Journal of Microscopy, Vol. 259, Issue 1 2015, pp. 59–65
doi: 10.1111/jmi.12248
Received 21 November 2014; accepted 24 February 2015
Simulation of differential interference contrast microscopy and influence of aberrations MORITZ ESSLINGER∗ & HERBERT GROSS†
∗ Fraunhofer-Institut f¨ ur Angewandte Optik und Feinmechanik IOF, Jena, Germany
†Institut f¨ur Angewandte Physik, Friedrich-Schiller-Universit¨at Jena, Germany
Key words. Aberrations, differential interference contrast, microscopy, partial coherence. We model differential interference contrast (DIC) for microscopes with residual aberrations. The model presented allows to predict the DIC performance of objectives directly from its bright field point spread function. We numerically simulate partially coherent illumination and discuss the influence of individual aberrations on the image quality. For the recently proposed PlasDIC setup, that comes without any condenser prism, we find that under coherent illumination the contrast reaches the performance of DIC. We present a rule for objective correction to drastically improve PlasDIC contrast also for partially coherent illumination. Introduction Interferometric microscopy methods allow to access both the optical amplitude and phase of a sample transmission (Zernike, 1955). Maybe the most popular interferometric microscopy technique today is differential interference contrast (DIC) microscopy that visualizes the derivative of the optical phase of a sample (Nomarski, 1960). DIC is widely used in label-free biological cell imaging (Allen et al., 1969), where samples often exhibit low absorption contrast and the phase retardation of the sample becomes a key quantity for measurements. The small depth of focus allows for some kind of sectioning even in wide field setups, a feature exploited, for example, in topography measurements in material research (Cogswell & Sheppard, 1992). Detailed considerations concerning the image formation in DIC microscopy was part of seminal works. Models to describe the image formation process were introduced (Murphy & Davidson, 2001; Mertz, 2009) and combined with numerical simulations (Holmes & Levy, 1987; Mehta & Sheppard, 2008). These models were further extended to a whole deconvolution framework to solve the inverse imaging problem in a three-dimensional fashion with (Preza, 2000) and without ¨ Angewandte Physik, FriedrichCorrespondence to: Herbert Gross, Institut fur Schiller-Universit¨at Jena, Germany. Tel: + 49 3641 9 47992; fax: +49 3641 9 47802; e-mail: [email protected]
C 2015 The Authors C 2015 Royal Microscopical Society Journal of Microscopy
a condenser prism (Preza et al., 1999). Moreover, a vectorial diffraction model has been presented to be able to describe the image formation for high-NA objectives under coherent illumination (Munro & T¨or¨ok, 2005). In practice, DIC is often used in conjunction with partially coherent illumination. In the most general representation of partial coherence (Hopkins, 1953; Born & Wolf, 1959), the numerical effort for DIC simulations drastically increases compared to the fully coherent case (Mehta & Sheppard, 2008). There is an excellent simulation framework for numerical evaluation of partially coherent imaging, including DIC (Mehta & Oldenbourg, 2014). The code of this framework is published as free software (Mehta, 2013). Recently, a technique similar to DIC, named PlasDIC by the Zeiss company (Carl Zeiss AG, Oberkochen, Baden¨ Wurttemberg, Germany), was proposed (Wehner, 2003; Danz et al., 2004). The PlasDIC setup consists of fewer components and allows faster alignment than DIC. However, up to now it is not used as commonly as DIC. PlasDIC is known for images with reduced contrast when using illumination with a low degree of coherence. New approaches propose similar contrast to DIC by structuring the illumination asymmetrically instead of employing birefringent prisms, reducing constructive effort, while avoiding detriments as they appear in PlasDIC (Mehta & Sheppard, 2009). Generalizing the image formation description of virtually all interference imaging techniques, advanced approaches allow to measure the whole phase space of a sample transmission (Mehta & Sheppard, 2010). However, the number of studies on image formation theory is comparably small with respect to the practical relevance of DIC microscopy techniques. Most approaches assume diffraction limited optics or do not explicitly discuss aberration dependence of imaging performance. Although it is empirically known that only objectives with high image quality are applicable for interference contrast measurements, in practice the suitability of objectives for DIC is often determined experimentally after manufacture. In this report, we investigate the influence of aberrations on both DIC and PlasDIC performance under partially coherent illumination. Within the
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assumptions of our model, we can predict the DIC performance of microscopes directly from its bright field point spread function (PSF).
optics from the incoherent source, a mostly coherent sample illumination has a narrow angular spectrum in the sample plane and a correspondingly small σ value, while incoherent illumination is represented by large σ values.
The imaging model In DIC microscopy, two birefringent Wollaston or Nomarski prisms are inserted into a bright field microscope with K¨ohler illumination (see Fig. 1). The first prism is located in the rear focal plane of the objective, and the second one in the front focal plane of the condenser. The prisms tilt ordinary and extraordinary rays with respect to each other, resulting in a real space shear in the sample plane. We assume that both the sample and objective do not turn the polarization of light. The PlasDIC setup is similar to the DIC setup, but has no birefringent prism in the condenser path. Our imaging model for both DIC and PlasDIC microscopy and coherent illumination is E i m (r ) = d 2 r E r +w cond )tsample (r )h(r − r + w ob j ) illu ( cond )tsample (r )h(r −r − w ob j ), (1) + ξ · E illu (r − w where h is the coherent bright field PSF and tsample is the sample transmission. As sample transmission, we consider a sinusoidal grating tsample (r ) = t0 + a sin(kG r ).
(2)
Using complex valued coefficients t0 and a, both the amplitude and phase gratings can be modelled. Thin samples of more complex shape can easily be represented as superposition of several spatial frequencies kG . w cond and w ob j are the real space shears of the Wollaston prisms in the condenser and objective path, respectively. We assume the prisms to be ideal, which means they do not introduce aberrations and both paths can be treated by the same PSF h. ξ is a factor denoting relative strength and phase retardation between ordinary and extraordinary path. We are interested in a DIC configuration where the sum of ordinary and extraordinary path extincts for uniform samples, that is, ξ = −1. We represent the illumination by plane waves E illu as E illu (r ) = E 0 exp(i killu r ).
