Abstract:
An optical-axis deflection type laser interferometer includes a reference ball serving as a measurement reference, a retroreflecting means disposed at an object to be measured, a laser interferometer length measuring apparatus outputting a measurement value between the apparatus and the retroreflecting means, and a rotational mechanism that rotates a beam projected from the laser interferometer length measuring apparatus around the reference ball. The laser interferometer additionally includes a displacement gauge that measures an error caused by a relative motion between the reference ball and the laser interferometer length measuring apparatus, and a rectilinear movement mechanism that displaces these in a direction of an optical axis of a measurement beam with respect to the reference ball while maintaining a relative positional relationship between the laser interferometer length measuring apparatus and the displacement gauge. With this structure, calibrating the displacement gauge without preparing a special calibrating device becomes possible, thereby securing traceability thereby securing traceability.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
   The disclosure of Japanese Patent Application No. 2006-229164 filed on Aug. 25, 2006 including specifications, drawings and claims is incorporated herein by reference in its entirety. 
   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to an optical-axis deflection type laser interferometer that includes a reference ball serving as a measurement reference, a retroreflecting means disposed at an object to be measured, a laser interferometer length measuring apparatus that outputs a measurement value according to an increase or a decrease in distance between the apparatus and the retroreflecting means, and a rotational mechanism that rotates a beam projected from the laser interferometer length measuring apparatus around the reference ball, the laser interferometer measuring a distance between the laser interferometer and the retroreflecting means that makes an optical axis of an outgoing beam projected from the laser interferometer length measuring apparatus mounted on the rotational mechanism and an optical axis of a return beam returned to the laser interferometer length measuring apparatus parallel to each other, based on center coordinates of the reference ball. The present invention relates to a calibration method of the laser interferometer, relates to a correcting method of the laser interferometer, and relates to a measuring method of the laser interferometer. More specifically, the present invention relates to an optical-axis deflection type laser interferometer that can secure traceability by the length standard without using a special device, relates to a calibration method thereof, relates to a correcting method thereof, and relates to a measuring method thereof. 
   2. Description of the Related Art 
   An optical-axis deflection type laser interferometer (also called a “tracking laser interferometer”) is known. This laser interferometer operates in the following way. As shown in  FIG. 1 , a laser beam (hereinafter, referred to as a “measurement beam”)  22  is projected from a laser interferometer length measuring apparatus  20  toward a retroreflecting means (also called a “retroreflector”)  12  disposed at a to-be-measured object  10 , and is reflected in the backward direction by the retroreflecting means  12 . The displacement of the retroreflecting means  12  is detected by use of the interference of the laser beam reflected therefrom, and a tracking operation is performed by use of a change in position of the optical axis of the measurement beam  22  by a two-axis rotational mechanism  30 . In  FIG. 1 , reference numeral  24  designates a light source, and reference numeral  32  designates a deflective plane mirror of the two-axis rotational mechanism  30 . 
   According to this optical-axis deflection type laser interferometer, based on information about the return beam returning from the retroreflecting means  12  that serves as a target, the distance from the apparatus to the target can be measured by the laser interferometer length measuring apparatus  20  with high accuracy. 
   However, the optical-axis deflection type laser interferometer has technical difficulties in making the precision of the two-axis rotational mechanism  30  as high as the precision in measurement of the laser interferometer. Especially, as in the optical system shown in  FIG. 1  in which the direction of the measurement beam  22  is deflected by, for example, the plane mirror  32  attached to the two-axis rotational mechanism  30 , the two-axis rotational mechanism  30  has a precision limit, and, in addition, it is difficult to allow a beam to accurately strike the rotational center of the plane mirror  32 . These make it difficult to achieve high precision. 
   Therefore, as shown in  FIG. 2 , European Patent EP0919830A2 (hereinafter, referred to as Patent Document 1) discloses that a relative displacement between the surface of a reference ball  34  disposed at the center of the two-axis rotational mechanism  30  and the laser interferometer length measuring apparatus  20  is measured with a laser beam  26  projected in the exactly opposite direction with respect to the target ( 12 ), in addition to the measurement of the retroreflecting means  12  serving as a target. In  FIG. 2 , reference numeral  36  designates a carriage used to mount the laser interferometer length measuring apparatus  20  on the two-axis rotational mechanism  30 , and reference numeral  40  designates a support used to fasten the reference ball  34 . 
