Invited Review Paper
OPTICAL REVIEW Vol. 21, No. 6 (2014) 849–861
Spatial Statistical Optics and Spatial Correlation Holography: A Review Mitsuo TAKEDA1;3 , Wei WANG2 , Dinesh N. NAIK3 , and Rakesh K. SINGH4 1
Center for Optical Research and Education (CORE), Utsunomiya University, Utsunomiya 321-8585, Japan Department of Mechanical Engineering, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, U.K. 3 Institute fu¨r Technische Optik, Universita¨t Stuttgart, 70569 Stuttgart, Germany 4 Department of Physics, Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala 695547, India 2
(Received July 1, 2014; Accepted August 8, 2014) Classical statistical optics is revisited with the aim of introducing the concept of spatial statistical optics that places particular focus and special emphasis on spatial statistics of the optical field, rather than on their temporal statistics. The principles of emerging technology of statistical correlation holography based on spatial statistical optics are reviewed, and their unique capabilities for coherence and polarization shaping as well as synthesizing stochastic optical fields with desired statistical properties are introduced. # 2014 The Japan Society of Applied Physics Keywords: statistical optics, coherence, holography, polarization, interferometry, correlation
1.
Introduction
Highly Frequencystabilized Laser Light
Be it classical or quantum, optical fields are intrinsically of statistical nature. Statistical optics deals with stochastic optical fields that vary randomly in time and/or in space. This review revisits classical statistical optics, and introduces the concept of spatial statistical optics and its applications for correlation holography that can create synthetic optical fields with desired statistical properties in space. Spatial statistical optics, which may be an unfamiliar term for most readers, is a system of statistical optics that places particular focus and special emphasis on spatial statistics of the optical field rather than on their temporal statistics. To give an idea about what is spatial statistical optics, let us start with its counter concept, namely temporal statistical optics, in conventional statistical optics, which may be familiar to most readers. The ensemble average forms the mathematical basis of statistical optics.1,2) Since the ensemble average is a conceptual quantity defined mathematically in terms of the probability assigned to an ensemble, it needs to be associated with a physical quantity that is observable by experiment. It is common practice to replace the ensemble average with the time average, assuming that the statistical field is stationary and ergodic in time.1–3) This assumption is justifiable in many cases of practical interest, such as partially coherent illumination in microscopy and thermal light analyzed by Fourier spectroscopy. Such a system of traditional statistical optics that is premised on temporal stationarity and temporal ergodicity may be called here temporal statistical optics to distinguish it from spatial statistical optics to be introduced in this review. With the advancement of laser technology, increasing use has been made of extreme optical fields that are strongly confined either on the time axis or on the optical frequency axis in the time-frequency phase space as shown in Fig. 1. For such optical fields, traditional temporal
ω
δ (ω −ω 0 )
Phase Space
t
Ultra-short Light Pulse
δ (t − t0 )
Fig. 1. (Color online) Optical fields that are temporally or spectrally confined in phase space.
statistical optics can no longer serve as a useful tool. For example, an ultra-short optical pulse, with its optical field strongly confined on the time axis, does not have temporal stationarity required for temporal statistical optics to be valid. On the other hand, for ideally stabilized monochromatic continuous wave (CW) laser light with its optical frequency spectrum tightly confined on the optical frequency axis, time averaging does not make sense because the optical field does not fluctuate with time (at least in classical sense). Thus the traditional framework of temporal statistical optics is not applicable to such extreme optical fields. Turning now our attention to the spatial variation of the optical fields, we note that such light, irrespective of the temporal waveform being a CW or a short optical pulse, creates spatially random optical fields when scattered by a diffusive object. Spatial statistics of the scattered fields provides useful information about the object, as best exemplified by speckle phenomena applied to optical metrology.4–6) This motivates this review to shed light on spatial statistical optics in which the ensemble average is replaced by the spatial average, which
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permits the statistical analysis of instantaneous or timefrozen optical fields that cannot be dealt with by temporal statistical optics. From the formal symmetry of space and time in the wave equation, this exchange between space and time in statistical optics appears to be a natural course of extension in the sense of analogy. Nonetheless, spatial statistical optics is much less common than temporal statistical optics. This is because the assumption of spatial stationarity and spatial ergodicity for spatially distributed random optical fields involves subtle problems that are yet to be clarified. Recently two kinds of techniques of unconventional holography, called coherence holography7–11) and photon correlation holography12) have been proposed and experimentally demonstrated. Coherence holography reconstructs a holographically recorded 3D object from the second-order spatial field correlation (or the spatial coherence function), and photon correlation holography reconstructs the 3D object from the forth-order field correlation (or the spatial intensity correlation) of the scattered optical fields. More recently, the principle of coherence holography for scalar optical fields has been extended to polarized vectorial fields by the techniques called vectorial coherence holography13) and Stokes holography.14) They all together may be referred to as correlation holography. Implicit in the principle of correlation holography is that the scattered optical fields are spatially stationary. In this review, we first address the issues of spatial stationarity of scattered optical fields to form the theoretical basis for correlation holography.15) We identify the optical geometry that permits ensemble average to be replaced by space average, which will be the first step toward the establishment of more general spatial statistical optics. Next we will review the principle of coherence holography and its extension to vector fields. Finally we introduce the technique of photon correlation holography that is based on speckle intensity correlation. Admittedly it is beyond the ability of the authors to present a well-balanced review that covers all the work relevant to the subject without bias. We therefore abandon such efforts from the outset, and restrict ourselves to a review that focuses on some of our recent efforts toward the establishment of a new realm of spatial statistical optics. 2.
