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Super resolution imaging using interferometric masking technique
Optik 126 (2015) 5629–5632
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Super resolution imaging using interferometric masking technique Anwar Hussain ∗ COMSATS Institute of Information Technology, Chak Shahzad, Islamabad, Pakistan
a r t i c l e
i n f o
Article history: Received 24 September 2014 Accepted 6 September 2015 Keywords: Super resolution Interferometry Mach Zehnder and Sagnac interferometer
a b s t r a c t This paper describes the super resolution technique which is based on the Torlado di Francia imaging technique. The super resolution has been achieved using two masks: phase and amplitude, separately in the same setup. In one case the Mach Zehnder interferometer was used to synthesize the shifted phase mask while in the second case the negative of amplitude mask was produced. The interferometric multiplication of the original aperture with the synthesized amplitude and phase masks separately gives super resolution in both cases. The achieved Rayleigh resolution is 85% of the conventional system. The side lobes reduction was also obtained in this technique. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction 1.1. Interferometric imaging Achieving high resolution beyond the classical limit is called super resolution. Different super resolution techniques like grating based [1], fringe illumination [2,3] and interferometry based [4,5] have been mentioned in literature to achieve high resolution. The point spread function (PSF) of the system reflects the resolution of the system. Narrower the width of PSF greater is the resolution. If two points lying close to each other are imaged through the system having wider PSF result in a single point. But if the same objects are imaged through an optical system of narrower PSF result in two separate objects, mean highly resolved. To synthesize an aperture who’s PSF has small central lob called super-resolution. In literature some techniques are reported to create PSF with narrow width. The first idea of mask was proposed by Toraldo di Francia in 1952 [6]. According to this idea an aperture pupil mask composed of concentric transparent rings with phases of 0 and is used for imaging. The point spread function produced by this mask having central spot very narrow as required for super-resolution. The drawback of this method was outer bright rings that carrying larger part of energy. The advantage of this method was that no limit on resolution using this method in view of Sparrow and Rayleigh resolution criterion. Due to its unlimited resolution nature Hegedus and Sarafis [7] prepared a phase mask consist of three rings to demonstrate super-resolution, later on implemented in confocal
∗ Corresponding author. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.09.008 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
microscopy. To implement the idea in far field Leizerson [8] further extended the idea by eliminating the bright rings by interferometric technique. The synthesized mask is obtained as result of interferometric superposition of two amplitude masks. The PSF of the synthesized mask is narrow compared to the PSFs of other masks. The next step is to eliminate the outer rings of PSF to make it more ideal. For this purpose another PSF which has zero value at the outer region is multiplied with the PSF of synthesized mask using interferometric method [7–10]. Here similar idea reported in [6–10] is followed with better resolution result for phase masking and setup is also modified for amplitude masking. We have established that better resolution can be achieved using a modified Mach Zehnder interferometer. Both interferometric phase and amplitude masking techniques, applied in the single experimental setup. A removable mirror (RM) when removed from the set up converts the system into amplitude interferometric masking which otherwise acts as an interferometric phase masking setup. We are presenting the simulation results to establish that the super resolution can be achieved either with the use of interferometric phase masks or with interferometric amplitude masks and that the same resolution can be achieved in both cases which 85% of the Rayleigh limit. The previous work [8] reports a maximum of 55% Rayleigh limit achievement. The detailed description of work is given below. 1.2. Detail of experimental setup The proposed experimental setup for implementing both the interferometric phase masking and the amplitude masking is shown in Fig. 1. The set up uses two lasers 1 and 2 which are mutually incoherent. In the present analysis we are assuming that both the lasers are operating on a single wavelength which is
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Fig. 1. Experimental setup for interferometric imaging, BC, B0, B2, B3 are beam splitters, MK1, is open square mask, MK2 and MK3 are used to formed synthesized mask. L1 and L2 are lenses, Ms are mirrors and RM represents removable mirror. BM is beam stopper. Set up B is used in the formation of amplitude mask. For detail see text.
