- Email: [email protected]
Broadband THz, extended depth of focus imaging based on step phase mask aided interferometry
Optics Communications 309 (2013) 1–5
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Broadband THz, extended depth of focus imaging based on step phase mask aided interferometry Assaf Bitman a,b,n, Inon Moshe a, Zeev Zalevsky b a b
Applied Physics Division, Soreq NRC, Yavne 81800, Israel Faculty of Engineering, Bar Ilan University, Ramat-Gan 52900, Israel
art ic l e i nf o
a b s t r a c t
Article history: Received 7 May 2013 Received in revised form 1 July 2013 Accepted 3 July 2013 Available online 12 July 2013
This work describes the realization of an extended depth of field (EDOF) in pulsed THz imaging systems using a step phase mask (SPM) attached to the objective lens. The SPM was designed to generate an EDOF compared to Gaussian broadband sources. This imaging property is demonstrated using a resolution target illuminated by broadband THz beams. An imaging depth improvement factor of 1.5 is demonstrated. In this paper we present the element design method together with numerical and experimental results. & 2013 Elsevier B.V. All rights reserved.
Keywords: THz Imaging OTF EDOF Coded aperture
1. Introduction Pulsed terahertz (THz) radiation imaging systems have the advantage of obtaining depth information in both transparent and opaque materials [1–5]. Therefore extending the depth of field of broadband THz imaging systems is a very crucial core technology. Extended depth of field (EDOF) enables one to overcome conventional optical imaging systems’ tradeoff between axial and lateral resolution. It ensures continuous lateral resolution over an extended axial length. McLeod was the first to show extended focal range using a conical lens (also known as “axicon”), with correlation to input beam radius and the axicon's base angle [6]. Much research followed, using invariant Bessel beams created by axicons, in optical imaging systems [7–15]. In this paper the achieved EDOF is based on a different element: a step phase mask (SPM) inserted at the entrance pupil of an imaging lens. These kinds of elements create interference plans before and after the focal plane. In return, these interference plane-zones cause to enlargement of the Rayleigh range, hence to EDOF. Unlike axicons, which are refractive optical phase elements, the step optical masks are mainly phase manipulating elements that define a low spatial frequency phase transition to code the lens aperture. One can implement the SPM in other spectral ranges. However, since the spectral region of THz has the potential
n Corresponding author at: Bar Ilan University, Faculty of Engineering, 52900 Ramat-Gan, Israel. Tel.: +972 506 292 251. E-mail addresses: [email protected], [email protected] (A. Bitman).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.008
to image objects which are optically opaque, it is intriguing to implement methods such as SPM to increase the focal depth. In addition, one of the challenges that this manuscript deals with is implementation of phase-mask to enable EDOF using broadband source, such as pulsed THz source. Since these binary phase elements contain low spatial frequencies; they are much less wavelength sensitive compared with diffractive optical elements. Such an approach can be very suitable for dealing with broadband THz radiation. Moreover, the phase affecting elements scatters little energy towards the outer regions of the field of view. The goal of this paper is to prove that a step phase element can be combined with an imaging lens to create EDOF using pulsed THz illumination sources. The optical element is made of step phase rings that modulate the entrance pupil of the imaging lens. The optimization of the SPM design is based on maximizing the image contrast, which is a factor of the optical transfer function (OTF). In Refs. [16–18] Zalevsky et al. investigated the theory of the proposed design concept for the optical visible regime, and for spatially incoherent radiation. They approximated the OTF which led to an analytic solution. In this paper we implement and adjust these concepts to a pulsed THz radiation. These THz pulses are highly broadband; and the spectral bandwidth share phase relation rising from pulse nature, i.e. this radiation is spatially coherent. Furthermore, we solved the OTF numerically without any approximations. These calculations led to a different result and to a different element design. In Section 2 the theoretical derivation is presented. Section 3 presents numerical simulations and the experimental results. Finally, conclusions are given in Section 4.