Spatial Stationarity of Scattered Optical Fields
Since the principles of correlation holography are based on the spatial stationarity of the correlation function of quasi-monochromatic optical fields, we restrict our discussions to wide-sense spatial stationarity, i.e., spatial stationarity in the correlation of the quasi-monochromatic optical fields,15) which is a less strict condition than spatial ergodicity. Figure 2 shows geometry of a source plane (or a diffusive object plane), a linear optical system, and an observation plane. Light vibrations from point sources [each having complex amplitude uð^rÞ at point r^ on the source plane] are passed through the optical system and are superposed to create an optical field
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G (r1 , rˆ1)
u (rˆ1 )
u (r1 )
rˆ1
r1
ˆ O
O
rˆ2
u (rˆ2 )
Source Plane
z
r2 u (r2 )
G (r2 , rˆ2)
Optical System
Observation Plane
Fig. 2. (Color online) Geometry of source plane, optical system, and observation plane.
Z uðrÞ ¼
Gðr; r^Þuð^rÞ d^r
ð1Þ
at point r on the observation plane, where Gðr; r^Þ is the Green’s function or the impulse response of the optical system. In all equations, the area of integration for the variable r^ is over the entire source plane unless otherwise stated, and the time dependence of the field is not shown explicitly with time variable t unless needed. The coherence function (or the second-order correlation) at the observation plane is given by ZZ G ðr1 ; r^ 1 ÞGðr2 ; r^2 Þhu ð^r1 Þuð^r2 Þi d^r1 d^r2 ; hu ðr1 Þuðr2 Þi ¼ ð2Þ where h i and denote, respectively, ensemble average and complex conjugate. 2.1 Wide-sense stationary source Let us first assume that the source (or the field passed by a ground glass) is wide-sense stationary in space, i.e., the spatial correlation function is invariant to the shift of the source point by r^s such that hu ð^r1 þ r^ s Þuð^r2 þ r^s Þi ¼ hu ð^r1 Þuð^r2 Þi:
ð3Þ
We also assume that the optical system is shift invariant such that Gðr þ rs ; r^ þ rs Þ ¼ Gðr; r^Þ;
ð4Þ
where the coordinates are scaled to make magnification unity so that we can put r^s ¼ rs . Then the shift of a pair of the observation points by rs gives hu ðr1 þ rs Þuðr2 þ rs Þi ZZ ¼ G ðr1 þ rs ; r^1 ÞGðr2 þ rs ; r^ 2 Þhu ð^r1 Þuð^r2 Þi d^r1 d^r2 ZZ ¼ G ðr1 ; r^ 1 rs ÞGðr2 ; r^2 rs Þ hu ð^r1 rs Þuð^r2 rs Þi d^r1 d^r2 ZZ ¼ G ðr1 ; r~ 1 ÞGðr2 ; r~2 Þhu ð~r1 Þuð~r2 Þi d~r1 d~r2 ¼ hu ðr1 Þuðr2 Þi; ð5Þ where we have put r~ ¼ r^ rs , and use has been made of the relations Eqs. (3) and (4). Thus we have shown that the combination of a source with wide-sense spatial stationarity
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and a system with shift invariance produces optical fields that have wide-sense spatial stationarity. This result is general in the sense that no specific form of the Green’s function is assumed other than its shift invariance. However, the condition of wide-sense spatial stationarity of the source is too restrictive to be applied for a wide class of practical source structures such as those used in correlation holography. For example, in correlation holography to be explained in the next section, the source plane is formed by a hologram illuminated by spatially incoherent light such that hu ð^r1 Þuð^r2 Þi ¼ Ið^r1 Þð^r1 r^2 Þ:
ð6Þ
In correlation holography, Ið^r1 Þ is made proportional to the intensity transmittance of the hologram, which generally has a complicated spatial structure formed by a fringe pattern and has a limited spatial extent. This means that such a source of practical interest is a typical non-stationary source. 2.2 Non-stationary source Now our interest is whether we can or cannot generate spatially stationary optical fields from a spatially nonstationary source. We will next show that this is possible by choosing an appropriate Green’s function. Substituting Eq. (6) into Eq. (2), we arrive at a general form of the van Cittert–Zernike theorem ZZ hu ðr1 Þuðr2 Þi ¼ G ðr1 ; r^1 ÞGðr2 ; r^2 ÞIð^r1 Þð^r1 r^2 Þ d^r1 d^r2 Z ¼ G ðr1 ; r^ÞGðr2 ; r^ÞIð^rÞ d^r: ð7Þ
2.2.1 Diffraction by Fresnel kernel To be more specific, let us consider free space propagation. Referring to Fig. 1, the Green’s function is given by 1 expðikjz þ r r^ jÞ Gðr; r^Þ ¼ i jz þ r r^j expðikzÞ jrj2 2r r^ þ j^rj2 exp ik ; ð8Þ iz 2z where and k ¼ 2= are, respectively, the mean wavelength and the mean wavenumber of the quasimonochromatic light, and a common Fresnel approximation has been made assuming that the distance z between the source and observation plane is large compared with the size of the source and the observation area of interest. Substituting Eq. (8) in Eq. (7), we have hu ðr1 Þuðr2 Þi Z 1 jr2 j2 jr1 j2 2ðr2 r1 Þ r^ 2 2 Ið^rÞ exp ik d^r; 2z z ð9Þ which has a more familiar form of the van Cittert–Zernike theorem similar to the Fresnel diffraction formula. For convenience of the further discussions, we introduce the change of variables r ¼ ðr1 þ r2 Þ=2; r ¼ r2 r1 , r1 ¼ r r=2; r2 ¼ r þ r=2;
Let us examine if the field correlation given by Eq. (11) has spatial stationarity. We can readily show Z 1 r ðr þ rs Þ i2r r^ Ið^rÞ exp hu ðr1 þ rs Þuðr2 þ rs Þi 2 2 exp ik d^r z z z r rs ¼ exp ik hu ðr1 Þuðr2 Þi: z
2.2.2 Diffraction by Fourier kernel Is there an optical system that can realize perfect widesense spatial stationarity in the scattered field with the non-stationary phase factor in Eq. (12) being removed? To
ð10Þ
and rewrite Eq. (9) as
hu ðr1 Þuðr2 Þi ¼ hu ðr r=2Þuðr þ r=2Þi Z 1 jr þ r=2j2 jr r=2j2 2r r^ 2 2 Ið^rÞ exp ik d^r 2z z Z 1 r r i2r r^ Ið^rÞ exp d^r: ¼ 2 2 exp ik z z z
This result shows that, apart from the phase factor expðikr rs =zÞ, the fields in the observation plane have the coherence function invariant to the shift, and one may regard the fields as wide-sense stationary if one is observing only the modulus of the coherence function, which is frequently the case in coherence holography. Strictly, however, the field created by Fresnel diffraction is not wide-sense stationary because of this non-stationary phase factor.
