Abstract:
A laboratory system has demonstrated the measurement of three degrees of vibrational freedom simultaneously using a single beam through heterodyne speckle imaging. The random interference pattern generated by the illumination of a rough surface with coherent light can be exploited to extract information about the surface motion. The optical speckle pattern is heterodyne mixed with a coherent reference. The recorded optical data is then processed to extract three dimensions of surface motion. Axial velocity is measured by demodulating the received time-varying intensity of high amplitude pixels. Tilt, a gradient of surface velocity, is calculated by measuring speckle translation following reconstruction of the speckle pattern from the mixed signal.

Description:
GOVERNMENT INTEREST 
     The invention described herein may be manufactured, used, sold, imported, and/or licensed by or for the Government of the United States of America. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to coherent optical vibration sensing, specifically for observation of three-degrees of freedom using temporal heterodyne speckle imaging. 
     STATEMENT REGARDING PRIOR DISCLOSURES BY THE CO-INVENTORS 
     An article co-authored by James Perea and Brad Libbey, (the co-inventors), entitled, “Development of a heterodyne speckle imager to measure 3-degrees of vibrational freedom,” was submitted to The Optical Society earlier in 2016 for publication in an Optical Express publication in 2016, the publication date TBD. 
     BACKGROUND OF THE INVENTION 
     Coherent optical vibration sensors have been investigated for use in numerous applications including strain measurements, equipment diagnostics, medical imaging, and seismic sensing. Various techniques are utilized to observe surface motion. These techniques include, but are not limited to heterodyne laser Doppler vibrometry (e.g., U.S. Pat. No. 4,834,111 A Khanna et al.) for observation of surface velocity in the axial dimension of the interrogation beam, shearography (e.g., U.S. Pat. No. 5,011,280 A Hung) for observation of the gradient of displacement in two dimensions, electronic speckle pattern interferometry (e.g., U.S. Pat. No. 4,018,531 A Leendertz) for in-plane or out of plane displacement or out of plane displacement gradients, and speckle pattern imaging for out of plane displacements. The techniques listed are generally used for observation of one or two degrees of freedom. Variations using multiple coherent beams (e.g., U.S. Pat. No. 7,242,481 B2 Shpantzer et al.) have been used to observe three degrees of freedom. 
     It is of interest to simultaneously observe three-degrees of freedom using a single coherent beam. 
     The current invention combines elements of heterodyne Doppler vibrometry and digital speckle photography along with signal processing routines to simultaneously observe velocity in the axial dimension of the interrogation beam and out of plane tilts of the illuminated region. This provides the ability to observe three-degrees of freedom using a single coherent beam. 
     SUMMARY OF THE INVENTION 
     The disclosure relates to measuring the motion of a surface plane by imaging optical speckle that has been modulated by a heterodyne process. A process takes the sequence of heterodyne images and extracts motion with three degrees of freedom. The device takes advantage of a probing optical field that illuminates the rough surface. The spatial optical field is described based on previous work. (See, e.g., P. Hariharan, Basics of Interferometry, Academic Press, 2007; P. Jacquot and J. M. Fournier, Interferometry in Speckle Light: Theory and Applications, Springer, 2000; H. J. Tiziani, “Analysis of Mechanical Oscillations by Speckling,” Applied Optics, vol. 11, no. 12, pp. 2911-2917, 1972; and R. Jones and C. Wykes, “Holographic and Speckle Interferometry,” Cambridge University Press, 1983.) These models are typically independent of time, but this invention takes advantage of surface motion that affects the probe beam by causing a phase shift due to path length change over time. The invention adds an optical frequency shift to observe these phase and amplitude shifts creating a heterodyne system. 
     The moving surface alters the path length of a probe beam in time. To aid in understanding this invention, it is useful to define the geometry of this moving surface. This surface is described in a {ξ,η,z} coordinate system, where ξ and η are in the surface plane and z is orthogonal to the plane. The sensor can determine two types of motion; out of plane pistoning in the axial direction z, and tilting of the plane in two directions. Out of plane motion results in a Doppler phase shift and tilting results in a translation of the speckle image. (See, e.g., R. Jones and C. Wykes, “Holographic and Speckle Interferometry,” Cambridge University Press, 1983, incorporated herein by reference.) A single dimension of tilting θ is presented to simplify illustration, however, the addition of a second tilting degree-of-freedom φ parallels the picture below and is superimposed in the section detailing the invention. The moving surface has a pivot point that is allowed to translate in the z direction, but is otherwise located at a fixed position ξ=ξ 0 . The position of a scattering point P 1 , originally on the z=0 plane at {ξ,0,0}, will be moved to a new location P 2 ,  FIG. 1 . This motion extends the path length of reflected light; the path length is used to determine the speckle image resulting from these two locations in a method paralleling Tiziani. (See, e.g., H. J. Tiziani, “Analysis of Mechanical Oscillations by Speckling,” Applied Optics, vol. 11, no. 12, pp. 2911-2917, 1972, incorporated herein by reference.) 
     In one aspect, an exemplary heterodyne speckle imaging sensor system is disclosed. In another aspect, an exemplary method for simultaneous observation of three-degrees of vibrational freedom is disclosed based on a heterodyne speckle imaging sensor system. 
     Finally, an exemplary signal processing method is disclosed for computing three-degrees of vibrational freedom using a heterodyne beam. Such a signal processing method comprises the steps of accessing a complete measurement of temporal spatial irradiance data obtained using a heterodyne beam as stored as pixel frames of image data on a computer; high-pass filtering each pixel independently to isolate heterodyne signal from the complete measurement to output heterodyne information at a local oscillator frequency; applying a computation process for sequential estimates of a detected probe speckle pattern amplitude to compute two dimensions of tilt based on time sequences; applying a parallel demodulation process based on an arctangent demodulation for Doppler shifted heterodyne signals applied to each pixel to calculate z-axis velocity; and displaying a representation of the computed tilt based on time sequences and the calculated z-axis velocity on a data display device. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Additional advantages and features will become apparent as the subject invention becomes better understood by reference to the following detailed description when considered in conjunction with the accompanying drawings wherein: 
         FIG. 1  shows an exemplary incident beam and moving surface geometry used to describe the three-dimensional vibration sensing method. 
         FIG. 2  shows an exemplary heterodyne speckle imaging sensor system with a dynamic diffuse-scatterer for the target. 
         FIG. 3  shows an exemplary method for simultaneous observation of three-degrees of vibrational freedom using a heterodyne beam, e.g., of a heterodyne speckle imaging sensor system shown in  FIG. 2 . 
         FIG. 4  shows an exemplary process for signal processing a sensor observation of three-degrees of vibrational freedom using a heterodyne beam as exemplified in  FIG. 3 . 
     