the same for both lasers. Two lasers sources are used to compare the super-resolved PSF of the system. The two lasers sources are like two points separated by some distance to image through mask-aperture, and also act like illumination source to produce the synthetic aperture. The lens L1 collimates the two spatially filtered beams and the polarizer just after the lens L1 polarizes the laser beams. The second polarizer just before the lens L2 acts as an analyzer. The linear polarizer is used to control the contrast of the image and polarization of the lasers sources. The path of the two lasers beams has been indicated with different colors shown in Fig. 1. B0 is a polarizing beam splitter which transmits the light from laser1 and reflects the light from laser2. The set up consists of eight mirrors including one removable mirror (RM) and six beam splitters. The beam splitters B2 and B3 and the two mirrors M5 and M6 form a Mach Zehnder interferometer. The beam splitter Bo, Mach Zehnder interferometer (B2, B3, M5, M6) and two mirrors RM and M1 form a Sagnac interferometer. With the existence of RM in the set up, the beam2 is stopped and when the RM is removed the beam1 is stopped. With the removal of RM two Mach Zehnder interferometers comprising components (B2, B3, M5, M6 Mach Zehnder 1) and (B1, B4, M7, M4 along with components of Mach Zehnder 1 Mach Zehnder 2) are formed. Another Sagnac interferometer is also formed comprising of the components (B0, M1, M2, M3 along with two Mach Zehnder interferometers). The two Mach Zehnder interferometers and the two Sagnac interferometers have been labeled with MZ1, MZ2, SAG1, SAG2 respectively and is shown in Fig. 1. Two masks have been placed in two arms of the MZ1 labeled with MK2 and MK3 shown respectively in Fig. 2(b) and (c) respectively. The open aperture labeled with MK1 is shown in Fig. 2(a) and its position has been indicated at two different locations in Fig. 1. The two beams from the MZ1 exiting toward RM will be phase shifted by whereas the two beams exiting toward B4 will be in phase. The interferometric addition of two beams toward RM will give the addition of two masks MK3 and MK2 and the resultant is shown in Fig. 2d. The white area in the two masks has a value of 1
Fig. 3. Comparison of three filters and their PSF. (a) Open aperture with PSF in one dimension in (d), (b) mask MK2 with the corresponding PSF in one dimension in (e), and (c) synthesized phase mask and the corresponding PSF in one dimension in (f).
and the black area has a value of 0 as assumed in the simulation. The open aperture mask MK1 is (placed between B0 and M1) shown in Fig. 2(a). The laser beam from laser source1 will pass through MK1 and will route through a portion of MZ1. The BM (beam stopper) is placed in MZ1 to stop the beam coming from B3 toward M5. Just before the lens L2 there will be an interferometric multiplication between the open aperture shown in Fig. 2(a) and the interferometric mask shown in Fig. 2(d). The resultant of this multiplication is shown in Fig. 3(c). The point spread functions in one dimension are shown at the bottom of each two dimensional masks both in Fig. 2 and Fig. 3 for easy comparison. The point spread function (PSF) in one dimension of the open aperture mask MK1 is shown in Fig. 3(d) and the synthesized PSF is shown in Fig. 3(f). The comparison reveals that the PSF in Fig. 3(f) has a narrower main lobe and reduced side lobes. The reduction in the main lobe width means that a super-resolved image is possible with the use of synthesized mask shown in Fig. 3(c). While in the case of amplitude mask the RM is removed from the setup thus only allowing the interferometric beam of MZ1 toward B4 with zero phases. The second beam from MZ2 with the same phase superimposes with the MZ1 beam after passing through B4. That interferometric beam traveled through SG1 and SG2 with zero phases. The beam from source1 pass through MK1 gives interferometric multiplication with beam of source 2 before the CCD result in synthesized mask. Quantitative data about the masks and simulation results is as follows. The open aperture MK1 shown in Fig. 2(a) has a dimension of 128 by 128 with the transparent region having dimensions of 60 by 60 pixels. The masks MK2 and MK3 shown in Fig. 2(b) and (c) respectively have the same dimensions as the open aperture. Each transparent (white) strip is 6 pixels wide. The central transparent region in MK3 is 12 by 12 pixels in dimension. The mask MK2 and MK3 are the negatives of each other. The normalized PSF of the system with open aperture MK1 is shown in Fig. 2(e) which has FWHM (full width half maximum) of 1.7 pixels with the first side lobes height equal to 0.051. The synthesized aperture is shown in
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Fig. 5. Convolution results of input object (left side) with open aperture (middle top) and phase mask (middle bottom) shown respectively on the right (top) and right (bottom) respectively. (a) Results with conventional system (b) result with synthesized phase mask.