2
A. Bitman et al. / Optics Communications 309 (2013) 1–5
2. Theoretical derivation The OTF of an imaging system can be expressed as an autocorrelation operation between the pupil function of the lens [19]: Hðμx ; μy ; Z i Þ ¼
λZ μ λZ μ þ1 P x þ λZ2i μx ; y þ 2i y P n x λZ2i μx ; y 2i y dx dy ∬1 þ1 jPðx; yÞj2 dxdy ∬1
:
ð1Þ P(x,y) is the pupil function (the lens aperture) which equals to “1” within the pupil and to “0” outside, and the asterisk means complex conjugate. This pupil function can be described as: β(ω)¼ n(ω)ω/c0. When aberrations are introduced, the generalized pupil function can be described as: Pðx; yÞ ¼ jPðx; yÞj exp
ðik Wðx; yÞÞ:
ð2Þ
where W(x,y) is the wave aberration, k¼2π/λ, and λ is the optical wavelength. If the aberrations are caused only by defocusing, W(x,y) has the form: Wðx; yÞ ¼ W m
ðx2 þ y2 Þ α2
ð3Þ
where α is the radius of the aperture P. The coefficient Wm determines the error's misalignment sensitivity, and can also be written as: Wm ¼
ψλ 2π
ð4Þ
Ψ is a phase-factor representing the out of focus's sensitivity: πα2 1 1 1 ψ¼ : ð5Þ þ Zi Zo f λ Zo is the distance between the object and the imaging lens; Zi is the distance between the imaging lens and the object plan (sensor) and f is the focal length. When imaging condition is fulfilled one has: 1 1 1 þ ¼ 0: Zi Zo f
ð6Þ
And thus the distortion phase Ψ equals zero. For the sake of simplicity we will perform a 1-D analysis. The autocorrelation function of an arbitrary function g(x) is [10]: Z þ1 x′ n x′ Rgg′ðx′Þ ¼ g x dx: ð7Þ g xþ 2 2 1 For step phase rings modulation and an imaging lens g(x) can be written as: " # n ¼ þN=2 xnΔx gðxÞ ¼ PðxÞ exp i ∑ an rect ð8Þ Δx n ¼ N=2 where an are binary coefficients equal to zero or to a certain phase modulation depth Δϕ: an ¼(0,Δϕ). Note that the binary coefficients are wavelength independent, i.e. an≠an(λ), in our THz domain. Δx represents the spatial segment of the element. It should be mentioned that since we do not want to create a diffractive optical element, we force the designed spatial segments to fulfill Δx≫λmax. N/2 equals to the number of rings (for example for 2 rings N ¼4). Using phase rings rather than phase lines (grid) lead the binary coefficients to satisfy a|n| ¼a |n|. In this case Eq. (8) can be rewritten as: x gðxÞ ¼ PðxÞ exp ia0 rect Δx " # n ¼ þN=2 x þ nΔx xnΔx þ rect : ð9Þ exp ∑ ian rect Δx Δx n¼1
R þ1 Since 1 PðxÞdx ¼ 2α, i.e. it equals to the element's diameter, the OTF of the system using Eqs. (1)–(9) can be expressed as: Z þ1 n h x i 1 1 Pðμx ; Z i Þ ¼ dx ðPðx1 Þ ≤exp ia0 rect 2α 1 Δx " #! n ¼ N=2 x1 þ nΔx x1 nΔx þ rect exp i ∑ an rect Δx Δx n¼1 " h x i m ¼ N=2 2 exp i ∑ am frect P n ðx2 Þ exp ia0 rect Δx m¼1 #!) x2 þ mΔx x2 mΔx þ rect : ð10Þ Δx Δx where x1 ¼x+(λZiμx)/2 and x2 ¼ x-(λZiμx)/2. Eq. (10) describes the general OTF of a system combining step phase rings with an imaging lens. The last expression allows extracting the derivative of OTF in respect to Δx or the phase depth modulation Δϕ. In Ref. [7] Zalevsky et al. presented an approximation to Eq. (10) for EDOF of an optical visible system. We, on the other hand, solve the exact equation to satisfy pulsed spatially coherent THz optical system. The mathematical formulation for the optimization of ring-configuration will be as follow: Compute the only rings phase combination that will provide a maximum for the minimum value of the OTF within the desired spatial spectrum region. This will ensure a continuous focus region (constant like OTF), i.e. a continuous contrast over an extended depth. Note that μx is the spatial frequency's coordinate of the OTF in the x-direction. We will use: μx ¼ 0:8μcutoff :
ð11Þ
The cut off frequency μcutoff is given by Ref. [19]: μcutoff ¼ 2α=λZ i :
ð12Þ
3. Numerical simulations and experimental results In order to produce the optimal EDOF element, numerical calculations were performed. Each calculation contained all the permutation combinations of rings. The optical-window size in our system was limited to 100 mm. In order to avoid diffraction effects we required that each width of each ring will be much larger than the maximum typical wavelength, i.e. Δx≫λmax (3 mm). That forces the SPM to be composed of three phase rings and one central lobe. Under these conditions the number of permutations will be 16 (¼ 24) as each ring is a step phase element. The phase modulation of each ring (and the central lobe) was determined and remained fixed during each calculation. In every permutation we calculated the OTF based on Eq. (10). Finally we derived the optimal ring configuration. Different values of phase