で、Λ(ｇ)は適切化のために新たに加えられた関数である。

＜２−２＞最適化の手法
この最適化問題の数値解法および、具体的な先験情報の利用法については、以下のように説明する。
＜２−２−１＞準ニュートン法
本実施形態では、光位相分布測定方法において、上記数４６で表す目的関数の最小化をするにあたり、最適化問題に対する数値解法の中で、非特許文献１１に開示されている『準ニュートン法』を用いる。

Here, w is a weighting factor for optimization, and Λ (g) is a function newly added for optimization.

<2-2> Optimization Method The numerical solution of this optimization problem and the specific usage of a priori information will be described as follows.
<2-2-1> Quasi-Newton Method In the present embodiment, in the optical phase distribution measurement method, in minimizing the objective function expressed by the above equation 46, among the numerical solutions for the optimization problem, Non-Patent Document 11 The “Quasi-Newton method” disclosed in the above is used.

  First, the quasi-Newton method will be briefly described as follows.

Here, it is assumed that there is an optimization problem of finding xεR ⁿ that minimizes the objective function f (x) of n variables.

In Newton's method, an objective function is approximated by a quadratic model, and the model function is locally minimized. Assuming that the Hessian matrix ∇ ² f (x), the gradient vector ∇f (x), and the search direction d, the model function is expressed by the following equation 47.

ヘッセ行列が正定値対称行列ならば、ｑ(ｄ)を最小にする点は、∇ｑ(ｄ)＝０を満たすベクトルｄを求めればよい。つまり、下記数４８で表す連立１次方程式を解けばよいことになる。

If the Hessian matrix is a positive definite symmetric matrix, the point that minimizes q (d) may be a vector d satisfying ∇q (d) = 0. That is, it is only necessary to solve the simultaneous linear equations expressed by the following equation (48).

上記数４８で表すこの方程式をニュートン方程式といい、解ｄをニュートン方向という。

This equation expressed by the above equation 48 is called a Newton equation, and the solution d is called a Newton direction.

However, there is no guarantee that the Hessian matrix is positive definite at any time, and the Newton direction is not always the descent direction of f (x). Therefore, the quasi-Newton method is to approximate the Hessian matrix with an appropriate positive definite symmetric matrix B _k and determine the search direction.

  The quasi-Newton method has an advantage that it has super-first-order fast convergence and does not require the calculation of the second derivative.

Next, the algorithm of the quasi-Newton method is shown as follows.
Step 1 Initial Setting An initial point x _{0 of} an approximate solution and a positive definite symmetrical initial matrix B ₀ are given, and k = 0 is set.
Step 2 solves the simultaneous linear equations represented by the decision following Expression 49 search direction d _k, determine the search direction d _k.

ステップ３ 直線探索
ｄ_ｋ方向でのステップ幅α_ｋを決定する
ステップ４ 近似解の更新
下記数５０のように、近似解を更新する。

Step 3 Line search d Step width α _k in the _k direction is determined Step 4 Approximate solution update The approximate solution is updated as shown in Equation 50 below.

ステップ５ 収束判定
収束条件が満たされていれば、ｘ_ｋ＋１を解とみなし、終了する。さもなくば、ステップ６へ行く。
ステップ６ Ｂ_ｋの更新
更新公式によりＢ_ｋ＋１を計算する。
ステップ７ ｋ＋１をｋに代入して（つまり、ｋ：＝ｋ＋１とし）、ステップ２へ

また、直線探索法には様々な手法が存在するが、実用上には次のようなArmijoの基準を用いた簡単な探索法で十分である。

Step 5 Convergence Determination If the convergence condition is satisfied, x _{k + 1} is regarded as a solution, and the process ends. Otherwise go to step 6.
By updating the update official Step 6 _{B k} to calculate the _{B k + 1.}
Step 7 Substitute k + 1 into k (that is, k: = k + 1) and go to Step 2

Various methods exist for the line search method, but the following simple search method using the Armijo criterion is sufficient for practical use.

  For a constant ξ such that 0 <ξ <1, α satisfying the following formula 51 is selected.

Armijoの基準の幾何学的意味は図７で示される。つまり、図７は目的関数ｆ(ｘ)を探索方向ｄ_ｋで切った断面図であり、横軸がステップ幅αに対応する。原点におけるｆ(ｘ_ｋ＋αｄ_ｋ)の接線ｙ＝ｆ(ｘ_ｋ)＋α∇ｆ(ｘ_ｋ)^Ｔｄ_ｋの傾きを緩和して得られた直線ｙ＝ｆ(ｘ_ｋ)＋ξα∇ｆ(ｘ_ｋ)^Ｔｄ_ｋよりも関数値が低くなる区間からステップ幅αを選ぶことになる。

The geometric meaning of Armijo's criteria is shown in FIG. That is, FIG. 7 is a sectional view taken along the objective function f (x) in the search direction d _k, the horizontal axis corresponds to the step width alpha. The straight line y = f (x _k ) + ξα∇f (x obtained by relaxing the slope of the tangent line y = f (x _k ) + α∇f (x _k ) ^T d _k of the origin f (x _k + αd _k ) _k ) The step width α is selected from the interval where the function value is lower than ^T d _k .