(3)
In a K¨ohler-type setup, each plane wave killu corresponds to a point in the front focal plane of the condenser. To improve efficiency of calculation, in this report we restrict ourselves to incoherent sources. After propagation to the sample plane, however, the illumination may show a certain degree of coherence. We model partially coherent illumination by an Abbe-type coherence model, coherently propagating the light emitted from a point source to the detector, and summing up the detected intensities from each individual point source. As a measure of coherence, we use the relative pupil filling factor σ according to van Cittert and Zernike, σ = NA cond /NA ob j for circular sources (Born & Wolf, 1959). Passing the condenser
DIC microscopy Coherently illuminated DIC In DIC microscopy, both birefringent prisms are chosen to shift the rays by the same amount in the sample plane, w ob j = w cond . We assume a single plane wave as illumination. Evaluating the integral in the image equation (Eq. (1)), we find the transmission offset t0 vanishes as a result of destructive interference between ordinary and extraordinary path. E coh,D I C (r ) = A D I C ± · exp ±i kG r .
(4)
The detected electric field in the detector plane, E coh,D I C , consists of a sinusoidal wave with the grating period kG and zero offset. The amplitude coefficients A depend on the illumination plane wave direction and objective aberrations. Their explicit form is shown in the Appendix. For an objective without apodisation, the intensity image created by a single plane illumination wave is of the form Icoh,D I C (r ) = B0 + Re B2 exp(2i kG r) . (5) The sinusoidal electric field turns into a sin2 intensity image that is modulated at the double grating frequency 2kG . The 2kG signal results from interference between the grating diffraction orders −1 and 1. If both orders are transmitted by the objective without apodisation, we find B0 = |B2 |, and B2 = 0 if one diffraction order is blocked by the pupil (see the Appendix). In the transmission case, the minima are zero and the visibility of the modulation is 100%. The perfect imaging is independent of lens aberrations, given the assumptions about the ideal prisms hold. Partially coherent DIC To obtain a partially coherent image, we incoherently add up the intensities from illuminating plane waves inside the condenser angular spectrum. I par coh,D I C (r ) = d 2 killu Icoh,D I C r, killu , E 0 (killu ) . (6) The unmodulated offset part of the intensity B0 is real valued and positive. It adds up constructively when integrating over killu . For aberration-free objectives, the modulated terms also add up constructively. A loss by contrast can only be caused by the diffraction limit, where a diffraction order of the sample cannot be transmitted, by apodisation or by aberrations. For systems with aberrations, the image intensities originating from different illumination directions are shifted in image C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
SIMULATION OF DIFFERENTIAL INTERFERENCE CONTRAST MICROSCOPY
(A)
(B)
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(C)
Fig. 1. Schematic of microscope setups in (A) DIC and (B) PlasDIC configuration. Light from a single point in the source plane creates a plane wave on the sample. (C) The illumination angular spectrum is built up by light emitted from all points of the finite sized source.
from t0 is not perfectly suppressed, but manifests itself in an electric field offset term A Plas D I C 0 (t0 ). E coh,Plas D I C (r ) = A Plas D I C ± · exp(i kG r) + A Plas D I C 0 . (7)
(B)
(A)
(C)
Fig. 2. (A) In a K¨ohler setup, partially coherent illumination is built up from plane waves, that are incoherently superimposed. (B and C) DIC image intensities of individual illuminating plane waves for (B) an aberration-free objective and (C) an objective with aberrations and without apodisation.
space with respect to each other (see Fig. 2). The summation is no longer fully constructive, which results in a slower growth of the modulated sum than the sum of offsets B0 . The DIC contrast is reduced. PlasDIC microscopy Coherently illuminated PlasDIC In PlasDIC microscopy, there is no birefringent prism in the condenser path w cond = 0. As a result, the offset originating C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
Being able to adjust the retardation ξ between ordinary and extraordinary path, for illumination with a single plane wave one can always find a retardation for which the offset vanishes. In this case, the contrast in the image intensity is perfect, like in a coherently illuminated DIC microscope. However, in general it is not possible to find a single value ξ that ensures suppression of offsets from multiple illumination directions at the same time. Therefore, we choose ξ = −1, where the central plane wave killu = 0 offset is suppressed. For all other directions, there is a finite offset, and we find for the intensity Icoh,PlasDIC (r ) = C 0 +Re(C 1 exp(i kG r)+C 2 exp(2i kG r)). (8) The coefficients C are found in the Appendix. In the intensity image, there are two modulation frequencies: the desired 2kG signal and the offset-related single frequency artefact at kG . Partially coherent PlasDIC Similarly to DIC, aberrations cause a loss of contrast when using partially coherent illumination. In most scenarios we show later, contrast in PlasDIC is less robust against aberrations than DIC. It is well known that for PlasDIC, even a perfect objective suffers from contrast reduction when illuminated with increasingly incoherent illumination. In real objectives with
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Fig. 3. Interference contrast versus illumination coherence. The subplots display the signal at the double grating spatial frequency in (A) a PlasDIC setup and (B) a DIC setup. Plot (C) shows the imaging artefact at the fundamental grating frequency in PlasDIC microscopy. The sample is a phase grating of period 1 μm. Each line represents an objective with one Zernike fringe coefficient of 0.25 waves.
aberrations, additionally the generation of the single grating frequency artefact kG may turn out as a challenge in image interpretation. From the image of an object with multiple spatial frequencies, real sample geometry and artefact-related signal can in general not be distinguished. In the following, we investigate which aberrations influence the artefact signal strength and whether the artefact can be avoided.