851
ð11Þ
ð12Þ
answer this question, let us consider an ideal optical Fourier transform system formed by the so called f-f geometry of an aberration-free Fourier transform lens with a focal length f , and assume that the lens has a sufficiently large aperture so that the effect of the limited aperture is negligible for the range of the spatial frequency and the field of view of our interest. A point source on the source plane creates a plane wave in the observation plane expðikf Þ r r^ Gðr; r^Þ ¼ exp ik : ð13Þ if f Note that the Green’s function with the Fourier kernel in Eq. (13) is not shift invariant by itself. However, the
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complex conjugate kernel product in the correlation integral does have shift invariance such that 1 ðr2 r1 Þ r^ G ðr1 ; r^ÞGðr2 ; r^Þ ¼ 2 2 exp ik f f ¼ G ðr1 þ rs ; r^ÞGðr2 þ rs ; r^Þ: ð14Þ Therefore from Eq. (5) we have hu ðr1 þ rs Þuðr2 þ rs Þi ZZ ¼ G ðr1 þ rs ; r^1 ÞGðr2 þ rs ; r^2 Þhu ð^r1 Þuð^r2 Þi d^r1 d^r2 ZZ ¼ G ðr1 ; r^1 ÞGðr2 ; r^ 2 Þhu ð~r1 Þuð~r2 Þi d~r1 d~r2 ¼ hu ðr1 Þuðr2 Þi:
ð15Þ
Thus we have shown that, for the optical Fourier transform system or the Fraunhofer diffraction geometry, the fields in the observation plane have perfect wide-sense spatial stationarity despite the fact that the source lacks in spatial stationarity and the Green’s function does not have shift invariance. Note that this result is obtained from Eq. (5) without assuming incoherent illumination expressed by Eq. (6). For the case of spatially incoherent illumination, we have from Eqs. (7) and (14) Z 1 2ir r^ hu ðr1 Þuðr2 Þi ¼ 2 2 Ið^rÞ exp d^r f f ¼ ðrÞ;
ð16Þ
which gives the van Cittert–Zernike theorem for Fraunhofer diffraction with the Fourier kernel. 3.
Replacing Ensemble Average with Spatial Average
Starting from Eq. (2), our discussions so far have been based entirely on the ensemble average. For a wide class of optical fields (except those strongly confined in the phase space), it is reasonable to assume stationarity and ergodicity in time and replace the ensemble average with the time average. As we have seen in the previous section, most sources of practical interest are not spatially stationary and their diffraction fields are generally not stationary in space. However, we have found that the Fourier kernel can create a wide-sense spatially stationary field even from a spatially non-stationary source. It is therefore of interest to examine the possibility of replacing the ensemble average with the spatial average for the Fourier kernel, even though we are aware that the wide-sense spatial stationarity does not guarantee spatial ergodicity. For this purpose, we again use the change of variables of Eq. (10) and modify Eq. (2) as hu ðr1 Þuðr2 ÞiS Z ¼ u ðr r=2Þuðr þ r=2Þ dr Z Z Z ¼ G ðr r=2; r^ 1 ÞGðr þ r=2; r^2 Þ dr u ð^r1 Þuð^r2 Þ d^r1 d^r2 ;
ð17Þ
where h iS denotes spatial averaging performed by integration over the observation plane.
