Abstract

We have developed a new table-top technique for collecting wide-field Fourier transform infrared (FTIR) microscopic images by combining a femtosecond pulse shaper with a mid-IR focal plane array. The pulse shaper scans the delay between a pulse pair extremely rapidly for high signal-to-noise, while also enabling phase control of the individual pulses to under-sample the interferograms and subtract background. Infrared absorption images were collected for a mixture of W(CO)6 or Mn2(CO)10 absorbed polystyrene beads, demonstrating that this technique can spatially resolve chemically distinct species. The images are sub-diffraction limited, as measured with a USAF test target patterned on CaF2 and verified with scalar wave simulations. We also find that refractive, rather than reflective, objectives are preferable for imaging with coherent radiation. We discuss this method with respect to conventional FTIR microscopes.

© 2015 Optical Society of America

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2015 (4)

K. Yeh, S. Kenkel, J.-N. Liu, and R. Bhargava, “Fast infrared chemical imaging with a quantum cascade laser,” Anal. Chem. 87(1), 485–493 (2015).
[Crossref] [PubMed]

C. Hughes, A. Henderson, M. Kansiz, K. M. Dorling, M. Jimenez-Hernandez, M. D. Brown, N. W. Clarke, and P. Gardner, “Enhanced FTIR bench-top imaging of single biological cells,” Analyst (Lond.) 140(7), 2080–2085 (2015).
[Crossref] [PubMed]

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. U.S.A. 112(5), 1304–1309 (2015).
[Crossref] [PubMed]

Y. Yu, B. Zhang, X. Gai, C. Zhai, S. Qi, W. Guo, Z. Yang, R. Wang, D.-Y. Choi, S. Madden, and B. Luther-Davies, “1.8-10 μm mid-infrared supercontinuum generated in a step-index chalcogenide fiber using low peak pump power,” Opt. Lett. 40(6), 1081–1084 (2015).
[Crossref] [PubMed]

2014 (3)

C. R. Baiz, D. Schach, and A. Tokmakoff, “Ultrafast 2D IR microscopy,” Opt. Express 22(15), 18724–18735 (2014).
[Crossref] [PubMed]

D. P. Kise, D. Magana, M. J. Reddish, and R. B. Dyer, “Submillisecond mixing in a continuous-flow, microfluidic mixer utilizing mid-infrared hyperspectral imaging detection,” Lab Chip 14(3), 584–591 (2014).
[Crossref] [PubMed]

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

2013 (3)

D. R. Skoff, J. E. Laaser, S. S. Mukherjee, C. T. Middleton, and M. T. Zanni, “Simplified and economical 2D IR spectrometer design using a dual acousto-optic modulator,” Chem. Phys. 422, 8–15 (2013).
[Crossref] [PubMed]

K. L. A. Chan and S. G. Kazarian, “Correcting the effect of refraction and dispersion of light in FT-IR spectroscopic imaging in transmission through thick infrared windows,” Anal. Chem. 85(2), 1029–1036 (2013).
[Crossref] [PubMed]

R. K. Reddy, M. J. Walsh, M. V. Schulmerich, P. S. Carney, and R. Bhargava, “High-definition infrared spectroscopic imaging,” Appl. Spectrosc. 67(1), 93–105 (2013).
[Crossref] [PubMed]

2012 (3)

K. Kim and Y. Park, “Fourier transform light scattering angular spectroscopy using digital inline holography,” Opt. Lett. 37(19), 4161–4163 (2012).
[Crossref] [PubMed]

R. Bhargava, “Infrared spectroscopic imaging: the next generation,” Appl. Spectrosc. 66(10), 1091–1120 (2012).
[Crossref] [PubMed]

M. R. Kole, R. K. Reddy, M. V. Schulmerich, M. K. Gelber, and R. Bhargava, “Discrete frequency infrared microspectroscopy and imaging with a tunable quantum cascade laser,” Anal. Chem. 84(23), 10366–10372 (2012).
[Crossref] [PubMed]

