Coherence requirement in digital holography - OSA Publishing

Coherence requirement in digital holography Daniel Claus,1,* Daciana Iliescu,2 and John M. Rodenburg1 1

Kroto Research Centre, University of Sheffield, Sheffield S3 7HQ, UK

2

School of Engineering, University of Warwick, Coventry CV4 7AL, UK *Corresponding author: [email protected]

Received 13 August 2012; revised 24 October 2012; accepted 24 October 2012; posted 24 October 2012 (Doc. ID 174276); published 5 December 2012

In this paper the coherence requirement for different holographic setups (Fresnel hologram, Fourier hologram, and image-plane hologram) is compared. This analysis is based on the investigation of the recorded interference pattern from the superposition of reference wave and object wave in in-line and off-axis mode. The outcome of this investigation can support the choice of light source needed for certain digital holographic setups, as well as the selection of the best applicable setup to take advantage of new short coherence light sources. Moreover, as a byproduct of this investigation, the minimum required recording distance (focal length) to enable Nyquist sampling of the recorded hologram is obtained. © 2013 Optical Society of America OCIS codes: 090.1995, 030.1640, 030.6140, 100.3175, 110.1650, 110.4980.

1. Introduction

Over the last two decades digital holography has emerged as one of the most promising imaging techniques. As a coherent diffractive imaging technique, it is very attractive for many nondirectly related disciplines such as medicine and engineering due to the retrieval of amplitude and phase information simultaneously, as first demonstrated in [1]. This is enabled in a nontactile manner, resulting in a wide field reconstruction with diffraction-limited resolution. However, due to the coherent nature of the light sources used in digital holography, the reconstruction is subjected to the speckle effect. Efforts have been made, especially for digital holographic microscopy, to use short coherence light sources that result in speckle reduction, hence less noisy reconstructions (see [2]). However, the coherence requirement imposed by different digital holographic arrangements has not been investigated yet and is, therefore, the subject of this paper. The three most commonly used digital holographic setups (the Fresnel hologram, the Fourier hologram, and the image-plane hologram) 1559-128X/13/01A326-10$15.00/0 © 2013 Optical Society of America A326

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are comparatively evaluated in in-line and off-axis configurations. The reconstructed hologram for an in-line setup is characterized by the overlap of the DC-term (undiffracted light), the real image, and the virtual image. In order to suppress the DC-term and twin-image in the reconstruction of an in-line setup, phase stepping is implemented in the experiment. However, this results in an additional experimental effort, since it is necessary to record at least three phase stepped holograms. Temporal phase stepping using a piezo mounted mirror in the reference arm is commonly applied. This restricts the setup to quasistatic events (or slow dynamic events with respect to the camera frame rate). Compared to the off-axis configurations, in-line holograms result in the recovery of more object information (increased optical resolution and/or increased field of view) as highlighted in [3]. In the off-axis setup, the various components of the reconstructed hologram, such as the DC-term and the real and virtual images, are separated in the Fourier domain. In this manner, the desired component can be isolated, e.g., the real image. In the off-axis configuration, a single recorded hologram reveals the complex object information. The numerical reconstruction of the single hologram can be speeded up significantly by running the algorithm on a

graphics card. This enables the application of realtime measurement for the investigation of dynamic events, as discussed in [4]. Moreover, it enables the investigation of the object when it is illuminated simultaneously by different wavelengths, different illumination angles, and different polarization states of the incident light beam, as discussed in [5,6]. A hologram is successfully recorded if the spherical wave propagating from each object point is recorded on the sensor. This means that it is not a strict requirement that all waves emergent from each object point have to interfere with one another. The requirement for recording a hologram is already fulfilled if the object wave and reference wave interfere. In fact a spatially incoherent light source can be used in the recording process, which has been successfully demonstrated and published in several papers [7–9]. Therefore this paper studies only the temporal coherence requirement since it has the most influence on the recording process. The results of this study can then be used in the choice of a setup and light source that satisfy the minimum coherence requirement imposed by the object dimensions. Because of the resulting reduced temporal coherence the speckle effect and the influence of other objects located within the optical path (dust particles) from which light is diffracted can be reduced or even eliminated, which has also been discussed in [10]. Other applications where the limited temporal coherence length has been taken into account may benefit from this analysis, such as angular multiplexing [6], imaging through multiple scattering samples [11,12], or imaging of a topography map [13]. 2. Methodology