3.1 Space averaging for Fourier kernel For the Fourier kernel in Eq. (13), we have Z G ðr r=2; r^1 ÞGðr þ r=2; r^2 Þ dr 1 2r ð^r2 þ r^1 Þ=2 ¼ 2 2 exp i f f Z 2r ð^r2 r^1 Þ exp i dr f 2r ð^r2 þ r^ 1 Þ=2 ¼ exp i ð^r2 r^1 Þ; f
ð18Þ
where ð^r2 r^1 Þ is a Dirac delta function originating from the integration over the observation plane. Substituting Eq. (18) into Eq. (17), and noting that the delta function chooses r^1 ¼ r^2 ¼ r^ through integration with variables r^ 1 and r^2 over the source plane, we arrive at Z hu ðr1 Þuðr2 ÞiS ¼ u ðr r=2Þuðr þ r=2Þ dr ZZ 2r ð^r2 þ r^1 Þ=2 ¼ exp i f ð^r2 r^1 Þu ð^r1 Þuð^r2 Þ d^r1 d^r2 Z 2r r^ ¼ Ið^rÞ exp i d^r; ð19Þ f where Ið^rÞ ¼ u ð^rÞuð^rÞ is the instantaneous intensity distribution over the source plane. Equation (19), which has the same form as Eq. (16), is the spatial-average version of the van Cittert–Zernike theorem in which the ensemble average is replaced by the spatial average over the observation plane. One may note that Eq. (19) can also be interpreted as the Wiener–Khinchin theorem16) based on spatial averaging. Whereas Eq. (16) was derived from the ensemble average plus the assumption of spatially incoherent illumination expressed by Eq. (6), Eq. (19) has been obtained from the spatial average without assuming incoherent illumination. 3.2 Space averaging for Fresnel kernel In Sect. 2.2.1, we have seen that the fields of Fresnel diffraction from a non-stationary source generally do not have wide-sense spatial stationarity because of the nonstationary phase factor expðikr rs =zÞ in Eq. (12). Therefore we cannot replace the ensemble average with the space average. However, let us note the anisotropic characteristic of this non-stationary phase factor. This phase factor is made of the inner product r rs that disappears if the direction of the shift vector rs is orthogonal to the correlation vector r. In other words, the fields are wide-sense stationary in the particular direction of the shift that satisfies r rs ¼ 0. Making use of this anisotropic characteristic of the nonstationary phase factor, we introduce a directional spatial average by integration along the direction for which r rs ¼ 0. We decompose r into two orthogonal components r ¼ rs þ r0 (with r0 and rs being parallel and normal to the correlation vector r, respectively), and modify Eq. (17) as
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hu ðr1 Þuðr2 ÞiS ¼
M. TAKEDA et al.
Z
u ðr r=2Þuðr þ r=2Þ dr Z Z Z ^ ^ ¼ G ðrs þ r0 r=2; r1 ÞGðrs þ r0 þ r=2; r2 Þ drs u ð^r1 Þuð^r2 Þ d^r1 d^r2 ;
853
ð20Þ
where spatial averaging is carried out by integration performed along the vector rs . For the Fresnel kernel defined by Eq. (8), we have Z G ðrs þ r0 r=2; r^1 ÞGðrs þ r0 þ r=2; r^ 2 Þ drs Z 1 r ð^r2 þ r^1 Þ ðj^r2 j2 j^r1 j2 Þ þ 2r0 ð^r2 r^1 rÞ 2rs ð^r2 r^1 Þ exp i ¼ 2 2 exp ik drs ð21Þ 2z z z r ð^r2 þ r^1 Þ ðj^r2 j2 j^r1 j2 Þ þ 2r0 ð^r2 r^ 1 rÞ ¼ exp ik ð^r2 r^1 Þ: 2z Substituting Eq. (21) into Eq. (20), we have Z hu ðr1 Þuðr2 ÞiS ¼ u ðr r=2Þuðr þ r=2Þ dr ZZ r ð^r2 þ r^1 Þ ðj^r2 j2 j^r1 j2 Þ þ 2r0 ð^r2 r^1 rÞ ¼ exp ik ð^r2 r^ 1 Þu ð^r1 Þuð^r2 Þ d^r1 d^r2 2z Z Z r r0 2r r^ r r 2r r^ Ið^rÞ exp i Ið^rÞ exp i ¼ exp ik d^r ¼ exp ik d^r; z z z z
ð22Þ
where we have made use of the relation r rs ¼ 0, which makes r r0 ¼ r r. Equation (22) gives the spatialaverage version of van Cittert–Zernike theorem for the Fresnel kernel, and has the same form as Eq. (11) that has been derived from the ensemble average. This means that for diffraction by the Fresnel kernel, the ensemble average can be replaced by a directional space average based on the spatial integration along the direction normal to the correlation vector. 4.
Coherence Holography
Coherence holography7–11) is an unconventional holographic technique in which an object recorded in a hologram is reconstructed as the distribution of a complex spatial coherence function, rather than as the complex amplitude distribution of the optical field that represents the reconstructed image in conventional holography. Just as a computer-generated hologram (CGH) can generate a desired optical field, a computer-generated coherence hologram (CGCH) can synthesize a statistical field with a desired coherence function or a correlation function. Principle of coherence holography is based on formal analogy between the diffraction integral and the formula of van Cittert– Zernike theorem as found in Eqs. (9) and (16). As an illustrative example, Fig. 3(a) shows a coherence function that is visualized as a fringe contrast by using a radial shearing interferometer.17) From Eq. (16), the field on the Fourier plane, produced by a spatially incoherent circular source with uniform intensity Ið^rÞ, has a coherence function ðrÞ / 2J1 ðarÞ=ar, with J1 and a being the first order Bessel function of the first kind and an appropriate scale factor, respectively. The modulus of the coherence function jðrÞj is detected from the fringe amplitude by using
(a)
(b) Fig. 3. Airy-pattern coherence function generated by a spatially incoherent circular source. (a) A coherence function visualized as a fringe contrast. (b) Modulus of the coherence function. (After Wang et al.17) with permission from SPIE.)
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M. TAKEDA et al. Fourier Hologram
Recording Lensless Fourier Hologram Fourier
(a)
Conventional Holography Reconstruction with Coherent Illumination
Optical Field
Fourier
Fourier
(b)
Fourier Transform Lens
Fig. 4. (Color online) (a) Recording of lensless Fourier-transform hologram. (b) Reconstruction with coherent illumination in conventional holography. (After Takeda18) with permission from the Japan Society of Applied Physics.)