Suppression of the single grating frequency artefact We assume a symmetric light source | E 0 (killu )| = | E 0 (−killu )|. A typical grating investigated by PlasDIC is a phase grating, where transmission offset t0 and modulation amplitude a are orthogonal in the complex plane. The intensity sum at the single grating frequency for each pair of opposing points in the light source C 1 (killu ) + C 1 (−killu ) is fully destructive for = P∗ (−k). P(k)
(9)
A bright field pupil function P that is equal to the complex conjugate of its counterpart at the opposing direction ensures fully artefact-free images of phase gratings. Interpreting the magnitude of P as apodisation and its phase as wave aberration, we find this condition fulfilled for systems with symmetric apodisation and antisymmetric wave aberration. In rotationally symmetric objectives, the apodisation is symmetric at least for the on-axis field point. We represent the wave aberration in terms of Zernike polynomials. For a desired antisymmetric wave aberration, only Zernike polynomials with angle dependence sin((2n + 1)φ), cos((2n + 1)φ) may have nonzero coefficients. That is, defocus (field curvature), spherical aberration
and astigmatismus must be corrected. Coma does not produce artefacts. Furthermore, for objectives that fulfill this condition, opposing plane illumination waves constructively build up the 2kG image intensity: C 2 (killu ) + C 2 (−killu ) = 2C 2 (killu ).
(10)
This relation holds for both DIC and PlasDIC. Although it does not mean yet that all summands C 2 (killu ) in the intensity image are constructive, one might expect a more robust behaviour of the double grating frequency signal and increased signal strength. Numeric simulations We perform numeric integrations over the illumination angular spectrum (Eq. (6)) with a circular, uniform source. The simulation code can be found elsewhere (Esslinger, 2015). The simulations address the dependence of imaging contrast on aberrations and on illumination coherence. Apodisation is neglected. We assume a series of synthetic objectives with 0.6 NA at a wavelength of 550 nm. The diffraction limit for transverse resolution of gratings in bright field mode is x = 2π/kG = λ/N A = 917 nm. The objectives have a pupil function with all Zernike coefficients except for one being zero. To investigate the dependence of DIC performance on individual aberrations, we create one objective for each primary aberration Zernike fringe coefficient 4 to 9, that is, defocus, astigmatism, coma and spherical aberration. The √transmission√coefficients of the phase grating are t0 = 1/ 2 and a = i / 2 and the Wollaston prism shear along the y-axis is w y = λ/(8NA ). We define visibility as the absolute value of C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
SIMULATION OF DIFFERENTIAL INTERFERENCE CONTRAST MICROSCOPY
(A)
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(B)
Fig. 4. (A) Modulation transfer function in a DIC setup. Each line represents an objective with one Zernike fringe coefficient of 0.25 waves. (B) Visibility versus aberration strength. The grating period is 1 μm. The intersection of both plots in parameter space is indicated by vertical grey lines. Illumination coherence is σ = 0.2
Fig. 5. Simulation of contrast versus grating period and illumination coherence. Aberration Zernike coefficients are 0.25 waves each.
C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
the ratio of the Fourier component at the respective spatial frequency and zero frequency in the intensity image. To compare DIC and PlasDIC performance at high resolutions, we choose a phase grating with a period of 1 μm, close to the diffraction limit. Figure 3 shows the visibilities for different degrees of coherence. For coherent illumination, the visibility is 100% for all objectives. At finite coherence, the signal strengths at the double grating frequency are lower, with similar performance of DIC and PlasDIC. However, PlasDIC shows a finite artefact visibility at the fundamental grating period for objectives with defocus, spherical aberration or astigmatism. These artefact-creating aberrations reduce the contrast at the double spatial frequency more rapidly than coma for both DIC and PlasDIC. The dependence of DIC contrast on grating constant and aberration strength is illustrated in Figure 4. The DIC contrast of a diffraction limited objective stays 100% up to the point where the sum of grating and illumination wavevector lay outside the numerical aperture cone. Objectives with aberrations show reduced contrast already at lower spatial frequencies (see Fig. 4A). For very small gratings, even a diffraction limited objective has finite contrast for partially coherent illumination (see Fig. 4B). With increasing aberration coefficients, we find the contrast is quite stable against coma, both for Zernike fringe coefficients c 7 and c 8 . The two Zernike coefficients representing astigmatism, however, differ in their behaviour. Objectives with c 5 -astigmatism perform much better than c 6 of the same magnitude. For rotationally symmetric objectives, the DIC performance on x- and y-axis is better than at field points 45◦ to the axes. Comparing the two types of astigmatism in PlasDIC, again c 5 -astigmatism shows higher signal strength than c 6 at the double grating frequency (see Fig. 5). In contrast, c 5 creates artefacts, whereas the symmetry of the sixth Zernike polynomial mismatches constructive summation of artefact intensities.
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Table 1. Overview over interference contrast performance in different scenarios. The term perfect denotes a visibility of 100%, if the sum of illumination and grating wavevectors lay inside the numerical aperture cone of the objective. We assume ideal Wollaston prisms, no apodisation, a symmetric light source with ideal condensor and a phase grating
Illumination
Aberrations
DIC
PlasDIC coarse grating
Plane wave
Diff. limited
Perfect
Perfect
Perfect
aberrations
Perfect
Perfect
Perfect
Diff. limited
Perfect
Losses
Converges to DIC
Part. coherent
PlasDIC fine grating
Coma
Minor losses
Losses
Converges to DIC
Defocus Spherical ab. astigmatism
Severe losses
Severe losses, obscured by artefact
Signal converges to DIC, obscured by artefact
Comparing DIC and PlasDIC performance versus coherence and grating period (see Fig. 5), PlasDIC in general shows lower contrast than DIC for partially coherent illumination. Even a diffraction limited objective in a PlasDIC setup cannot achieve contrast values comparable with DIC microscopy for partially coherent illumination. At high spatial frequencies (see righthand side of each subplot), DIC and PlasDIC performance of the same objective at the double grating frequency converge. The only difference between DIC and PlasDIC is then the occurrence of the artefact signal. The findings of this section are aggregated in Table 1. Summary We simulated DIC and PlasDIC microscopy with K¨ohler-type illumination and destructive interference between both paths. For systems without apodisation, coherent illumination creates perfect contrast for both DIC and PlasDIC, independent of aberrations. Considering partially coherent illumination, PlasDIC setups in general may exhibit an imaging artefact at half the spatial frequency where signal is expected for diffractionlimited optics. The artefact that occurs when imaging phase gratings can be suppressed by proper objective design, correcting field curvature, spherical aberration and astigmatism. The contrast in both DIC and PlasDIC is very sensitive to these aberrations, while even strong coma results only in a minor contrast reduction. Disregarding the possible occurrence of the artefact, the PlasDIC signal strength reaches that of DIC microscopy at high spatial frequencies. The results presented are a guide to objective designs with high DIC performance. It also may help in the optimal choice of objectives and operation parameters in microscopic applications.
Acknowledgement M.E. thanks S¨oren Schmidt, Manuel Tessmer, Sven Schr¨oder and M´eabh Garrick for valuable input. We acknowledge the German Bundesministerium f¨ur Bildung und Forschung for financial support of the project KoSimO (FKZ:031PT609X). Appendix We approximate the complex valued pupil function P as the Fourier transform of the PSF as = 1 r ), P(k) d 2 r h(r ) exp(i k (A1) 2π where the PSF h is normalized so that P(0) = 1. To allow fluent reading of the main part, we move the explicit forms of the coefficients in the imaging equations to this section. They are directly obtained by integrating the imaging Eq. (1). A D I C + = αP(kG + killu ),
(A2)
A D I C − = αP(−kG + killu ),
(A3)
exp(i killu r), α = −a E 0 sin(kG w)
(A4)
B0 = β(|P|2 (kG + killu ) + |P|2 (−kG + killu )),
(A5)
B2 = 2βP(kG + killu )P∗ (−kG + killu ),
(A6)
a 2 β = E 0 sin2 (kG w), 2
(A7)
2 (killu ) C 0 = 4|t0 E 0 |2 sin2 (killu w)|P| 2 (kG + killu ) + 2|a E 0 |2 sin2 ((kG + killu )w)|P| 2 + 2|a E 0 |2 sin2 ((killu − kG )w)|P|
× (−kG + killu ),
(A8)
C 1 = 4i t0 a ∗ | E 0 |2 sin(killu w) sin(kG w − killu w)P( killu ) × P∗ (−kG + killu ) + sin(kG w + killu w) − 4i t0∗ a| E 0 |2 sin(killu w) × P∗ (killu )P(kG + killu ),
(A9)
− cos(2kG w)) C 2 = 2|a E 0 |2 (cos(2killu w) × P(kG + killu )P∗ (−kG + killu ).
(A10)
Suppression of the fundamental grating period. We assume a symmetric light source | E 0 (killu )|2 = | E 0 (−killu )|2 . We rewrite the total image intensity at the fundamental grating period as the sum over image intensity of pairs of plane waves of two C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
SIMULATION OF DIFFERENTIAL INTERFERENCE CONTRAST MICROSCOPY
opposing directions C 1 (killu ) + C 1 (−killu ). We rearrange the terms to C 1 (killu ) + C 1 (−killu ) = 4i sin(killu w)| E 0 (killu )|2 ×(sin(kG w − killu w)ρ 1 − sin(kG w + killu w)ρ 2 ), with ρ1 = t0 a ∗ P(killu )P∗ (−kG + killu ) + t0∗ aP∗ (−killu )P(kG − killu ), ρ2 = t0∗ aP∗ (killu )P(kG + killu ) + t0 a ∗ P(−killu )P∗ (−kG − killu ). In the special case of an aberration-free objective P = 1, both terms ρ1 and ρ2 vanish for t0∗ a = −t0 a ∗ only. This condition is true for phase gratings, where a and t0 are orthogonal in the complex plane. Although in DIC configuration an objective without apodisation creates no artefact for both amplitude and phase gratings, in PlasDIC even an aberration-free objective cannot image an amplitude grating without artefacts in extinction mode ξ = −1. Theoretically, the fundamental grating period of amplitude gratings can be suppressed in PlasDIC by an objective with P(killu ) = −P∗ (−killu ). References Allen, R.D., David, G.B. & Nomarski, G. (1969) The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy. Z. wiss. Mikrosk. 69, 193–221. Born, M. & Wolf, E. (1959) Principles of Optics. Pergamon Press, Oxford, UK. Cogswell, C.J. & Sheppard, C.J.R. (1992) Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging. J. Microsc. 165, 81–101. Danz, R., Vogelgsang, A. & K¨athner, R. (2004) PlasDIC: a useful modification of the differential interference contrast according to Smith/Nomarski in transmitted light arrangement. Photonik. Esslinger, M. (2015) DIC and Aberrations, https://github.com/ mess42/dic_and_aberrations/. Accessed February 13, 2015.
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Holmes, T.J. & Levy, W.J. (1987) Signal-processing characteristics of differential-interference-contrast microscopy. Appl. Opt. 26, 3929– 3939. Hopkins, H.H. (1953) On the diffraction theory of optical images. P. R. Soc. Lond. 217, 408–432. Mehta, S.B. & Oldenbourg, R. (2014) Image simulation for biological microscopy: microlith Biomed. Opt. Exp. 5, 1822–1838. Mehta, S.B. & Sheppard, C.J. (2008) Partially coherent image formation in differential interference contrast (dic)microscope. Opt. Exp. 16, 19462– 19479. Mehta, S.B. & Sheppard, C.J.R. (2009) Quantitative phase-gradient imaging at high resolution with asymmetric illumination-based differential phase contrast. Opt. Lett. 34, 1924–1926. Mehta, S.B. & Sheppard, C.J. (2010) Using the phase-space imager to analyze partially coherent imaging systems: bright-field, phase contrast, differential interference contrast, differential phase contrast, and spiral phase contrast. J. Mod. Optic. 57, 718–739. Mehta, S.B. (2013) microlith Image simulation for microscopy and lithography systems, https://code.google.com/p/microlith/. Accessed March 28, 2014. Mertz, J. (2009) Introduction to Optical Microscopy. Roberts and Company Publishers, Greenwood Village, Colorado. Munro, P. & T¨or¨ok, P. (2005) Vectorial, high numerical aperture study of Nomarski’s differential interference contrast microscope. Opt. Exp. 13, 6833–6847. Murphy, D.B. & Davidson, M.W. (2001) Fundamentals of Light Microscopy and Electronic Imaging. Wiley-Blackwell, Hoboken, New York. Nomarski, G. (1960). Interferential polarizing device for study of phase objects. Patent US2924142. Preza, C., Snyder, D.L. & Conchello, J.A. (1999) Theoretical development and experimental evaluation of imaging models for differentialinterference-contrast microscopy. J. Opt. Soc. Am. A 16, 2185–2199. Preza, C. (2000) Rotational-diversity phase estimation from differentialinterference-contrast microscopy images. J. Opt. Soc. Am. A 17, 415– 424. Wehner, E. (2003) PlasDIC, an innovative relief contrast for routine observation in cell biology. Imag. Microsc. 4, 23. Zernike, F. (1955) How I discovered phase contrast. Science 121, 345–349.
doi: 10.1111/jmi.12248
Received 21 November 2014; accepted 24 February 2015
Simulation of differential interference contrast microscopy and influence of aberrations MORITZ ESSLINGER∗ & HERBERT GROSS†
∗ Fraunhofer-Institut f¨ ur Angewandte Optik und Feinmechanik IOF, Jena, Germany
†Institut f¨ur Angewandte Physik, Friedrich-Schiller-Universit¨at Jena, Germany
Key words. Aberrations, differential interference contrast, microscopy, partial coherence. We model differential interference contrast (DIC) for microscopes with residual aberrations. The model presented allows to predict the DIC performance of objectives directly from its bright field point spread function. We numerically simulate partially coherent illumination and discuss the influence of individual aberrations on the image quality. For the recently proposed PlasDIC setup, that comes without any condenser prism, we find that under coherent illumination the contrast reaches the performance of DIC. We present a rule for objective correction to drastically improve PlasDIC contrast also for partially coherent illumination. Introduction Interferometric microscopy methods allow to access both the optical amplitude and phase of a sample transmission (Zernike, 1955). Maybe the most popular interferometric microscopy technique today is differential interference contrast (DIC) microscopy that visualizes the derivative of the optical phase of a sample (Nomarski, 1960). DIC is widely used in label-free biological cell imaging (Allen et al., 1969), where samples often exhibit low absorption contrast and the phase retardation of the sample becomes a key quantity for measurements. The small depth of focus allows for some kind of sectioning even in wide field setups, a feature exploited, for example, in topography measurements in material research (Cogswell & Sheppard, 1992). Detailed considerations concerning the image formation in DIC microscopy was part of seminal works. Models to describe the image formation process were introduced (Murphy & Davidson, 2001; Mertz, 2009) and combined with numerical simulations (Holmes & Levy, 1987; Mehta & Sheppard, 2008). These models were further extended to a whole deconvolution framework to solve the inverse imaging problem in a three-dimensional fashion with (Preza, 2000) and without ¨ Angewandte Physik, FriedrichCorrespondence to: Herbert Gross, Institut fur Schiller-Universit¨at Jena, Germany. Tel: + 49 3641 9 47992; fax: +49 3641 9 47802; e-mail: [email protected]
C 2015 The Authors C 2015 Royal Microscopical Society Journal of Microscopy
a condenser prism (Preza et al., 1999). Moreover, a vectorial diffraction model has been presented to be able to describe the image formation for high-NA objectives under coherent illumination (Munro & T¨or¨ok, 2005). In practice, DIC is often used in conjunction with partially coherent illumination. In the most general representation of partial coherence (Hopkins, 1953; Born & Wolf, 1959), the numerical effort for DIC simulations drastically increases compared to the fully coherent case (Mehta & Sheppard, 2008). There is an excellent simulation framework for numerical evaluation of partially coherent imaging, including DIC (Mehta & Oldenbourg, 2014). The code of this framework is published as free software (Mehta, 2013). Recently, a technique similar to DIC, named PlasDIC by the Zeiss company (Carl Zeiss AG, Oberkochen, Baden¨ Wurttemberg, Germany), was proposed (Wehner, 2003; Danz et al., 2004). The PlasDIC setup consists of fewer components and allows faster alignment than DIC. However, up to now it is not used as commonly as DIC. PlasDIC is known for images with reduced contrast when using illumination with a low degree of coherence. New approaches propose similar contrast to DIC by structuring the illumination asymmetrically instead of employing birefringent prisms, reducing constructive effort, while avoiding detriments as they appear in PlasDIC (Mehta & Sheppard, 2009). Generalizing the image formation description of virtually all interference imaging techniques, advanced approaches allow to measure the whole phase space of a sample transmission (Mehta & Sheppard, 2010). However, the number of studies on image formation theory is comparably small with respect to the practical relevance of DIC microscopy techniques. Most approaches assume diffraction limited optics or do not explicitly discuss aberration dependence of imaging performance. Although it is empirically known that only objectives with high image quality are applicable for interference contrast measurements, in practice the suitability of objectives for DIC is often determined experimentally after manufacture. In this report, we investigate the influence of aberrations on both DIC and PlasDIC performance under partially coherent illumination. Within the
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assumptions of our model, we can predict the DIC performance of microscopes directly from its bright field point spread function (PSF).
optics from the incoherent source, a mostly coherent sample illumination has a narrow angular spectrum in the sample plane and a correspondingly small σ value, while incoherent illumination is represented by large σ values.