modulation Δϕ were tested and filtered based on the criterion of minimum OTF modulation around the focal region that met the additional constraint of OTF ≥0.1. Plotting graphs of OTF amplitude versus DOF for different phase rings combination (example for that can be seen in Fig. 2), revealed that an optimum is obtained when the central lobe has Δϕ≈7π/4 and the outer ring has Δϕ≈3π/4. The other rings have no phase (Δϕ¼0), as depicted in Fig. 1. As previously mentioned the segment width of each ring Δx was designed to be more than three times larger than λmax ¼3 mm. The optical-window size in our system was limited to 100 mm. Therefore, the rings phase element used for the experiment was designed to have Δx¼14 mm in order to fulfill the requirement of Δx≫λmax, and to avoid significant diffraction phenomenon. This phase mask is different in respect to the principle presented in Refs. [16,17] and it is also expressed in Eqs. (8)–(10) which differ from the corresponding approximated equations presented in Ref. [16]. Fig. 2 depicts the modulate transfer function (MTF) of the lens with (dash line) and without (solid line) EDOF element for optical
A. Bitman et al. / Optics Communications 309 (2013) 1–5
wavelengths of 0.3 cm and 0.03 cm, respectively. It can be seen, that the MTF of the imaging system without any mask on the lens aperture, behaves as a sinc-function. However, for both wavelengths an EDOF appears while using the phase element. Recall that the MTF is the real part of the OTF [19]. The MTF criterion mentioned above was used to choose the optimized phase ring design. The increase in the focal depth as depicted by the MTFs in Fig. 2 can be used as the basis for predicting the broadband EDOF performance of the imaging system. Note that the relationship between the MTF curves and the imaging system broadband viewing contrast is not direct. While the calculated MTF curves describe the imaging performance for a discrete wavelength and specific spatial frequencies, in the experiment a broadband THz source has been used having a range of spatial frequencies. In order to experimentally compare the EDOF performances of the optical system, with and without the phase element, a resolution target was used. The target used was a metal plate with three
Fig. 1. A cross section of the optimal SPM designed to maximize depth of field.
3
rectangular slits 5 mm wide each (y direction) and 20 mm long (x direction). These apertures were separated in the y direction by 5 mm wide metal (duty cycle of 0.5). The slits were transversally x–y scanned at several points along the optical direction-z. Fig. 3 shows the optical setup where the resolution target was scanned. For experimental exhibits a transmission mode THz system was used as described in Ref. [15]. The THz transmitter based on a photoconductive switch antenna was attached to a hemispherical lens to produce broadband Gaussian THz source. A lens of 76 mm focal length was added to the transmitter at distance of 76 mm from the hemispherical lens to collimate the output radiation (fS in Fig. 3). Next four high density poly ethylene planoconvex lenses of two types were used: one with focal length f2 ¼ 30 mm; and three with focal length f3 ¼ 125 mm. First we measured the broadband Gaussian beam performance without phase element (B in Fig. 3), by scanning the resolution target at various distances from the objective lens (A in Fig. 3), along z direction. We repeated these measurements with the step phase element inserted in front of f3 (B in Fig. 3). Fig. 4(a) shows the image intensity distribution measured when placing the previously described resolution target at a distance of 125 mm from the objective lens, without the phase element. Fig. 4(c) and (e) depicts the measured image at distances of 170 mm and 210 mm from the objective lens, respectively. For comparison Fig. 4(b), (d) and (f) shows the resolution target image measured for the same locations, respectively; but with use of the SPM. Fig. 5 summarizes the total contrast along the z direction for the two cases of broadband Gaussian with and without the phase element. The contrast was calculated from the averaged cross section profile by using the following equation: contrast ¼
I max I min : I max þ I min
ð13Þ
Fig. 2. Simulation of the MTF with (red dashed line) and without phase rings (blue solid line) for: (a) λ ¼ 0.3 cm (b) λ¼ 0.03 cm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Setup sketch of the imaging system for scanning the resolution target with a broadband Gaussian beam, with and without the interference EDOF SPM (B).