Reconstruction througha Moving Ground Glass
Uniform Field with Temporal Fluctuation
No Information?
(a)
Information Retrieval by Field Correlation
(b)
Time Av.
Correlation by Time Averaging
Fourier
Fourier
Γ 12 = u1* u 2
Fig. 5. (Color online) (a) Reconstruction of a hologram with spatially incoherent illumination in coherence holography. (b) Retrieving image information from the coherence function with an interferometer. (After Takeda18) with permission from the Japan Society of Applied Physics.)
Fourier fringe analysis.18) As shown in Fig. 3(b), jðrÞj has the same Airy pattern as that created by diffraction from a circular aperture. This demonstrates the analogy between the van Cittert–Zernike theory and the diffraction formula. One step further, let us now replace the circular source (or aperture) with a Fourier hologram in which letters ‘‘Fourier’’ are recorded, for example, by lensless Fourier transform geometry, as shown in Fig. 4(a). In conventional holography, the hologram is coherently illuminated with a collimated beam, and the recorded image is reconstructed through a Fourier transform optical system (often referred to as f-f geometry), as shown in Fig. 4(b). In the Fourier transform plane, the reconstructed twin (original and conjugate) images with a bright central spot are directly
observable as the field intensity distribution. In coherence holography, the hologram is illuminated with a spatially incoherent quasi-monochromatic light, which is generated, for example, by destroying spatial coherence of a laser beam with a moving ground glass, as shown in Fig. 5(a). Then the twin images will disappear and will be replaced by a nearly uniform intensity distribution without structure. Does this mean that the information about the recorded object has been lost? Of course, it is not. As we see in Eq. (16), the object information recorded in the intensity transmittance Ið^rÞ of the Fourier hologram is encoded into the coherence function ðrÞ through the van Citttert–Zernike theorem. Therefore, we can reconstruct the recorded image by detecting the coherence function (or the correlation function) in the
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BS
L1
α=
L1
L2
(b)
L2
f1 f2
855
f2
f1
Sagnac radial shearing interferometer
(a)
(c)
Fig. 6. (Color online) Sagnac radial shear interferometer for the detection of a full-field spatial coherence function. (a) Schematic geometry of the interferometer. (b) Telescopic system introduces magnification and demagnification to counter propagating beams. (c) Radial shear proportional to the position vector.
hu ðrÞuð1 rÞi ¼ ðr; 1 rÞ ¼ ðr 1 rÞ ¼ ½ð 1 Þr ¼ ðrÞ:
Γ (Δ r)
I (rˆ) ∝ H (rˆ) (a)
(b)
Fig. 7. (Color online) (a) Intensity transmittance of the computer-generated coherence hologram in which a Greek letter is recorded. (b) The modulus of the coherence function that gives the image reconstructed from the coherence hologram. (After Naik et al.8) with permission from the Optical Society.)
observation plane with the aid of an appropriate interferometer, as shown in Fig. 5(b). One can reasonably assume that the moving ground glass introduces stationary quasi-ergodic time fluctuation. This permits one to replace the ensemble average in Eq. (16) with the time average carried out by the time integration over the response time of the detector. This is the basic principle of coherence holography. Shown in Fig. 6(a) is a Sagnac radial shearing interferometer that can detect the full-field 2D coherence function ðrÞ at a time.8) The field uðrÞ to be correlated is introduced into a beam splitter BS and is split into two counter propagating beams. Inside the Sagnac interferometer is a telescopic optical system composed of a pair of lenses L1 and L2 with different focal lengths f1 and f2 , which images the entering field uðrÞ onto its original location inside the beam splitter with the different magnifications ¼ f1 =f2 and 1 ¼ f2 =f1 for the counter propagating beams, as shown in Fig. 6(b). The different magnifications introduce a radial shear r ¼ ð 1 Þr between the two beams exiting from the beam splitter, which is proportional to the radial vector r as shown in Fig. 6(c). From the interference between the radially sheared beams uðrÞ and uð1 rÞ, we can detect the coherence function
ð23Þ
Remember that Ið^rÞ in Eq. (16) is proportional to the intensity transmittance of the Fourier transform hologram so that its Fourier transform ðrÞ gives the image reconstructed from the hologram. When writing ðr; 1 rÞ ¼ ðr 1 rÞ in Eq. (23), we have made use of the wide sense spatial stationarity of the optical field transformed by the Fourier kernel. In Fig. 7, (a) is the intensity transmittance of the computer-generated coherence hologram in which a Greek letter is recorded, and (b) is the modulus of the coherence function that gives the image reconstructed from the coherence hologram. Experimental details are found in Ref. 8. 5.
Spatial Statistical Optics for Polarized Optical Fields
5.1
Polarization speckles and spatial degree of polarization We review some of the spatial statistical properties of polarization-related speckle phenomena, with an introduction of a less known concept of polarization speckles and their spatial degree of polarization.17,20,21) As shown in Fig. 8, a linearly polarized fully coherent laser beam, with the vibration direction of electric field making an angle of 45 degrees to the x-axis, enters a polarization beam splitter PBS. The incident beam is split into two mutually orthogonal linearly polarized beams, referred to as X- and Y-polarized beams, which impinge on moving random phase diffusers GG1 and GG2. The lights scattered from the diffusers are Fourier transformed by lenses L1 and L2, and combined by a non-polarization beam splitter BS to form a paraxial beam propagating in the z-direction. The scattered field at location r on the Fourier plane and at time t is given by Eðr; tÞ ¼ Ex ðr; tÞ^x þ Ey ðr; tÞ^y;
ð24Þ
where Ex ðr; tÞ and Ey ðr; tÞ are x- and y-components of the scattered field, and x^ and y^ are unit vectors. If we detect the
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Y-polarized beam PBS Mirror 1 45 degree linearly polarized laser beam
Random phase diffuser (GG1) X-polarized beam
Lens L1 Polarization Speckles
Random phase diffuser (GG2)
Mirror 2 BS Lens L2 Fig. 8.
