## Experimental evidence of the theoretical spatial frequency response of cubic phase mask wavefront coding imaging systems |

Optics Express, Vol. 20, Issue 2, pp. 1878-1895 (2012)

http://dx.doi.org/10.1364/OE.20.001878

Acrobat PDF (3025 KB)

### Abstract

The optical transfer function of a cubic phase mask wavefront coding imaging system is experimentally measured across the entire range of defocus values encompassing the system’s functional limits. The results are compared against mathematical expressions describing the spatial frequency response of these computational imagers. Experimental data shows that the observed modulation and phase transfer functions, available spatial frequency bandwidth and design range of this imaging system strongly agree with previously published mathematical analyses. An imaging system characterization application is also presented wherein it is shown that the phase transfer function is more robust than the modulation transfer function in estimating the strength of the cubic phase mask.

© 2012 OSA

## 1. Introduction

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**(11), 1859–1866 (1995). [CrossRef] [PubMed]

18. S. Chen, Z. Fan, H. Chang, and Z. Xu, “Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask,” Appl. Opt. **50**(19), 3337–3345 (2011). [CrossRef] [PubMed]

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**(11), 1859–1866 (1995). [CrossRef] [PubMed]

7. J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. **30**(13), 1647–1649 (2005). [CrossRef] [PubMed]

19. H. B. Wach, E. R. Dowski Jr, and W. T. Cathey, “Control of chromatic focal shift through wave-front coding,” Appl. Opt. **37**(23), 5359–5367 (1998). [CrossRef] [PubMed]

8. M. Somayaji and M. P. Christensen, “Form factor enhancement of imaging systems using a cubic phase mask,” in *Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM*, Technical Digest (Optical Society of America, 2005), paper CMB4.

10. M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. **45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

20. M. Somayaji and M. P. Christensen, “Improving photon count and flat profiles of multiplex imaging systems with the odd-symmetric quadratic phase modulation mask,” Appl. Opt. **46**(18), 3754–3765 (2007). [CrossRef] [PubMed]

21. G. E. Johnson, E. R. Dowski Jr, and W. T. Cathey, “Passive ranging through wave-front coding: information and application,” Appl. Opt. **39**(11), 1700–1710 (2000). [CrossRef] [PubMed]

22. E. R. Dowski Jr, R. H. Cormack, and S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” Proc. SPIE **4041**, 114–120 (2000). [CrossRef]

23. K. Kubala, E. Dowski, and W. T. Cathey, “Reducing complexity in computational imaging systems,” Opt. Express **11**(18), 2102–2108 (2003). [CrossRef] [PubMed]

24. R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. **44**(5), 701–712 (2005). [CrossRef] [PubMed]

25. S.-H. Lee, N.-C. Park, and Y.-P. Park, “Breaking diffraction limit of a small f-number compact camera using wavefront coding,” Opt. Express **16**(18), 13569–13578 (2008). [CrossRef] [PubMed]

26. M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express **17**(8), 6118–6127 (2009). [CrossRef] [PubMed]

8. M. Somayaji and M. P. Christensen, “Form factor enhancement of imaging systems using a cubic phase mask,” in *Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM*, Technical Digest (Optical Society of America, 2005), paper CMB4.

10. M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. **45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

11. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. **30**(20), 2715–2717 (2005). [CrossRef] [PubMed]

10. M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. **45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

28. S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. **36**(35), 9157–9166 (1997). [CrossRef] [PubMed]

30. E. R. Dowski Jr and G. E. Johnson, “Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems,” Proc. SPIE **3779**, 137–145 (1999). [CrossRef]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

## 2. Overview of the frequency response of cubic phase mask wavefront coding systems

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

### 2.1. The pupil function

*θ*(

*x*,

*y*), into a rectangular clear pupil to modify the incoming wavefront (and hence the PSF). Here, (

*x*,

*y*) represent the normalized pupil plane coordinates. For a cubic phase mask, this function is given by

*θ*(

*x*,

*y*) = α(

*x*

^{3}+

*y*

^{3}), where α is a positive constant. When considered in conjunction with the wavefront error introduced by defocus, the unit-power, generalized pupil function of a CPM imaging system may be expressed along one dimension (1D) as [1

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**(11), 1859–1866 (1995). [CrossRef] [PubMed]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

32. S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. **36**(35), 9157–9166 (1997). [CrossRef] [PubMed]

*in Eq. (1) is explained in [10*

_{min}**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

**34**(11), 1859–1866 (1995). [CrossRef] [PubMed]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

*L*is the aperture width,

*f*is the focal length, and

*z*and

_{o}*z*are the distances of the object and image capture planes respectively, from the pupil.

