Advances  in  Phase  Contrast  Microscopy  

Colin  Sheppard  Nano-­‐Physics  Department  

Italian  Ins;tute  of  Technology  (IIT)  Genoa,  Italy  

colinjrsheppard@gmail.com  

Perfect  imaging  

�

t(x, y) = a(x, y)eiφ (x ,y )

�

a(x, y)

�

φ (x, y)

�

I (x, y) = a(x, y)eiφ (x ,y ) 2= a2 (x, y)

Object amplitude transmission

is modulus (amplitude), real

is phase, real

Perfect image

• No phase information in perfect image!

To see phase, we need to have imperfect imaging system!

• Introduce phase (aberration) • Introduce asymmetry

Methods  of  phase  contrast  •   Dark  field  •  Zernike  phase  contrast  •  Defocus  •  Transport  of  intensity  equa;on  (TIE)  •  Offset  illumina;on  (Schlieren)    

-­‐  (Hoffmann  modula;on  contrast)  -­‐  Differen;al  phase  contrast  (DPC)  -­‐  Wavefront  sensing  (Shack-­‐Hartmann)  

•  Interference  microscopy  •  Differen;al  Interference  Contrast  (DIC)  •  Digital  holographic  microscopy  (DHM)  

Divide  into  to  two  approaches  • 1  Coherent  methods  (Digital  holographic  microscopy)  

–  Spa;al  frequencies  only  on  Ewald  sphere  –  Limited  3D  imaging  performance  

–  But  can  improve  by  holographic  tomography  –  Limited  spa;al  resolu;on  

–  Can  reconstruct  with  Rytov  approxima;on  

• 2  Par;ally  coherent  methods  

–  Improved  image  bandwidth  –  No  speckle  – More  difficult  to  extract  quan;ta;ve  informa;on  

Dark field microscope

• Direct light blocked • Only see changes • Partially-coherent imaging

Encyclopedia of Modern Optics, RD Guenther, DG Steel, L Bayvel, eds, Elsevier, Oxford, 3, pp. 103-110

Zernike phase contrast

• Direct light changed in phase • Partially-coherent imaging • Phase imaged by imaginary part of transfer function

Encyclopedia of Modern Optics, RD Guenther, DG Steel, L Bayvel, eds, Elsevier, Oxford, 3, pp. 103-110

Weak  phase  object,  φ  small  

Direct light

Weak phase object Total almost unchanged

t = a + iaφ

Brightfield

No direct light to reduce contrast

Weak phase object Darkfield

Direct light

Phase of phase object changed

Change in total much larger Zernike

�

t(x, y) = a(x, y)eiφ (x ,y )

Zernike  phase  contrast  • Phase  contrast  amplified  by  transmiZance  of  phase  ring  

• Can  make  +ve  or  –ve  phase  contrast  from  phase  of  phase  ring  

• Difficult  to  get  quan;ta;ve  informa;on  

• Haloes  around  phase  changes  

Coherent  imaging  

�

I(x,y) = c(m,n)T(m,n)exp 2πj(mx + ny{ }dmdn−∞

∞

∫−∞

∞

∫2

= c(m,n)c * (p,q)T(m,n)T * (p,q)exp 2πj (m − p)x + (n − q)y[ ]{ }−∞

∞

∫−∞

∞

∫−∞

∞

∫−∞

∞

∫ dmdndpdq

For partially coherent, C(m, n; p, q) does not separate

T(m, n) is object spatial frequency content, FT of t(x, y) c(m, n) is coherent transfer function

Imaging  depends  on  coherence  Fluorescence behaves as incoherent imaging

Brightfield etc. S = 0, coherent illumination S = 1, full or complete illumination S -> ∞, incoherent illumination

Resolution depends on coherence

Coherence parameter S =

�

nc sinαc

no sinαo

Par;ally  coherent  image  forma;on  

Proc. R. Soc. Lond. A 217, 408 (1953)

C(m,n;p,q) = transmission cross-coefficient (TCC)

effective source pupil

Image intensity

m, p are both spatial frequencies in x direction n, q are both spatial frequencies in y direction

Propagate mutual intensity through the system:

• System and object separated. • Although Hopkins propagated mutual intensity, he did not give mutual intensity of the image.

Weak  object  t = a + iaφ =  a0  +  Δa  +  ia0  φ

Amplitude part: a0δ(m)δ(n) + TΔa is Hermitian, Phase part: a0Tp is skew-Hermitian

Neglect interference of scattered light with scattered light (weak) Phase part imaged by: Im (and even) part of C(m,n;0,0), gives phase contrast or odd (and real) part of C(m,n;0,0), gives differential phase contrast

a and φ real Δa, φ small

�

I(xs,ys) = a02C(0,0;0,0) + a0 [TΔa∫ (m,n) + ja0Tp (m,n)]C(m,n;p,q)exp[2πj(mxs + nys)]dmdn

+ a0 [TΔa∗∫ (m,n) − ja0Tp

∗(m,n)]C∗(m,n;p,q)exp[−2πj(pxs + qys)]dpdq +

Phase  Gradient  Transfer  Func;on  

Hamilton  DK,  Sheppard  CJR,  Wilson  T,  Improved  imaging  of  phase  gradients  in  scanning  op;cal  microscopy    Journal  of  Microscopy  153,  275-­‐286  (1984)  