The imaging model In DIC microscopy, two birefringent Wollaston or Nomarski prisms are inserted into a bright field microscope with K¨ohler illumination (see Fig. 1). The first prism is located in the rear focal plane of the objective, and the second one in the front focal plane of the condenser. The prisms tilt ordinary and extraordinary rays with respect to each other, resulting in a real space shear in the sample plane. We assume that both the sample and objective do not turn the polarization of light. The PlasDIC setup is similar to the DIC setup, but has no birefringent prism in the condenser path. Our imaging model for both DIC and PlasDIC microscopy and coherent illumination is E i m (r ) = d 2 r E r +w cond )tsample (r )h(r − r + w ob j ) illu ( cond )tsample (r )h(r −r − w ob j ), (1) + ξ · E illu (r − w where h is the coherent bright field PSF and tsample is the sample transmission. As sample transmission, we consider a sinusoidal grating tsample (r ) = t0 + a sin(kG r ).
(2)
Using complex valued coefficients t0 and a, both the amplitude and phase gratings can be modelled. Thin samples of more complex shape can easily be represented as superposition of several spatial frequencies kG . w cond and w ob j are the real space shears of the Wollaston prisms in the condenser and objective path, respectively. We assume the prisms to be ideal, which means they do not introduce aberrations and both paths can be treated by the same PSF h. ξ is a factor denoting relative strength and phase retardation between ordinary and extraordinary path. We are interested in a DIC configuration where the sum of ordinary and extraordinary path extincts for uniform samples, that is, ξ = −1. We represent the illumination by plane waves E illu as E illu (r ) = E 0 exp(i killu r ).
(3)
In a K¨ohler-type setup, each plane wave killu corresponds to a point in the front focal plane of the condenser. To improve efficiency of calculation, in this report we restrict ourselves to incoherent sources. After propagation to the sample plane, however, the illumination may show a certain degree of coherence. We model partially coherent illumination by an Abbe-type coherence model, coherently propagating the light emitted from a point source to the detector, and summing up the detected intensities from each individual point source. As a measure of coherence, we use the relative pupil filling factor σ according to van Cittert and Zernike, σ = NA cond /NA ob j for circular sources (Born & Wolf, 1959). Passing the condenser
DIC microscopy Coherently illuminated DIC In DIC microscopy, both birefringent prisms are chosen to shift the rays by the same amount in the sample plane, w ob j = w cond . We assume a single plane wave as illumination. Evaluating the integral in the image equation (Eq. (1)), we find the transmission offset t0 vanishes as a result of destructive interference between ordinary and extraordinary path. E coh,D I C (r ) = A D I C ± · exp ±i kG r .
(4)
The detected electric field in the detector plane, E coh,D I C , consists of a sinusoidal wave with the grating period kG and zero offset. The amplitude coefficients A depend on the illumination plane wave direction and objective aberrations. Their explicit form is shown in the Appendix. For an objective without apodisation, the intensity image created by a single plane illumination wave is of the form Icoh,D I C (r ) = B0 + Re B2 exp(2i kG r) . (5) The sinusoidal electric field turns into a sin2 intensity image that is modulated at the double grating frequency 2kG . The 2kG signal results from interference between the grating diffraction orders −1 and 1. If both orders are transmitted by the objective without apodisation, we find B0 = |B2 |, and B2 = 0 if one diffraction order is blocked by the pupil (see the Appendix). In the transmission case, the minima are zero and the visibility of the modulation is 100%. The perfect imaging is independent of lens aberrations, given the assumptions about the ideal prisms hold. Partially coherent DIC To obtain a partially coherent image, we incoherently add up the intensities from illuminating plane waves inside the condenser angular spectrum. I par coh,D I C (r ) = d 2 killu Icoh,D I C r, killu , E 0 (killu ) . (6) The unmodulated offset part of the intensity B0 is real valued and positive. It adds up constructively when integrating over killu . For aberration-free objectives, the modulated terms also add up constructively. A loss by contrast can only be caused by the diffraction limit, where a diffraction order of the sample cannot be transmitted, by apodisation or by aberrations. For systems with aberrations, the image intensities originating from different illumination directions are shifted in image C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
SIMULATION OF DIFFERENTIAL INTERFERENCE CONTRAST MICROSCOPY
(A)
(B)
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(C)
Fig. 1. Schematic of microscope setups in (A) DIC and (B) PlasDIC configuration. Light from a single point in the source plane creates a plane wave on the sample. (C) The illumination angular spectrum is built up by light emitted from all points of the finite sized source.
from t0 is not perfectly suppressed, but manifests itself in an electric field offset term A Plas D I C 0 (t0 ). E coh,Plas D I C (r ) = A Plas D I C ± · exp(i kG r) + A Plas D I C 0 . (7)
(B)
(A)
(C)
Fig. 2. (A) In a K¨ohler setup, partially coherent illumination is built up from plane waves, that are incoherently superimposed. (B and C) DIC image intensities of individual illuminating plane waves for (B) an aberration-free objective and (C) an objective with aberrations and without apodisation.
space with respect to each other (see Fig. 2). The summation is no longer fully constructive, which results in a slower growth of the modulated sum than the sum of offsets B0 . The DIC contrast is reduced. PlasDIC microscopy Coherently illuminated PlasDIC In PlasDIC microscopy, there is no birefringent prism in the condenser path w cond = 0. As a result, the offset originating C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
Being able to adjust the retardation ξ between ordinary and extraordinary path, for illumination with a single plane wave one can always find a retardation for which the offset vanishes. In this case, the contrast in the image intensity is perfect, like in a coherently illuminated DIC microscope. However, in general it is not possible to find a single value ξ that ensures suppression of offsets from multiple illumination directions at the same time. Therefore, we choose ξ = −1, where the central plane wave killu = 0 offset is suppressed. For all other directions, there is a finite offset, and we find for the intensity Icoh,PlasDIC (r ) = C 0 +Re(C 1 exp(i kG r)+C 2 exp(2i kG r)). (8) The coefficients C are found in the Appendix. In the intensity image, there are two modulation frequencies: the desired 2kG signal and the offset-related single frequency artefact at kG . Partially coherent PlasDIC Similarly to DIC, aberrations cause a loss of contrast when using partially coherent illumination. In most scenarios we show later, contrast in PlasDIC is less robust against aberrations than DIC. It is well known that for PlasDIC, even a perfect objective suffers from contrast reduction when illuminated with increasingly incoherent illumination. In real objectives with
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Fig. 3. Interference contrast versus illumination coherence. The subplots display the signal at the double grating spatial frequency in (A) a PlasDIC setup and (B) a DIC setup. Plot (C) shows the imaging artefact at the fundamental grating frequency in PlasDIC microscopy. The sample is a phase grating of period 1 μm. Each line represents an objective with one Zernike fringe coefficient of 0.25 waves.