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Fig. 4. Convolution results with three masks. (a) Conventional results using open aperture PSF in the middle and convolution result on the right, (b) convolution result with mask MK2 PSF, (c) convolution result with synthesized phase mask PSF.
Fig. 3(c) and the corresponding PSF is shown in Fig. 3(f). The FWHM of the synthesized PSF is 1.1 pixels and the first side lobes height is 0.042. The quantitative comparison of the two PSFs (open aperture and synthesized) indicates that the synthesized mask (Fig. 3(c)) when used will give a super-resolved image. The reduction in main lobe of the synthesized aperture is 64.7% of the open aperture mask measured at FWHM. 2. Simulation results
Fig. 6. (a) Amplitude mask obtained by the addition of MK1, MK2, MK3. (b) Synthesized mask obtained by interferometric multiplication mask in (a) with the open aperture.
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To further elaborate the importance of the synthesized mask, we performed simulation with two point sources in one dimension. The object consisting of two point sources represented by two delta functions with a separation of 2 pixels in shown in Fig. 4(a) (left hand image). It is convolved with the open aperture PSF shown in the middle of Fig. 4(a) and the result is on the right side of Fig. 4(a). The dip is 0.12. The result in Fig. 4(b) is with the mask MK2 multiplied with the mask MK1 (when MK3 is not present in the setup of Fig. 1). The dip is 0.67. The result in Fig. 4(c) is obtained with the synthesized mask which is shown in the middle and the convolved image is shown on the right side with a dip of 0.97. The dip comparison indicates that the synthesized image (Fig. 4(c) right side) is better by 85% with respect to the open aperture image (Fig. 4(a) right side). To further elaborate the concept we reduced the size of the open aperture and other mask to the dimension of 128 by 128 with a transparent window of the size 50 by 50 pixels. With the reduction of the open aperture and other masks the PSF would be broader. The PSF with open aperture mask is shown in the middle of Fig. 5(a) and the synthesized PSF is shown in the middle of Fig. 5(b). Input object consisting of two delta functions with a separation of two pixels is shown on the left side of Fig. 5 and the result images with the open aperture PSF and with synthesized PSF are shown on the right side (top and bottom images respectively) of Fig. 5. The comparison of the two images reveals that with the open aperture the input object is unresolved whereas with the synthesized aperture the input object is well resolved. The dip is zero in the open aperture case whereas the dip is 0.75 or 75% in the synthesized case. In the above paragraph we have described the use of phase masks in producing a narrower synthesized PSF. Here we are describing the amplitude synthesized mask for which the mirror RM is removed from the setup. With the removal of RM the beam1 does not reach the CCD plane. Now beam2 contributes at the CCD plane. There is an interferometric addition of the three masks MKI
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Fig. 7. Convolution result of two delta functions with amplitude PSF.