   However, the optical system disclosed by Patent Document 1 has a disorder of an interference wavefront or a reduction in light quantity with respect to a rotational error (i.e., runout) in a direction perpendicular to the optical axis. Additionally, it has become apparent that the optical system is not necessarily robust in respect of a change in the optical path length. Therefore, high-cost specifications are required for the rotational accuracy of the two-axis rotational mechanism  30 . Additionally, the problem of the light quantity ratio of the laser interferometer length measuring apparatus  20  is included therein. In more detail, the laser interferometer length measuring apparatus  20  measures a phase difference between a reference beam whose optical path is unchangeably fixed there inside and a measurement beam  22  that travels toward the target ( 12 ) and that returns backwardly therefrom. Therefore, if there is a great difference in light quantity between the reference beam of light and the measurement beam of light, the contrast of interference fringes cannot be sufficiently secured, and, as a result, interference measurement cannot be performed with high accuracy. However, in the optical system disclosed by Patent Document 1, the measurement beam is first projected and reflected to and from the reference ball  34 , and is then projected and reflected to and from the target ( 12 ). The light quantity of the measurement beam that has returned in this way is remarkably attenuated, and hence a difference in light quantity with respect to the reference beam that is generally regarded as not being easily attenuated causes a large problem. 
   On the other hand, the present applicant proposes an optical system shown in  FIG. 3  in Japanese Published Unexamined Patent Application No. 2007-57522 (not yet published at the priority date thereof: hereinafter referred to as Patent Document 2). In this prior invention, a motion error of the two-axis rotational mechanism  30  with respect to the reference ball  34  is measured with a displacement gauge  50  separately mounted on a carriage  38  formed integrally with the carriage  36 , not with a measurement beam of the laser interferometer length measuring apparatus  20 . According to this prior invention, a highly accurate optical-axis deflection type laser interferometer can be, of course, realized with reference to the reference ball  34 , and the precision requirement against the two-axis rotational mechanism  30  can be greatly eased without sacrificing robustness with respect to an error component in the direction perpendicular to the optical axis. Additionally, according to this prior invention, the measurement beam of the laser interferometer length measuring apparatus  20  is projected only onto the target ( 12 ), and hence the disadvantage of the attenuation of the measurement beam that becomes a problem in the technique of Patent Document 1 is greatly reduced. 
   However, in this prior invention, a motion error in the direction of the optical axis of the two-axis rotational mechanism  30  with respect to the reference ball  34  is detected, and hence there is a need to additionally provide the displacement gauge  50 . Since the optical-axis deflection type laser interferometer has been expected to perform highly accurate measurement that is traceable to the length standard, the technique to secure the traceability of the displacement gauge  50  has been expected to be achieved. 
   SUMMARY OF THE INVENTION 
   The present invention has been made to solve the conventional problems. It is therefore an object of the present invention to construct a structure capable of calibrating a displacement gauge used to correct an error in the direction of the optical axis resulting from the motion accuracy of a two-axis rotational mechanism with respect to a reference ball of an optical-axis deflection type laser interferometer, without preparing a special calibrating device, on the operating instrument, thus securing traceability. 
   In an optical-axis deflection type laser interferometer, the present invention employs a structure in which a laser interferometer length measuring apparatus that measures the amount of positional change in a target, an optical axis thereof, a displacement gauge to be calibrated, and the direction of detection sensitivity thereof are aligned together. To achieve the object, an optical-axis deflection type laser interferometer of the present invention includes a reference ball serving as a measurement reference, a retroreflecting means disposed at an object to be measured, a laser interferometer length measuring apparatus that outputs a measurement value according to an increase or a decrease in distance between the apparatus and the retroreflecting means, and a rotational mechanism that rotates a beam projected from the laser interferometer length measuring apparatus around the reference ball, and the optical-axis deflection type laser interferometer measures a distance