    
    
     DETAILED DESCRIPTION 
     Before entering into the detailed description of one embodiment of the present invention according to the accompanying Figures, the theory of the present invention will be explained hereinafter. 
       FIG. 1  shows an exemplary incident beam and moving surface geometry  100  used to describe the three-dimensional vibration sensing method. The object plane, z=0, is allowed to rotate about a pivot { 0 ,η,0} and translate along the z-axis. An incident electromagnetic beam contacts {ξ P ,0,0} at angle β before tilting. As the object plane rotates by angle θ, the incident electromagnetic beam impinges on a new spatial location, located distance d 1  away along the axis of the beam. The image plane is located a sufficient distance from the object plane to be considered in the far field. 
     The variation in path length ΔP due to tilt and translation is a combination of the incoming and outgoing light. The incoming light is assumed to be coherent and planar at an angle β relative to the surface plane. The image plane is assumed to be sufficiently far such that outgoing light is orthogonal to the surface plane. The path variation is geometrically determined,
 
Δ P=d   1   +d   2   −d   3   −d   4  
 
Δ P =(ξ−ξ 0 )[sin(β)(1−cos(θ))+(1+cos(β))sin(θ)]− d   4 (1+cos(β))
 
where θ is angular change due to the tilting surface to which a small angle assumption is applied sin(θ)≈θ,
 
                 cos   ⁡     (   θ   )       ≅     1   -       θ   2     2         ,         
and θ 2 &lt;&lt;θ. This assumption leads to a simplified path length.
 