Armijo's standard line search algorithm is as follows.
Step A Initial Setting The current approximate solution x _k , parameters 0 <ξ <1, 0 <τ <1 are given.
Step B Search In the search direction d _k , a step width α _k that satisfies the Armijo criterion is _obtained .

Step B1 β _{k, 0} = 1, i = 0.

  Step B2 If the criteria of Armijo represented by the following equation 52 are satisfied, go to Step C, otherwise go to Step B3.

ステップＢ３ β_{ｋ,ｉ＋１}＝τβ_ｋ,ｉ、ｉ：＝ｉ＋１とおいて（つまり、ｉ＋１をｉに代入して）、ステップＢ２へいく。
ステップＣ ステップ幅決定
α_ｋ＝β_ｋ,ｉとおく。

また、Ｂ_ｋの更新には種々の公式が考えられているが、代表的なものは、下記数５３で表すＢ公式のＢＦＧＳ（Broyden Fletcher Goldfarb Shanno）更新公式である。

Step B3 β _{k, i + 1} = τβ _{k, i} , i: = i + 1 (that is, i + 1 is substituted for i), and the process goes to Step B2.
Step C Step width determination α _k = β _{k, i}

Various formulas are considered for updating B _k , but a representative one is a B formula BFGS (Broyden Fletcher Goldfarb Shanno) update formula expressed by the following equation (53).

なお、Ｂ_０は単位行列とするのが一般的である。すなわち、初回の探索方向は最急降下方向である。また、ｓ_ｋ,ｙ_ｋは、下記数５４で表されるベクトルである。

In general, B ₀ is a unit matrix. That is, the initial search direction is the steepest descent direction. Further, s _k , y _k is a vector represented by the following formula 54.

ヘッセ行列の逆行列を直接行列Ｈ_ｋで近似することも考えられている。このとき、探索方向は連立方程式を解くこと無く、下記数５５として求めることができる。

Is also considered to be approximated by the inverse matrix directly matrix H _k Hessian. At this time, the search direction can be obtained as Equation 55 below without solving simultaneous equations.

Ｈ_ｋの更新公式は、上記数５３の逆行列を計算することによって得られる。具体的には、Ｈ公式のＢＦＧＳ更新公式は、下記数５６のような形になる。

The update formula for H _k is obtained by calculating the inverse matrix of the above equation (53). Specifically, the H-form BFGS update formula takes the form

Ｂ公式およびＨ公式を用い、実際にそれぞれ最適化の計算を行ったところ、収束性に違いはほとんど見られなかった。Ｈ公式は連立方程式を解く必要がないため、計算コスト上有利である。本実施形態では、最適化問題ではＨ公式を採用した。

＜２−２−２＞被測定光に関する先験情報の利用
被測定光の光波の強度および位相の空間分布は、滑らかに分布していることが予測された場合に、つまり、「複素振幅（つまり、位相）は滑らかに分布しているであろう」ということが先験情報として利用できる。この先験情報を具体的に表現すると、『ある点における複素振幅とその隣り合う点における複素振幅との差は、小さいはずである』という情報である。これは最適化問題の目的関数に隣り合う点の差を最小化するという目的を加えることで実現できる。また、被測定光の複素ベクトルの分布が滑らかであるという先験情報は、最適化問題の目的関数に被測定光の複素ベクトルｇのノルムを最小化する目的を加えても実現できる。

When the optimization calculation was actually performed using the B formula and the H formula, there was almost no difference in convergence. Since the H formula does not need to solve simultaneous equations, it is advantageous in terms of calculation cost. In this embodiment, the H formula is adopted for the optimization problem.

<2-2-2> Use of a priori information regarding light to be measured When the spatial distribution of the intensity and phase of the light wave of the light to be measured is predicted to be smoothly distributed, that is, “complex amplitude ( That is, “phase) will be distributed smoothly” can be used as a priori information. To express this a priori information concretely, it is information that “the difference between the complex amplitude at a certain point and the complex amplitude at the adjacent point should be small”. This can be realized by adding the objective of minimizing the difference between adjacent points to the objective function of the optimization problem. Further, a priori information that the distribution of the complex vector of the measured light is smooth can be realized by adding the purpose of minimizing the norm of the complex vector g of the measured light to the objective function of the optimization problem.

When the light to be measured g distributed in one dimension is discretized into n pieces, that is, when the design variable is gεC ⁿ , the objective function representing the minimization of the difference between adjacent points is expressed by the following Expression 57. Can be expressed.

上記数５７のような目的関数を先験情報として、数４６の適切化関数とする。つまり、光波が一次元で分布している、設計変数がｇ∈Ｃ^ｎの場合では、下記数５８のような適切化関数を用いる。

The objective function as shown in Equation 57 is used as a priori information, and the appropriate function as shown in Equation 46 is used. That is, in the case where the light wave is distributed in one dimension and the design variable is gεC ⁿ , an appropriate function as shown in the following equation 58 is used.

