Space–bandwidth conditions for efficient phase-shifting ... - CiteSeerX

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J. Opt. Soc. Am. A / Vol. 25, No. 3 / March 2008

A. Stern and B. Javidi

Space–bandwidth conditions for efficient phase-shifting digital holographic microscopy Adrian Stern1,* and Bahram Javidi2 1 Electro Optical Unit, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel Department of Electrical and Computer Engineering, University of Connecticut, Storrs, Connecticut 06269-1157, USA *Corresponding author: [email protected]

2

Received August 13, 2007; revised December 17, 2007; accepted January 7, 2008; posted January 10, 2008 (Doc. ID 86435); published February 21, 2008 Microscopy by holographic means is attractive because it permits true three-dimensional (3D) visualization and 3D display of the objects. We investigate the necessary condition on the object size and spatial bandwidth for complete 3D microscopic imaging with phase-shifting digital holography with various common arrangements. The cases for which a Fresnel holographic arrangement is sufficient and those for which object magnification is necessary are defined. Limitations set by digital sensors are analyzed in the Wigner domain. The trade-offs between the various holographic arrangements in terms of conditions on the object size and bandwidth, recording conditions required for complete representation, and complexity are discussed. © 2008 Optical Society of America OCIS codes: 090.1995, 180.6900.

1. INTRODUCTION Microscopy with digital holography [1–20]offers the unique advantage of simultaneously capturing complete 3D information about the specimen. In contrast to conventional microscopy, no depth scanning and focusing at various depths of the 3D specimen are necessary [21]. Rather, the 3D data is captured simultaneously. For visualizing the 3D specimen, the captured data is numerically processed to display images appropriate to desired focusing distances. In this paper we consider phase-shifting digital holography (PSDH) in the realm of microscopy [8–13]. Our purpose is to clarify fundamental limitations and trade-offs of various PSDH setups. Understanding the trade-offs is necessary for optimizing the PSDH microscopy system for a specific task. PSDH follows in-line (also called “on-axis”) architecture, therefore exploits more efficiently the sensor pixels than off-axis holography, which needs to separate spatially the object signal from undesirable terms (the conjugate and background field). A schematic of the PSDH digital holographic microscope is shown in Fig. 1. Typically, four exposures are taken with the reference beam phase shifted by ␸R = 0, ␲ / 2, ␲, and 3␲ / 2 by means of retarder plates or piezoelectric transducer mirror. The complex field amplitude recorded on the CCD is given by * 共x,y兲u 共x,y兲 H共x,y; ␸R兲 = 兩uH共x,y兲兩2 + 兩uR共x,y兲兩2 + uH R * 共x,y兲 + uH共x,y兲uR 2 + 2ARAH = 关AH共x,y兲兴2 + AR

⫻cos关⌽H共x,y兲 − ␸R兴,

共1兲

where uH共x , y兲 = AH共x , y兲ej⌽H共x,y兲 is the object complex field amplitude at the hologram plane and uR共x , y兲 = ARej␸R is 1084-7529/08/030736-6/$15.00

the complex field amplitude of the reference beam. The phase of the object field at the hologram plane can be calculated from the four exposures by ⌽R共x,y兲 =

H共x,y; ␸R = 3␲/2兲 − H共x,y; ␸R = ␲/2兲 H共x,y; ␸R = 0兲 − H共x,y; ␸R = ␲兲

.

共2兲

Note that the obtained phase is uncoupled from the conjugate field. In principle it is possible to capture the phase distribution with only two exposures [12] or, with some compromise in the quality, with only one exposure [14–17]. The real amplitude of the object wave can also be derived from the intensity that results from blockage of the reference beam. Hence the complete complex field amplitude of the object at the sensor plane can be obtained [8]. Then the original object complex field amplitude can be calculated numerically by applying the inverse transforms of that describing the wave propagation process from the object plane to the hologram plane. Given the object complex field amplitude, the 3D object can be displayed by holographic means, or, various depth images and various perspectives can be displayed on 2D screens [1,5]. In this work we investigate the space–bandwidth conditions on the microscopic objects for distortionless PSDH imaging. Two common types of holograms are considered: “image plane holograms” and Fresnel holograms. With image plane holograms [18,19] an objective lens is used to precondition the object wave whereas with Fresnel holography the object field propagates in free space until it reaches the sensor. Figure 1 shows the schematic setup for image plane holography. Fresnel holography is obtained with the lens removed in Fig. 1. Fresnel holography is analyzed in Section 2 and imaging holography in Section 3. The limitations imposed by the sensor on both types of holography are investigated. We derive the maxi© 2008 Optical Society of America