Δ P ≈(ξ−ξ 0 )θ(1+cos(β))− d   4 (1+cos(β))
 
     In the description below the direction of illumination, and hence the Doppler velocity, are assumed to be aligned with the z-axis, β=0.
 
Δ P ≈(ξ−ξ 0 )2 sin(θ)−2 d   4  
 
If we maintained the cos(β) term through the subsequent equations, the velocity would be measured along an axis that is acute with the plane. This increases the utility of the method to horizontal translation as well as pistoning.
 
     DETAILED DESCRIPTION OF THE INVENTION 
     An exemplary heterodyne speckle imaging system is represented in  FIG. 2 . Such an exemplary heterodyne speckle imaging sensor system  200  is shown comprising a laser source  201 ; a beam splitter  202 ; 750 mm bi-convex lens  203 ; a first mirror  204 ; a dynamic diffuse scatterer  205 ; 500 mm bi-convex lens  206 ; a second mirror  207 ; 250 mm plano-convex lens  208 ; a third mirror  209 ; a polarizing filter  210 ; a first acousto-optic modulator AOM1  211 , 80.01 MHz upshift; a second acousto-optic modulator AOM2 ( 212 ), 80 MHz downshift; a fourth mirror  213 ; 50 mm plano-convex lens  214 ; 50 μm diameter spatial filter  215 ; 250 mm plano-convex lens  216 ; another polarizing filter  217 ; another beam splitter  218 ; 64×64 pixel detector  219 ; a trans impedance amplifier  220  sampling at 30 kHz; and a signal processor and data display  221 . 
       FIG. 3  shows an exemplary method for simultaneous observation of three-degrees of vibrational freedom  300  using a heterodyne beam, e.g., of a heterodyne speckle imaging sensor system shown in  FIG. 2 . Referring now to  FIGS. 2 and 3 , the source  201  is a linearly polarized laser  301 . A splitter  202  divides the source into a probe and reference beams  302 . In the probe beam path, a single lens provides focus adjustment for  303  the beam  203  and mirror  204  redirects the beam to a dynamic object  205 ; the object has a diffuse surface. The electric field scatters  305  from the moving object  205  and now has a random spatial phase or amplitude. The purpose of the invention is to determine changes to the surface&#39;s position by tracking changes in the random, scattered electric field. As the object is tilted and translated, a phase shift is imparted due to the optical path change near the object plane  205 . Thus, the optical field at the object is modified by a phase associated with the new position of the object. 
                       u   O     ⁡     (     ξ   ,   η     )       →         u   O     ⁡     (     ξ   ,   η     )       ⁢     ⅇ       j2πΔ   ⁢           ⁢   P     λ                 (   1   )               
Where λ is the optical wavelength.
 
     Some of the resulting scattered radiation is collected  306  by a single lens  206 . The radiation is redirected  307  using a mirror  207  and propagates through a secondary optic  208  which defocuses  308  the field relative to the image plane  219 . An additional mirror  209  redirects the beam  309  through a dichroic polarizing filter  210  where the horizontally polarized radiation is absorbed  310 . The radiation from the probe beam propagates to a second beam splitter  218  where half the radiation propagates through the splitter and half is redirected towards the focal plane array  219 . The probe-leg electric field at the image plane  219  can be described as U P (x,y,t 1 ), at one time, and U P (x,y,t 2 ) at a later time. Where U P (x,y,t 2 ) is a phase modified version of U P (x,y,t 1 ), the field at a previous frame. The relationship between the these fields is 
                       U   p     ⁡     (     x   ,   y   ,     t   2       )       =       ⅇ       j4π   ⁡     (       θξ   O     +     d   4       )       λ       ⁢     ⅇ       -   j2     ⁢           ⁢   kx   ⁢           ⁢   θ       ⁢     ⅇ     j   ⁢           ⁢   2   ⁢   kz   ⁢           ⁢     θ   2         ⁢       U   p     ⁡     (       x   -     2   ⁢   θ   ⁢           ⁢   z       ,     y   -     2   ⁢   ϕ   ⁢           ⁢   z       ,     t   1       )                 (   2   )               
Where k is the optical wavenumber. The invention makes use of this shifted and phase modified version of the original field to determine the object&#39;s motion; the spatial shift corresponds to tilting of the surface  205  while the phase corresponds to the axial motion of the surface  205 .
 