_{a}### 2.2. The modulation transfer function

**34**(11), 1859–1866 (1995). [CrossRef] [PubMed]

*u*is the spatial frequency normalized to the diffraction-limited cutoff frequency η

_{o}such that –1 ≤

*u*≤ 1. In Eq. (4), the MTF appears to be invariant to the defocus parameter ψ. However, as was subsequently shown in [8

8. M. Somayaji and M. P. Christensen, “Form factor enhancement of imaging systems using a cubic phase mask,” in *Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM*, Technical Digest (Optical Society of America, 2005), paper CMB4.

11. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. **30**(20), 2715–2717 (2005). [CrossRef] [PubMed]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

*C*() and

*S*() denote the Fresnel cosine integral and Fresnel sine integral respectively, whose operands are [10

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

*M*(

_{t}*u*,ψ) = |

*H*(

_{t}*u*,ψ)| may be expressed as [10

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

### 2.3. Available spatial frequency bandwidth

*M*(

_{t}*u*,ψ) drops to one-half the defocus-independent approximate MTF

*M*(

_{a}*u*,ψ) = |

*H*(

_{a}*u*,ψ)|. Defocus thus has a low-pass filtering effect on the frequency response by reducing the extent of the MTF to a cutoff frequency whose value is [10

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

11. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. **30**(20), 2715–2717 (2005). [CrossRef] [PubMed]

*u*obtained from experimental data closely follows the trend predicted by the theoretical description of Eq. (8). It is also clear from Eq. (8) that

_{c}*u*must obviously remain within the interval [0, 1] which means that two conditions must be satisfied, namely (a) |ψ| ≤ 3α and (b) α > 0. In practical scenarios however, the utility of an imaging system is typically exhausted before

_{c}*u*becomes zero, that is, before |ψ| reaches 3α. A more practical definition of the limit of the defocus magnitude at which the system ceases to usefully operate is hence reviewed next.

_{c}### 2.4. The ambiguity function and the design range

*u*axis of the radial line in the AF plot whose slope is 2ψ/π and intercept is zero. The 2D AF hence contains information about the 1D OTF for all defocus values. The AF is a useful tool to illustrate the design range, i.e., the range of defocus values within which an imaging system can effectively operate. This design range encompasses all defocus values ψ that satisfy the condition –|ψ

*| ≤ ψ ≤ + |ψ*

_{m}*|, where |ψ*

_{m}*| is a maximum defocus magnitude given by [10*

_{m}**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

*t*in the above expression is a threshold parameter whose magnitude is typically taken to be between zero and unity. The value of

*t*is used to test the maximum magnitude of either

*a*(

*u*) or

*b*(

*u*) of Eq. (6) depending on whether ψ is positive or negative respectively, to identify values of ψ under which the Fresnel integrals in Eq. (7) fail to achieve a stationary value of ±½. The significance of

*t*has been further elaborated in [10

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

### 2.5. The phase transfer function

*u*= 0, where it becomes zero. It is apparent from Eq. (11) that the PTF exhibits a strong dependence on defocus. Variations in the PTF as a function of defocus thus cause disparities in spatial frequency shifts for objects at different distances in a scene. This disparity in turn results in image artifacts when the captured image is restored with a single restoration filter. These effects have been previously studied in [16

16. M. Demenikov and A. R. Harvey, “Image artifacts in hybrid imaging systems with a cubic phase mask,” Opt. Express **18**(8), 8207–8212 (2010). [CrossRef] [PubMed]

**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

## 3. Experimental setup

*x*and

*y*axes), the end-to-end variation along one dimension was one-half of this value. It is recalled that the extent of the 1D variation across the open aperture is used in calculating the strength of the phase mask. The maximum surface sag ξ′ = ξ/λ of the phase mask along 1D across its 12.7mm face was therefore 30 waves for λ = 550nm in free space. Since the aperture width was 7.5mm and the surface profile of the phase mask varied as a cubic function, the effective maximum sag along one dimension in multiples of wavelengths across the face of this aperture was ξ/λ = 30 × (7.5/12.7)

^{3}= 6.179 waves. Multiplying this quantity with 2π as per Eq. (2) then yielded the effective strength of the phase mask as α = 38.822 for this experiment.