The discussion of off-axis geometries in this paper is based on the spatial separation of image and twin image in the reconstructed hologram. It is furthermore assumed that the autocorrelation terms from object wave and reference wave have been removed from the digital hologram prior to the reconstruction. Commonly applied methods to suppress the autocorrelation terms involve recording separately the reference wave and object wave, subtracting the average intensity or recording several speckle decorrelated holograms; a more detailed description of different autocorrelation terms suppression techniques in digital holography can be found in [14–16]. A measure for the temporal coherence is the coherence length. Our investigation assumes that the optical path lengths of the reference beam and object beam along the optical axis are matched (based on a two-dimensional object). The sensor and object planes are parallel to each other and perpendicular to the optical axis (see Fig. 1). Hence any differences in the optical path length are due to the object thickness, its refractive index (in transmission), light from off-axis object points, and the angle of the reference wave relative to the object wave. The paraxial approximation, which describes the propagation of light from the object to the sensor, is used to obtain

Fig. 1. (Color online) Nomenclature of coordinates and projection of most extremely located three-dimensional object point onto the x axis.

the required temporal coherence, so that u0
x0 

Z Z exp
ikd ∞ ∞  u
x iλd −∞ −∞
 iπ 0 2 0 2

x − x 
y − y  dxdy; (1) × exp λd

where u0
x0  is the complex amplitude in the hologram plane, u
x is the complex amplitude in the object plane, k  2π∕λ is the wave number, λ is the wavelength, and d is the distance between object and hologram planes, as shown in Fig. 1. The derivation of the required coherence length is split in this section. At first, we derive the coherence requirement due to the object’s longitudinal dimension (ΔLlong ). Second, the coherence requirement due to the object’s and sensor’s lateral extension is calculated (ΔLlat ). Both need to be added in order to obtain the required coherence length of a three-dimensional object. Considering the entire three-dimensional object is important for applications where a threedimensional sparse volume is investigated, such as in digital holographic particle image velocimetry or submersible digital holography [17,18]. However, for axial sectioning of a highly scattering object, only the lateral coherence length should be taken into account. In this case, a light source with short coherent length can offer coherence gating, as discussed in [11–13]. The influence of the longitudinal dimension can be obtained by calculating the phase difference for the most distant sections of the three-dimensional volume along the optical axis. This means that the x0 and x coordinates in Eq. (1) become zero. Let us assume that the object surface closest to the sensor is arranged so that its light path matches the reference wave’s light path (ΔLlong  0). The phase difference recorded for the two different sections (see Fig. 1) along the optical axis hence becomes Δφlong 

2π
d − d1 ; λ 2

(2)

which results in a required longitudinal coherence length of 1 January 2013 / Vol. 52, No. 1 / APPLIED OPTICS

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ΔLlong 

Δφlong λ  d2 − d1 : 2π

(3)

The calculation of the lateral required coherence length is based on the object plane, which results in the largest angle ε (consequently largest phase departure) between the optical axis and a line drawn from the center of the sensor to the most off-axis positioned object point, as shown in Fig. 1. The largest phase departure is important to calculate the required coherence length and the required recording distance, which will also be discussed in this context. This plane is usually defined by the object point positioned closest to the camera sensor. If this is not the case the object point within the entire three-dimensional object introducing the largest phase departure in the recording plane is projected via angle ε along this object plane, as shown in Fig. 1 (x0  −xmax ). Without loss of generality, only one of the two lateral dimensions is treated in this paper (lying along the x direction). The coherence requirement when adding the second lateral dimension can be found in a similar manner. In addition, constant factors and phase terms such as encountered in the prefactor in Eq. (1) (exp
ikd) are discarded since they can be compensated by the alignment of reference arm and object arm distance. Hence in Eq. (1) the parabolic phase term within the integral as a function of the object x and y coordinates is reduced to a single value given by
 iπ 0 u00
x0   A0 exp
x − x0 2 : (4) λd In the next pages the required lateral coherence length for different holographic setup configurations is discussed. The required longitudinal coherence length [see Eq. (3)] remains the same for all setup arrangements. 3. Fresnel Hologram