ここで、マトリックスＣ∈Ｒ^ｎ×ｎは、下記数５９で表される。

Here, the matrix CεR ^{n × n} is expressed by the following equation 59.

同様に、被測定光ｇが二次元（ｎ×ｎ）で分布している場合には、隣り合う点の差の最小化を表す目的関数は、次のように表現できる。まず、簡単のため、設計変数を二次元配列[ｇ]∈Ｃ^ｎ×ｎのまま表記すると、下記数６０のようになる。

Similarly, when the light to be measured g is distributed two-dimensionally (n × n), the objective function representing the minimization of the difference between adjacent points can be expressed as follows. First, for simplicity, design variables are expressed as two-dimensional arrays [g] εC ^{n × n} as shown in the following Expression 60.

上記数６０を設計変数がベクトル

の表記で書き直すと、下記数６１のようになる。

The number 60 above is a design variable vector

When rewritten with the notation, the following formula 61 is obtained.

ここで、上記数６１において、第一項は図８(Ａ)で表される行方向の、第二項は図８(Ｂ)で表される列方向の隣り合う点の差を表している。図８はｎ＝４の場合を示している。よって、上記数５８と同様に表せば、被測定光が二次元（ｎ×ｎ）で分布している場合の適切化関数は、下記数６２のようになる。

Here, in the above equation 61, the first term represents the difference between adjacent points in the row direction represented by FIG. 8A, and the second term represents the column direction represented by FIG. 8B. . FIG. 8 shows a case where n = 4. Therefore, if expressed in the same manner as in the above equation 58, the appropriate function when the light to be measured is distributed two-dimensionally (n × n) is represented by the following equation 62.

Ｃ_１は、図８で表されるように、二次元の複素振幅分布の行方向の差をとるマトリックスであり、Ｃ_２は列方向の差をとるマトリックスである。つまり、マトリックス

は、数５９で表すマトリックスＣを用い、下記数６３で表される。

As shown in FIG. 8, C ₁ is a matrix that takes a difference in the row direction of a two-dimensional complex amplitude distribution, and C ₂ is a matrix that takes a difference in the column direction. That is, the matrix

Is expressed by the following Expression 63 using the matrix C expressed by Expression 59.

また、マトリックス

は、１≦ｉ≦ｎ(ｎ−１)としたときに、（ｉ,ｉ）成分および（ｉ,ｉ＋ｎ）成分は、下記数６４のようになり、その他の成分は０となる。

Also matrix

When 1 ≦ i ≦ n (n−1), the (i, i) component and the (i, i + n) component are represented by the following expression 64, and the other components are 0.

具体的に、マトリックスで書き表してみると、下記数６５のような形となる。

Specifically, when written in a matrix, the following formula 65 is obtained.

また、被測定光の複素ベクトルｇのノルムを最小化するという先験情報を利用する場合には、マトリックスＣは単位マトリックスで表される。

When a priori information for minimizing the norm of the complex vector g of the light to be measured is used, the matrix C is represented by a unit matrix.

In addition, when the measurement of the optical phase distribution is used in the field such as surface shape measurement, the rough shape of the surface shape may be known in advance. Information such as the rough shape of the surface shape is equivalent to the fact that the rough shape of the optical phase distribution is known, and can be used as a priori information regarding the light to be measured. That is, when the reference variable g _ref of the design variable is given, the following equation 66 can be used as an appropriate function.

本発明では、上述したような被測定光に関する先験情報を利用することが可能である。位相分布同定逆問題の設定により、適切化関数を使い分けたり、組み合わせることで、位相分布同定逆問題の悪条件性を克復することができる。

In the present invention, it is possible to use a priori information regarding the light to be measured as described above. Depending on the setting of the phase distribution identification inverse problem, it is possible to overcome the ill-conditioned property of the phase distribution identification inverse problem by properly using or combining appropriate functions.

Needless to say, the a priori information that can be used in the present invention is not limited to the two types of information described above, and other a priori information regarding the light to be measured can be used.

<2-2-3> Size Conversion Method for Optimization Problem In the present invention, light wave intensity distribution information obtained by a light wave detection sensor (CCD) as an observation amount becomes enormous data. For this reason, a very large-scale optimization problem is solved, and a problem that enormous calculation time is required arises. In order to overcome this problem, the optical phase distribution measurement method of the present invention uses the following size conversion method of the optimization problem in order to cope with the set large-scale optimization problem.

In other words, the design variables G∈C ^N, observables | f | when the ∈R ^N, optics transformation matrix [H] ∈C ^{N ×} N, when there is an optimization problem expressed by the following Expression 67, this optimization The problem (hereinafter referred to as the original optimization problem) is not directly solved, and as shown in FIG. 9, the design variable g of the original optimization problem is represented by design variables g representing several adjacent points. It is expressed by 'εC ⁿ (where n <N).