For a single phase gradient, object spectrum m0 is independent of x

For a slowly varying phase gradient, m0 is a function of x. m0(x) is the instantaneous frequency

m0(x)

x

Image intensity:

WOTF  C(m;0)  and  PGTF  C(m;m)  for  conven;onal  microscope  

0.5 1 1.5 2-0.2

0.2

0.4

0.6

0.8

1

m

C (m)

0.5 1 1.5 2-0.2

0.2

0.4

0.6

0.8

1

m

C (m)

Weak object transfer function Phase gradient transfer function

C(m;0) C(m;m) S = 0

S = 1

S = 0

S = 1

Partially coherent imaging is complicated, but becomes simpler for 2 cases: •Weak object •Slowly varying phase gradient

Defocus  WOTF,  S  =  0.01  (nearly  coherent)  

Like  cos  or  sin  (ul2/2)  

l is radial spatial frequency, l = (m2+n2)1/2 Sheppard CJR

Defocused transfer function for a partially coherent microscope, J. Opt. Soc. Am. A, 21, 828-831(2004)

Phase imaged by imaginary part

WOTF,  S  =  0.5  

Sheppard CJR Defocused transfer function for a partially coherent microscope,

J. Opt. Soc. Am. A, 21, 828-831(2004)

Real Imaginary

WOTF,  S  =  0.99  

Sheppard CJR Defocused transfer function for a partially coherent microscope,

J. Opt. Soc. Am. A, 21, 828-831(2004)

Real Imaginary (very weak)

Small  defocus:  analy;c  expression  

Sheppard CJR Defocused transfer function for a partially coherent microscope,

J. Opt. Soc. Am. A, 21, 828-831(2004)

I(Δu)  –  I(–Δu)  gives  phase  contrast  image  (amplitude  image  cancels)    

Parabolic for small l

Sheppard CJR Defocused transfer function for a partially coherent microscope,

J. Opt. Soc. Am. A, 21, 828-831(2004)

Small  defocus,  aler  inverse  Laplacian  

Phase restored up to l = 1 – S

Sheppard CJR Defocused transfer function for a partially coherent microscope,

J. Opt. Soc. Am. A, 21, 828-831(2004)

WOTF  

Kou SS, Waller L, Barbastathis G, Marquet P, Depeursinge C, Sheppard CJR Quantitative phase restoration by direct inversion using the optical transfer function, Opt. Lett. 36, 2671-2673 (2011).

Transport  of  Intensity  Equa;on  (TIE)  

• Teague, JOSA A 1434, 73 (1983) • Streibl, Opt. Commun. 6, 49 (1985) • Barty, Nugent, Paganin, Roberts, Opt. Lett. 817, 23 (1998)

�

∂I∂z

= −∇T ⋅ I∇Tφ( )

�

∂ ln I∂z

= −∇T2φ −∇T ln I ⋅ ∇Tφ

Amplitude in image space satisfies paraxial wave equation

Logarithmic  deriva;ve  image  

Testicle of rat, Streibl, Opt. Commun. 6, 49 (1984)

Barty,  Nugent,  Paganin,  Roberts  Opt.  Le9,  23,  817  (1998)  

TIE phase image DIC

Quan;ta;ve  phase  imaging  

• IATIA system: measure φ using TIE equation • Can then simulate Zernike, DIC, etc. images

Problems  with  TIE  imaging  

• Measures phase of image not object • Similar to defocus method for weak object, but not limited to weak phase • Not enough information to directly recover object phase for strong object • Problem with 3D imaging: Measure so no information on zero axial spatial frequency

�

∂I /∂z

Differential phase contrast (DPC)

Encyclopedia of Modern Optics, RD Guenther, DG Steel, L Bayvel, eds, Elsevier, Oxford, 3, pp. 103-110

Differen;al  phase  contrast  

Extrac;ng  phase  

Two detectors, A and B

DPC  image  of  a  cheek  cell  

Hamilton DK, Sheppard CJR (1984), J. Microsc. 133, 27-39 (1984)

DPC  image  of  an  integrated  circuit  

DPC DIC

DPC  image  of  a  single  monolayer  

Hamilton  DK,  Sheppard  CJR,  Wilson  T,  Journal  of  Microscopy  153,  275-­‐286  (1984)  

DPC with an annular split detector

a = 1 a = 0.7

Asymmetric Illumination DPC (AI-DPC)

Arrows reversed, source from each semicircle

Asymmetric  illumina;on  DPC  (AI-­‐DPC)  

Condenser pupil structures (top row), partially coherent transfer function in direction of differentiation (middle row), and experimental images (bottom row) obtained with AIDPC. The sample is skin H&E stained section courtesy Graham Wright, TLL and Declan Lunny, IMB.

Phase  measurement  using  DPC  

Measure yx ∂∂φ∂∂φ /,/

φ(x, y) = F−1

F ∂φ∂x

+ i ