## Graded-field microscopy with white light

Optics Express, Vol. 14, Issue 12, pp. 5191-5200 (2006)

http://dx.doi.org/10.1364/OE.14.005191

Acrobat PDF (888 KB)

### Abstract

We present a general imaging technique called graded-field microscopy for obtaining phase-gradient contrast in biological tissue slices. The technique is based on introducing partial beam blocks in the illumination and detection apertures of a standard white-light widefield transillumination microscope. Depending on the relative aperture sizes, one block produces phase-gradient contrast while the other reduces brightfield background, allowing a full operating range between brightfield and darkfield contrast. We demonstrate graded-field imaging of neurons in a rat brain slice.

© 2006 Optical Society of America

## 1. Introduction

2. F. Zernike, “How I discovered phase contrast,” Science **121**, 345–349 (1955). [CrossRef] [PubMed]

4. R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. **69**, 193–221 (1969). [PubMed]

5. B. Kachar, “Asymmetric illumination contrast: a method of image formation for video microscopy,” Science **227**, 766–768 (1985). [CrossRef] [PubMed]

6. W. B. Piekos, “Diffracted-light contrast enhancement: A re-examination of oblique illumination,” Micros. Res. Tech. **46**, 334–337 (1999). [CrossRef]

8. H. U. Dodt, M. Eder, A. Frick, and W. Zieglgänsberger, “Precisely localized LTD in the neocortex revealed by infrared-guided laser stimulation”, Science **286**, 111–113 (1999). [CrossRef]

11. J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Classification scheme and properties of schlieren techniques,” Appl. Opt. **16**,18, 3338–3341 (1979). [CrossRef]

12. J. G. Dodd, “Interferometry with Schlieren microscopy,” Appl. Opt. **16**,16, 470–472 (1977). [CrossRef] [PubMed]

13. D. Axelrod, “Zero-cost modification of bright field microscopes for imaging phase gradient on cells: Schlieren optics,” Cell Biophys. **3**, 167–173 (1981). [PubMed]

14. S. Lowenthal and Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. **11**, 49–51 (1967). [CrossRef]

15. R. Hoffman and L. Gross, “Modulation contrast microscopy,” Appl. Opt. **14**, 1169–1176 (1975). [CrossRef] [PubMed]

11. J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Classification scheme and properties of schlieren techniques,” Appl. Opt. **16**,18, 3338–3341 (1979). [CrossRef]

6. W. B. Piekos, “Diffracted-light contrast enhancement: A re-examination of oblique illumination,” Micros. Res. Tech. **46**, 334–337 (1999). [CrossRef]

13. D. Axelrod, “Zero-cost modification of bright field microscopes for imaging phase gradient on cells: Schlieren optics,” Cell Biophys. **3**, 167–173 (1981). [PubMed]

## 2. Formalism

*fa*(equivalently, this source can be located exactly at the back focal plane rather than imaged onto it, as depicted in Fig. 1). We refer to the condenser back focal plane as the illumination aperture. The transilluminated light emanating from the sample, located at plane 1, is then imaged onto a CCD camera, located at plane 3. For simplicity, we consider unit magnification only, though our formalism may easily be generalized to arbitrary magnification. The back aperture of the objective lens

*fb*is located at plane 2. We refer to the objective back aperture as the detection aperture. In both the illumination and detection apertures we introduce translatable light blocks that partially block these apertures to arbitrary fill factors. In Fig. (1) these blocks are introduced from the same side, along the transverse

*x*-axis. Again for simplicity, we consider square apertures which allows us to treat the

*x*and

*y*axes independently, Henceforth, we will consider only the

*x*axis, with the understanding that conventional brightfield imaging occurs along the

*y*axis.

*E*

_{0}(

*x*0), then the field amplitude incident on the sample is

_{0}=

*kx*

_{0}/

*f*

_{a}(

*k*=2

*π*/

*λ*and

*λ*is the average illumination wavelength) and α

_{1,2}=

*ka*

_{1,2}/

*f*

_{a}, where

*a*

_{1}and

*a*

_{2}represent the physical limits of the illumination aperture as prescribed by the position of its partial-beam block (i.e.

*a*

_{1,2}are spatial coordinates whereas α

_{1,2}are Fourier coordinates). If the block is removed then

*a*

_{1}→-

*a*and

*a*

_{2}→

*a*, meaning that the fully opened aperture is a square of spatial dimension 2

*a*. We omitted the prefactor in front of the integral in Eq. (1) since this plays no role in our discussion.

*x*

_{1d}=

*x′*

_{1}-

*x*

_{1}and

*d*to denote the difference between two variables and

*c*to denote their average (or center). In other words,

*α*

_{d}denotes the width of the illumination aperture and

*α*

_{c}its offset (in Fourier coordinates).

*x*

_{1c}, meaning that the intensity incident on the sample is spatially uniform independently of the aperture block position. This follows from the general principle of Köhler illumination. Moreover, because the aperture size is finite, the incident illumination is now partially coherent over an approximate range 2

*π*/

*α*

_{d}, as prescribed by the sinc function in Eq. (5). Finally, the effect of an offset in the aperture is to introduce a complex phase in the (otherwise real) incident mutual coherence. This phase will play a critical role in producing a gradient-field contrast.

*I*

_{3}(

*x*

_{3}) at the CCD camera given an arbitrary mutual coherence

*x*

_{1},

*x′*

_{1}) emanating from the sample. Following the same line of reasoning as above, we obtain

_{2}=

*kx*

_{2}/

*f*

_{b}and

*β*

_{1,2}=

*kb*

_{1,2}/

*f*

_{b}, where

*b*

_{1}and

*b*

_{2}represent the physical (spatial) limits of the detection aperture as prescribed by the position of its partial-beam block. As above, if the beam block is removed, then the detection aperture is a square of spatial dimension 2

*b*.

*I*

_{3}(

*x*

_{3})=

*J*

_{3}(

*x*

_{3},

*x*

_{3}). We have also recast mutual coherence in the form

*J*

_{1}(

*x*

_{1c},

*x*

_{1d})=

*G*

_{13}in Eq. (10) can be thought of as a kind of point spread function (PSF), however instead operating on an incoherent intensity, as does a normal PSF, it operates on a partially coherent intensity. As in the illumination case, an offset in the detection aperture introduces a complex phase in

*G*

_{13}. As will be seen below, the interplay between this phase and the phase in Eq. (5) will ultimately determine the contrast level in our graded-field microscope.

*t*(

*x*

_{1}) such that

*t*(

*x*

_{1}). From the definition of mutual coherence, we obtain the relation

*I*

_{3}at the camera and the phase variations in the sample manifest in

*T*. This relationship will be explicitly derived below.

## 3. Symmetry properties

*t*(

*x*

_{1}), which we assume to be arbitrary and hence impose no conditions on its profile. By construction, however, the associated mutual coherence transmittivity obeys the following condition

*T*(

*x*

_{1c},

*x*

_{1d})=

*T*

_{r}(

*x*

_{1c},

*x*

_{1d})+

*iT*

_{i}(

*x*

_{1c},

*x*

_{1d}), and note that the real functions

*T*

_{r}(

*x*

_{1c},

*x*

_{1d}) and

*T*

_{i}(

*x*

_{1c},

*x*

_{1d}) are respectively even and odd in

*x*

_{1d}.

*J*

_{1}(

*x*

_{1d}) into its real and imaginary components such that

*J*

_{1}(

*x*

_{1d})=

*J*

_{r}(

*x*

_{1d})+

*iJ*

_{i}(

*x*

_{1d}), where

*J*

_{r}(

*x*

_{1d}) and

*J*

_{i}(

*x*

_{1d}) are respectively even and odd in

*x*

_{1d}.

*G*

_{13}(

*x*

_{3}+

*x*

_{1c},

*x*

_{1d}) into its real and imaginary components

*x*

_{1d}.

*x*

_{1d}. Through simple trigonometric relations we find

*T*(

*x*

_{1c},

*x*

_{1d}), to the final intensity detected by the CCD camera. The functions

*K*

_{e}(

*x*

_{3}+

*x*

_{1c},

*x*

_{1d}) and

*K*

_{o}(

*x*

_{3}+

*x*

_{1c},

*x*

_{1d}) play dual roles. They serve as transfer functions for the coordinate

*x*

_{1c}whereas they serve as window functions for the coordinate

*x*

_{1d}. As transfer functions, they essentially image a sample point

*x*

_{1c}to a detector point

*x*

_{3}.with an attendant loss of resolution characterized by the width of

*K*(

*x*

_{3}+

*x*

_{1c},

*x*

_{1d}). As window functions their role is more subtle in that they reveal non-local features in the sample transmittivity. The roles of

*K*

_{e}and

*K*

_{o}as window functions are shown in Fig. (2) for various states of aperture blocking.