     Following initial propagation through the beam splitter  202 , the reference beam propagates through two acousto-optic modulators AOM1 ( 211 ) and AOM2 ( 212 ). AOM1  211  upshifts the frequency of the electromagnetic field  311  by a specified amount AOM2  212  downshift the frequency  312  by an amount less than the upshift producing a modest frequency offset of the reference field. A mirror  213  redirects the reference beam  313  where it propagates through a focusing lens  214  to focus the beam  314 . An aperture  215  is placed at the focal point of  214 , acting as a spatial filter to spatial filter the beam  315 . The diverging beam propagates to another lens  216  which collimates the expanded beam  316 . The resulting electromagnetic radiation propagates through a dichroic polarizing filter  217  where the horizontally polarized radiation is absorbed  317 . The reference field propagates to the beam splitter  218  where half the radiation transmits through the splitter. 
     The device uses the same propagation distance, the distance between splitter mirror  202  and beam splitter  218 , on both probe and reference fields. The beam splitter  218  additively combines the probe and reference electric fields  318  in the region between splitter  218  and the focal plane array  219 . The total field U t  at the image plane is the superposition of the probe and reference fields.
 
 U   t ( x,y,t )= U   r ( x,y,t )+ U   p ( x,y,t )  (3)
 
The focal plane array  219  transduces the irradiance of these two fields  319  into an electrical charge proportional to the irradiance.
 
 I ( x,y,t )= U   t   U   t   *=U   r   U   r   *+U   m   U   m   *+U   r   U   m   *+U   r   *U   m   (4)
 
                     I   ⁡     (     x   ,   y   ,   t     )       =       1       λ   2     ⁢     z   2         [       R   2     +          P        2     +     2   ⁢           ⁢   R   ⁢        P        ⁢   cos   ⁢           ⁢     (         ω   LO     ⁢   t     +         4   ⁢   π     λ     ⁢     (         θ   ⁡     (   t   )       ⁢     ξ   0       +       ϕ   ⁡     (   t   )       ⁢     η   0       +       d   4     ⁡     (   t   )         )       -     α   p       ]                   (   5   )               
Where R and |P| are the magnitudes of the reference and probe field respectively. The combined irradiance is the sum of three terms: 1) irradiance of the reference R 2 , 2) irradiance of the probe |P| 2  and 3) the mix term R|P|.
 
     The system relies on a heterodyne component where the reference field strength amplifies the probe field. This is possible since the reference field is frequency shifted at a frequency equal to ω LO , the difference frequency of AOM1  211  and AOM2  212 . The magnitude 2R|P| of the heterodyne irradiance shifts spatially in direct proportion to the surface tilt  205 . The phase terms associated with cos(ω LO t+ψ) contain sufficient information to compute the z-axis velocity of the surface. This phase term modulates the local oscillator frequency, ω LO . The phase is 
                     ψ   ⁡     (   t   )       =           4   ⁢   π     λ     ⁢     (         θ   ⁡     (   t   )       ⁢     ξ   O       +       ϕ   ⁡     (   t   )       ⁢     η   O       +       d   4     ⁡     (   t   )         )       -     α   p               (   6   )               
where the term containing d 4  is the Doppler shift due to translation of the object in the z dimension. The terms containing θ and φ are Doppler shift due to the tilting motion. It occurs in this expression because the pivots do not intersect the origin and the illumination is assumed to be centered at the origin. In other words, this term represents motion in the z direction based on the amount of tilt and the relative distance between the pivot point and the center of illumination. The two terms just described do not depend on spatial position on the image plane. In contrast α p (x,y) is the phase of the probe speckle pattern and will shift in a manner similar to the magnitude of the speckle pattern.
 