2011 (2)

M. J. Nasse, M. J. Walsh, E. C. Mattson, R. Reininger, A. Kajdacsy-Balla, V. Macias, R. Bhargava, and C. J. Hirschmugl, “High-resolution Fourier-transform infrared chemical imaging with multiple synchrotron beams,” Nat. Methods 8(5), 413–416 (2011).
[Crossref] [PubMed]

L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express 19(17), 16410–16417 (2011).
[Crossref] [PubMed]

2010 (4)

A. M. Beale, S. D. M. Jacques, and B. M. Weckhuysen, “Chemical imaging of catalytic solids with synchrotron radiation,” Chem. Soc. Rev. 39(12), 4656–4672 (2010).
[Crossref] [PubMed]

P. B. Petersen and A. Tokmakoff, “Source for ultrafast continuum infrared and terahertz radiation,” Opt. Lett. 35(12), 1962–1964 (2010).
[Crossref] [PubMed]

M. D. Thomson, V. Blank, and H. G. Roskos, “Terahertz white-light pulses from an air plasma photo-induced by incommensurate two-color optical fields,” Opt. Express 18(22), 23173–23182 (2010).
[Crossref] [PubMed]

J. T. King, C. R. Baiz, and K. J. Kubarych, “Solvent-dependent spectral diffusion in a hydrogen bonded “vibrational aggregate”,” J. Phys. Chem. A 114(39), 10590–10604 (2010).
[Crossref] [PubMed]

2009 (1)

P. Bassan, H. J. Byrne, F. Bonnier, J. Lee, P. Dumas, and P. Gardner, “Resonant Mie scattering in infrared spectroscopy of biological materials--understanding the ‘dispersion artefact’,” Analyst (Lond.) 134(8), 1586–1593 (2009).
[Crossref] [PubMed]

2006 (3)

2002 (2)

T. Witte, D. Zeidler, D. Proch, K. L. Kompa, and M. Motzkus, “Programmable amplitude- and phase-modulated femtosecond laser pulses in the mid-infrared,” Opt. Lett. 27(2), 131–133 (2002).
[Crossref] [PubMed]

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418(6897), 512–514 (2002).
[Crossref] [PubMed]

2001 (1)

G. L. Carr, “Resolution limits for infrared microspectroscopy explored with synchrotron radiation,” Rev. Sci. Instrum. 72(3), 1613 (2001).
[Crossref]

2000 (1)

1999 (1)

1998 (2)

Stelzer, “Contrast, resolution, pixelation, dynamic range and signal-to-noise ratio: fundamental limits to resolution in fluorescence light microscopy,” J. Microsc. 189(1), 15–24 (1998).
[Crossref]

N. Jamin, P. Dumas, J. Moncuit, W. H. Fridman, J. L. Teillaud, G. L. Carr, and G. P. Williams, “Highly resolved chemical imaging of living cells by using synchrotron infrared microspectrometry,” Proc. Natl. Acad. Sci. U.S.A. 95(9), 4837–4840 (1998).
[Crossref] [PubMed]

1997 (1)

1995 (1)

E. N. Lewis, P. J. Treado, R. C. Reeder, G. M. Story, A. E. Dowrey, C. Marcott, and I. W. Levin, “Fourier transform spectroscopic imaging Using an infrared focal-plane array detector,” Anal. Chem. 67(19), 3377–3381 (1995).
[Crossref] [PubMed]

1971 (1)

Baiz, C. R.