A. Stern and B. Javidi

Fig. 1.

Vol. 25, No. 3 / March 2008 / J. Opt. Soc. Am. A

(Color online) Schematic of PSDH microscope. BS1, BS2: beam splitters; M1, M2: mirrors; RP: retarder plates.

mum specimen size and bandwidth that is captured and can be completely reconstructed. Also we compare the capturing space–bandwidth product (SBP) requirements for both holography setups. Our investigation is carried out in the Wigner domain, which is particularly useful to holography exploration [19]. The object complex field amplitude u0共x兲 can be represented uniquely in the space–spatial-frequency domain by applying the Wigner–Ville distribution (WVD) [19,20]:

冕 冉 冊冉 冊 ⬁

Wu0共x, ␯兲 =

737

u0 x +

−⬁

x⬘ 2

u0* x −

x⬘ 2

exp共− j2␲x⬘␯兲dx⬘ , 共3兲

where ␯ denotes the spatial frequency. In Eq. (3) and in the following we adopt a 1D notation. The extension to two dimensions is straightforward. The WVD describes the local spatial-frequency distribution of the optical fields involved in the holographic process.

2. FRESNEL PHASE-SHIFTING DIGITAL HOLOGRAPHY A. Object Space and Bandwidth Limitations with Fresnel Arrangement For Fresnel holography, we assume that there is no lens in Fig. 1 so that the object complex field amplitude propagates the distance z = z1 + z2 until it impinges on the CCD. Assuming that the object width is W0 and its essential

bandwidth is B0, the support of its WVD distribution is as shown in Fig. 2(a). The WVD of the propagated field uH共x兲 is given by [19,20] WuH共x, ␯兲 = WuH共x − ␭z␯, ␯兲,

共4兲

which represents the shearing of the object field’s Wigner chart in the x direction, as shown in Fig. 2(b). The intensity of uH共x兲 interfered with that of the reference field uR共x兲 is captured by the CCD [Eq. (1)]. As explained in Section 1 the complex field amplitude of uH共x兲 can be numerically reconstructed. However in order to make it possible to fully reconstruct the object field from the captured interfered pattern, certain conditions set by the recording sensor should be fulfilled. The mathematical operation of a pixelated sensor such as a CCD can be modeled as [22] us共x兲 = 关uH共x兲 * rect共x/␣⌬兲兴comb共x/⌬兲rect共x/Ws兲,

共5兲

where us共x兲 is the recorded field, ⌬ is the pixels size, ␣ 苸 共0 , 1兴 is the pixel fill factor and Ws is the sensor size. The rect共x / Ws兲 in Eq. (5) accounts for the finite size of the sensor. It limits the acceptance range in the Wigner chart to the range −Ws / 2 ⱕ x ⱕ Ws / 2 as shown in Fig. 2(c). The convolution with the rect共x / ␣⌬兲 performs low-pass filtering accounting for the spatial integration of the field over the effective pixel size ␣⌬. It limits the acceptance (double-sided) bandwidth to Bs ⱕ 1 / ␣⌬. A further bandwidth limitation may occur due the sampling process [the comb function, Eq. (5)]. According to Shannon sampling theorem, the object field bandwidth above 1 / ⌬ causes

Fig. 2. (Color online) (a) Support of the object field in the Wigner domain, (b) the support of the object field at the sensor plane, (c) the sensor acceptance range (bold rectangle). The entire object information is captured if the propagated object field fits the acceptance range of the sensor.