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A. Bitman et al. / Optics Communications 309 (2013) 1–5
Fig. 4. Images of the resolution target at different distances from the objective lens: (a)–(b) 120 mm from lens, (c)–(d) 170 mm from lens, (e)–(f) 210 mm from lens. (a), (c) and (e) were measured without the EDOF phase element while (b), (c) and (f) were measured with the phase element.
From Fig. 5 one can observe, as expected, that both configurations get their maximum contrast at the image plane, i.e. at a distance of 125 mm from the objective lens. However, imaging with SPM shows enhancement in focal depth, and better image contrast along wider range around the image plane. One can see that using SPM leads to an increase of the focal depth by a factor of 1.5.
4. Summary and conclusion
Fig. 5. Comparison between imaging-contrast of a resolution target using broadband Gaussian with (red circles) and without (black squares) SPM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In this paper we demonstrated an interference based EDOF concept in broadband THz imaging system. This idea was first introduced in Ref. [16–18] for visible incoherent imaging optics. Here, the EDOF principle was implemented in a pulsed spatially coherent THz imaging system, and a factor of 1.5 improvements in the depth of field was shown. This super resolved image was achieved by using a phase element composed of low spatial frequency step phase rings at the entrance pupil of the THz imaging lens. The designed element was based on calculation of the OTF and was optimized with the goal of extending the depth of field. The optimized SPM was produced and then tested experimentally
A. Bitman et al. / Optics Communications 309 (2013) 1–5
in a broadband THz imaging setup. As this element contained only low spatial frequencies it was straight-forward to fabricate. Its influence on the light phase and energy efficiency was high compared to refractive elements, such as the axicons previously used to extend the focal depth [15]. Extending depth of field using an element with high energy efficiency and it is also easy to fabricate, is important, especially when imaging with spectrally broad radiation (the case under investigation here). Based on the results presented in this paper, one can deduce the great potential for integrating easily manufactured phase elements into THz imaging systems. In order to further increase the EDOF of THz imaging systems, additional research should be performed in the fabrication of more phase rings or even the use of a binary phase grid to modulate the entrance pupil of the imaging lens. References [1] W.L. Chan, J. Deibel, D.M. Mittleman, Reports on Progress in Physics 70 (2007) 1325. [2] P.F. Taday, D.A. Newnham, Spectroscopy Europe 16 (2004) 20. [3] J.A. Zeitler, P.F. Taday, D.A. Newnham, M. Pepper, K.C. Gordon, T. Rades, Journal of Porphyrins and Phthalocyanines (2006) 209.