(from MZ2), MK2 and MK3 (from MZ1) which is shown in Fig. 6(a). The resultant of these will now multiply with the MK1 outside the two Mach Zehnder interferometers and the resultant is shown in Fig. 6(b). This now gives us the synthesized amplitude mask. The PSF corresponding to this synthesized amplitude mask is shown in the middle of Fig. 7. The FWHM of the PSF is 1.1. The input object is shown in the left side of Fig. 7 and is the same as in Fig. 4 and Fig. 5. The resultant image after convolution of the synthesized amplitude PSF with the input object is shown on the right hand side of Fig. 7. The dip in this case is 97% which is identical with the case of phase masks discussed earlier. The dip in the case of open aperture is 12% as shown in Fig. 4(a). The analysis with the phase and amplitude masks when compared with the open aperture case, establish that the results are 85% better. 3. Conclusion This paper has described the formation of phase and amplitude masks and their application in super-resolved images. Simulation results clearly establish that the resolution with the proposed masks is better than the conventional open aperture by 85%. Experimental set up for implementing both types of masks has also been described. The performance of amplitude mask is found identical to the phase mask as far as the resolution is concerned. We believe that the technique will give promising results for transmission mode super resolution application.
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Acknowledgement I acknowledge Higher education Commission (HEC) of Pakistan for funding the Project. References [1] V. Mico, O. Limon, A. Gur, Z. Zalevsky, J. García, Transverse resolution improvement using rotating grating time-multiplexing approach, J. Opt. Soc. Am. A 25 (5) (2008). [2] A. Mudassar, A.R. Harvey, A.H. Greenaway, J. Jones, Band pass active aperture synthesis using spatial frequency heterodyning, J. Phys.: Conf. Ser. 15 (2005) 290–295. [3] A. Asloob Mudassar, A. Hussain, Super-resolution of active spatial frequency heterodyning using holographic approach, Appl. Opt. 49 (17) (2010).
[4] V. Mico, Z. Zalevsky, P. García-Martínez, J. García, Synthetic aperture superresolution with multiple off-axis holograms, J. Opt. Soc. Am. A 23 (12) (2006). [5] Y. Kuznetsova, A. Neumann, S.R.J. Brueck, Imaging interferometric microscopy approaching the linear systems limits of optical resolution, Opt. Express 15 (11) (2007). [6] I.G. Toraldo di Francia, Super-gain antennas and optical resolving power, Nuovo Cimento Suppl. 9 (1952) 426–438. [7] Z. Hegedus, V. Sarafis, Superresolving filters in confocally scanned imaging systems, J. Opt. Soc. Am. A 3 (1986) 1892–1896. [8] I. Leizerson, S.G. Lipson, V. Sarafis, Improvement of optical resolution in far-field imaging by optical multiplication, Micron 34 (6–7) (2003) 301–307. [9] I. Leizerson, S.G. Lipson, V. Sarafis, Super resolution in far-field imaging, Opt. Express 25 (4) (2000). [10] I. Leizerson, S.G. Lipson, V. Sarafis, Super resolution in far-field imaging, J. Opt. Soc. Am. A 19 (3) (2002).
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Super resolution imaging using interferometric masking technique Anwar Hussain ∗ COMSATS Institute of Information Technology, Chak Shahzad, Islamabad, Pakistan
a r t i c l e
i n f o
Article history: Received 24 September 2014 Accepted 6 September 2015 Keywords: Super resolution Interferometry Mach Zehnder and Sagnac interferometer
a b s t r a c t This paper describes the super resolution technique which is based on the Torlado di Francia imaging technique. The super resolution has been achieved using two masks: phase and amplitude, separately in the same setup. In one case the Mach Zehnder interferometer was used to synthesize the shifted phase mask while in the second case the negative of amplitude mask was produced. The interferometric multiplication of the original aperture with the synthesized amplitude and phase masks separately gives super resolution in both cases. The achieved Rayleigh resolution is 85% of the conventional system. The side lobes reduction was also obtained in this technique. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction 1.1. Interferometric imaging Achieving high resolution beyond the classical limit is called super resolution. Different super resolution techniques like grating based [1], fringe illumination [2,3] and interferometry based [4,5] have been mentioned in literature to achieve high resolution. The point spread function (PSF) of the system reflects the resolution of the system. Narrower the width of PSF greater is the resolution. If two points lying close to each other are imaged through the system having wider PSF result in a single point. But if the same objects are imaged through an optical system of narrower PSF result in two separate objects, mean highly resolved. To synthesize an aperture who’s PSF has small central lob called super-resolution. In literature some techniques are reported to create PSF with narrow width. The first idea of mask was proposed by Toraldo di Francia in 1952 [6]. According to this idea an aperture pupil mask composed of concentric transparent rings with phases of 0 and is used for imaging. The point spread function produced by this mask having central spot very narrow as required for super-resolution. The drawback of this method was outer bright rings that carrying larger part of energy. The advantage of this method was that no limit on resolution using this method in view of Sparrow and Rayleigh resolution criterion. Due to its unlimited resolution nature Hegedus and Sarafis [7] prepared a phase mask consist of three rings to demonstrate super-resolution, later on implemented in confocal