_{o}= 180cyc/mm and η

_{N}= 227.27cyc/mm respectively, thus ensuring an absence of aliasing. To keep out stray light as the sensor was incrementally moved forward, a telescoping baffle was used between the imaging lens and the sensor, which was designed to fold as the working distance was reduced.

**M**of approximately –0.01. The observed image of this point source yielded the PSF, which was then directly used to measure the OTF. This point source was realized by back-illuminating a 100μm pinhole by a white-light LED (Thorlabs model# MCWHL2). Given that theoretical analyses of the spatial frequency response assumed a single wavelength, a color filter (Thorlabs model# FB550-10) with center wavelength and full-width-half maximum passband of 550nm and ±10nm respectively was included in the imaging chain. This filter helped emulate monochromatic illumination while assuring near-diffraction-limited imaging performance by the lone imaging lens.

*z*was then incrementally reduced by moving the sensor towards the lens in steps of 10μm across a defocus range far exceeding the imaging system’s operating limits. The range of working distance variation was 27.285mm which, when expressed relative to the in-focus working distance

_{a}*z*was –15.7mm <

_{i}*z*–

_{i}*z*< 11.6mm. In terms of defocus, this range of distances from best focus corresponded to 4.685α > ψ > –4.937α. A linear motorized actuator (Zaber actuator model# T-LA28A) with a 28mm maximum travel length was employed to move the sensor.

_{a}*u*≤ 1.

*M*(

_{a}*u*) = |

*H*(

_{a}*u*)| and

*M*(

_{t}*u*) = |

*H*(

_{t}*u*)| in Fig. 2 exclusively pertain to the imaging system’s optics, whereas the measured MTF

*M*(

_{m}*u*) encompasses the modulation transfer function of the imaging system as a whole. Therefore, a more accurate comparison would be of the latter with the theoretical MTF lowered by factors such as the pixel MTF, effective optical SNR and other sources of noise that contribute to the overall system response. While an extended treatise on the effective optical SNR and other noise factors is beyond the scope of this paper, the pixel MTF

*M*(

_{p}*u*) is relatively straightforward to evaluate and is hence included while comparing theoretical versus observed data in subsequent discussions. For a square pixel of pitch and size

*p*, the pixel MTF is given by [34

34. Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE **3950**, 49–56 (2000). [CrossRef]

*u*η

_{o}

*p*) = sin(π

*u*η

_{o}

*p*)/(π

*u*η

_{o}

*p*). It is noted that the pixel MTF is not a function of defocus. The MTF data in Fig. 2 shows

*M*(

_{p}*u*) for this experiment, as well as the product

*M*(

_{t}*u*) ×

*M*(

_{p}*u*). From Fig. 2(a), it is seen that

*M*(

_{m}*u*) is slightly lower than

*M*(

_{t}*u*) ×

*M*(

_{p}*u*), as naturally expected of the aggregate system MTF.

*Θ*(

_{a}*u*) and the exact theoretical PTF

*Θ*(

_{t}*u*) are nearly identical, whereas the measured PTF

*Θ*(

_{m}*u*) deviates from the first two curves. This deviation is primarily due to the fact that the ROI window was centered about the peak intensity pixel of the in-focus PSF. However, it has been shown that for an on-axis point object, the peak intensity of the CPM imaging system’s PSF is offset from the optical axis even when the defocus is zero [18

18. S. Chen, Z. Fan, H. Chang, and Z. Xu, “Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask,” Appl. Opt. **50**(19), 3337–3345 (2011). [CrossRef] [PubMed]

35. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express **19**(24), 24609–24626 (2011). [CrossRef] [PubMed]

*Θ*(

_{c}*u*) =

*mu*η

_{o}applied to the measured PTF, where

*m*is the slope of this correction term. In this experiment, this slope was estimated to be

*m*= 0.04 radian-mm/cyc or in the spatial domain, a global PSF shift of 6.366μm or 2.89 pixels based on the analysis in [35

35. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express **19**(24), 24609–24626 (2011). [CrossRef] [PubMed]

## 4. Experimental results and discussion

*u*as a function of working distance (i.e., defocus). Experimental findings supporting the validity of the theoretical design range defocus limit |ψ

_{c}*| are also included. Additionally, theoretical versus measured PTFs are compared across the design range and an imaging system characterization scenario is presented wherein the value of α is estimated from the MTF as well as the PTF.*

_{m}### 4.1. The modulation transfer function

*u*and distance from best focus

*z*–

_{i}*z*. It is herein recalled that negative values of the defocus parameter ψ correspond to positive values of the distance from best focus, that is, the value of ψ decreases as the sensor is moved towards the lens.