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[4] D. Dragonman, M. Dragonman, Progress in Quantum Electronics 28 (2004) 1. [5] D. Mittleman, Sensing with Terahertz Radiation, Springer Series in Opt. Sci, New-York, 2003. [6] J.H. McLeod, Journal of the Optical Society of America 44 (1954) 592. [7] J. Durnin, Journal of the Optical Society of America A: Optics, Image Science, and Vision 4 (1987) 651. [8] J. Lloyd, K. Wang, A. Barkan, D.M. Mittleman, Optics Communications 219 (2003) 289. [9] Y. Yu, W. Dou, Optics Express 17 (2009) 888. [10] S. Winnerl, B. Zimmermann, F. Peter, H. Schneider, M. Helm, Optics Express 17 (2009) 1571. [11] M.U. Shaukat, P. Dean, S.P. Khanna, M. Lachab, S. Chakraborty, E.H. Linfield, A.G. Davies, Optics Letters 34 (2009) 1030. [12] Y. Yu, In: Progress in Electromagnetics Research Symposium Proceedings, Xi'an, China, March 22–26,2010, pp. 1471–1474. [13] J. Liu, L. Wang, J. Li, W. Wang,Z. Hong, In: Proceeding of the SPIE Infrared, Millimeter Wave, and Terahertz Technologies 7854 Beijing, China, 2010, pp. 1–6. [14] Z. Zhang,T. Buma, In: Proceeding SPIE Terahertz Technology and Applications IV 7938 2011, pp. 1–8. [15] A. Bitman, I. Moshe, Z. Zalevsky, Optics Letters 37 (2012) 4164. [16] Z. Zalevsky, A. Shemer, A. Zlotnik, E.B. Eliezer, E. Marom, Optics Express 7 (2006) 2631. [17] I. Raveh, Z. Zalevsky, Optics Express 15 (2007) 17912. [18] Z. Zalevsky, US Patent Application 10/97494 (August 2004). [19] J.W. Goodman, Introduction to Fourier Opticssecond ed., McGraw-Hill, New-York, 1996.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Broadband THz, extended depth of focus imaging based on step phase mask aided interferometry Assaf Bitman a,b,n, Inon Moshe a, Zeev Zalevsky b a b
Applied Physics Division, Soreq NRC, Yavne 81800, Israel Faculty of Engineering, Bar Ilan University, Ramat-Gan 52900, Israel
art ic l e i nf o
a b s t r a c t
Article history: Received 7 May 2013 Received in revised form 1 July 2013 Accepted 3 July 2013 Available online 12 July 2013
This work describes the realization of an extended depth of field (EDOF) in pulsed THz imaging systems using a step phase mask (SPM) attached to the objective lens. The SPM was designed to generate an EDOF compared to Gaussian broadband sources. This imaging property is demonstrated using a resolution target illuminated by broadband THz beams. An imaging depth improvement factor of 1.5 is demonstrated. In this paper we present the element design method together with numerical and experimental results. & 2013 Elsevier B.V. All rights reserved.
Keywords: THz Imaging OTF EDOF Coded aperture
1. Introduction Pulsed terahertz (THz) radiation imaging systems have the advantage of obtaining depth information in both transparent and opaque materials [1–5]. Therefore extending the depth of field of broadband THz imaging systems is a very crucial core technology. Extended depth of field (EDOF) enables one to overcome conventional optical imaging systems’ tradeoff between axial and lateral resolution. It ensures continuous lateral resolution over an extended axial length. McLeod was the first to show extended focal range using a conical lens (also known as “axicon”), with correlation to input beam radius and the axicon's base angle [6]. Much research followed, using invariant Bessel beams created by axicons, in optical imaging systems [7–15]. In this paper the achieved EDOF is based on a different element: a step phase mask (SPM) inserted at the entrance pupil of an imaging lens. These kinds of elements create interference plans before and after the focal plane. In return, these interference plane-zones cause to enlargement of the Rayleigh range, hence to EDOF. Unlike axicons, which are refractive optical phase elements, the step optical masks are mainly phase manipulating elements that define a low spatial frequency phase transition to code the lens aperture. One can implement the SPM in other spectral ranges. However, since the spectral region of THz has the potential
n Corresponding author at: Bar Ilan University, Faculty of Engineering, 52900 Ramat-Gan, Israel. Tel.: +972 506 292 251. E-mail addresses: [email protected], [email protected] (A. Bitman).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.008
to image objects which are optically opaque, it is intriguing to implement methods such as SPM to increase the focal depth. In addition, one of the challenges that this manuscript deals with is implementation of phase-mask to enable EDOF using broadband source, such as pulsed THz source. Since these binary phase elements contain low spatial frequencies; they are much less wavelength sensitive compared with diffractive optical elements. Such an approach can be very suitable for dealing with broadband THz radiation. Moreover, the phase affecting elements scatters little energy towards the outer regions of the field of view. The goal of this paper is to prove that a step phase element can be combined with an imaging lens to create EDOF using pulsed THz illumination sources. The optical element is made of step phase rings that modulate the entrance pupil of the imaging lens. The optimization of the SPM design is based on maximizing the image contrast, which is a factor of the optical transfer function (OTF). In Refs. [16–18] Zalevsky et al. investigated the theory of the proposed design concept for the optical visible regime, and for spatially incoherent radiation. They approximated the OTF which led to an analytic solution. In this paper we implement and adjust these concepts to a pulsed THz radiation. These THz pulses are highly broadband; and the spectral bandwidth share phase relation rising from pulse nature, i.e. this radiation is spatially coherent. Furthermore, we solved the OTF numerically without any approximations. These calculations led to a different result and to a different element design. In Section 2 the theoretical derivation is presented. Section 3 presents numerical simulations and the experimental results. Finally, conclusions are given in Section 4.