図９では、設計変数ｇの隣り合う４点を代表するものを設計変数ｇ´としている。まず、この設計変数ｇ´で表現されたサイズの小さな最適化問題を解く。観測量は元の最適化問題の|ｆ|の隣り合ういくつかの点を平均化して、サイズの小さな最適化問題に対応する観測量|ｆ´|∈Ｒ^ｎを作るものとする。図９では、|ｆ|の隣り合う４点づつの平均をとって並べたものを|ｆ´|∈Ｒ^ｎとしている。また、光学系の変換マトリックスは、サイズの小さな最適化問題に対応するように、前述した光学系による変換マトリックスの計算式により、新たに[Ｈ´]∈Ｃ^ｎ×ｎを用意する。つまり、元の最適化問題より、サイズの小さな下記数６８で表す最適化問題を解く。

In FIG. 9, the design variable g ′ is representative of four adjacent points of the design variable g. First, an optimization problem with a small size expressed by the design variable g ′ is solved. The observed quantity is obtained by averaging several adjacent points of | f | of the original optimization problem to create an observed quantity | f ′ | ∈R ⁿ corresponding to the small-sized optimization problem. In FIG. 9, an average of four adjacent | f | points is arranged as | f ′ | ∈R ⁿ . Further, the conversion matrix of the optical system newly prepares [H ′] εC ^{n × n} by the above-described calculation formula of the conversion matrix by the optical system so as to cope with the optimization problem with a small size. That is, the optimization problem represented by the following formula 68, which is smaller in size than the original optimization problem, is solved.

上記数６８で表すこのサイズの小さな最適化問題の推定解を、元の最適化問題の設計変数の初期値（つまり、初期推定解）として用い、元の最適化問題を解く。

The original optimization problem is solved by using the estimated solution of the small optimization problem represented by the above formula 68 as the initial value of the design variable of the original optimization problem (that is, the initial estimated solution).

  The procedure of the above-described optimization problem size conversion method will be described with reference to the drawings. For example, when there is a one-dimensional 100-point optimization problem (that is, the original optimization problem), 10 points are put together, and first, a 10-point optimization problem (that is, an optimization problem with a small size). To obtain an estimated solution of the design variable g ′ as shown in FIG. Next, this estimated solution is expanded to 100 points of the original size, and as shown in FIG. 11, an initial estimated solution (initial value) of the design variable g of the 100-point optimization problem is set. Finally, by using this initial estimated solution (initial value) and solving the 100-point optimization problem, an estimated solution of the design variable g of the 100-point optimization problem is obtained as shown in FIG. .

  In short, in the present invention, by using such a size conversion method of the optimization problem, in the original optimization problem, a search for a solution can be started from a position closer to the true solution, compared to a direct solution. It can be expected to converge with a small number of iterations.

  In the present invention, by using such a size conversion method of the optimization problem, a rough estimation is performed with a small optimization problem that does not require a calculation time, and the original optimization problem that requires a large calculation cost is used. The calculation time can be shortened.

Further, in the size conversion method of the optimization problem of the present invention, not only the two-stage estimation as described above, but also several optimization problems with different sizes are prepared, and the optimization problem is gradually increased from the small size optimization problem. It is also possible to perform several stages of estimation for large optimization problems.

<3> Optical Phase Distribution Measurement Result to which the Present Invention is Applied In order to confirm the effectiveness of the optical phase distribution measuring system and the optical phase distribution measuring method according to the present invention, a numerical experiment was performed. Using the optical phase distribution measuring system and optical phase distribution measuring method of the present invention, several measurement results (identification examples) of the optical phase distribution are shown as follows.

<3-1> In the case where the light to be measured has a one-dimensional distribution The measurement results (that is, Examples 1 to 8) in the case where the light to be measured is one-dimensionally distributed are shown, and the use and optimization of a priori information in the present invention Check the effectiveness of the size conversion method in question.

<3-1-1> Phase distribution identification inverse problem setting First, two types of optical systems are used for the optical phase distribution measurement system of the present invention, that is, a diffraction system and a lens focus system are used. Here, it is assumed that the lens is an ideal lens having an infinite diameter.

When the input light and the observation light are discretized into n pieces, the (i, j) component of the transformation matrix [H ₁ ] of the diffraction system is expressed by the following Equation 69 as described in <1> Mathematical Model. The

ここで、ｋは波数で、Ｒは光の入力面と観測面との距離である。被測定光の波長λ＝０.５×１０^−６(ｍ)、つまり、波数ｋ＝２π／λ＝１.２５６×１０^−７(１／ｍ)とする。また、Ｒ＝０.１(ｍ)とし、入力面ｘおよび観測面Ｘの離散点の間隔Δｘ,ΔＸ＝１.０×１０^−５(ｍ)とする。

Here, k is the wave number, and R is the distance between the light input surface and the observation surface. The wavelength λ of the light to be measured is set to 0.5 × 10 ⁻⁶ (m), that is, the wave number k = 2π / λ = 1.256 × 10 ⁻⁷ (1 / m). Also, R = 0.1 (m), and the distances Δx, ΔX = 1.0 × 10 ⁻⁵ (m) between the discrete points of the input surface x and the observation surface X.