*K*and govern image resolution. The aperture offsets, on the other hand, govern the relative weights of

*K*

_{e}and

*K*

_{o}. If neither aperture is offset (

*α*

_{c}=

*β*

_{c}=0) then

*K*

_{e}→

*K*and

*K*

_{o}→0. This corresponds to a standard brightfield configuration and the

*K*

_{e}component in Eq. (25) can be thought of as performing brightfield imaging. When aperture offsets are introduced, then

*K*

_{o}begins to play a role at the expense of

*K*

_{e}. Because of the asymmetry in

*x*

_{1d}, this

*K*

_{o}component can be thought of as performing gradient imaging. As we will see below,

*K*

_{e}produces amplitude contrast whereas

*K*

_{o}produces phase contrast.

## 4. Graded field tuning

*K*

_{e}and

*K*

_{o}is gained by considering their net contributions. We quantify these contributions by evaluating the integrals of

*K*

_{e}and

*K*

_{o}over

*x*

_{1c}and

*x*

*1*

_{d}. It is clear that the integral of

*K*

_{o}vanishes because

*K*

_{o}is odd in

*x*

_{1d}. In general, however, the integral of

*K*

_{e}does not vanish and we define the net brightfield contribution by

*K*

_{total}corresponds to the total brightfield contribution when the aperture blocks are removed. In other words, the parameter

*B*characterizes the extent to which the microscope contrast is graded. If

*B*=0 then the contrast is fully darkfield; if

*B*=1 then the contrast is fully brightfield.

*B*can be tuned within this range simply by adjusting the aperture offsets. We note that the integrals over

*x*

_{1c}in Eq. (26) are straightforward from:

*B*is shown in Fig. (2).

*B*can be thought of as the relative amount of ballistic (i.e. unscattered) illumination light that arrives at the detector. When the illumination and detection apertures are fully open then all the ballistic illumination arrives at the detector and

*B*=1. When beam blocks are introduced, then

*B*is reduced in proportion to the to the intersection of the aperture areas. Finally, when the beam blocks cover exactly half of their respective apertures, then

*B*=0. In this case, there is no brightfield background and the contrast is purely darkfield. The illumination light that directly traverses the sample (i.e. ballistic light) is completely blocked and the only light impinging on the detector arises from sample scattering.

## 5. Phase versus amplitude contrast

*t*(

*x*

_{1}) into real and imaginary components

*q*(

*x*

_{1}) and

*p*(

*x*

_{1}) respectively impart amplitude and phase variations to the transmitted light. We assume that the sample is thin, meaning both

*q*(

*x*

_{1}) and

*p*(

*x*

_{1}) are much smaller than 1. To first order, we then write

*T*(

*x*

_{1c},

*x*

_{1d}) respectively.

*K*

_{e}in Eq. (10) produces amplitude contrast while the gradient component

*K*

_{o}produces phase contrast. More precisely,

*K*

_{e}reveals

*q*(

*x*

_{1c}) only when

*B*>0 (if

*B*=0 then

*K*

_{e}reveals the local curvature in

*q*(

*x*

_{1c}), or the second derivative

*q″*(

*x*

_{1c})). In contrast,

*K*

_{o}reveals the local slope in

*p*(

*x*

_{1c}), or the first derivative

*p′*(

*x*

_{1c}), thereby further refining its role as performing “phase gradient” imaging (the possibility of a partial aperture block to reveal phase gradients has been previously recognized[11

11. J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Classification scheme and properties of schlieren techniques,” Appl. Opt. **16**,18, 3338–3341 (1979). [CrossRef]

*p′*(

*x*

_{1c}). Positive phase gradients lead to an increase in detected intensity whereas negative phase gradients lead to a decrease in detected intensity. We note that in the case of darkfield imaging (

*B*=0) then the detected intensity cannot be less than zero and our phase gradient contrast becomes one-sided. That is, positive phase gradients are detected whereas negative phase gradients are not. In particular, one can verify that when

*B*=0 (i.e.

*α*

_{c}=

*α*

_{d}/2 and

*β*

_{c}=

*β*

_{d}/2) and when the sample complex transmittivity is a pure phase (i.e.

*t*(

*x*

_{1})=exp[

*i*

*ϕ*(

*x*

_{1})]) then the

*K*

_{e}and

*K*

_{o}integrals in Eq. (25) exactly cancel for negative phase gradients. However once a small brightfield background is introduced by slightly opening one or both aperture blocks (

*B*≲0), then negative phase gradients become observable and the image takes on a 3D relief appearance, much like a DIC image. Experimental examples of graded-field imaging are shown in Fig. (3).