2
A. Bitman et al. / Optics Communications 309 (2013) 1–5
2. Theoretical derivation The OTF of an imaging system can be expressed as an autocorrelation operation between the pupil function of the lens [19]: Hðμx ; μy ; Z i Þ ¼
λZ μ λZ μ þ1 P x þ λZ2i μx ; y þ 2i y P n x λZ2i μx ; y 2i y dx dy ∬1 þ1 jPðx; yÞj2 dxdy ∬1
:
ð1Þ P(x,y) is the pupil function (the lens aperture) which equals to “1” within the pupil and to “0” outside, and the asterisk means complex conjugate. This pupil function can be described as: β(ω)¼ n(ω)ω/c0. When aberrations are introduced, the generalized pupil function can be described as: Pðx; yÞ ¼ jPðx; yÞj exp
ðik Wðx; yÞÞ:
ð2Þ
where W(x,y) is the wave aberration, k¼2π/λ, and λ is the optical wavelength. If the aberrations are caused only by defocusing, W(x,y) has the form: Wðx; yÞ ¼ W m
ðx2 þ y2 Þ α2
ð3Þ
where α is the radius of the aperture P. The coefficient Wm determines the error's misalignment sensitivity, and can also be written as: Wm ¼
ψλ 2π
ð4Þ
Ψ is a phase-factor representing the out of focus's sensitivity: πα2 1 1 1 ψ¼ : ð5Þ þ Zi Zo f λ Zo is the distance between the object and the imaging lens; Zi is the distance between the imaging lens and the object plan (sensor) and f is the focal length. When imaging condition is fulfilled one has: 1 1 1 þ ¼ 0: Zi Zo f
ð6Þ
And thus the distortion phase Ψ equals zero. For the sake of simplicity we will perform a 1-D analysis. The autocorrelation function of an arbitrary function g(x) is [10]: Z þ1 x′ n x′ Rgg′ðx′Þ ¼ g x dx: ð7Þ g xþ 2 2 1 For step phase rings modulation and an imaging lens g(x) can be written as: " # n ¼ þN=2 xnΔx gðxÞ ¼ PðxÞ exp i ∑ an rect ð8Þ Δx n ¼ N=2 where an are binary coefficients equal to zero or to a certain phase modulation depth Δϕ: an ¼(0,Δϕ). Note that the binary coefficients are wavelength independent, i.e. an≠an(λ), in our THz domain. Δx represents the spatial segment of the element. It should be mentioned that since we do not want to create a diffractive optical element, we force the designed spatial segments to fulfill Δx≫λmax. N/2 equals to the number of rings (for example for 2 rings N ¼4). Using phase rings rather than phase lines (grid) lead the binary coefficients to satisfy a|n| ¼a |n|. In this case Eq. (8) can be rewritten as: x gðxÞ ¼ PðxÞ exp ia0 rect Δx " # n ¼ þN=2 x þ nΔx xnΔx þ rect : ð9Þ exp ∑ ian rect Δx Δx n¼1
R þ1 Since 1 PðxÞdx ¼ 2α, i.e. it equals to the element's diameter, the OTF of the system using Eqs. (1)–(9) can be expressed as: Z þ1 n h x i 1 1 Pðμx ; Z i Þ ¼ dx ðPðx1 Þ ≤exp ia0 rect 2α 1 Δx " #! n ¼ N=2 x1 þ nΔx x1 nΔx þ rect exp i ∑ an rect Δx Δx n¼1 " h x i m ¼ N=2 2 exp i ∑ am frect P n ðx2 Þ exp ia0 rect Δx m¼1 #!) x2 þ mΔx x2 mΔx þ rect : ð10Þ Δx Δx where x1 ¼x+(λZiμx)/2 and x2 ¼ x-(λZiμx)/2. Eq. (10) describes the general OTF of a system combining step phase rings with an imaging lens. The last expression allows extracting the derivative of OTF in respect to Δx or the phase depth modulation Δϕ. In Ref. [7] Zalevsky et al. presented an approximation to Eq. (10) for EDOF of an optical visible system. We, on the other hand, solve the exact equation to satisfy pulsed spatially coherent THz optical system. The mathematical formulation for the optimization of ring-configuration will be as follow: Compute the only rings phase combination that will provide a maximum for the minimum value of the OTF within the desired spatial spectrum region. This will ensure a continuous focus region (constant like OTF), i.e. a continuous contrast over an extended depth. Note that μx is the spatial frequency's coordinate of the OTF in the x-direction. We will use: μx ¼ 0:8μcutoff :