E ( r , t ) = E x ( r , t ) xˆ + E y ( r , t ) yˆ
(Color online) Optical geometry for generating polarization speckles.
speckle pattern17,20,21) to distinguish it from the conventional intensity speckle pattern observed by the polarization-blind detection scheme. Generally, the statistical properties of the two orthogonal field components are described by a 2 2 polarization matrix defined by " # hEx ðr; tÞEx ðr; tÞi hEx ðr; tÞEy ðr; tÞi J¼ ; ð26Þ hEy ðr; tÞEx ðr; tÞi hEy ðr; tÞEy ðr; tÞi where h i denotes the ensemble average. It is also a standard procedure to define the degree of polarization by P ¼ ½1 4 detðJÞ=ðtrðJÞÞ2 1=2 ;
Fig. 9. (Color online) Polarization speckles with random spatial variation of the state of polarization. (After Takeda et al.21) with permission from SPIE.)
intensity of the scattered field without using an analyzer, we have Iðr; tÞ ¼ E ðr; tÞ Eðr; tÞ ¼ jEx ðr; tÞj2 þ jEy ðr; tÞj2 ;
ð25Þ
which does not give any polarization-specific information other than the intensity superposition of the two time varying speckle patterns arising from the mutually orthogonal field components. However, if we detect the complex amplitudes of the two orthogonal field components, Ex ðr; tÞ and Ey ðr; tÞ, by using an appropriate polarimetric interferometer,22–24) we can visualize spatio-temporal variations of the state of polarization. When we stop the movement of the diffusers, we will observe a static speckle-like random pattern that represents the spatial variation of the polarization state as shown in Fig. 9. We call this spatial random pattern of polarization states (which becomes visible only with a polarization-sensitive detection scheme) as a polarization
ð27Þ
where ‘‘tr’’ and ‘‘det’’ are the trace and determinant operation, respectively. It is often the case that the ensemble average is replaced by the time average h iT with the assumption that the field is stationary and ergodic in time. The conventional degree of polarization, defined by Eq. (27) for the polarization matrix JT on the basis of the time average, will be called temporal degree of polarization PT to distinguish it from spatial degree of polarization PS to be introduced here. Because the two moving diffusers are statistically independent, we have hEx Ey iT ¼ hEy Ex iT ¼ 0, and, because of the 45 degree orientation of the polarizer, we have hEx Ex iT ¼ hEy Ey iT ¼ hIiT =2. Thus we have we have PT ¼ 0, which means that the field created by the two statistically independent moving diffusers is unpolarized. When the movement of the diffusers is stopped in order to observe polarization speckles, the fields become frozen in time to their states at the moment of the stop, and the time average operation on the elements of the polarization matrix in Eq. (26) will be removed. In this case, we have detðJT Þ ¼ Ex Ex Ey Ey Ex Ey Ey Ex ¼ 0, so that the temporal degree of polarization becomes unity PT ¼ 1, which means fully polarized. Even if the diffusers are moving, a similar situation can occur if the incident laser light is a short optical pulse during which the movement of the diffuser is negligible. Polarization speckles manifest themselves as spatial random variations of the state of polarization of such
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Fig. 10. (Color online) Spatial degree of polarization PS and distribution of polarization states. (a) Polarization map and (b) distribution of polarization states on Poincare sphere for PS ¼ 0:11. (c) Polarization map and (d) distribution of polarization states on Poincare sphere for PS ¼ 0:95. (After Wang et al.20) with permission from SPIE.)
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fully polarized static (or time-frozen) fields. Our interest here is in the static (or instantaneous) random spatial distribution of the state of polarization of the fully polarized optical fields. To describe the statistical property of such fields, we replace the ensemble average h i in the polarization matrix with the spatial average h iS , and denote the spatial average version of the polarization matrix by JS . This is based on the grounds that the field transformed by the Fourier kernel is at least wide-sense stationary in space though not necessarily spatially ergodic. In Fig. 10, (a) and (b) show, respectively, part of the polarization map and the distribution of polarization states on the Poincare sphere for the spatial degree of polarization PS ¼ 0:11; (c) and (d) show the corresponding figures for PS ¼ 0:95. The spatial degree of polarization can be controlled by changing power ratio of the x- and y-polarized beams through the variation of the vibration angle of the linearly polarized light incident on the polarization beam splitter. As seen from the figure, spatial degree of polarization is related to the degree of order or disorder of the spatial distribution of polarization states. With the help of knowledge established for temporal statistics of Stokes parameters,25,26) spatial statistics of Stokes parameters has been studied both theoretically27) and by experiments.28) 5.2
Vectorial coherence holography based on spatial average Coherence holography in Sect. 4 is based on time averaging of scalar fields generated by a moving diffuser. The principle can be extended to spatial averaging of polarized vector fields generated by a static diffuser.13) This
yˆ
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(b) Fig. 11. (Color online) Two holograms for vectorial coherence holography. (a) Fourier transform CGH Hx ð^rÞ assigned for the x-component of the vector field. (b) Fourier transform CGH Hy ð^rÞ assigned for the y-component of the vector field. (After Singh et al.13) with permission from the Optical Society.)