_{a}*M*(

_{a}*u*,ψ) and

*M*(

_{t}*u*,ψ) respectively, are lowered by the pixel MTF

*M*(

_{p}*u*,ψ) =

*M*(

_{p}*u*) before comparing them to the measured MTF

*M*(

_{m}*u*,ψ). The plots in Fig. 3 are shown for a defocus range of 3α > ψ > –3α, corresponding to a distance from best focus of roughly –9.34mm <

*z*–

_{i}*z*< 7.5mm. This set of working distances encompassed the entire design range of this imaging system.

_{a}*M*derived from Eq. (4) does not reveal the effects of defocus on the spatial frequency response of a CPM imaging system. On the other hand, the exact MTF

_{a}*M*given by Eq. (7) and represented by Fig. 3(b) predicts a reduction in spatial frequency bandwidth when the magnitude of defocus is increased. Inspecting Fig. 3(c), it is seen that apart from noise and a slight overall contrast reduction, the measured MTF

_{t}*M*accurately follows the theoretically predicted frequency response embodied by Fig. 3(b). As noted in the previous section, the contrast reduction seen in

_{m}*M*as relative to

_{m}*M*×

_{t}*M*is to be expected, given that the quantity being measured was the overall system MTF. The experimental evidence presented in Fig. 3 hence corroborates the theoretical prediction of the modulation transfer function of a cubic phase mask wavefront coding imaging system as given by Eq. (7) and previously presented in [10

_{p}**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

### 4.2. Available spatial frequency bandwidth

*u*as given by Eq. (8) versus its experimentally estimated counterpart. In order to better understand this comparison, the approach used to estimate the value of

_{c}*u*from the experimental data is explained herein.

_{c}*u*is defined as the location on the spatial frequency axis where the exact theoretical MTF

_{c}*M*drops to one half the approximate MTF

_{t}*M*[10

_{a}**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

*M*however, does not accurately reflect this fact as seen in Fig. 2(a) and Fig. 3(a). On the other hand, at ψ = 0, the entire diffraction-limited bandwidth is in principle available to the imaging system, and

_{a}*u*as given by Eq. (8) is unity. Nonetheless, the exact MTF

_{c}*M*drops to zero at the optical cutoff as expected, rather than reduce to one half the height of

_{t}*M*. Hence for very low defocus magnitudes, a more accurate approach would be to identify the height of

_{a}*M*×

_{t}*M*at the theoretical value of

_{p}*u*and then locate the spatial frequency at which

_{c}*M*drops to this threshold value. This approach continues to apply as |ψ| is increased and hence may be employed across the entire design range. The measured values of

_{m}*u*shown in Fig. 4 were thus obtained using this method.

_{c}*u*for a given distance from best focus exhibit very similar trends, thereby strongly supporting the validity of Eq. (8). However, inspecting the left-hand side of Fig. 4 reveals that the computed values of

_{c}*u*from measured data are somewhat lower than its theoretical counterpart. This is because for a given aperture, the effective optical SNR is lower for longer working distances and therefore

_{c}*M*is observed to be slightly lower than

_{m}*M*×

_{t}*M*. The value of

_{p}*M*therefore drops to the threshold value of

_{m}*M*×

_{t}*M*evaluated at the theoretical prediction of

_{p}*u*, at a spatial frequency that was fractionally lower than that predicted by theory. When |

_{c}*z*–

_{i}*z*| and hence |ψ| approaches zero, both

_{a}*M*×

_{t}*M*and

_{p}*M*converge to the optical cutoff frequency and therefore the theoretical and estimated values of

_{m}*u*converge to unity at

_{c}*z*–

_{i}*z*= 0, as seen in Fig. 4. As

_{a}*z*–

_{i}*z*increases, the peak PSF intensity and hence the optical SNR increases, thereby causing

_{a}*M*to closely follow

_{m}*M*×

_{t}*M*especially at lower spatial frequency values of

_{p}*u*, as seen on the right-hand side of Fig. 4. In the next subsection, this observation is illustrated via the magnitude of the ambiguity function as obtained from experimental data.