between the laser interferometer and the retroreflecting means that makes an optical axis of an outgoing beam projected from the laser interferometer length measuring apparatus mounted on the rotational mechanism and an optical axis of a return beam returned to the laser interferometer length measuring apparatus parallel to each other, based on center coordinates of the reference ball. The laser interferometer is characterized by further including a displacement gauge that measures an error caused by a relative motion between the reference ball and the laser interferometer length measuring apparatus, and a rectilinear movement mechanism that displaces these in a direction of an optical axis of a measurement beam with respect to the reference ball while maintaining a relative positional relationship between the laser interferometer length measuring apparatus and the displacement gauge. 
   The displacement gauge maybe disposed on both sides of the reference ball. 
   The displacement gauge may be an electrical capacitance type displacement gauge or an eddy current type displacement gauge. 
   The laser interferometer length measuring apparatus may be a Michelson interferometer. 
   The reference ball may be a commercially available metallic ball. 
   The present invention additionally provides a calibration method of the optical-axis deflection type laser interferometer characterized in that, referring only to a measurement value of the laser interferometer length measuring apparatus that is traceable to the length standard, calibration data of the displacement gauge is acquired by comparative measurement with the laser interferometer length measuring apparatus. 
   In more detail, the calibration method can include a step of fixing the retroreflecting means near the optical-axis deflection type laser interferometer, a step of allowing a measurement beam of the optical-axis deflection type laser interferometer to strike the retroreflecting means, a step of operating the rectilinear movement mechanism and giving a displacement for calibration, a step of recording a measurement value of the laser interferometer length measuring apparatus and a measurement value of the displacement gauge obtained at that time, and a step of calculating a calibration value of the displacement gauge in accordance with acquired data when data acquisition is completed and storing or recording the calibration value. 
   The present invention additionally provides a correcting method of the optical-axis deflection type laser interferometer characterized in that a correcting operation is performed by using the calibration data of the displacement gauge obtained by the comparative measurement with the laser interferometer length measuring apparatus. 
   The present invention additionally provides a measuring method of the optical-axis deflection type laser interferometer characterized in that a distance between the apparatus and the to-be-measured object or a displacement is measured traceably by the length standard by performing a correcting operation according to the above correcting method. 
   In more detail, the measuring method can include a step of fixing the retroreflecting means near the optical-axis deflection type laser interferometer, a step of allowing a measurement beam of the optical-axis deflection type laser interferometer to strike the retroreflecting means, a step of operating the rectilinear movement mechanism and giving a displacement for calibration, a step of recording a measurement value of the laser interferometer measuring apparatus and a measurement value of the displacement gauge obtained at that time, a step of calculating a calibration value of the displacement gauge in accordance with acquired data when data acquisition is completed and storing or recording the calibration value, and a step of attaching the retroreflecting means to an object to be measured and starting a measuring operation. 
   Alternatively, the measuring method can include a step of fixing the retroreflecting means near the optical-axis deflection type laser interferometer, a step of allowing a measurement beam of the optical-axis deflection type laser interferometer to strike the retroreflecting means, a step of operating the rectilinear movement mechanism and giving a displacement for calibration, a step of storing a measurement value of the laser interferometer length measuring apparatus and a measurement value of the displacement gauge obtained at that time, a step of calculating a calibration value of the displacement gauge in accordance with acquired data when data acquisition is completed and storing or recording the calibration value, a step of attaching the retroreflecting means to an object to be measured and starting a measuring operation, and a step of correcting the measurement value by using the calibration value of the displacement gauge. 
   According to the present invention, it is possible to calibrate the displacement gauge used to correct an error in the direction of the optical axis resulting from the motion accuracy of the rotational mechanism with respect to the reference ball of the optical-axis deflection type laser interferometer. A calibrating operation for this can be easily performed in an operating state of the optical-axis deflection type laser interferometer, and a special device is not required to be prepared to calibrate the displacement gauge. As a result, a highly accurate optical-axis deflection type laser interferometer can be realized at low cost. 