C. R. Baiz, D. Schach, and A. Tokmakoff, “Ultrafast 2D IR microscopy,” Opt. Express 22(15), 18724–18735 (2014).
[Crossref] [PubMed]

J. T. King, C. R. Baiz, and K. J. Kubarych, “Solvent-dependent spectral diffusion in a hydrogen bonded “vibrational aggregate”,” J. Phys. Chem. A 114(39), 10590–10604 (2010).
[Crossref] [PubMed]

Baker, M. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Bassan, P.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

P. Bassan, H. J. Byrne, F. Bonnier, J. Lee, P. Dumas, and P. Gardner, “Resonant Mie scattering in infrared spectroscopy of biological materials--understanding the ‘dispersion artefact’,” Analyst (Lond.) 134(8), 1586–1593 (2009).
[Crossref] [PubMed]

Beale, A. M.

A. M. Beale, S. D. M. Jacques, and B. M. Weckhuysen, “Chemical imaging of catalytic solids with synchrotron radiation,” Chem. Soc. Rev. 39(12), 4656–4672 (2010).
[Crossref] [PubMed]

Beaurepaire, E.

Bhargava, R.

K. Yeh, S. Kenkel, J.-N. Liu, and R. Bhargava, “Fast infrared chemical imaging with a quantum cascade laser,” Anal. Chem. 87(1), 485–493 (2015).
[Crossref] [PubMed]

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

R. K. Reddy, M. J. Walsh, M. V. Schulmerich, P. S. Carney, and R. Bhargava, “High-definition infrared spectroscopic imaging,” Appl. Spectrosc. 67(1), 93–105 (2013).
[Crossref] [PubMed]

R. Bhargava, “Infrared spectroscopic imaging: the next generation,” Appl. Spectrosc. 66(10), 1091–1120 (2012).
[Crossref] [PubMed]

M. R. Kole, R. K. Reddy, M. V. Schulmerich, M. K. Gelber, and R. Bhargava, “Discrete frequency infrared microspectroscopy and imaging with a tunable quantum cascade laser,” Anal. Chem. 84(23), 10366–10372 (2012).
[Crossref] [PubMed]

M. J. Nasse, M. J. Walsh, E. C. Mattson, R. Reininger, A. Kajdacsy-Balla, V. Macias, R. Bhargava, and C. J. Hirschmugl, “High-resolution Fourier-transform infrared chemical imaging with multiple synchrotron beams,” Nat. Methods 8(5), 413–416 (2011).
[Crossref] [PubMed]

Blank, V.

Bonnier, F.

P. Bassan, H. J. Byrne, F. Bonnier, J. Lee, P. Dumas, and P. Gardner, “Resonant Mie scattering in infrared spectroscopy of biological materials--understanding the ‘dispersion artefact’,” Analyst (Lond.) 134(8), 1586–1593 (2009).
[Crossref] [PubMed]

Brown, M. D.

C. Hughes, A. Henderson, M. Kansiz, K. M. Dorling, M. Jimenez-Hernandez, M. D. Brown, N. W. Clarke, and P. Gardner, “Enhanced FTIR bench-top imaging of single biological cells,” Analyst (Lond.) 140(7), 2080–2085 (2015).
[Crossref] [PubMed]

Butler, H. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Byrne, H. J.

P. Bassan, H. J. Byrne, F. Bonnier, J. Lee, P. Dumas, and P. Gardner, “Resonant Mie scattering in infrared spectroscopy of biological materials--understanding the ‘dispersion artefact’,” Analyst (Lond.) 134(8), 1586–1593 (2009).
[Crossref] [PubMed]

Cao, H.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. U.S.A. 112(5), 1304–1309 (2015).
[Crossref] [PubMed]

Carney, P. S.

Carr, G. L.

G. L. Carr, “Resolution limits for infrared microspectroscopy explored with synchrotron radiation,” Rev. Sci. Instrum. 72(3), 1613 (2001).
[Crossref]

N. Jamin, P. Dumas, J. Moncuit, W. H. Fridman, J. L. Teillaud, G. L. Carr, and G. P. Williams, “Highly resolved chemical imaging of living cells by using synchrotron infrared microspectrometry,” Proc. Natl. Acad. Sci. U.S.A. 95(9), 4837–4840 (1998).
[Crossref] [PubMed]

Cerjan, A.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. U.S.A. 112(5), 1304–1309 (2015).
[Crossref] [PubMed]

Chan, K. L. A.