_{c}### 4.3. The design range

*| as given in Eq. (10) would be helpful in determining the design range, i.e., the system’s useful operating limits. Depending on the imaging application at hand, it may be desirable to limit bandwidth reduction to an acceptable lower bound of*

_{m}*u*. This choice of

_{c}*u*in turns specifies the value of |ψ

_{c}*| and by extension, the threshold parameter*

_{m}*t*in Eq. (10). For purposes of the following illustration, the value of

*t*was chosen as 0.25, yielding |ψ

*| = 2.575α.*

_{m}*|/π mark the extent of the operating region of such a system. Figure 5 demonstrates this concept in the form of AF magnitude plots derived from*

_{m}*M*×

_{t}*M*as well as from

_{p}*M*.

_{m}*|. The measured AF shows a nearly identical result. To quantify the estimate of the design range, the percentage of power contained within the region bounded by the design range of –|ψ*

_{m}*| ≤ ψ ≤ + |ψ*

_{m}*| with respect to the overall plot was calculated for the theoretical prediction as well as the experimental data. To ensure a fair comparison, only those regions were included in the theoretical plot for which experimental data was collected. The resulting calculations showed that for the theoretical data, 95.38% of power was contained within the radial lines marking the defocus values of ±|ψ*

_{m}*|. This ratio was 93.00% in the experimental data, resulting in a difference of 2.38% or an error of 2.49%. As a benchmark, the percentage of power within ±|ψ*

_{m}*| in the theoretical case with respect to the entire plot was 93.37%. The experimental results thus validate the expression for the defocus design range as previously presented in [10*

_{m}**45**(13), 2911–2923 (2006). [CrossRef] [PubMed]

*u*= 0 on account of the Hermitian symmetry of the OTF, it follows that the AF magnitude plot is symmetric about

*u*= 0 along a radial line of arbitrary slope 2ψ/π. Furthermore, in systems where the MTF as a function of

*u*is symmetric about ψ = 0, the AF magnitude plot is also symmetric about the horizontal axis, that is, about the line 2

*u*ψ/π = 0 or ψ = 0. In theory, the CPM imaging system exhibits both the above traits as may be seen in Fig. 5(a). In practice however, factors such as variations in the peak intensity of the PSF as a function of working distance cause the MTFs at ψ < 0 to not exactly match their counterparts for ψ > 0. In such a situation, the resulting AF magnitude plot exhibits an asymmetry about the horizontal axis, i.e., about ψ = 0, while maintaining symmetry about

*u*= 0 along a radial line. This effect is observed in Fig. 5(b). Regions where ψ < 0 (second and fourth quadrant) correspond to working distances where

*z*–

_{i}*z*> 0, whereas those areas where ψ > 0 (first and third quadrant) mark working distances where the sensor lay between the lens and the in-focus plane.

_{a}*u*from the first quadrant as revealed by the left-hand side of Fig. 4.

_{c}### 4.4. The phase transfer function

35. V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express **19**(24), 24609–24626 (2011). [CrossRef] [PubMed]

16. M. Demenikov and A. R. Harvey, “Image artifacts in hybrid imaging systems with a cubic phase mask,” Opt. Express **18**(8), 8207–8212 (2010). [CrossRef] [PubMed]

*u*and

*z*–

_{i}*z*, while Fig. 6(b) presents the corresponding experimentally measured PTF. In order to offset the effects of centering the ROI window to the peak intensity location of the PSF, a linear PTF function has been added to the measured data, whose slope is

_{a}*m*= 0.04 radian-mm/cyc as described in section 3. This compensation function is independent of defocus and hence constant across all working distances.

**19**(24), 24609–24626 (2011). [CrossRef] [PubMed]

*z*–

_{i}*z*= 0 and along

_{a}*u*. In other words, the measured PTF at working distances where

*z*–

_{i}*z*is positive has an overall trend that is higher than its theoretical counterpart; whereas the opposite is the case for locations corresponding to

_{a}*z*–

_{i}*z*< 0. One reason for this effect may be due to the aforesaid lateral drift of the sensor as the working distance is varied. That is, when misalignment causes the longitudinal travel of the sensor to occur along a path that is not perfectly parallel to the optical axis, a linear phase error accumulates due to the resulting PSF shift within the fixed ROI window. The slope of this phase error monotonically increases as

_{a}*z*–

_{i}*z*increases. It is expected that experimental setups with enhanced motion precision capabilities could help mitigate this error.