The conversion matrix [H ₂ ] of the lens focal system is a unit matrix for connecting a complete image with no blur at the focal point as shown in the following equation 70 when an ideal lens having an infinite diameter is used. It becomes.

次に、ある正解の入力光ｇを設定し、これを元に、数４３で表す光位相分布測定系の観測方程式を用いて、模擬的に測定された強度分布|ｆ_１|（回折系）、|ｆ_２|（レンズ焦点系）を作成する。これを有効数字３桁に丸め、誤差を含ませたものを観測量とする。

＜３−１−２＞本発明の光位相分布測定方法を用いた測定結果（同定結果）
まず、本発明において、先験情報の利用の有効性を確認する。

Next, a certain correct input light g is set, and based on this, the intensity distribution | f ₁ | (diffraction system) measured in a simulated manner using the observation equation of the optical phase distribution measurement system expressed by Equation 43 , | F ₂ | (lens focus system). This is rounded to 3 significant figures and the error is included in the observed quantity.

<3-1-2> Measurement result (identification result) using the optical phase distribution measurement method of the present invention
First, in the present invention, the effectiveness of using a priori information is confirmed.

Example 1 in which the input surface and the observation surface are discretized to 100 points is shown in FIGS. 13 to 15, Example 2 is shown in FIGS. 16 to 18, Example 3 is shown in FIGS. 19 to 21, and Example 4 is shown in FIGS. . FIGS. 13, 16, 19, and 22 show the observed quantities | f ₁ |, | f ₂ |, which are the intensities of the light under measurement, and FIGS. 14, 17, 20, and 23 show a priori information. FIG. 15, FIG. 18, FIG. 21 and FIG. 24 show the a priori information that “the complex amplitude distribution (that is, the phase) is smoothly distributed”. The results identified using the present invention are shown.

  Examples 1, 2, and 3 are examples in which the intensity distribution of the light to be measured is a Gaussian distribution. 14, 17, and 20, which are the results of identification using the present invention without using a priori information, the identification accuracy is relatively good in the central portion, but both ends where the intensity of the light to be measured is weak It can be seen that the accuracy is not so good in the part. Next, referring to FIGS. 15 and 18, which are the results of identification using the present invention using the a priori information that “the complex amplitude distribution is distributed smoothly”, it is often found that the identification was successful throughout. As can be seen, the effectiveness of the use of a priori information has been confirmed in the present invention.

  However, in Example 3, as shown in FIG. 21, which is a result of identification using the present invention using the a priori information that “the complex amplitude distribution is distributed smoothly”, a region where the intensity of the light to be measured is weak is shown. When there is a large phase difference, the two ends cannot be identified well. Since the behavior of the design variable value in the weak and weak region does not significantly affect the objective function value, it can be assumed that the solution is difficult to converge.

  Accordingly, as in Example 4, the intensity distribution of the light to be measured as shown in FIG. 22 was made uniform, and the optical phase distribution identification was performed using the optical phase distribution measurement method of the present invention. Then, as shown in FIG. 23 and FIG. 24, it was possible to identify with high accuracy throughout. 24, which is the result of identification using the present invention using a priori information, seems to be slightly more accurate.

  Thus, even when the intensity of the light to be measured is uniform, the identification light in the case where the light to be measured and the observation light in Example 5 are discretized into 1000 points and the phase distribution has a complicated shape is shown in FIG. As can be seen from FIG. 26, when the identification is performed using the present invention without using the a priori information, a completely out of order solution is obtained. Even in such an unfavorable phase distribution identification inverse problem, it is preferable that the identification can be performed well by suppressing the fluctuation of the solution when looking at FIG. 27 which is the result of identification using the present invention using a priori information. I understand. Even in the case where the light intensity has a uniform distribution, the effectiveness of using a priori information could be confirmed in the present invention.

  Next, the effectiveness of using a priori information is confirmed in the present invention from the aspect of the number of iterations until the optimization calculation converges. Table 1 below shows the number of iterations required for the optimization calculations in Examples 1 to 5 to converge when the a priori information is not used and when the a priori information is used.

表１から、例１、例２、例３及び例５では、先験情報の利用は、同定精度の向上と計算時間の短縮に貢献していることが分かる。また、例４では、先験情報を用いた場合も先験情報を用いない場合も最終的な同定精度はほぼ同程度であるが、やはり先験情報の利用は計算時間を短縮させている。このことから、計算時間の面でも、本発明において、先験情報の利用の有効性の確認ができた。

From Table 1, it can be seen that in Examples 1, 2, 3, and 5, the use of a priori information contributes to an improvement in identification accuracy and a reduction in calculation time. In Example 4, the final identification accuracy is almost the same whether the prior information is used or not used, but the use of the prior information also shortens the calculation time. From this, it was possible to confirm the effectiveness of the use of a priori information in the present invention in terms of calculation time.