K. L. A. Chan and S. G. Kazarian, “Correcting the effect of refraction and dispersion of light in FT-IR spectroscopic imaging in transmission through thick infrared windows,” Anal. Chem. 85(2), 1029–1036 (2013).
[Crossref] [PubMed]

Cheong, F. C.

Choi, D.-Y.

Choma, M. A.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. U.S.A. 112(5), 1304–1309 (2015).
[Crossref] [PubMed]

Clarke, N. W.

C. Hughes, A. Henderson, M. Kansiz, K. M. Dorling, M. Jimenez-Hernandez, M. D. Brown, N. W. Clarke, and P. Gardner, “Enhanced FTIR bench-top imaging of single biological cells,” Analyst (Lond.) 140(7), 2080–2085 (2015).
[Crossref] [PubMed]

Débarre, D.

Dixon, L.

Dorling, K. M.

C. Hughes, A. Henderson, M. Kansiz, K. M. Dorling, M. Jimenez-Hernandez, M. D. Brown, N. W. Clarke, and P. Gardner, “Enhanced FTIR bench-top imaging of single biological cells,” Analyst (Lond.) 140(7), 2080–2085 (2015).
[Crossref] [PubMed]

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Dowrey, A. E.

E. N. Lewis, P. J. Treado, R. C. Reeder, G. M. Story, A. E. Dowrey, C. Marcott, and I. W. Levin, “Fourier transform spectroscopic imaging Using an infrared focal-plane array detector,” Anal. Chem. 67(19), 3377–3381 (1995).
[Crossref] [PubMed]

Dudovich, N.

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418(6897), 512–514 (2002).
[Crossref] [PubMed]

Dumas, P.

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A. M. Beale, S. D. M. Jacques, and B. M. Weckhuysen, “Chemical imaging of catalytic solids with synchrotron radiation,” Chem. Soc. Rev. 39(12), 4656–4672 (2010).
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N. Jamin, P. Dumas, J. Moncuit, W. H. Fridman, J. L. Teillaud, G. L. Carr, and G. P. Williams, “Highly resolved chemical imaging of living cells by using synchrotron infrared microspectrometry,” Proc. Natl. Acad. Sci. U.S.A. 95(9), 4837–4840 (1998).
[Crossref] [PubMed]

Witte, T.

Woerner, M.

Wood, B. R.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
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Wurm, M.

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K. Yeh, S. Kenkel, J.-N. Liu, and R. Bhargava, “Fast infrared chemical imaging with a quantum cascade laser,” Anal. Chem. 87(1), 485–493 (2015).
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Yu, Y.

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Anal. Chem. (4)

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K. Yeh, S. Kenkel, J.-N. Liu, and R. Bhargava, “Fast infrared chemical imaging with a quantum cascade laser,” Anal. Chem. 87(1), 485–493 (2015).
[Crossref] [PubMed]

M. R. Kole, R. K. Reddy, M. V. Schulmerich, M. K. Gelber, and R. Bhargava, “Discrete frequency infrared microspectroscopy and imaging with a tunable quantum cascade laser,” Anal. Chem. 84(23), 10366–10372 (2012).
[Crossref] [PubMed]

K. L. A. Chan and S. G. Kazarian, “Correcting the effect of refraction and dispersion of light in FT-IR spectroscopic imaging in transmission through thick infrared windows,” Anal. Chem. 85(2), 1029–1036 (2013).
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Appl. Spectrosc. (3)

Chem. Phys. (1)

D. R. Skoff, J. E. Laaser, S. S. Mukherjee, C. T. Middleton, and M. T. Zanni, “Simplified and economical 2D IR spectrometer design using a dual acousto-optic modulator,” Chem. Phys. 422, 8–15 (2013).
[Crossref] [PubMed]