A Fresnel hologram is recorded when a plane reference wave interferes with the spherical object wave [see Fig. 2]. The plane reference wave can be described mathematically by

u0r
x0   Ar exp
−i2πvr x0 ;

where Ar and vr  sin α∕λ correspond to the amplitude and carrier frequency, respectively, of the reference wave. The interference pattern obtained can be described by i hπ
x0 − x0 2  2πvr x0 : I
x0   A20  A2r  2A0 Ar cos λd (6) The largest phase difference for the recorded interference pattern in the hologram plane is obtained for x0  −xmax , as shown in Fig. 2. The corresponding phase expressed in discrete values is π φ 
lΔx0  xmax 2  2πvr lΔx0 ; (7) λd where Δx0 is the pixel size of the sensor and l is an index variable corresponding to the pixel position (ranging from −N∕2 to N∕2; N-is the pixel number of the sensor). In order to examine the phase difference, the phase has to be differentiated with respect to the pixel position l, which results in Δφ π  2
lΔx0  xmax Δx0  2πvr Δx0 : Δl λd

This corresponds to a lateral required coherence length ΔLlat 

Δφλ NΔx0
NΔx0  X:  2d 2π

APPLIED OPTICS / Vol. 52, No. 1 / 1 January 2013

(10)

In order to validate this equation, the required coherence length was then calculated using the geometric relations shown in Fig. 2. For an in-line configuration (α  0; L2  0) the following expression is obtained: X 2d 2L1 NΔx0 X : ⇒ ΔL   c NΔx0 2d

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(8)

Next, the phase difference Δφ which is accumulated over the entire sensor, is calculated. This means that the difference Δl for the pixel position would have to be the number of pixels N and l either −N∕2 or N∕2. Inserting these parameters for the in-line case (vr  0) results in   2πNΔx0 NΔx0 πNΔx0  xmax 
NΔx0  X: Δφ  λd 2 λd (9)

ΔLlat  2L1

Fig. 2. Graphical representation of required lateral coherence length ΔLlat resulting from maximum object extent and inclination angle of reference wave.

(5)

with tan β 

(11)

Hence the result obtained in Eq. (10) gives rise to a much larger required coherence length [from the additional term N 2 Δx2 ∕
2d in Eq. (10)]. The required coherence length calculated in Eq. (10) relates to the path difference that would have been obtained when

recording with the entire sensor from N∕2 in positive x direction (N∕2…3N∕2) or −N∕2 in negative x direction (−3N∕2… − N∕2), which does not correspond with the region we are interested in (−N∕2…N∕2). Therefore, the calculation of the required coherence length is split into two steps: first we calculate the phase difference from the center of the sensor to the upper edge (l  l, Δl  N∕2 − 1) and then the phase difference from the center to the lower edge (l  l, Δl  l  N∕2). Both phase terms are then added up, which simplifies the calculation to give the phase difference for (l  l, Δl  N): 

 Δx0  xmax Δφ  2πNΔx  vr : λd 0

(12)

The corresponding coherence length for an in-line Fresnel hologram (νr  0) is hence given by ΔLlat 

Δφλ NΔx0
2Δx0  X:  2d 2π

(13)

Also the minimum recording distance dmin (according to the Nyquist criterion) for the in-line setup can be calculated using Eq. (8) by substituting Δφ ≤ π for l  N∕2 and Δl  1 (see [3] for a more detailed derivation), which results in dmin 


X  NΔx0 Δx0 : λ

(14)

Inserting dmin in Eq. (13) results in a Nyquist adapted coherence length ΔLlat 

Nλ
2Δx0  X : 2
X  NΔx0 

(15)

In the off-axis case the carrier frequency needs to be large enough to enable the separation of the image and the twin image in the reconstructed hologram (vr 
X  NΔx0 ∕
2λd). This results in the required coherence length ΔLlat

NΔx0
2Δx0  2X  NΔx0 :  2d

(16)