extension provides a new technique to control both spatial coherence and polarization properties of the field based on spatial averaging. To this end, we use two separate coherence holograms, each of which is assigned to one of the orthogonal polarization components of the vectorial fields, as shown in Fig. 11. Equation (19), which gives a correlation function for scalar fields by spatial averaging, can be straightforwardly generalized to the spatial-average version of a 2 2 polarization coherence matrix W S for vectorial fields with elements given by: WijS ðrÞ ¼ hEi ðr1 ÞE j ðr2 ÞiS Z ¼ Ei ðr r=2ÞE j ðr þ r=2Þ dr ZZ 2r ð^r2 þ r^1 Þ=2 ¼ exp i f ð^r2 r^1 ÞEi ð^r1 ÞE j ð^r2 Þ d^r1 d^r2 Z 2r r^ ¼ Jij ð^rÞ exp i d^r; f
ð28Þ
where Jij ð^rÞ ¼ Ei ð^rÞE j ð^rÞ are elements of polarization matrix without being time averaged, and i and j represent
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one of the field components either x or y. Equation (28) is the spatial-average version of the van Cittert–Zernike theorem for vector fields,29) and serves as the basis of vectorial coherence holography for synthesizing a vectorial field with a desired coherence-polarization matrix. Two computer-generated Fourier holograms Hx ð^rÞ and Hy ð^rÞ, of which one records a numeral ‘‘0’’ and the other records a numeral ‘‘1’’ as an off-axis object, are created. The holograms are illuminated, respectively, with an x-polarized beam and a y-polarized beam, and are imaged onto a static birefringence-free ground glass in such a manner that the two holograms with orthogonal field components are superposed at the same location on the static ground glass, which corresponds to the source plane in Fig. 1. Thus the field immediately behind the ground glass is made to have the field intensities Jxx ð^rÞ ¼ Hx ð^rÞ for the x-component and Jyy ð^rÞ ¼ Hy ð^rÞ for the y-component. Because the Fourier holograms Hx ð^rÞ and Hy ð^rÞ serve as the source intensities Jxx ð^rÞ and Jyy ð^rÞ for the van Cittert–Zernike theorem in Eq. (28), we can reconstruct the recorded objects by S S detecting the field correlations Wxx ðrÞ and Wyy ðrÞ. If we were to choose time averaging using a rotating ground glass as in Sect. 4, we could reconstruct the image by direct field correlation using a radial shearing interferometer together with an analyzer. However, our interest is reconstruction based on spatial average using a static ground glass. For this purpose, we take an alternative approach of numerical field correlation. The elements of the coherence-polarization S S S S matrix, Wxx ðrÞ, Wyy ðrÞ, Wxy ðrÞ, and Wyx ðrÞ, are computed numerically from the field components Ex ðrÞ and Ey ðrÞ, which are detected with a spatial carrier frequency multiplex polarization interferometer22–24) combined with Fourier fringe analysis.18) In Fig. 12, (a), (b), (c) and (d) show, respectively, the moduli of the elements of S S the coherence-polarization matrix, jWxx ðrÞj, jWyy ðrÞj, S S jWxy ðrÞj, and jWyx ðrÞj, which shows that the coherencepolarization matrix of the scattered field has been controlled as intended, and thereby the use of spatial averaging is justified. 6.
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Fig. 12. (Color online) Images reconstructed from the element of coherence polarization matrix based on spatial averaging. (a) S S S S jWxx ðrÞj, (b) jWyy ðrÞj, (c) jWxy ðrÞj, and (d) jWyx ðrÞj. (After 13) Singh et al. with permission from the Optical Society.)
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Fig. 13. (Color online) Fourier transform CGH with intensity transmittance Hð^rÞ in which two objects, letters ‘‘’’ and ‘‘O’’ axially separated by small distance z ¼ 5 mm, are recorded. (After Naik et al.12) with permission from the Optical Society.)
Photon-Correlation Holography
Whereas the principle of coherence holography is based the second order field correlation (or the coherence function), photon correlation holography12) makes use of the fourth order correlations (or intensity-based photon correlations) of the optical field to reconstruct the object using an intensity interferometer. The relation between the two techniques is analogous to that between Michelson stellar interferometer30) and Hanbury Brown–Twiss intensity interferometer.31) To demonstrate 3-D reconstruction capability of photon-correlation holography, we create a Fourier transform CGH with intensity transmittance Hð^rÞ, in which two objects, letters ‘‘’’ and ‘‘O’’ axially separated by a small distance z ¼ 5 mm, are recorded as shown in Fig. 13. As shown in Fig. 14, the hologram is illuminated by laser light through a static or moving ground glass, and the field behind the hologram is optically Fourier transformed by lens L to create a stationary random field uðrÞ on the detection plane.
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Fig. 14. (Color online) Geometry for image reconstruction by photon-correlation holography. (After Naik et al.12) with permission from the Optical Society.)
From the central limit theorem, we may reasonably assume that the field uðrÞ obeys a complex circular Gaussian process. This permits us to express the fourth order field correlation
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Fig. 15. (Color online) Speckle patters created near the observation plane. (m) Full-field speckle intensity distribution recorded at z ¼ 0. The yellow square in the center indicates the location of the image sub area shown in (a)–(l). The images (a)–(l) are recorded at different plane locations ranging from z ¼ 6 to 5 mm with steps of 1 mm. (After Naik et al.12) with permission from the Optical Society.)