   Additionally, the calibration of the displacement gauge can be performed by referring only to the measurement value of the laser interferometer length measuring apparatus that is provided in the optical-axis deflection type laser interferometer and that is traceable to the length standard. Therefore, the to-be-calibrated displacement gauge automatically becomes traceable to the length standard, and hence it becomes possible to also perform a calibrating operation that is traceable to the length standard, not only to correct the sensitivity and the nonlinear error of the displacement gauge. Therefore, it is possible to secure traceability by the length standard, which has not easily been achieved by the conventional technique. 
   At this time, for example, if a He—Ne wavelength-stabilizing laser having an emission wavelength of 633 nm is employed as a light source, and if the emission wavelength (emission frequency) thereof is calibrated by the comparative measurement with a wavelength-stabilizing laser by an iodine absorption line that is an upper standard, the calibration of the displacement gauge that is traceable to the length standard can be automatically achieved. Additionally, this makes it possible to secure traceability as the whole of the optical-axis deflection type laser interferometer system with ease and at low cost. 
   These and other novel features and advantages of the present invention will become apparent from the following detailed description of preferred embodiments. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The preferred embodiments will be described with reference to the drawings, wherein like elements have been denoted throughout the figures with like reference numerals, and wherein; 
       FIG. 1  is an optical path diagram showing an example of an optical system of an optical-axis deflection type laser interferometer; 
       FIG. 2  is an optical path diagram showing an optical system of an optical-axis deflection type laser interferometer disclosed by Patent Document 1; 
       FIG. 3  is an optical path diagram showing an optical system of an optical-axis deflection type laser interferometer proposed by the present applicant in Patent Document 2; 
       FIG. 4  shows an embodiment of the present invention; and 
       FIG. 5  is a flowchart that shows calibrating steps of a displacement gauge in the embodiment. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   An embodiment of the present invention will be hereinafter described in detail with reference to the accompanying drawings. 
   In this embodiment, a rectilinear movement mechanism  52  that enables a movement in the direction of a measurement optical axis (i.e., in the direction indicated by arrow “A”) is provided between a two-axis rotational mechanism  30  and carriages  36  and  38  that are mechanical components formed integrally with a laser interferometer length measuring apparatus  20  and with displacement gauges  50 R and  50 L as shown in  FIG. 4 , in addition to the prior invention that has been proposed by the present applicant in Patent Document 2 as shown in  FIG. 3 . A movement in the direction of the measurement optical axis by the rectilinear movement mechanism  52  is performed together with the laser interferometer length measuring apparatus  20  and the displacement gauges  50 R and  50 L. 
   Therefore, once the displacement gauges  50 R and  50 L are calibrated, the system of  FIG. 4  can operate as an optical-axis deflection type laser interferometer in which the central coordinate of a reference ball  34  is regarded as a fixed reference. 
   For example, a Michelson interferometer can be used as the laser interferometer length measuring apparatus  20 . 
   For example, a commercially available metallic ball can be used as the reference ball  34 . This ball is industrially widely used, and is low in cost. A ceramic ball, a semiconducting ball, a glass ball, or a metal-coated ball can be used as the reference ball  34 , besides such a metallic ball. However, if an eddy current sensor is used as the displacement gauge, a metallic ball or a metal-coated ball must be used as the reference ball  34 . 
   For example, an electrical capacitance type displacement gauge or an eddy current type displacement gauge can be used as the displacement gauges  50 R and  50 L. Each of these displacement gauges has a sensor having a larger effective area than a dust size or a flaw size and having relatively low lateral resolution of the sensor, and hence is insusceptible to dust or flaws existing on the surface of the reference ball  34 . A fiber sensor or various contact type displacement sensors can be used as the displacement gauge. As in the present embodiment, this displacement gauge can be disposed at both sides of the reference ball  34  so as to reduce the influence of a temperature change. 
   Displacement measurement in this embodiment is performed as follows. 
   In detail, the amount ΔL of change in length (i.e., displacement) between the fixed reference ball  34  used as a positional reference point and the retroreflecting means  12  is calculated by the following equation.
 