∗ Corresponding author. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.09.008 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
microscopy. To implement the idea in far field Leizerson [8] further extended the idea by eliminating the bright rings by interferometric technique. The synthesized mask is obtained as result of interferometric superposition of two amplitude masks. The PSF of the synthesized mask is narrow compared to the PSFs of other masks. The next step is to eliminate the outer rings of PSF to make it more ideal. For this purpose another PSF which has zero value at the outer region is multiplied with the PSF of synthesized mask using interferometric method [7–10]. Here similar idea reported in [6–10] is followed with better resolution result for phase masking and setup is also modified for amplitude masking. We have established that better resolution can be achieved using a modified Mach Zehnder interferometer. Both interferometric phase and amplitude masking techniques, applied in the single experimental setup. A removable mirror (RM) when removed from the set up converts the system into amplitude interferometric masking which otherwise acts as an interferometric phase masking setup. We are presenting the simulation results to establish that the super resolution can be achieved either with the use of interferometric phase masks or with interferometric amplitude masks and that the same resolution can be achieved in both cases which 85% of the Rayleigh limit. The previous work [8] reports a maximum of 55% Rayleigh limit achievement. The detailed description of work is given below. 1.2. Detail of experimental setup The proposed experimental setup for implementing both the interferometric phase masking and the amplitude masking is shown in Fig. 1. The set up uses two lasers 1 and 2 which are mutually incoherent. In the present analysis we are assuming that both the lasers are operating on a single wavelength which is
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Fig. 1. Experimental setup for interferometric imaging, BC, B0, B2, B3 are beam splitters, MK1, is open square mask, MK2 and MK3 are used to formed synthesized mask. L1 and L2 are lenses, Ms are mirrors and RM represents removable mirror. BM is beam stopper. Set up B is used in the formation of amplitude mask. For detail see text.
the same for both lasers. Two lasers sources are used to compare the super-resolved PSF of the system. The two lasers sources are like two points separated by some distance to image through mask-aperture, and also act like illumination source to produce the synthetic aperture. The lens L1 collimates the two spatially filtered beams and the polarizer just after the lens L1 polarizes the laser beams. The second polarizer just before the lens L2 acts as an analyzer. The linear polarizer is used to control the contrast of the image and polarization of the lasers sources. The path of the two lasers beams has been indicated with different colors shown in Fig. 1. B0 is a polarizing beam splitter which transmits the light from laser1 and reflects the light from laser2. The set up consists of eight mirrors including one removable mirror (RM) and six beam splitters. The beam splitters B2 and B3 and the two mirrors M5 and M6 form a Mach Zehnder interferometer. The beam splitter Bo, Mach Zehnder interferometer (B2, B3, M5, M6) and two mirrors RM and M1 form a Sagnac interferometer. With the existence of RM in the set up, the beam2 is stopped and when the RM is removed the beam1 is stopped. With the removal of RM two Mach Zehnder interferometers comprising components (B2, B3, M5, M6 Mach Zehnder 1) and (B1, B4, M7, M4 along with components of Mach Zehnder 1 Mach Zehnder 2) are formed. Another Sagnac interferometer is also formed comprising of the components (B0, M1, M2, M3 along with two Mach Zehnder interferometers). The two Mach Zehnder interferometers and the two Sagnac interferometers have been labeled with MZ1, MZ2, SAG1, SAG2 respectively and is shown in Fig. 1. Two masks have been placed in two arms of the MZ1 labeled with MK2 and MK3 shown respectively in Fig. 2(b) and (c) respectively. The open aperture labeled with MK1 is shown in Fig. 2(a) and its position has been indicated at two different locations in Fig. 1. The two beams from the MZ1 exiting toward RM will be phase shifted by whereas the two beams exiting toward B4 will be in phase. The interferometric addition of two beams toward RM will give the addition of two masks MK3 and MK2 and the resultant is shown in Fig. 2d. The white area in the two masks has a value of 1
Fig. 3. Comparison of three filters and their PSF. (a) Open aperture with PSF in one dimension in (d), (b) mask MK2 with the corresponding PSF in one dimension in (e), and (c) synthesized phase mask and the corresponding PSF in one dimension in (f).