aberrations, additionally the generation of the single grating frequency artefact kG may turn out as a challenge in image interpretation. From the image of an object with multiple spatial frequencies, real sample geometry and artefact-related signal can in general not be distinguished. In the following, we investigate which aberrations influence the artefact signal strength and whether the artefact can be avoided.
Suppression of the single grating frequency artefact We assume a symmetric light source | E 0 (killu )| = | E 0 (−killu )|. A typical grating investigated by PlasDIC is a phase grating, where transmission offset t0 and modulation amplitude a are orthogonal in the complex plane. The intensity sum at the single grating frequency for each pair of opposing points in the light source C 1 (killu ) + C 1 (−killu ) is fully destructive for = P∗ (−k). P(k)
(9)
A bright field pupil function P that is equal to the complex conjugate of its counterpart at the opposing direction ensures fully artefact-free images of phase gratings. Interpreting the magnitude of P as apodisation and its phase as wave aberration, we find this condition fulfilled for systems with symmetric apodisation and antisymmetric wave aberration. In rotationally symmetric objectives, the apodisation is symmetric at least for the on-axis field point. We represent the wave aberration in terms of Zernike polynomials. For a desired antisymmetric wave aberration, only Zernike polynomials with angle dependence sin((2n + 1)φ), cos((2n + 1)φ) may have nonzero coefficients. That is, defocus (field curvature), spherical aberration
and astigmatismus must be corrected. Coma does not produce artefacts. Furthermore, for objectives that fulfill this condition, opposing plane illumination waves constructively build up the 2kG image intensity: C 2 (killu ) + C 2 (−killu ) = 2C 2 (killu ).
(10)
This relation holds for both DIC and PlasDIC. Although it does not mean yet that all summands C 2 (killu ) in the intensity image are constructive, one might expect a more robust behaviour of the double grating frequency signal and increased signal strength. Numeric simulations We perform numeric integrations over the illumination angular spectrum (Eq. (6)) with a circular, uniform source. The simulation code can be found elsewhere (Esslinger, 2015). The simulations address the dependence of imaging contrast on aberrations and on illumination coherence. Apodisation is neglected. We assume a series of synthetic objectives with 0.6 NA at a wavelength of 550 nm. The diffraction limit for transverse resolution of gratings in bright field mode is x = 2π/kG = λ/N A = 917 nm. The objectives have a pupil function with all Zernike coefficients except for one being zero. To investigate the dependence of DIC performance on individual aberrations, we create one objective for each primary aberration Zernike fringe coefficient 4 to 9, that is, defocus, astigmatism, coma and spherical aberration. The √transmission√coefficients of the phase grating are t0 = 1/ 2 and a = i / 2 and the Wollaston prism shear along the y-axis is w y = λ/(8NA ). We define visibility as the absolute value of C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
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(A)
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(B)
Fig. 4. (A) Modulation transfer function in a DIC setup. Each line represents an objective with one Zernike fringe coefficient of 0.25 waves. (B) Visibility versus aberration strength. The grating period is 1 μm. The intersection of both plots in parameter space is indicated by vertical grey lines. Illumination coherence is σ = 0.2
Fig. 5. Simulation of contrast versus grating period and illumination coherence. Aberration Zernike coefficients are 0.25 waves each.
C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
the ratio of the Fourier component at the respective spatial frequency and zero frequency in the intensity image. To compare DIC and PlasDIC performance at high resolutions, we choose a phase grating with a period of 1 μm, close to the diffraction limit. Figure 3 shows the visibilities for different degrees of coherence. For coherent illumination, the visibility is 100% for all objectives. At finite coherence, the signal strengths at the double grating frequency are lower, with similar performance of DIC and PlasDIC. However, PlasDIC shows a finite artefact visibility at the fundamental grating period for objectives with defocus, spherical aberration or astigmatism. These artefact-creating aberrations reduce the contrast at the double spatial frequency more rapidly than coma for both DIC and PlasDIC. The dependence of DIC contrast on grating constant and aberration strength is illustrated in Figure 4. The DIC contrast of a diffraction limited objective stays 100% up to the point where the sum of grating and illumination wavevector lay outside the numerical aperture cone. Objectives with aberrations show reduced contrast already at lower spatial frequencies (see Fig. 4A). For very small gratings, even a diffraction limited objective has finite contrast for partially coherent illumination (see Fig. 4B). With increasing aberration coefficients, we find the contrast is quite stable against coma, both for Zernike fringe coefficients c 7 and c 8 . The two Zernike coefficients representing astigmatism, however, differ in their behaviour. Objectives with c 5 -astigmatism perform much better than c 6 of the same magnitude. For rotationally symmetric objectives, the DIC performance on x- and y-axis is better than at field points 45◦ to the axes. Comparing the two types of astigmatism in PlasDIC, again c 5 -astigmatism shows higher signal strength than c 6 at the double grating frequency (see Fig. 5). In contrast, c 5 creates artefacts, whereas the symmetry of the sixth Zernike polynomial mismatches constructive summation of artefact intensities.