_{a}### 4.5. Estimation of α

*u*> 0 are considered.

*M*and the spatial frequency

*u*. The second term is a scaled function of Fresnel integrals which oscillates rapidly about unity as a function of

*u*. For a low-order polynomial curve fitting approach, this term may hence be ignored. As a result, only the stationary phase component of the overall MTF as given by the modulus of Eq. (4) is utilized in this application. The MTF in terms of this component is then

*M*= [π/(24α

*u*)]

^{½}, which may be rewritten as

*u*, namely a straight line, the resulting coefficient of the first degree exponent yields the slope of the fit, which is also the value of α. It is apparent from Eq. (13) that the above polynomial in

*u*is highly sensitive to the MTF. It is further noted that larger values of α serve to lower the MTF. If the MTF used in the estimation is then lower than that predicted by Eq. (7) due to a host of system or environmental factors, this reduced MTF would result in a higher but erroneous estimate of α. Given that the system MTF is almost always lower that its theoretical optical-only counterpart, using the measured MTF to estimate this parameter would virtually assure incorrect results.

*u*

^{3}, is a cubic exponent with respect to

*u*and only contains the parameter α within its coefficient. The second term 2ψ

^{2}

*u*/(3α) is a linear function of

*u*and a quadratic function of ψ. It has been shown that the quadratic dependence on ψ may be exploited to locate the in-focus plane of a CPM imaging system [36]. The third term contributes to a rapid oscillation of the PTF about π/4 radians within the available spatial frequency bandwidth of the system. This term may therefore be ignored in a low-order polynomial estimation problem. The equation used for curve-fitting would then be

*u*is independent of defocus or other linear effects. Thus, in the absence of optical aberrations specifically contributing to the cubic exponent in the above equation, the coefficient of this exponent directly yields an estimate of α while remaining largely unaffected by noise or other measurement errors. Practical utility of this approach is herein demonstrated via theoretical and experimental data.

*M*,

_{t}*M*×

_{t}*M*, and

_{p}*Θ*, and three experimental sources, namely

_{t}*M*,

_{m}*M*/

_{m}*M*, and

_{p}*Θ*. It is noted that no linear or other phase correction was employed on the experimentally measured PTF

_{m}*Θ*prior to its use in the estimation exercise. The value of α was estimated for each data source corresponding to a given defocus value within the range –20 ≤ ψ ≤ 20 and then averaged. The curve-fitting approach used normalized spatial frequencies

_{m}*u*within the range 0 ≤

_{est}*u*≤

_{est}*u*as the base variables whose coefficients were to be estimated. To ensure consistency across all ψ, the value of

_{cc}*u*was kept constant throughout this range. The value of

_{cc}*u*was determined by inspecting the measured MTF at ψ = 20 and identifying the actual cutoff frequency

_{cc}*u*at this defocus value. The upper limit

_{ca}*u*was then chosen to be 0.95

_{cc}*u*|

_{ca}_{ψ = 20}. Figure 7 presents the results of the above exercise. Each colored curve represents the estimated value of α, namely α

*, from one of the abovementioned data sources and across the specified range of defocus values. Estimates from theoretical data are shown in Fig. 7(a) while those from experimental data are illustrated in Fig. 7(b).*

_{est}*M*and

_{t}*Θ*yield reasonably close estimates of α as expected, whereas the result obtained by evaluating

_{t}*M*×

_{t}*M*shows a significant error amounting to a deviation of roughly 33% from the design value. The fact that visual inspection of Fig. 2(a) reveals only small differences between

_{p}*M*and

_{t}*M*×

_{t}*M*underscores the high sensitivity of the estimation method to variations in MTF height. This sensitivity is further highlighted upon inspecting Fig. 2(a) while examining α

_{p}*in Fig. 7(b) obtained from the measured MTF*

_{est}*M*. It is seen that the estimate of α from

_{m}*M*produced an error of nearly 120%. Attempting to raise the measured MTF by the pixel MTF does little to help as seen in Fig. 7(b), since

_{m}*M*/

_{m}*M*would remain below

_{p}*M*considering that

_{t}*M*×

_{t}*M*is higher than

_{p}*M*, especially at higher frequencies as attested by Fig. 2(a).