and the black area has a value of 0 as assumed in the simulation. The open aperture mask MK1 is (placed between B0 and M1) shown in Fig. 2(a). The laser beam from laser source1 will pass through MK1 and will route through a portion of MZ1. The BM (beam stopper) is placed in MZ1 to stop the beam coming from B3 toward M5. Just before the lens L2 there will be an interferometric multiplication between the open aperture shown in Fig. 2(a) and the interferometric mask shown in Fig. 2(d). The resultant of this multiplication is shown in Fig. 3(c). The point spread functions in one dimension are shown at the bottom of each two dimensional masks both in Fig. 2 and Fig. 3 for easy comparison. The point spread function (PSF) in one dimension of the open aperture mask MK1 is shown in Fig. 3(d) and the synthesized PSF is shown in Fig. 3(f). The comparison reveals that the PSF in Fig. 3(f) has a narrower main lobe and reduced side lobes. The reduction in the main lobe width means that a super-resolved image is possible with the use of synthesized mask shown in Fig. 3(c). While in the case of amplitude mask the RM is removed from the setup thus only allowing the interferometric beam of MZ1 toward B4 with zero phases. The second beam from MZ2 with the same phase superimposes with the MZ1 beam after passing through B4. That interferometric beam traveled through SG1 and SG2 with zero phases. The beam from source1 pass through MK1 gives interferometric multiplication with beam of source 2 before the CCD result in synthesized mask. Quantitative data about the masks and simulation results is as follows. The open aperture MK1 shown in Fig. 2(a) has a dimension of 128 by 128 with the transparent region having dimensions of 60 by 60 pixels. The masks MK2 and MK3 shown in Fig. 2(b) and (c) respectively have the same dimensions as the open aperture. Each transparent (white) strip is 6 pixels wide. The central transparent region in MK3 is 12 by 12 pixels in dimension. The mask MK2 and MK3 are the negatives of each other. The normalized PSF of the system with open aperture MK1 is shown in Fig. 2(e) which has FWHM (full width half maximum) of 1.7 pixels with the first side lobes height equal to 0.051. The synthesized aperture is shown in
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Fig. 5. Convolution results of input object (left side) with open aperture (middle top) and phase mask (middle bottom) shown respectively on the right (top) and right (bottom) respectively. (a) Results with conventional system (b) result with synthesized phase mask.
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Fig. 4. Convolution results with three masks. (a) Conventional results using open aperture PSF in the middle and convolution result on the right, (b) convolution result with mask MK2 PSF, (c) convolution result with synthesized phase mask PSF.
Fig. 3(c) and the corresponding PSF is shown in Fig. 3(f). The FWHM of the synthesized PSF is 1.1 pixels and the first side lobes height is 0.042. The quantitative comparison of the two PSFs (open aperture and synthesized) indicates that the synthesized mask (Fig. 3(c)) when used will give a super-resolved image. The reduction in main lobe of the synthesized aperture is 64.7% of the open aperture mask measured at FWHM. 2. Simulation results
Fig. 6. (a) Amplitude mask obtained by the addition of MK1, MK2, MK3. (b) Synthesized mask obtained by interferometric multiplication mask in (a) with the open aperture.