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Table 1. Overview over interference contrast performance in different scenarios. The term perfect denotes a visibility of 100%, if the sum of illumination and grating wavevectors lay inside the numerical aperture cone of the objective. We assume ideal Wollaston prisms, no apodisation, a symmetric light source with ideal condensor and a phase grating
Illumination
Aberrations
DIC
PlasDIC coarse grating
Plane wave
Diff. limited
Perfect
Perfect
Perfect
aberrations
Perfect
Perfect
Perfect
Diff. limited
Perfect
Losses
Converges to DIC
Part. coherent
PlasDIC fine grating
Coma
Minor losses
Losses
Converges to DIC
Defocus Spherical ab. astigmatism
Severe losses
Severe losses, obscured by artefact
Signal converges to DIC, obscured by artefact
Comparing DIC and PlasDIC performance versus coherence and grating period (see Fig. 5), PlasDIC in general shows lower contrast than DIC for partially coherent illumination. Even a diffraction limited objective in a PlasDIC setup cannot achieve contrast values comparable with DIC microscopy for partially coherent illumination. At high spatial frequencies (see righthand side of each subplot), DIC and PlasDIC performance of the same objective at the double grating frequency converge. The only difference between DIC and PlasDIC is then the occurrence of the artefact signal. The findings of this section are aggregated in Table 1. Summary We simulated DIC and PlasDIC microscopy with K¨ohler-type illumination and destructive interference between both paths. For systems without apodisation, coherent illumination creates perfect contrast for both DIC and PlasDIC, independent of aberrations. Considering partially coherent illumination, PlasDIC setups in general may exhibit an imaging artefact at half the spatial frequency where signal is expected for diffractionlimited optics. The artefact that occurs when imaging phase gratings can be suppressed by proper objective design, correcting field curvature, spherical aberration and astigmatism. The contrast in both DIC and PlasDIC is very sensitive to these aberrations, while even strong coma results only in a minor contrast reduction. Disregarding the possible occurrence of the artefact, the PlasDIC signal strength reaches that of DIC microscopy at high spatial frequencies. The results presented are a guide to objective designs with high DIC performance. It also may help in the optimal choice of objectives and operation parameters in microscopic applications.
Acknowledgement M.E. thanks S¨oren Schmidt, Manuel Tessmer, Sven Schr¨oder and M´eabh Garrick for valuable input. We acknowledge the German Bundesministerium f¨ur Bildung und Forschung for financial support of the project KoSimO (FKZ:031PT609X). Appendix We approximate the complex valued pupil function P as the Fourier transform of the PSF as = 1 r ), P(k) d 2 r h(r ) exp(i k (A1) 2π where the PSF h is normalized so that P(0) = 1. To allow fluent reading of the main part, we move the explicit forms of the coefficients in the imaging equations to this section. They are directly obtained by integrating the imaging Eq. (1). A D I C + = αP(kG + killu ),
(A2)
A D I C − = αP(−kG + killu ),
(A3)
exp(i killu r), α = −a E 0 sin(kG w)
(A4)
B0 = β(|P|2 (kG + killu ) + |P|2 (−kG + killu )),
(A5)
B2 = 2βP(kG + killu )P∗ (−kG + killu ),
(A6)
a 2 β = E 0 sin2 (kG w), 2
(A7)
2 (killu ) C 0 = 4|t0 E 0 |2 sin2 (killu w)|P| 2 (kG + killu ) + 2|a E 0 |2 sin2 ((kG + killu )w)|P| 2 + 2|a E 0 |2 sin2 ((killu − kG )w)|P|
× (−kG + killu ),
(A8)
C 1 = 4i t0 a ∗ | E 0 |2 sin(killu w) sin(kG w − killu w)P( killu ) × P∗ (−kG + killu ) + sin(kG w + killu w) − 4i t0∗ a| E 0 |2 sin(killu w) × P∗ (killu )P(kG + killu ),
(A9)
− cos(2kG w)) C 2 = 2|a E 0 |2 (cos(2killu w) × P(kG + killu )P∗ (−kG + killu ).
(A10)
Suppression of the fundamental grating period. We assume a symmetric light source | E 0 (killu )|2 = | E 0 (−killu )|2 . We rewrite the total image intensity at the fundamental grating period as the sum over image intensity of pairs of plane waves of two C 2015 The Authors C 2015 Royal Microscopical Society, 259, 59–65 Journal of Microscopy
SIMULATION OF DIFFERENTIAL INTERFERENCE CONTRAST MICROSCOPY
opposing directions C 1 (killu ) + C 1 (−killu ). We rearrange the terms to C 1 (killu ) + C 1 (−killu ) = 4i sin(killu w)| E 0 (killu )|2 ×(sin(kG w − killu w)ρ 1 − sin(kG w + killu w)ρ 2 ), with ρ1 = t0 a ∗ P(killu )P∗ (−kG + killu ) + t0∗ aP∗ (−killu )P(kG − killu ), ρ2 = t0∗ aP∗ (killu )P(kG + killu ) + t0 a ∗ P(−killu )P∗ (−kG − killu ). In the special case of an aberration-free objective P = 1, both terms ρ1 and ρ2 vanish for t0∗ a = −t0 a ∗ only. This condition is true for phase gratings, where a and t0 are orthogonal in the complex plane. Although in DIC configuration an objective without apodisation creates no artefact for both amplitude and phase gratings, in PlasDIC even an aberration-free objective cannot image an amplitude grating without artefacts in extinction mode ξ = −1. Theoretically, the fundamental grating period of amplitude gratings can be suppressed in PlasDIC by an objective with P(killu ) = −P∗ (−killu ). References Allen, R.D., David, G.B. & Nomarski, G. (1969) The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy. Z. wiss. Mikrosk. 69, 193–221. Born, M. & Wolf, E. (1959) Principles of Optics. Pergamon Press, Oxford, UK. Cogswell, C.J. & Sheppard, C.J.R. (1992) Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging. J. Microsc. 165, 81–101. Danz, R., Vogelgsang, A. & K¨athner, R. (2004) PlasDIC: a useful modification of the differential interference contrast according to Smith/Nomarski in transmitted light arrangement. Photonik. Esslinger, M. (2015) DIC and Aberrations, https://github.com/ mess42/dic_and_aberrations/. Accessed February 13, 2015.
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