  From the above, it was confirmed that the a priori information that “the complex amplitude is smoothly distributed” is effective for stabilizing the solution, improving the identification accuracy, and shortening the calculation time.

  Next, in the present invention, the effectiveness of using the size conversion method (hereinafter also referred to as a two-step identification method) for the optimization problem is confirmed.

  Example 6 to which the two-stage identification method is applied is shown in FIGS. 28 to 30, Example 7 is shown in FIGS. 31 to 33, and Example 8 is shown in FIGS. 34 to 36. In Example 6, Example 7, and Example 8, the input surface and the observation surface are each discretized at 1000 points. In Example 6, Example 7 and Example 8, the two-stage identification technique is applied as follows.

  First, the 1000-point optimization problem is converted into a 100-point optimization problem, and the small-size 100-point optimization problem is solved. Next, the estimated solution for the 100-point optimization problem is used as the initial estimated solution for the 1000-point optimization problem, and the 1000-point optimization problem is solved.

28, 31, and 34 show the intensity distribution of the light to be measured, which is an observation amount, and FIGS. 29, 32, and 35 show an estimated solution (that is, a two-stage solution) of the 100-point optimization problem. FIG. 30, FIG. 33, and FIG. 36 show the estimated solution of the original 1000-point optimization problem (that is, the second stage of the two-stage identification technique). The phase distribution obtained in (1) is shown.
From Examples 6, 7, and 8, it can be seen that the solution is roughly identified by the optimization problem of 100 points, and then the solution is identified in detail by the original optimization problem of 1000 points.

  Table 2 below shows the number of iterations required for the optimization calculation to converge when the two-step identification method is used and when the two-step identification method is not used.

表２では、二段階の同定手法を用いた場合の最適化の計算が収束するまでにかかる反復回数は、一段階目(１００点の最適化問題)と二段階目(１０００点の最適化問題)と分けて表示してある。なお、設計変数１００点の１回の反復計算にかかる時間は、１０００点のそれと比べ非常に短いため無視できる程度である。

In Table 2, the number of iterations required for the optimization calculation when the two-stage identification method is converged is the first stage (100-point optimization problem) and the second stage (1000-point optimization problem). ) And displayed separately. It should be noted that the time required for one iteration of 100 design variables is very short compared to 1000, so it can be ignored.

From Table 2, it can be clearly seen that the number of iterations until the optimization calculation converges is greatly reduced by using the two-step identification method. In the optical phase distribution measurement method of the present invention, it has been confirmed that the use of the two-step identification method is effective in reducing the calculation time. Of course, it is possible to cope with a large-scale optimization problem by extending this two-stage identification method to multiple stages.

<3-2> In the case where the light to be measured has a two-dimensional distribution Even if the light to be measured has a two-dimensional distribution, the measurement result (that is, Example 9 to Example 12) shows that the optical phase distribution measurement method according to the present invention is effective. ) To confirm.

<3-2-1> Phase distribution identification inverse problem setting First, two types of optical systems are used in the optical phase distribution measurement system of the present invention, that is, a diffraction system and a lens focus system are used. Here, it is assumed that the lens is an ideal lens having an infinite diameter.

When the input light and the observation light are discretized into n × n, the (n (i−1) + j, n (k−1) + l) component of the transformation matrix [H ₁ ] of the diffraction system is expressed by the following formula 71 It is represented by

ここで、ｋは波数で、Ｒは光の入力面と観測面との距離である。被測定光の波長λ＝０.５×１０^−６(ｍ)、つまり、波数ｋ＝２π／λ＝１.２５６×１０^−７(１／ｍ)とする。また、Ｒ＝０.１(ｍ)とし、入力面(ｘ,ｙ)および観測面(Ｘ,Ｙ)の離散点の間隔Δｘ,Δｙ,ΔＸ,ΔＹ＝１.０×１０^−５(ｍ)とする。

Here, k is the wave number, and R is the distance between the light input surface and the observation surface. The wavelength λ of the light to be measured is set to 0.5 × 10 ⁻⁶ (m), that is, the wave number k = 2π / λ = 1.256 × 10 ⁻⁷ (1 / m). Further, R = 0.1 (m), and the distances between discrete points of the input surface (x, y) and the observation surface (X, Y) Δx, Δy, ΔX, ΔY = 1.0 × 10 ⁻⁵ (m) And

Similarly to the case of the one-dimensional distribution, the conversion matrix [H ₂ ] of the lens focal system is also blurred at the focal point as expressed by Equation 70 when an ideal lens having an infinite diameter is used. It becomes a unit matrix to connect complete images without.

Next, a certain correct input light g is set, and based on this, the intensity distribution | f ₁ | (diffraction system) measured in a simulated manner using the observation equation of the optical phase distribution measurement system expressed by Equation 43 , | F ₂ | (lens focus system). This is rounded to 3 significant figures and the error is included in the observed quantity.