_{m}*Θ*yields a value of α

_{m}*that is very close to the design value, with an error of less than 2%. This is in spite of the fact that*

_{est}*Θ*is lower than

_{m}*Θ*as seen in Fig. 2(b). Given that the correction factor

_{t}*Θ*used in Fig. 2(b) and Fig. 6(b) is a strictly linear function, it does little to influence the estimation of α via the cubic exponent α

_{c}*u*

^{3}. It is therefore seen that the PTF is a far better property than the MTF for estimating the strength α of a cubic phase mask imaging system, at least in cases where a polynomial curve-fitting method is used. Another insight that may be gleaned from Fig. 7 is that oscillations due to the Fresnel integrals are more pronounced in the MTF than in the PTF. The values of α

*as a function of ψ thus exhibit greater deviations from their respective mean values when the MTFs are used as a data source.*

_{est}## 5. Conclusions

## Acknowledgments

## References and links

1. | E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. |

2. | J. van der Gracht, E. R. Dowski Jr, M. G. Taylor, and D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. |

3. | D. L. Marks, R. A. Stack, D. J. Brady, and J. van der Gracht, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. |

4. | W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. |

5. | A. Sauceda and J. Ojeda-Castañeda, “High focal depth with fractional-power wave fronts,” Opt. Lett. |

6. | S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE |

7. | J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. |

8. | M. Somayaji and M. P. Christensen, “Form factor enhancement of imaging systems using a cubic phase mask,” in |

9. | M. Somayaji and M. P. Christensen, “Form factor enhancement of imaging systems using a cubic phase mask,” in |

10. | M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. |

11. | G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. |

12. | P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. |

13. | M. Somayaji and M. P. Christensen, “Frequency analysis of the wavefront-coding odd-symmetric quadratic phase mask,” Appl. Opt. |

14. | S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A |

15. | G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. |

16. | M. Demenikov and A. R. Harvey, “Image artifacts in hybrid imaging systems with a cubic phase mask,” Opt. Express |

17. | S. Barwick, “Catastrophes in wavefront-coding spatial-domain design,” Appl. Opt. |

18. | S. Chen, Z. Fan, H. Chang, and Z. Xu, “Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask,” Appl. Opt. |

19. | H. B. Wach, E. R. Dowski Jr, and W. T. Cathey, “Control of chromatic focal shift through wave-front coding,” Appl. Opt. |

20. | M. Somayaji and M. P. Christensen, “Improving photon count and flat profiles of multiplex imaging systems with the odd-symmetric quadratic phase modulation mask,” Appl. Opt. |

21. | G. E. Johnson, E. R. Dowski Jr, and W. T. Cathey, “Passive ranging through wave-front coding: information and application,” Appl. Opt. |

22. | E. R. Dowski Jr, R. H. Cormack, and S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” Proc. SPIE |

23. | K. Kubala, E. Dowski, and W. T. Cathey, “Reducing complexity in computational imaging systems,” Opt. Express |

24. | R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. |

25. | S.-H. Lee, N.-C. Park, and Y.-P. Park, “Breaking diffraction limit of a small f-number compact camera using wavefront coding,” Opt. Express |

26. | M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express |

27. | M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in |

28. | S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. |

29. | R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE |

30. | E. R. Dowski Jr and G. E. Johnson, “Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems,” Proc. SPIE |

31. | M. Somayaji, V. R. Bhakta, and M. P. Christensen, “Experimental validation of exact optical transfer function of cubic phase mask wavefront coding imaging systems,” in |

32. | S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. |

33. | K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. |

34. | Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE |

35. | V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express |

36. | V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase transfer function of sampled imaging systems,” in |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(110.4850) Imaging systems : Optical transfer functions

(110.1758) Imaging systems : Computational imaging

(110.7348) Imaging systems : Wavefront encoding

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: November 22, 2011

Revised Manuscript: January 7, 2012

Manuscript Accepted: January 9, 2012

Published: January 12, 2012

**Citation**

Manjunath Somayaji, Vikrant R. Bhakta, and Marc P. Christensen, "Experimental evidence of the theoretical spatial frequency response of cubic phase mask wavefront coding imaging systems," Opt. Express **20**, 1878-1895 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1878

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### References

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