Inserting the Nyquist adapted minimum distance for the off-axis configuration we have 2
X  NΔx0 Δx0 ; λ

(17)

Nλ
2Δx0  2X  NΔx0  : 4
X  NΔx0 

(18)

dmin  giving ΔLlat 

compensation for the curvature of the object wave, due to the propagation of light to the hologram plane, is achieved. A lensless system uses a spherical reference wave of equal curvature (see Fig. 3), whereas a lens-based system employs a convergent lens. Two types of lens-based Fourier holograms coexist. The first is characterized by the object and hologram placed at the lens focal distances f and f 0 [see Fig. 4]. A plane reference wave then interferes with the lens projected object wave. The second lens-based Fourier hologram is characterized by a convergent beam focused on the hologram with the object placed in between lens and hologram [see Fig. 5]. A. Lensless Fourier Hologram

The complex reference wave can be described by
 iπ 0 u0r
x0   Ar exp
x − xr 2 ; (19) λdr where xr and dr are the coordinates of the reference source point. The phase of the recorded interference between object wave and reference wave is then given by π 2x0
xr − x0   x20 − x2r  λd π  2lΔx0
xr − x0   x20 − x2r : λd

φ

(20)

The maximum interference angle α does now depend on the lateral positions of the source points of object wave and reference wave. Applying a derivation to the discrete phase with respect to the pixel position l and inserting the same values (l  1, Δl  N, x0  −xmax ) results in Δφ 

2πNΔx0
xr  xmax : λd

(21)

If we again insert the minimum distance (see [16] for its derivation), this time for an in-line Fourier hologram (xr  0), XΔx0 ; (22) λ the required coherence length becomes (inserting for x0  xmax for the constant term) dmin 

4. Fourier Hologram

A Fourier hologram can be obtained using either a lens-based or a lensless system. In both cases

Fig. 3. Graphical representation of Fourier holographic setup (black line-object wave, gray line-reference wave). 1 January 2013 / Vol. 52, No. 1 / APPLIED OPTICS

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 iπ 2 iπ 0 2 u
x   u
~x exp − x~ exp
x − x~  d~x λf λd1 −∞ Z Z 
 ∞ ∞ iπ 02 iπ x u
x exp
~x − x2 dx  exp λd λd1 −∞ −∞    
iπ 2 1 1 i2π 0 x~ exp − − (27) x~ x d~x: × λ d2 f λd2 0

Fig. 4. Graphical representation of a lens-based Fourier hologram, black object wave, gray reference wave, and ε diffraction angle. α is the reference wave angle.

ΔLlat 

NΔx0 X Nλ  : 2d 2

(23)

In the off-axis setup the lateral location of the reference wave point source needs to be chosen to offset the highest spatial frequency introduced by the object (xr ≥ xmax ). This results in the same required coherence length of ΔLlat 

NΔx0 X Nλ  ; d 2

(24)

whereas the minimum distance for an off-axis Fourier-hologram is doubled compared to the in-line setup, dmin 

B.

2XΔx0 : λ

(25)

Z

∞

The exponential term (impulse response) in the first integral with regard to dx is shift invariant and can hence be implemented as a convolution integral. The last exponential term in Eq. (27) with regards to x~ x0 represents a Fourier transformation (ω  2πx0 ∕
λd2 ). The impulse response’s corresponding transfer function can be found with the help of [19], so that   2   p x  2a exp −aω2 ; (28) F exp − 4a where F stands for the Fourier transformation and a  iλd1 ∕
4π. With d2  d1  f and inserting p the transfer function (nonexponential term 2a has been discarded), Eq. (27) becomes     iπ 02 iπd1 02 0  Ffu
xg: u
x   exp x Ffu
xg exp − 2 x λf λf (29) Hence the diffraction pattern recorded in the hologram plane represents a perfect Fourier transformation of the object without any parabolic phase term as in the lensless case. Taking into account the plane reference wave of inclination angle α [see Eq. (5) for the representation of plane reference wave] the following interference phase is obtained: φ  2πx0
ν0  νr ;