ð11Þ
The cut off frequency μcutoff is given by Ref. [19]: μcutoff ¼ 2α=λZ i :
ð12Þ
3. Numerical simulations and experimental results In order to produce the optimal EDOF element, numerical calculations were performed. Each calculation contained all the permutation combinations of rings. The optical-window size in our system was limited to 100 mm. In order to avoid diffraction effects we required that each width of each ring will be much larger than the maximum typical wavelength, i.e. Δx≫λmax (3 mm). That forces the SPM to be composed of three phase rings and one central lobe. Under these conditions the number of permutations will be 16 (¼ 24) as each ring is a step phase element. The phase modulation of each ring (and the central lobe) was determined and remained fixed during each calculation. In every permutation we calculated the OTF based on Eq. (10). Finally we derived the optimal ring configuration. Different values of phase modulation Δϕ were tested and filtered based on the criterion of minimum OTF modulation around the focal region that met the additional constraint of OTF ≥0.1. Plotting graphs of OTF amplitude versus DOF for different phase rings combination (example for that can be seen in Fig. 2), revealed that an optimum is obtained when the central lobe has Δϕ≈7π/4 and the outer ring has Δϕ≈3π/4. The other rings have no phase (Δϕ¼0), as depicted in Fig. 1. As previously mentioned the segment width of each ring Δx was designed to be more than three times larger than λmax ¼3 mm. The optical-window size in our system was limited to 100 mm. Therefore, the rings phase element used for the experiment was designed to have Δx¼14 mm in order to fulfill the requirement of Δx≫λmax, and to avoid significant diffraction phenomenon. This phase mask is different in respect to the principle presented in Refs. [16,17] and it is also expressed in Eqs. (8)–(10) which differ from the corresponding approximated equations presented in Ref. [16]. Fig. 2 depicts the modulate transfer function (MTF) of the lens with (dash line) and without (solid line) EDOF element for optical
A. Bitman et al. / Optics Communications 309 (2013) 1–5
wavelengths of 0.3 cm and 0.03 cm, respectively. It can be seen, that the MTF of the imaging system without any mask on the lens aperture, behaves as a sinc-function. However, for both wavelengths an EDOF appears while using the phase element. Recall that the MTF is the real part of the OTF [19]. The MTF criterion mentioned above was used to choose the optimized phase ring design. The increase in the focal depth as depicted by the MTFs in Fig. 2 can be used as the basis for predicting the broadband EDOF performance of the imaging system. Note that the relationship between the MTF curves and the imaging system broadband viewing contrast is not direct. While the calculated MTF curves describe the imaging performance for a discrete wavelength and specific spatial frequencies, in the experiment a broadband THz source has been used having a range of spatial frequencies. In order to experimentally compare the EDOF performances of the optical system, with and without the phase element, a resolution target was used. The target used was a metal plate with three
Fig. 1. A cross section of the optimal SPM designed to maximize depth of field.