<3-2-2> Measurement result (identification result) using the optical phase distribution measurement method of the present invention
Example 9 in which the input surface and the observation surface are discretized into 40 × 40 points are shown in FIGS. 37 to 40, Example 10 is FIGS. 41 to 44, Example 11 is FIGS. 45 to 48, and Example 12 is FIGS. Each is shown. Example 9, Example 10, Example 11 and Example 12 do not apply the two-step identification method, and use the a priori information that “the complex amplitude is smoothly distributed”, and the optical phase distribution measurement method of the present invention. It is the measurement result (identification result) used.

37, 41, 45 and 49 show the distribution of the observed quantity | f ₁ | which is the intensity of the light under measurement. Note that the observed amount | f ₂ | has a uniform distribution with the observed amount | f ₁ |, and thus the distribution diagram thereof is omitted. 38, 42, 46, and 50 are the correct phase distributions of the light to be measured. 39, 43, 47 and 51 are phase distributions of the light to be measured identified using the optical phase distribution measurement method of the present invention. 40, FIG. 44, FIG. 48, and FIG. 52 show cross sections at y = 20 (points) of the phase distribution of the correct answer and the identification result according to the present invention.

  When the phase of the correct answer of the light under measurement is smoothly distributed as in Example 9 and Example 10, it can be clearly seen that the identification results using the present invention agree well with the correct answer in all regions. . Even when the correct phase of the light to be measured is distributed as in Example 11 or Example 12, even if the identification result using the present invention is seen, there is some variation. It can be clearly seen that is successfully identified in all regions. Moreover, such variation can be suppressed by increasing the number of discrete points and increasing the spatial frequency.

As described above, the effectiveness of the optical phase distribution measurement system and the optical phase distribution measurement method according to the present invention was confirmed through Examples 1 to 12.

<4> Summary of the Present Invention As described above, the optical phase distribution measurement method according to the present invention identifies the phase distribution of the light to be measured from the intensity images of the light to be measured that has passed through different optical systems. The biggest feature.

  In short, in the present invention, first, by using an optical phase distribution measurement system composed of a plurality of different optical systems and a light wave detection sensor, the same measured light is respectively transmitted to the plurality of optical systems whose optical characteristics are known. A plurality of intensity distributions of the light to be measured are measured by passing the light and modulating the intensity and phase. Then, based on the obtained intensity distribution information of the light to be measured and the optical characteristics of the optical system, an observation equation of the optical phase distribution measurement system is established, and the phase distribution for identifying the phase distribution of the light to be measured from this observation equation An identification inverse problem is set, the set phase distribution identification inverse problem is formulated as a nonlinear optimization problem, and the phase distribution of the light to be measured is identified by solving the optimization problem.

  As described above, it is preferable to minimize the objective function of the optimization problem of the present invention using the quasi-Newton method, but without being limited thereto, if it is a numerical solution to the optimization problem, Other methods can be used to solve the optimization problem of the present invention.

  Further, in the present invention, the optical phase distribution measurement system is configured by setting an arbitrary number of appropriate optical systems, or in the optical phase distribution measurement method, by giving a priori information on the phase distribution of the light to be measured, It is possible to overcome the ill-conditioning of the observation equation of the distribution measurement system, stabilize the solution, and obtain the only solution.

  Furthermore, in the phase distribution identification inverse problem set by the optical phase distribution measurement method of the present invention, the intensity distribution information of the light to be measured obtained by the light wave detection sensor (CCD) as the observation amount becomes enormous data. To solve large-scale optimization problems. In the optical phase distribution measuring method of the present invention, the size conversion method of the optimization problem is used to cope with such a large-scale optimization problem.