First Lens-Based Fourier Hologram

For a lens-based Fourier hologram, as shown in Fig. 4, the propagation from the object to the hologram plane is split into three steps. First there is free-space propagation from object to the first lens surface, then the passage of light through the lens, and finally free-space propagation from second lens surface to hologram plane. We now consider and take into account the entire range of possible object points since the relationship between the recorded interference pattern and the object point is not as clear as in the lensless case. The object wave reaching the first lens surface can be described by  iπ u
~x  u
x exp
~x − x2 dx: λd1 −∞ Z

∞




(26)

The object wave, which reaches the hologram plane after passage through the lens, can be described by A330

APPLIED OPTICS / Vol. 52, No. 1 / 1 January 2013

(30)

where νr and ν0 can be calculated as follows: ν0 

ω x0  ; 2π λf

and

νr 

sin α ≥ ν0 : λ

Hence the phase difference introduced can be calculated by ( 4πΔx0 2 l Δl for in-line λf : Δφ  8πΔx 02l Δl for off -axis λf In contrast to the lensless system the spatial frequency ν0 of a lens-based system does not depend on the coordinates of the object. Instead it is determined by the coordinates in the hologram plane. The maximum spatial frequency is recorded at the edge of the detector (x0  NΔx0 ∕2). This also affects the minimum focal length (with respect to the Nyquist criterion, l  N∕2, Δl  1):

f min ≥

 2NΔx0 2 λ 4NΔx0 2 λ

for in-line : for off -axis

(31)

Compared to the lensless system, the resulting required coherence length is reduced to ΔLlat  λ:

(32)

This difference arises from the time compensation characteristics of a lens, which means that light diffracted at a large angle suffers a smaller phase delay when traveling through the lens than light that is diffracted at a smaller angle. C.

Second Lens-Based Fourier Hologram

Now to the last of the three Fourier holographic arrangements, as shown in Fig. 5. In contrast to the previous case, no lens is used between object and hologram. Here the lens is only used to generate a convergent wave focused on the hologram, which illuminates the object. The propagation to the hologram plane can again been split into several sections. First the passage of light through the lens and free-space propagation to the object plane is evaluated as 


2  iπ 2 iπ  u
x  x − x~ exp − x~ exp d~x λf λd1 −∞ Z  
 ∞ iπ 2 iπ 2 1 1 x~ x exp −  exp λd1 λ d1 f −∞   i2π x~x d~x: × exp − λd1 Z

∞

 iπ 0 2  u
xA
x exp
x − x dx; λd2 −∞ Z

∞



iπ 02 x u0
x0   exp λd2

Z

3

2 ∞

iπ

x2 5A
x exp4  λ d1 − f     iπ 2 i2π 0 x exp − x x dx × exp λd2 λd2 2 0 13 Z  ∞ iπ 02 iπ d  d − f 1 A5 x A
x exp4 x2 @ 2  exp λd2 λ −∞ d −f d −∞

1

 i2π 0 x x dx: × exp − λd2 

2

(35)

The last exponential term in Eq. (35) once again represents a Fourier transformation, which for d2  d1  f results in a Fourier transformation of the complex object function only [with ω  2πx0 ∕
λd2 ]. However, this time the resulting complex object wave recorded in the hologram plane also possesses a parabolic phase as a function of the hologram plane coordinate x0 . This parabolic object phase can be suppressed using a reference wave of the same curvature (dr  d2 ). Hence the phase obtained from the interference of object and reference wave is π
2x0 xr − x2r  λdr 2π 02 π x 
2x0 xr − x2r :  λd2 λdr

φ  2πx0 ν0  (33)

Then free-space propagation from the object plane to the hologram plane: u0
x0 

The last term in Eq. (33) represents a Fourier transformation (with ω  2πx∕
λd1 ),




(34)

where A
x represents the complex object function (amplitude and phase) in the object plane.