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aliasing. More generally, we have shown in [23,24] that if the object field maximum local bandwidth (MLB) [24] is smaller than 1 / ⌬ [as for example in Fig. 2(b)] then the aliasing problem can be alleviated by proper postprocessing [23]. Thus, we may summarize the three spatial spatial-frequency limitation set by the sensor (Fig.3) by

MLB= B0. In such a case, the bandwidth limitation is dictated by condition (8) rather than condition (10), that is B0 ⱕ 1 / ⌬.

where WH and BH are the width and (double-sided) bandwidth of uH共x兲, respectively. Considering the Fresnel holography Wigner chart shown in Fig. 2, we can find from simple geometrical considerations [23] that BH = B0, WH = W0 + ␭zB0 and MLB= W0 / ␭z. Condition (8) is fulfilled for most practical cases involved in PSDH microscopy. For example, with ␭ = 0.5 ␮m, z = 0.2 m, ⌬ = 5 ␮m, ␣ = 50% condition (8) requires that MLB= W0 / ␭z ⬍ 1 / ⌬ = 105 l / m, which is fulfilled for any specimen smaller than 1 cm 共W0 ⬍ 1 cm兲. In this case, the dominant conditions for lossless imaging are set by conditions (6) and (7), imposing the following conditions on the object size and bandwidth:

B. Recording Capacity of Fresnel Arrangement The graphical representation of the WVD is helpful for evaluating the efficiency of the digital holographic recording process by comparing the SBP of the object field to that of the recording device. The object SBP indicates the number of real coefficients required to represent the object. Roughly speaking, the object SBP is the minimum number of pixels required to fully represent the object and the recording device SBP is typically the number of CCD pixels. For a 1D signal, the SBP denoted as N is given by the product of the signal extension in space with its overall bandwidth. In the Wigner space, N equals the area of the rectangle with extensions W and B in the x and ␯ directions, respectively. Therefore, we see from Fig. 2(a) that the object SBP is SBP0 = W0B0. The recording device SBP is SBPs = WsBs [Fig. 2(c)], where Ws and Bs denote the recording device width and bandwidth, respectively. From Figs. 2(b) and 2(c) we see that for lossless recording in the Fresnel holography the sensor needs to have a SBP of at least

W0 ⱕ Ws − ␭zB0 ,

SBPF = WsBs ⱖ WHBH = SBP0 + ␭zB02 .

WH ⱕ Ws ,

共6兲

BH ⱕ Bs = 1/␣⌬,

共7兲

MLB ⱕ 1/⌬,

共8兲

B0 ⱕ

1

␣⌬

,

共9兲

共10兲

respectively. Violation of these constrains results in low resolution, distorted, and truncated reconstructions. Note that although generally high sensor element fill factor ␣ is desired for efficient optical light throughput, we see from condition (10) that for high-resolution imaging we need low fill factor ␣, that is, quasi-point sampling with small sampling smear. Conditions (9) and (10) hold for the case that the Wigner chart of the propagated object field is as shown in Fig. 2(b), which happens if ␭zB0 ⬎ W0. This is the most common case in microscopy. However, if ␭zB0 ⬍ W0, then the object field Wigner chart undergoes a weaker x shearing than the example shown in Fig. 2(b), such that the

共11兲

One may recognize the overhead SBP, ␭zB02, as the Fresnel number [19]. We see that this overhead is proportional to the distance z indicating that larger propagation distances cause less efficient usage of the sensor capacity. In other words, large propagation distances z implies more CCD pixels for recording a given object.

3. IMAGE PLANE PHASE-SHIFTING DIGITAL HOLOGRAPHY A. Object Space and Bandwidth Limitations with Image Plane Holography From condition (10) we see that the maximum bandwidth that can be captured with Fresnel setup is 1 / 共␣⌬兲, meaning that the smallest resolvable detail is the effective pixel size ␣⌬, which is typically a few micrometers. If finer resolution is required, or if the object size W0 is not

Fig. 3. (Color online) Wigner chart of the (a) original field, (b) the field in the image plane, and (c) together with the sensor limitations.