 ΔL =(Δ L   2   −ΔL   3 )/2 +ΔL   1   (1)
 
   Herein, ΔL 1  is a relative displacement (on the supposition that the direction in which a distance is increased is positive +) between the laser interferometer length measuring apparatus  20  and the retroreflecting means  12 , which is measured by using the laser interferometer length measuring apparatus  20 . ΔL 2  is a relative displacement (on the supposition that the direction in which a distance is increased is positive +) between the displacement gauge  50 R, which is disposed between the laser interferometer length measuring apparatus  20  and the reference ball  34 , and the surface of the reference ball  34 . ΔL 3  a relative displacement (on the supposition that the direction in which a distance is increased is positive +) between the displacement gauge  50 L and the surface of the reference ball  34 . 
   The surface of the highly-accurate reference ball  34  is constant in the distance from the center of the reference ball  34  with high accuracy. Therefore, the displacement of the retroreflecting means  12  based on the center of the reference ball  34  can be measured with high accuracy even when the carriage  38  rotates around the center of the reference ball  34 . 
   The measurement of ΔL 1  can be performed according to a measuring method using the well-known Michelson interferometer in which the retroreflecting means  12  is an object to be measured, and is disclosed by Patent Documents 1 and 2, and hence a detailed description of this is omitted. 
   Additionally, the automatic tracking method of the retroreflecting means  12  is the same as the method disclosed by Patent Documents 1 and 2, and hence a description of this is omitted. 
   A series of steps for the calibration of the displacement gauge according to the present invention are performed as shown in  FIG. 5 . 
   These are hereinafter described in detail. 
   (1) As shown in  FIG. 4 , the retroreflecting means  12  that serves as a target is simply fixed near the optical-axis deflection type laser interferometer by use of, for example, a support  14  (step S 1 ). 
   (2) A measurement beam  22  of the optical-axis deflection type laser interferometer is allowed to strike the retroreflecting means  12 . 
   (3) The rectilinear movement mechanism  52 , which is a mechanism that generates a displacement used to calibrate the displacement gauge, is moved in the direction of arrow “A” shown in  FIG. 4  so as to give a displacement for calibration (step S 2 ). The measurement value of the laser interferometer length measuring apparatus  20  (step S 3 ) and the measurement values of the displacement gauges  50 R and  50 L obtained at this time are stored or recorded into, for example, a data memory (step S 4 ). 
   Steps S 2  to S 4  are repeatedly performed until data acquisition is completed. 
   (4) When it is determined that data acquisition has been completed at step S 5 , the calibration values of the displacement gauges  50 R and  50 L are calculated according to, for example, a least squares method based on the acquired data (step S 6 ), and are stored or recorded into, for example, a data memory (step S 7 ). 
   (5) The displacement gauge is corrected with reference to the calibration curve of the displacement gauge obtained above. 
   (6) As shown in  FIG. 3 , the retroreflecting means  12  serving as a target is attached to the to-be-measured object  10 , and measurement starts to be performed (step S 8 ). 
   In this embodiment, the displacement gauges  50 R and  50 L are disposed at both sides of the reference ball  34 , respectively, and hence the temperature drift of output projected from the displacement gauge can be compensated. In detail, if the right and left displacement gauges  50 R and  50 L are the same in the tendency of the temperature drift of output projected therefrom, the displacement ΔL obtained by Equation (1) will not be affected by the temperature drift of the output of the displacement gauge. For example, let it be supposed that an error ΔD caused by the temperature drift arises in the amount ΔL 2  of change measured by the right displacement gauge  50 R so as to come to ΔL 2 +ΔD. If the right and left displacement gauges  50 R and  50 L are the same in the tendency of the temperature drift, the same error as in the right displacement gauge  50 R which is caused by the temperature drift will arise in the amount ΔL 3  of change measured by the left displacement gauge  50 L at this time so as to come to ΔL 3 +ΔD. At this time, the displacement ΔL is expressed by the following equation without being affected by the temperature drift of the displacement gauge. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         L 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             { 
                             