## 6. Image resolution

*K*

_{e}and

*K*

_{o}serve as transfer functions for the coordinate

*x*

_{1c}and as window functions for the coordinate

*x*

_{1d}. Image resolution therefore depends on the ranges allowed for these coordinates. In general, these ranges are determined by the widths and offsets of the illumination and detection apertures, whose roles, according to Eqs. (22) and (23), can be analyzed separately.

*K*(Eq. (24)), their role is the same for both

*K*

_{e}and

*K*

_{o}. We consider two limits that characterize qualitatively different states of illumination, which we refer to as incoherent (

*a*

_{d}≫

*b*

_{d}) and coherent (

*a*

_{d}≪

*b*

_{d}).

*a*

_{d}≫

*b*

_{d}(incoherent illumination) we can make the approximation

*K*as transfer and window functions have become separated here, and the ranges of

*x*

_{1d}and

*x*

_{1c}are roughly defined by |

*x*

_{1d}≳ 2

*π*/

*α*

_{d}and |

*x*

_{3}+

*x*

_{1c}|≳

*π*/

*β*

_{d}. Because

*K*is localized to small

*x*

_{1d}’s, it effectively reduces to an incoherent PSF where the observed light intensity at the detector position

*x*

_{3}depends only on the transmitted light intensity in the vicinity of the sample position

*x*

_{1c}.

*a*

_{d}≪

*b*

_{d}(coherent illumination) we can make the approximation

*K*as a window and transfer function remain inseparable. The ranges of

*x*

_{1d}and

*x*

_{1c}are now roughly defined by |

*x*

_{1d}|≳2

*π*/

*β*

_{d}and |

*x*

_{3}+

*x*

_{1c}|≳

*π*/

*β*

_{d}, which are no longer independent. The observed light intensity at the detector position

*x*

_{3}depends on the distribution of transmitted light fields (as opposed to intensities) in the vicinity of the sample position

*x*

_{1c}. We note that the limit

*a*

_{d}≪

*b*

_{d}effectively corresponds to slit illumination, as is the case for both Schlieren and Hoffman contrast techniques[10, 12

12. J. G. Dodd, “Interferometry with Schlieren microscopy,” Appl. Opt. **16**,16, 470–472 (1977). [CrossRef] [PubMed]

15. R. Hoffman and L. Gross, “Modulation contrast microscopy,” Appl. Opt. **14**, 1169–1176 (1975). [CrossRef] [PubMed]

*x*

_{1c}to detector position

*x*

_{3}is determined by the detection aperture width

*b*

_{d}alone. When bd becomes large, then

*K*is non-zero only when

*x*

_{3}≈-

*x*

_{1c}and the transfer function resolution becomes high (note: the image is inverted relative to the object).

*x*

_{1d}, since ultimately this window function is responsible for revealing non-local transmission variations in the sample, such as phase gradients. The range allowed for |

*x*

_{1d}| depends not only on the aperture widths, as discussed above, but also on the aperture offsets. If the aperture offsets are too large, then the cosine and sine terms in Eqs. (22) and (23) oscillate too rapidly, and both

*K*

_{e}and

*K*

_{o}are ineffective at producing contrast. If instead the aperture offsets are too small, then the sine term in Eq. (23) oscillates too slowly, and

*K*

_{o}is ineffective at producing phase gradient contrast. As a rule of thumb, graded-field microscopy is effective at revealing phase gradients when

*α*

_{c}+

*β*

_{c}≈

*ηα*

_{d}for incoherent illumination, or

*α*

_{c}+

*β*

_{c}≈

*ηβ*

_{d}for coherent illumination, where

*η*ranges between

*η*leads to greater suppression of brightfield background). A simple conclusion may be drawn from this rule of thumb: If only a single aperture block is used to reveal phase gradients, then for incoherent illumination it is more effective to partially block the illumination aperture (as corroborated by Fig. (3)), whereas for coherent illumination it is more effective to partially block the detection aperture. In both cases, the effect of a second aperture block is then to suppress brightfield background. An advantage of incoherent over coherent illumination is that it allows higher phase gradient resolution since it makes use of a larger illumination aperture. Correspondingly, this is an advantage of graded-field microscopy, which allows the use of a large illumination aperture, over Schlieren or Hoffman contrast microscopies, both of which use slit illumination.