3
rectangular slits 5 mm wide each (y direction) and 20 mm long (x direction). These apertures were separated in the y direction by 5 mm wide metal (duty cycle of 0.5). The slits were transversally x–y scanned at several points along the optical direction-z. Fig. 3 shows the optical setup where the resolution target was scanned. For experimental exhibits a transmission mode THz system was used as described in Ref. [15]. The THz transmitter based on a photoconductive switch antenna was attached to a hemispherical lens to produce broadband Gaussian THz source. A lens of 76 mm focal length was added to the transmitter at distance of 76 mm from the hemispherical lens to collimate the output radiation (fS in Fig. 3). Next four high density poly ethylene planoconvex lenses of two types were used: one with focal length f2 ¼ 30 mm; and three with focal length f3 ¼ 125 mm. First we measured the broadband Gaussian beam performance without phase element (B in Fig. 3), by scanning the resolution target at various distances from the objective lens (A in Fig. 3), along z direction. We repeated these measurements with the step phase element inserted in front of f3 (B in Fig. 3). Fig. 4(a) shows the image intensity distribution measured when placing the previously described resolution target at a distance of 125 mm from the objective lens, without the phase element. Fig. 4(c) and (e) depicts the measured image at distances of 170 mm and 210 mm from the objective lens, respectively. For comparison Fig. 4(b), (d) and (f) shows the resolution target image measured for the same locations, respectively; but with use of the SPM. Fig. 5 summarizes the total contrast along the z direction for the two cases of broadband Gaussian with and without the phase element. The contrast was calculated from the averaged cross section profile by using the following equation: contrast ¼
I max I min : I max þ I min
ð13Þ
Fig. 2. Simulation of the MTF with (red dashed line) and without phase rings (blue solid line) for: (a) λ ¼ 0.3 cm (b) λ¼ 0.03 cm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Setup sketch of the imaging system for scanning the resolution target with a broadband Gaussian beam, with and without the interference EDOF SPM (B).
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Fig. 4. Images of the resolution target at different distances from the objective lens: (a)–(b) 120 mm from lens, (c)–(d) 170 mm from lens, (e)–(f) 210 mm from lens. (a), (c) and (e) were measured without the EDOF phase element while (b), (c) and (f) were measured with the phase element.
From Fig. 5 one can observe, as expected, that both configurations get their maximum contrast at the image plane, i.e. at a distance of 125 mm from the objective lens. However, imaging with SPM shows enhancement in focal depth, and better image contrast along wider range around the image plane. One can see that using SPM leads to an increase of the focal depth by a factor of 1.5.
4. Summary and conclusion
Fig. 5. Comparison between imaging-contrast of a resolution target using broadband Gaussian with (red circles) and without (black squares) SPM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In this paper we demonstrated an interference based EDOF concept in broadband THz imaging system. This idea was first introduced in Ref. [16–18] for visible incoherent imaging optics. Here, the EDOF principle was implemented in a pulsed spatially coherent THz imaging system, and a factor of 1.5 improvements in the depth of field was shown. This super resolved image was achieved by using a phase element composed of low spatial frequency step phase rings at the entrance pupil of the THz imaging lens. The designed element was based on calculation of the OTF and was optimized with the goal of extending the depth of field. The optimized SPM was produced and then tested experimentally
A. Bitman et al. / Optics Communications 309 (2013) 1–5
in a broadband THz imaging setup. As this element contained only low spatial frequencies it was straight-forward to fabricate. Its influence on the light phase and energy efficiency was high compared to refractive elements, such as the axicons previously used to extend the focal depth [15]. Extending depth of field using an element with high energy efficiency and it is also easy to fabricate, is important, especially when imaging with spectrally broad radiation (the case under investigation here). Based on the results presented in this paper, one can deduce the great potential for integrating easily manufactured phase elements into THz imaging systems. In order to further increase the EDOF of THz imaging systems, additional research should be performed in the fabrication of more phase rings or even the use of a binary phase grid to modulate the entrance pupil of the imaging lens. References [1] W.L. Chan, J. Deibel, D.M. Mittleman, Reports on Progress in Physics 70 (2007) 1325. [2] P.F. Taday, D.A. Newnham, Spectroscopy Europe 16 (2004) 20. [3] J.A. Zeitler, P.F. Taday, D.A. Newnham, M. Pepper, K.C. Gordon, T. Rades, Journal of Porphyrins and Phthalocyanines (2006) 209.
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