                               
                                 ( 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       L 
                                       2 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     D 
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 ( 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       L 
                                       3 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     D 
                                   
                                 
                                 ) 
                               
                             
                             } 
                           
                           / 
                           2 
                         
                         + 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             1 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   L 
                                   2 
                                 
                               
                               - 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   L 
                                   3 
                                 
                               
                             
                             ) 
                           
                           / 
                           2 
                         
                         + 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             1 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   Likewise, if the reference ball  34  is isotropically and thermally expanded, the thermal expansion of the reference ball  34  can be compensated. 
   Therefore, when the displacement gauges  50 R and  50 L are provided at both sides of the reference ball  34 , a system that is robust against the temperature deformation can be constructed. 
   Any of the amount ΔL 2 , the amount ΔL 3 , and a difference therebetween (i.e. ΔL 2 −ΔL 3 ) can be regarded as being calibrated according to the present invention. 
   Additionally, the apparatus in this embodiment is very robust also against the runout of the rotational mechanism. In detail, even if the entire carriage is positionally changed in the direction of the optical axis of a measurement beam of the laser interferometer length measuring apparatus  20  when the carriage  38  rotates around the reference ball  34 , the displacement ΔL calculated as above is not affected by this positional change. 
   For example, let it be supposed that the entire carriage has been displaced by ΔD in the direction of the retroreflecting means  12 . If so, ΔL 1  comes to ΔL 1 −ΔD, ΔL 2  comes to ΔL 2 +ΔD, and ΔL 3  comes to ΔL 3 −ΔD. Therefore, ΔL is expressed as follows. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         L 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             { 
                             
                               
                                 ( 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       L 
                                       2 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     D 
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 ( 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       L 
                                       3 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     D 
                                   
                                 
                                 ) 
                               
                             
                             } 
                           
                           / 
                           2 
                         
                         + 
                         
                           ( 
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 L 
                                 1 
                               
                             
                             - 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               D 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   L 
                                   2 
                                 
                               
                               - 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   L 
                                   3 
                                 
                               
                             
                             ) 
                           
                           / 
                           2 
                         
                         + 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             1 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
     
   
   Thus, even if the carriage  38  is positionally changed in the direction of the optical axis of a measurement beam, the displacement ΔL calculated as above is not affected by this positional change. 
   Additionally, even if the entire carriage is positionally changed (rectilinearly) in the direction perpendicular to the optical axis of a measurement beam when the carriage  38  rotates around the reference ball  34 , the displacement ΔL calculated as above is not affected by this positional change. First, ΔL 1  is not affected by this positional change. There is no change in the forward and backward optical path lengths between the laser interferometer length measuring apparatus  20  and the retroreflecting means  12  even if the laser interferometer measuring apparatus  20  is displaced in the direction perpendicular to the optical axis of a measurement beam. Therefore, ΔL 1  is not affected by this displacement (i.e. positional change). This results from the properties of the retroreflecting means  12 . Next, if the displacement gauge  50 R is displaced in the direction perpendicular to the optical axis of a measurement beam so that the value of ΔL 2  increases by ΔE, the value of ΔL 3  will increase by ΔE correspondingly. At this time, ΔL is expressed as follows. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         L 
                       
                       = 
                       
                         
                           
                             { 
                             
                               
                                 ( 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       L 
                                       2 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     E 
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 ( 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       L 
                                       3 
                                     
                                   
                                   - 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     E 
                                   
                                 
                                 ) 
                               
                             
                             } 
                           
                           / 
                           2 
                         
                         + 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             1 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           
                             ( 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   L 
                                   2 
                                 
                               
                               - 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   L 
                                   3 
                                 
                               
                             
                             ) 
                           
                           / 
                           2 
                         
                         + 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             1 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   Therefore, even if the entire carriage is positionally changed in the direction perpendicular to the optical axis of a measurement beam, the displacement ΔL calculated as above is not affected by this positional change. 
   As described above, even if the laser interferometer length measuring apparatus  20  is displaced in the direction of the optical axis of a measurement beam, and/or is displaced in the direction perpendicular to the optical axis of a measurement beam, ΔL is not affected by these positional changes in this embodiment. Therefore, the apparatus in this embodiment is very robust against the runout of the rotational mechanism. 
   It should be noted that ΔL can be measured, for example, by arranging the displacement gauge only on the side of the retroreflecting means  12 . In this case, the displacement ΔL is calculated according to the following equation.
 
 ΔL=ΔL   2   +ΔL   1   (5)
 
   Herein, ΔL 2  and ΔL 1  are defined in the same way as in Equation (1). 
   In this case, since the number of displacement gauges to be used is one, the apparatus can be produced at low cost. 
   It should be apparent to those skilled in the art that the above-described embodiments are merely illustrative which represent the application of the principles of the present invention. Numerous and varied other arrangements can be readily devised by those skilled in the art without departing from the spirit and the scope of the invention.