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Vol. 25, No. 3 / March 2008 / J. Opt. Soc. Am. A

sufficiently small to fulfill condition (9), then optical magnification or demagnification should be used using an objective lens (Fig. 1). With the notation as in Fig. 1 and assuming that the imaging condition 1 / z1 + 1 / z2 = 1 / f fulfilled, where f is the lens focal distance, the object field at the sensor is given by [25,26] 2

ui共x兲 = ␭z1ej共␲/␭Mf兲x u0共− x/M兲,

共12兲

where M = z2 / z1 is the absolute lateral magnification. In Eq. (12) we assume that the lens aperture is sufficiently large so the image is not diffraction limited. The effect of the aperture lens will be considered later. By substituting Eq. (12) in the WVD definition Eq. (3), it can be shown that the WVD of the field in the image plane is

冉

Wui共x, ␯兲 = Wu0 −

x M

,− M␯ −

x ␭fM

冊

.

共13兲

The corresponding Wigner chart is shown in Fig. 3(b). It can be seen that the object size is increased by M, so that WH = MW0. Therefore, since for lossless recording WH needs not to exceed Ws, the object size needs to fulfill W0 = WH/M ⱕ Ws/M.

共14兲

From geometrical considerations in Figs. 3(a) and 3(b), together with Eq. (13), we find that the overall bandwidth of the field at the sensor plane is BH = 共B0 / M兲 + 共W0 / ␭f兲. This, together with condition (7), yield B0 ⱕ 共M / ␣⌬兲 − 共MW0 / ␭f兲. In addition, we need to fulfill the sampling condition (8). Noticing from Fig. 3(b) that MLB= B0 / M, we may summarize the two conditions on the object bandwidth as B0 ⱕ

M ⌬

冉

min 1,

1

␣

−

⌬W0 ␭f

冊

.

共15兲

We see from the condition [Eq. (15)] that if ␣ and W0 are sufficiently small so that 共1 / ␣兲 − 共⌬W0 / ␭f兲 ⱖ 1, we obtain the maximum object bandwidth B0 max = M / ⌬. Note that with proper magnification M, imaging holography permits capturing higher bandwidth than Fresnel holography. If we are interested to visualize object details of size ␦, so that B0 ⬵ 1 / ␦, then according to the condition [Eq. (15)] the required magnification is

Mⱖ

⌬

␦ min共1,1/␣ − ⌬W0/共␭f兲兲

739

共16兲

.

In the above discussion, we assumed that the objective lens aperture is virtually infinite. Since objective with aperture A acts as a low-pass filter [25] with cutoff spatial frequency ␯c = A / ␭z1 in the object plane coordinates, finite aperture simply imposes an additional limitation to the relation [Eq. (15)]; B0 ⱕ 2␯c = 2A / ␭z1. B. Recording Capacity of Image Plane Arrangement From Fig. 3, we may see that the minimum SBP of the sensor required for completely recording the object signal is SBPi = WsBs ⱖ WHBH = SBP0 + M

W02 ␭f

.

共17兲

It can be seen that the SBP overhead is inversely proportional to the focal length of the objective lens and is proportional to the object size. For small size microscopic objects (with W0 typically approximately 1 mm or less) the overhead is much smaller than SBP0. For example, with ␭ = 0.5 ␮m, f = 0.2 m, ␣ = 50%, M = 5, W0 = 0.5 cm, and B0 = 106 1 / m the overhead is less than 5%. However if a larger field of view is required the overhead may be significant relative to the object SBP. C. Out-of-Focus Imaging Digital Holography In some cases, because of technical constrains, the sensor is placed further from the image plane, at some distance z2 + d. Assume, for example, that the object is located in plane O1 in Fig. 4(a) and the sensor is placed in plane O2⬘ rather than in the image plane O1⬘ . In such cases the field in the image plane O1⬘ propagates a distance d until it arrives at the sensor plane O2⬘ . Consequently, the Wigner distribution in the image plane O1⬘ undergoes the transform in Eq. (4) with z = d, corresponding to an x sharing similar to that in Fig. 2. The resulting Wigner chart is illustrated in Fig. 4(b). The x shearing maintains the bandwidth; therefore, the condition on the object bandwidth remains as in the inequality [Eq. (15)]. However, the width of the complex field amplitude increases by the

Fig. 4. (Color online) (a) Object planes O1 and O2 imaged in planes O1⬘ and O2⬘ , respectively. (b) The Wigner chart (shaded) of the field on plane O2⬘ due to object points in O1 is obtained by x shearing of Wigner chart of the field in O1⬘ due to the same object points (dashed lines).