Chem. Soc. Rev. (1)

A. M. Beale, S. D. M. Jacques, and B. M. Weckhuysen, “Chemical imaging of catalytic solids with synchrotron radiation,” Chem. Soc. Rev. 39(12), 4656–4672 (2010).
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[Crossref]

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J. Opt. Soc. Am. B (1)

J. Phys. Chem. A (1)

J. T. King, C. R. Baiz, and K. J. Kubarych, “Solvent-dependent spectral diffusion in a hydrogen bonded “vibrational aggregate”,” J. Phys. Chem. A 114(39), 10590–10604 (2010).
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Lab Chip (1)

D. P. Kise, D. Magana, M. J. Reddish, and R. B. Dyer, “Submillisecond mixing in a continuous-flow, microfluidic mixer utilizing mid-infrared hyperspectral imaging detection,” Lab Chip 14(3), 584–591 (2014).
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Nat. Methods (1)

M. J. Nasse, M. J. Walsh, E. C. Mattson, R. Reininger, A. Kajdacsy-Balla, V. Macias, R. Bhargava, and C. J. Hirschmugl, “High-resolution Fourier-transform infrared chemical imaging with multiple synchrotron beams,” Nat. Methods 8(5), 413–416 (2011).
[Crossref] [PubMed]

Nat. Protoc. (1)

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Nature (1)

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Opt. Express (5)

Opt. Lett. (6)

Proc. Natl. Acad. Sci. U.S.A. (2)

N. Jamin, P. Dumas, J. Moncuit, W. H. Fridman, J. L. Teillaud, G. L. Carr, and G. P. Williams, “Highly resolved chemical imaging of living cells by using synchrotron infrared microspectrometry,” Proc. Natl. Acad. Sci. U.S.A. 95(9), 4837–4840 (1998).
[Crossref] [PubMed]

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M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ Press, 1999).

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Figures (9)

Fig. 1
Fig. 1

Diagram of the experimental setup showing the (a) Ge-AOM based pulse shaper, (b) the optional diffuser/recollimating device used for incoherent imaging and (c) the microscope.

Fig. 2
Fig. 2

(a) Illustration of shaped pulse pairs with relative phase shifts of zero (blue) and π (green). (b) Simulated interferograms for the two cases depicted in (a), according to color, along with their difference (red). Subtracting the phase cycled interferograms doubles the interference signal while removing the laser background.

Fig. 3
Fig. 3

(a1-c1) Measured one-sided interferograms, generated using rotating frame frequencies of (a) 0 (b) 1800 and (c) 2000 cm−1, along with their respective Fourier transforms (a2-c2). Phase cycling shifts the apparent carrier frequency so that fewer data points are needed to sample the interferogram.

Fig. 4
Fig. 4

Images of a mixture of Mn2(CO)10 or W(CO)6 soaked polystyrene beads suspended in water. (a) Transmitted IR light image of bead sample. (b) Fourier transform infrared (FTIR) absorption spectra of different beads, as indicated by pixel numbers. (c) False color FTIR absorption image. The W(CO)6 (1971 cm−1) absorption is mapped to red while the Mn2(CO)10 (2004 cm−1) is colored in cyan.

Fig. 5
Fig. 5

FT-IR absorption images of a bead containing W(CO)6 measured at 1970 cm−1 with (a) no diffuser and (b) a rotating diffuser.

Fig. 6
Fig. 6

Experimental and simulated transmission images of group 7 of a USAF resolution pattern at λ = 5 μm. (a) Experimental image using a rotating diffuser and (b) a simulated image using incoherent illumination. (c) Cuts through the resolution elements in the experiment (blue) and simulation (red) labeled with element linewidths, as indicated by the dashed lines in (a) and (b).