     A trans-impedance amplifier circuit  220  converts charge from the focal plane array  219  into a digital representation  320  of the irradiance on a computer  221 . The acquisition is repeated sequentially in time at a frame rate sufficient to observe the intermediate carrier frequency resulting from the upshift and downshift from the acousto-optic modulators  211  and  212 . The focal plane array also has sufficient pixels to observe speckles caused by light reflected from a diffuse target. 
     The sequence of image frames captured on the computer  221  undergoes a process on the same computer  321  to calculate tilt in the x and y directions, Δθ and Δφ respectively, as well as axial velocity in the z direction. The process is outlined in  FIG. 4 , with two process sequences. 
       FIG. 4  shows an exemplary process  400  for signal processing a sensor observation of three-degrees of vibrational freedom using a heterodyne beam. The process to produce two dimensions of tilt is based on sequential estimates of the probe leg&#39;s speckle pattern amplitude. First, the image data stored on the computer  221  is accessed  401 . The heterodyne signal is isolated from the complete measurement by high-pass filtering  402  each pixel independently. This filter removes R 2  and |P| 2  from the camera&#39;s representation of the optical field, equation 5. The filter  402  outputs the heterodyne information at the local oscillator frequency ω LO . A temporal Hilbert transform  403  extracts the envelope 2R|P| at each pixel. The output of  403  is a speckle pattern that shifts spatially for each subsequent frame. Each frame is cross-correlated  404  with its prior time frame producing a sequence of new data frames each containing a spatial peak. A peak&#39;s position relative to center of the frame corresponds to the number of pixels shifted between two image frames. A peak&#39;s location is estimated using a two dimensional parabolic spatial fit  405  on a cross correlation frame. In this way the peak location can capture sub-pixel shifts in the image pairs and consequently small amplitudes of tilt. The output of the parabolic fit  405  is a temporal sequence of peak locations applied in two dimensions, in other words, two temporal sequences of spatial shifts s x  and s y  between subsequent image frames. The spatial shifts are converted  406  to units of object tilt difference Δθ and Δφ between subsequent image frames,
 
Δθ= s   x /2 z   (7a)
 
Δφ= s   y /2 z.   (7b)
 
The tilt can then be displayed as two time sequences on a data display device  407 .
 
     A parallel sequence of the processing is used to calculate z-axis velocity. This demodulation process uses a standard arctangent demodulation for Doppler shifted heterodyne signals applied to each pixel. (See, e.g., B. K. Park, O. Boric-Lubecke, and V. M. Lubecke, “Arctangent demodulation with DC offset compensation in quadrature Doppler radar receiver systems,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 5, pp. 1073-1079, May 2007, incorporated herein by reference.) The output of filter  402  provides the heterodyne information at the local oscillator frequency ω LO  necessary to begin the demodulation. In-phase and quadrature  408  are calculated by multiplying each pixel by the sine and cosine of the local oscillator frequency ω LO . The resulting in-phase I demod  and quadrature Q demod  time sequences are low pass filtered  409  to remove unwanted components at frequencies greater than ω LO .
 
 Q   demod ( t )= LP[HP[I ( t )] sin(ω LO   t )]  (8)
 
 I   demod ( t )= LP[HP[I ( t )] cos(ω LO   t )]  (9)
 
The in-phase and quadrature terms are then processed in block  410  that contains mathematical equation 10 and produces an estimate of the Doppler phase ψ at each pixel
 
                   ψ   ≈       unwrap   ⁡     [       tan     -   1       ⁡     (         Q   demod     ⁡     (   t   )           I   demod     ⁡     (   t   )         )       ]       .             (   10   )               
The velocity for each pixel is then estimated  411  using the Doppler phase.
 
                   v   =       λ     4   ⁢   π       ⁢       (         ψ   2     -     ψ   1           t   2     -     t   1         )     .               (   11   )               
The optical component  208  produces defocused images, as a result, estimates of velocity at each pixel should be equal. However, destructive interference from the diffuse scattering source  205  produces some low amplitude field strengths from the probe that are not accurately captured with the camera  219  and  220 . The velocity estimates associated with low amplitude pixels will not produce accurate estimates of velocity. Process step  412  removes low signal strength pixels. The remaining pixels are averaged  413  to reduce noise. The output is displayed  414  as a single velocity time series representing axial velocity in the z direction.
 
     It is obvious that many modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as described.