(36)

In the off-axis case, xr
νr 
xr − x0 ∕
λd2  needs to be chosen to suppress the largest spatial frequency introduced by ν0
xr  x0max  NΔx0 ∕2. The minimum distance d2 between object plane and hologram plane then becomes d2 min ≥

8
for off -axis : λ xr  2β0 This means that the required coherence length for an in-line image-plane hologram (xr  0) with a curvature adapted spherical reference wave is ΔLlat 

NΔx02  λ: d2 β0

(45)

The carrier frequency introduced in the off-axis setup needs to be larger than the largest angle of light emerging from the optical lens, which means that xr  D∕2. This results in the required coherence length ΔLlat 

NΔx0
2Δx0  Dβ0  Nλ
2Δx0  Dβ0  :  2β0 d2 2
NΔx0  Dβ0 

reference
νr  sin α∕λ 
NΔx0  D∕
2d2 λ wave is also calculated. A plane wave results in the following phase:   πx02 β0  1  2πx0 νr λd2 β0   0 πΔx02 l2 β0  1 0 D  NΔx  l :  πΔx λd2 λd2 β0

φ

The corresponding phase difference is


πΔx0   2πΔx02 l  0 0 Δφ  β 1  D  NΔx Δl: (48) λd2 λβ0 d2

is

The corresponding minimum imaging distance d2

d2min 

8 > < NΔx0 0 2
β0  1

for in-line

λβ

0 > : Δx λβ0

0

0

0

NΔx
2β  1  Dβ 

for off -axis

; (49)

which results in 8 02 0 < NΔx for in-line d2 β0
β  1  λ : (50) ΔLlat  Nλ2Δx0
β0 1β0
DNΔx0  : for off -axis NΔx0
2β0 1Dβ0  The required coherence length for an in-line and off-axis image-plane holograms with plane reference wave is significantly increased compared to an image-plane hologram spherically adapted reference wave. This is demonstrated in Fig. 8. Image-plane holograms are most commonly applied to magnify small objects using microscope objectives. The imaging distances d2 in Fig. 8 have therefore been chosen so that it covers the range of the tube lens focal length (infinity optical system) or the image side focal length (finite optical system) of different microscope objectives, as discussed in [22]. The diameter of the lens D and its magnification β0 have likewise been chosen to represent a typical values of a microscope objective (D  10 mm, β0  10). Very often the coherence length is not provided by the light source manufacturer. Instead the spectral response of the light source is more commonly supplied by the manufacturer. The coherence length can be calculated from the shape (rectangular, Lorentzian, or Gaussian) and the full width at half maximum
ΔλFWHM  of the spectral response of the light source. Rearranging the equations shown in [23] results in

(46)

To have a complete description and for comparison purposes, the required coherence length for a plane

(47)

ΔLlong  ΔLlat

8 2 λ ⋅0.66 > > < ΔλFWHM ; Gaussian λ2 ⋅0.32 ; Lorentzian :  Δλ FWHM > > : λ2 ; Rectangular ΔλFWHM

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Fig. 8. Required coherence length for image-plane holograms with plane reference wave and spherical reference wave for (a) in-line and (b) off-axis geometry (N  1000, Δx0  10 μm, λ  500 nm and D  10 mm).

tissue samples. This work in combination with the two papers previously published [3,24] results in a very comprehensive analysis of digital holographic setup arrangements. They will give a good guidance in the choice of setup for a specific task. Moreover, as a byproduct of the coherence investigation, the minimum recording distances (focal length) have been calculated, which provides a crucial criterion to enable the recording of a well sampled hologram. References

Fig. 9. Coherence length ΔLlong  ΔLlat versus spectral width ΔλFWHM ; the larger the required coherence length the smaller the required spectral width.

The required spectral width of the three spectral shapes as a function of the coherence length are displayed in Fig. 9. 6. Discussion and Conclusion

The most commonly applied digital holographic setup arrangements have been quantitatively analyzed and compared with respect to their required coherence length. For the lensless setups discussed here, it could be shown that the required temporal coherence is smaller for Fourier holograms compared to the equivalent Fresnel holograms (i.e., in-line or off-axis). The difference in the required coherence length between both setups decreases significantly the larger the object size is. The smallest temporal coherence required for image-plane configurations discussed here is obtained using a reference wave with an appropriately engineered curvature. The analysis performed is especially useful to reduce the impact of speckle noise for the reconstructed image and to obtain a good reconstruction through multiple scattering samples such as most biological A334

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