It is a schematic diagram for demonstrating the two-dimensional arrangement | sequence of the complex number of a light wave. It is a schematic diagram for demonstrating the measurement principle and mathematical model of the optical phase distribution measuring system which concern on this invention. It is a schematic diagram for demonstrating the mathematical model of the conversion of the light wave by a diffraction system. It is a schematic diagram for demonstrating the mathematical model of the conversion of the light wave by an imaging lens system. It is a figure which shows the point image response function PSF of an imaging optical system. It is a schematic diagram for demonstrating the mathematical model of the conversion of the light wave by a diffraction grating. It is a figure for demonstrating the geometrical meaning of the reference | standard of Armijo. It is a schematic diagram for demonstrating the adjacent point of a row direction or a column direction in case a light wave is distributed by two dimensions (4x4). It is a schematic diagram for demonstrating size conversion of an optimization problem. It is a figure which shows the estimated solution of the design variable g 'of an optimization problem with small size. It is a figure which shows the initial value of the design variable g of the original optimization problem. It is a figure which shows the estimated solution of the design variable g of the original optimization problem. In Example 1, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 1, it is a phase distribution map of the to-be-measured light identified using this invention, without utilizing a priori information. In Example 1, it is a phase distribution map of the to-be-measured light identified using this invention using a priori information. In Example 2, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 2, it is a phase distribution map of the to-be-measured light identified using this invention, without utilizing prior information. In Example 2, it is a phase distribution map of the to-be-measured light identified using this invention using a priori information. In Example 3, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 3, it is a phase distribution map of the to-be-measured light identified using this invention, without utilizing a priori information. In Example 3, it is a phase distribution map of the to-be-measured light identified using this invention using a priori information. In Example 4, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 4, it is a phase distribution map of the to-be-measured light identified using this invention, without utilizing prior information. In Example 4, it is a phase distribution map of the to-be-measured light identified using this invention using a priori information. In Example 5, it is an intensity distribution map of the to-be-measured light measured using the optical phase distribution measurement system. In Example 5, it is a phase distribution map of the to-be-measured light identified using this invention, without utilizing prior information. In Example 5, it is a phase distribution map of the to-be-measured light identified using this invention using a priori information. In Example 6, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 6, it is a phase distribution map obtained in the first step of the two-step identification method. In Example 6, it is a phase distribution map obtained by the 2nd step of the 2-step identification method. In Example 7, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 7, it is a phase distribution map obtained in the first step of the two-step identification method. In Example 7, it is a phase distribution map obtained in the second stage of the two-stage identification method. In Example 8, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 8, it is a phase distribution map obtained in the first step of the two-step identification method. In Example 8, it is a phase distribution map obtained by the 2nd step of the 2-step identification method. In Example 9, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 9, it is a phase distribution map of the correct answer of to-be-measured light. In Example 9, it is a phase distribution map of the to-be-measured light identified using this invention. In Example 9, it is sectional drawing in y = 20 (point) of the phase distribution map of the to-be-measured light identified using this invention. In Example 10, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 10, it is a phase distribution map of the correct answer of to-be-measured light. In Example 10, it is a phase distribution map of the to-be-measured light identified using this invention. In Example 10, it is sectional drawing in y = 20 (point) of the phase distribution map of the to-be-measured light identified using this invention. In Example 11, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 11, it is a phase distribution map of the correct answer of measured light. In Example 11, it is a phase distribution map of the to-be-measured light identified using this invention. In Example 11, it is sectional drawing in y = 20 (point) of the phase distribution map of the to-be-measured light identified using this invention. In Example 12, it is an intensity distribution figure of the to-be-measured light measured using the optical phase distribution measurement system. In Example 12, it is a phase distribution map of the correct answer of to-be-measured light. In Example 12, it is a phase distribution map of the to-be-measured light identified using this invention. In Example 12, it is sectional drawing in y = 20 (point) of the phase distribution map of the to-be-measured light identified using this invention. It is a schematic diagram for demonstrating the optical phase distribution measuring method by the conventional interferometer. It is a figure which shows the example of an interference fringe image. It is a schematic diagram for demonstrating the optical phase distribution measuring method by the conventional Shack Hartman sensor. It is a figure which shows the example of a Hartmann pattern.

Claims (8)

It has an optical system with a plurality of different optical characteristics and a light wave detection sensor,
The light to be measured is input to each optical system, the intensity and phase are modulated, the output light to be measured is detected by the light wave detection sensor, and the intensity distribution of the detected light to be measured is measured as an image. An optical phase distribution measuring system.   The optical phase distribution measurement system according to claim 1, wherein the light wave detection sensor is a CCD image sensor.   The optical phase distribution measurement system according to claim 1, wherein the plurality of optical systems includes a diffraction system and a lens focus system. An optical phase distribution measuring method for measuring a phase distribution of light to be measured,
An intensity distribution measuring step for measuring an intensity distribution of the light under measurement using the optical phase distribution measuring system according to any one of claims 1 to 3,
An observation equation setting step for setting an observation equation of the optical phase distribution measurement system based on the intensity distribution obtained in the intensity distribution measurement step and the optical characteristics of each optical system;
A phase distribution identification inverse problem setting step for setting a phase distribution identification inverse problem for identifying the phase distribution of the light under measurement from the observation equation;
An optimization problem formulation step for formulating the set phase distribution identification inverse problem as a nonlinear optimization problem;
A phase distribution identification step for identifying a phase distribution of the light under measurement by solving the optimization problem;
An optical phase distribution measurement method comprising:   5. The optimization problem according to claim 4, wherein in the optimization problem, a design variable is a complex vector representing the light to be measured, and an objective function includes an optimization function set based on a priori information regarding the phase distribution of the light to be measured. Optical phase distribution measurement method.   6. The optical phase distribution measuring method according to claim 5, wherein the a priori information is information that the phase of the light to be measured will be smoothly distributed.   5. The optical phase distribution measurement method according to claim 4, wherein the optimization function objective function is minimized using a quasi-Newton method.   The second design variable is a representative of a predetermined point adjacent to the design variable of the optimization problem, and the estimated solution of the second optimization problem with a small size expressed by the second design variable is The optical phase distribution measurement method according to claim 4, wherein the optimization problem is solved by using it as an initial estimated solution of the design variable of the optimization problem.

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