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To further elaborate the importance of the synthesized mask, we performed simulation with two point sources in one dimension. The object consisting of two point sources represented by two delta functions with a separation of 2 pixels in shown in Fig. 4(a) (left hand image). It is convolved with the open aperture PSF shown in the middle of Fig. 4(a) and the result is on the right side of Fig. 4(a). The dip is 0.12. The result in Fig. 4(b) is with the mask MK2 multiplied with the mask MK1 (when MK3 is not present in the setup of Fig. 1). The dip is 0.67. The result in Fig. 4(c) is obtained with the synthesized mask which is shown in the middle and the convolved image is shown on the right side with a dip of 0.97. The dip comparison indicates that the synthesized image (Fig. 4(c) right side) is better by 85% with respect to the open aperture image (Fig. 4(a) right side). To further elaborate the concept we reduced the size of the open aperture and other mask to the dimension of 128 by 128 with a transparent window of the size 50 by 50 pixels. With the reduction of the open aperture and other masks the PSF would be broader. The PSF with open aperture mask is shown in the middle of Fig. 5(a) and the synthesized PSF is shown in the middle of Fig. 5(b). Input object consisting of two delta functions with a separation of two pixels is shown on the left side of Fig. 5 and the result images with the open aperture PSF and with synthesized PSF are shown on the right side (top and bottom images respectively) of Fig. 5. The comparison of the two images reveals that with the open aperture the input object is unresolved whereas with the synthesized aperture the input object is well resolved. The dip is zero in the open aperture case whereas the dip is 0.75 or 75% in the synthesized case. In the above paragraph we have described the use of phase masks in producing a narrower synthesized PSF. Here we are describing the amplitude synthesized mask for which the mirror RM is removed from the setup. With the removal of RM the beam1 does not reach the CCD plane. Now beam2 contributes at the CCD plane. There is an interferometric addition of the three masks MKI
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Fig. 7. Convolution result of two delta functions with amplitude PSF.
(from MZ2), MK2 and MK3 (from MZ1) which is shown in Fig. 6(a). The resultant of these will now multiply with the MK1 outside the two Mach Zehnder interferometers and the resultant is shown in Fig. 6(b). This now gives us the synthesized amplitude mask. The PSF corresponding to this synthesized amplitude mask is shown in the middle of Fig. 7. The FWHM of the PSF is 1.1. The input object is shown in the left side of Fig. 7 and is the same as in Fig. 4 and Fig. 5. The resultant image after convolution of the synthesized amplitude PSF with the input object is shown on the right hand side of Fig. 7. The dip in this case is 97% which is identical with the case of phase masks discussed earlier. The dip in the case of open aperture is 12% as shown in Fig. 4(a). The analysis with the phase and amplitude masks when compared with the open aperture case, establish that the results are 85% better. 3. Conclusion This paper has described the formation of phase and amplitude masks and their application in super-resolved images. Simulation results clearly establish that the resolution with the proposed masks is better than the conventional open aperture by 85%. Experimental set up for implementing both types of masks has also been described. The performance of amplitude mask is found identical to the phase mask as far as the resolution is concerned. We believe that the technique will give promising results for transmission mode super resolution application.
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Acknowledgement I acknowledge Higher education Commission (HEC) of Pakistan for funding the Project. References [1] V. Mico, O. Limon, A. Gur, Z. Zalevsky, J. García, Transverse resolution improvement using rotating grating time-multiplexing approach, J. Opt. Soc. Am. A 25 (5) (2008). [2] A. Mudassar, A.R. Harvey, A.H. Greenaway, J. Jones, Band pass active aperture synthesis using spatial frequency heterodyning, J. Phys.: Conf. Ser. 15 (2005) 290–295. [3] A. Asloob Mudassar, A. Hussain, Super-resolution of active spatial frequency heterodyning using holographic approach, Appl. Opt. 49 (17) (2010).
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