## 7. Conclusion

## Acknowledgments

## References and links

1. | F. Zernike, “Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung [in German],” Z. Tech. Phys. |

2. | F. Zernike, “How I discovered phase contrast,” Science |

3. | G. Nomarski, “Microinterféromètre différentiel à ondes polarisées [in French],” J. Phys. Radium |

4. | R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. |

5. | B. Kachar, “Asymmetric illumination contrast: a method of image formation for video microscopy,” Science |

6. | W. B. Piekos, “Diffracted-light contrast enhancement: A re-examination of oblique illumination,” Micros. Res. Tech. |

7. | S. Inoue, |

8. | H. U. Dodt, M. Eder, A. Frick, and W. Zieglgänsberger, “Precisely localized LTD in the neocortex revealed by infrared-guided laser stimulation”, Science |

9. | C. F. Saylor, “Accuracy of microscopical methods for determining refractive index by immersion,” J. Res. US Natl. Bur. Stds. |

10. | E. H. Linfoot, |

11. | J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Classification scheme and properties of schlieren techniques,” Appl. Opt. |

12. | J. G. Dodd, “Interferometry with Schlieren microscopy,” Appl. Opt. |

13. | D. Axelrod, “Zero-cost modification of bright field microscopes for imaging phase gradient on cells: Schlieren optics,” Cell Biophys. |

14. | S. Lowenthal and Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. |

15. | R. Hoffman and L. Gross, “Modulation contrast microscopy,” Appl. Opt. |

16. | M. Born and E. Wolf, |

**OCIS Codes**

(110.0180) Imaging systems : Microscopy

(110.1220) Imaging systems : Apertures

(110.4980) Imaging systems : Partial coherence in imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: May 4, 2006

Revised Manuscript: May 30, 2006

Manuscript Accepted: May 30, 2006

Published: June 12, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Ran Yi, Kengyeh K. Chu, and Jerome Mertz, "Graded-field microscopy with white light," Opt. Express **14**, 5191-5200 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-12-5191

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### References

- F. Zernike, "Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung [in German],"Z. Tech. Phys. 16,454 (1935).
- F. Zernike, "How I discovered phase contrast,"Science 121,345-349 (1955). [CrossRef] [PubMed]
- G. Nomarski, "Microinterf´erom`etre diff´erentiel `a ondes polaris´ees [in French]," J. Phys. Radium 16,S9 (1955).
- R. D. Allen, G. B. David, G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmittedlight microscopy," Z. Wiss. Mikrosk. 69,193-221 (1969). [PubMed]
- B. Kachar, "Asymmetric illumination contrast: a method of image formation for video microscopy,"Science 227,766-768 (1985). [CrossRef] [PubMed]
- W. B. Piekos, "Diffracted-light contrast enhancement: A re-examination of oblique illumination," Micros. Res. Tech. 46,334-337 (1999). [CrossRef]
- S. Inoue, Video Microscopy (Plenum Press, New York, 1986).
- H. U. Dodt, M. Eder, A. Frick, W. Zieglg¨ansberger, "Precisely localized LTD in the neocortex revealed by infrared-guided laser stimulation", Science 286,111-113 (1999). [CrossRef]
- C. F. Saylor, "Accuracy of microscopical methods for determining refractive index by immersion,"J. Res. US Natl. Bur. Stds. 15,277 (1935).
- E. H. Linfoot, Recent advances in optics (Clarendon Press, Oxford, 1955).
- J. Ojeda-Castaneda, L. R. Berriel-Valdos, "Classification scheme and properties of schlieren techniques," Appl. Opt. 16,18, 3338-3341 (1979). [CrossRef]
- J. G. Dodd, "Interferometry with Schlieren microscopy," Appl. Opt. 16,16, 470-472 (1977). [CrossRef] [PubMed]
- D. Axelrod, "Zero-cost modification of bright field microscopes for imaging phase gradient on cells: Schlieren optics,"Cell Biophys. 3,167-173 (1981). [PubMed]
- S. Lowenthal, Y. Belvaux, "Observation of phase objects by optically processed Hilbert transform,"Appl. Phys. Lett. 11, 49-51 (1967). [CrossRef]
- R. Hoffman and L. Gross, "Modulation contrast microscopy," Appl. Opt. 14,1169-1176 (1975). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of optics (Cambridge University Press, Cambridge, UK, 1999).

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