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amount of ␭dBH, so that WH = MW0 + ␭dBH. This together with the requirement that the sensor size is to be larger than that of the object field, Ws ⱖ WH, and assuming no bandwidth loss 共BH = B0兲, sets the condition on the object size: W0 ⱕ

Ws − ␭dB0 M

共18兲

,

which is more restrictive than the condition [Eq. (14)]. Multiplying the maximum width WH = MW0 + ␭dBH by the maximum bandwidth BH = 共B0 / M兲 + 共W0 / ␭f兲 we find that the required sensor SBP is SBPd = SBP0 + M

W02 ␭f

+ ␭d

冉

B0 M

+

W0 ␭f

冊

2

.

共19兲

Equation (19) indicates that there is additional sensor SBP waste by the amount of ␭d关共B0 / M兲 + 共W0 / ␭f兲兴2. D. Object Space–Bandwidth Requirements for 3D Objects In the above analysis we assumed that the depth of the object is sufficiently small (typically less than approximately 100 ␮m兲 so it does not impose space–bandwidth limitations. Next, we consider a 3D object with nonnegligible depth D. Yet we assume that D is small relative to z1 and z2, which is the case in microscopy. Again we refer to Fig. 4(a) where planes O1⬘ and O2⬘ are the image planes of O1 and O2, respectively. The sensor is assumed to be in plane O2⬘ . With the notations as in Fig. 4(a), we may write d ⬇ 冑M1M2D, where M1 = z2 / z1 and M2 = 共z2 + d兲 / 共z1 + D兲 are the absolute lateral magnifications of planes O1 and O2, respectively. As shown in Subsection 3.D, out-of-focus imaging imposes more severe conditions on the object size. Therefore, object areas in the out-of-focus plane O1 set the limitation on the object size W0 according to Eq. (18) with M = M1 and d ⬇ 冑M1M2D, yielding W0 ⱕ

Ws − ␭冑M1M2DB0 M1

共20兲

.

The bandwidth conditions for object areas in planes O1 and O2 are according Eq. (15) with appropriate lateral magnification. Since M2 ⬎ M1, a more severe condition is set by Eq. (15) with M = M1, that is B0 ⱕ

M1 ⌬

冉

min 1,

1

␣

−

⌬W0 ␭f

冊

.

space–bandwidth requirements for out-of-focus image plane holography and those for 3D objects. By evaluating the SBP of the hologram field we can calculate the number of CCD pixels required to fully record the object information. We have found that the overhead SBP, indicating the capturing inefficiency, for Fresnel holography is dependent on the object size [Eq. (11)] and that of imaging holography is dependent on the object bandwidth according to Eq. (17). This suggests that Fresnel holography is more suitable for relatively large objects, whereas image plane holography is more appropriate for smaller objects with very fine details.

ACKNOWLEDGMENT This project was supported in part by Defense Advanced Projects Research Agency (DARPA).

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共21兲

4. CONCLUSIONS In conclusion, we have derived the space–bandwidth conditions for lossless phase-shift digital holography microscopy. The maximum object size and bandwidth in typical Fresnel configuration are given by conditions (9) and (10), respectively. We have shown that the maximum resolution achievable with Fresnel holography is ␣⌬. With the expense of a more complex setup, finer resolution can be obtained using a magnifying objective in an imaging holographic setup. The maximum object size and bandwidth for image plane configuration are given by the inequalities [Eqs. (14) and (15)]. We have investigated also the

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