(or the intensity correlation) by the second order correlation (or the coherence function). The cross-covariance of the intensity distribution on the observation plane is given by CðrÞ ¼ hIðrÞIðr þ rÞi ¼ hðIðrÞ IÞðIðr þ rÞ IÞi ¼ hIðrÞIðr þ rÞi I2 ¼ hu ðrÞuðrÞu ðr þ rÞuðr þ rÞi I2
CðrÞ ¼ hIðrÞIðr þ rÞiT
¼ hu ðrÞuðrÞihu ðr þ rÞuðr þ rÞi
þ hu ðrÞuðr þ rÞihuðrÞu ðr þ rÞi I2 ¼ jhu ðrÞuðr þ rÞij2 ;
modulus of the correlation function rather than the correlation function itself. This means that, in intensity correlation, the field has wide-sense spatial stationarity not only for the Fourier kernel but also for the Fresnel kernel because the non-stationary phase term expðikr rs =zÞ in Eq. (12) disappears due to the modulus operation. In either case of temporal or spatial averaging, noting Eqs. (16) and (19), we can rewrite Eq. (29) as
ð29Þ
where use has been made of the properties of the Gaussian process, and the stationarity of the field, i.e., I ¼ hIðrÞi ¼ hIðr þ rÞi ¼ hu ðrÞuðrÞi ¼ hu ðr þ rÞuðr þ rÞi. When the ground glass is moving, the light passed by the ground glass resembles quasi-monochromatic thermal light so that the ensemble average h i can be replaced by the time average h iT . Alternatively, we may replace the ensemble average with the space average h iS for a static ground glass. Note that the covariance is given by the squared
or S rÞiT or S j2
¼ jhu ðrÞuðr þ Z 2 2r r^ / Hð^rÞ exp i d^r ; f
ð30Þ
where we have made use of the fact that the field intensity immediately behind the hologram is proportional to the intensity transmittance of the hologram Ið^rÞ / Hð^rÞ. Thus we have shown that the covariance of the intensity distribution gives the squared modulus of the object image reconstructed from the Fourier hologram. When the hologram is illuminated with laser light through a static ground glass, it creates speckle intensity distributions
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Fig. 16. (Color online) 3-D images reconstructed from intensity cross-covariance functions. (a)–(l) show, respectively, the cross-covariance functions Cðr; zÞ ¼ Cðx; y; zÞ for z being varied from z ¼ 6 to 5 mm with steps of 1 mm. (After Naik et al.12) with permission from the Optical Society.)
near the observation plane, as shown in Fig. 15. The middle figure, Fig. 15(m), shows the full-field speckle intensity recorded at z ¼ 0 plane with a CCD camera having total image resolution of 2048 2048 pixels, and the yellow square in the center indicates the location of the image subarea shown in Figs. 15(a)–15(l). These series of images represent magnified images of the 128 128 pixel area selected from the center of the full-field image, which were recorded at different plane locations ranging from z ¼ 6 to 5 mm with steps of 1 mm. A careful look at Figs. 15(a)–15(l) reveals that the speckle intensity distribution changes as the CCD camera is translated in the longitudinal direction. To demonstrate 3-D reconstruction of the object, we spatially correlated the intensity distribution at the plane z ¼ 0
with those at the planes ranging from z ¼ 6 to 5 mm. Figures 16(a)–16(l) show, respectively, the cross-covariance functions Cðr; zÞ ¼ Cðx; y; zÞ for z being varied from z ¼ 6 to 5 mm with steps of 1 mm. Figure 16(b) represents the image ‘‘O’’ reconstructed from Cðx; y; 5 mmÞ, and Fig. 16(g) represents the image ‘‘’’ and its conjugate image reconstructed from Cðx; y; 0 mmÞ, whereas Fig. 16(l) represents the conjugate image of ‘‘O’’ reconstructed from Cðx; y; 5 mmÞ. Note that the function of focusing can be achieved by intensity correlation and 3-D depth information is preserved like conventional holography even though the phase information about the object is not present.
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7.
Conclusion
We have introduced the concept of spatial statistical optics and reviewed the principles of correlation holography. As stated in the introduction, this review is not intended to provide the readers with a well-balanced and unbiased overview on this somewhat old and yet new subject of statistical optics. Rather, we restricted ourselves to introducing some of our recent efforts to develop unconventional holographic techniques that can control spatial statistics of optical fields and synthesize stochastic optical fields with desired statistical properties. Admitting that this review is too narrow in its scope and too subjective in its view, we hope it will serve as a clue for shedding light on the importance of the synthetic aspect of spatial statistical optics, and also as a small first step toward the establishment of the new realm of spatial statistical optics. Acknowledgments Part of this research was supported by JSPS KAKENHI Grant Number 25390090. Mitsuo Takeda and Dinesh N. Naik are thankful to Wolfgang Osten of Universita¨t Stuttgart and Alexander von Humboldt Foundation for the opportunity of staying at ITO, Universita¨t Stuttgart, which has been instrumental to the preparation of this paper. References 1) L. Mandel and E. Wolf: Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995) p. 41. 2) J. W. Goodman: Statistical Optics (Wiley, New York, 1985) p. 60. 3) E. Wolf: Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, U.K., 2007) p. 11. 4) J. W. Goodman: Speckle Phenomena in Optics, Theory and Applications (Roberts & Co., Greenwood Village, CO, 2007). 5) Advances in Speckle Metrology and Related Techniques, ed. G. H. Kaufmann (Wiley-VCH, Weinheim, 2011). 6) Speckle Metrology, ed. R. S. Sirohi (Marcel Dekker, New York, 1993).
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