Fig. 7
Fig. 7

(a) A cut through the simulated intensity point-spread-function for a refractive (blue) and a Schwarzchild (green) objective, the later using a central obscuration 0.2 times the size of the outer diameter. Simulated images of Group 6 element 1 for (b) a refractive and (c) Schwarzchild objective.

Fig. 8
Fig. 8

Two overlapping Airy patterns set to the distance defined by the Rayleigh criterion, convoluted with windows of different size to simulate pixilation

Fig. 9
Fig. 9

Scheme of the simulated optical system.

Equations (21)

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I(τ)= | E 1 | 2 + | E 2 | 2 +2| E 1 || E 2 |cos[ φ 1 φ 2 (τ)+2π ν ˜ τ ]
I( τ, φ 1 =0 )I( τ, φ 1 =π )=4| E 1 || E 2 |cos[ 2π ν ˜ τ φ 2 ( τ ) ]
| E z = d 2 ν| ν e i2π ν z ( ν )z ν | E z=0
| E z = d 2 ν | ν e iπλ ν ν z ν | E z=0 = U ^ z | E z=0
| E z = d 2 r 1 d 2 ν | r 1 r 1 | ν e iπλ ν ν z ν | E z=0 = d 2 r 1 | r 1 d 2 ν ν | E z=0 e iπλ ν ν z+i2π ν r 1
| E z d 2 r 1 | r 1 d 2 r 0 r 0 | E z=0 e iπ( r 1 r 0 )( r 1 r 0 ) λz
U z = d 2 ν | ν e iπλ ν ν z ν |
U z = d 2 r 1 | r 1 d 2 r 0 e iπ( r 1 r 0 )( r 1 r 0 ) λz r 0 |
L ^ f = d 2 r | r A( r ) e iπ r r /λf r |
E D ( r )= r | U ^ i L ^ f 2 U ^ o S ^ U ^ d U ^ f L ^ f U ^ f | E z=0
r | E 2f = r | U ^ f L ^ f U ^ f | E z=0 = d 2 r 1 e iπ( r r 1 )( r r 1 )/λf r 1 | d 2 r 0 | r 0 A( r 0 ) e iπ r 0 r 0 /λf U ^ f | E z=0 = d 2 r 1 A( r 1 ) r 1 | U ^ f | E z=0 e iπ r r /λf e 2π r r 1 /λf
r | E 2f = d 2 r 1 A( r 1 ) r 1 | U ^ f | E z=0 e iπ r r /λf e 2π r r 1 /λf A ˜ ( ν ){ e iπ r r /λf d 2 r 1 r 1 | U ^ f | E z=0 e 2π r 1 ν } A ˜ ( ν ){ e iπ r r /λf ν | U ^ f | E z=0 }
E 2f ( r )= A ˜ ( r / λf ) E ˜ z=0 ( r / λf )
r | E o+i = r | U ^ i L ^ f U ^ o | E samp
r | E o+i = d 2 r 1 e iπ( r r 1 )( r r 1 )/λ d i A( r ) d 2 r 0 e iπ( r 1 r 0 )( r 1 r 0 )/λ d o r 0 | E samp
e iπ r r /λ d i d 2 r 0 e iπ r 0 r 0 /λ d o r 0 | E samp d 2 r 1 A( r 1 ) e i2π r 1 ( r 0 / λ d o + r / λ d i )
E D ( r )= e iπ r r /λ d i [ { e iπ r r / M 2 λ d o E samp ( r /M ) } A ˜ ( r / λ d i ) ]
E D ( r )= e iπ r r /λ d i [ { e iπ r r / M 2 λ d o S( r /M ) r | U ^ d | E 2f } A ˜ 2 ( r / λ d i ) ]
I( r )= | E D ( r ,t ) | 2 dt
I( r ) | [ e iπ r r / M 2 λ d o E samp ( r /M ) ] A ˜ ( r / λ d i ) | 2
I( r ) | E samp ( r /M ) | 2 | A ˜ ( r / λ d i ) | 2

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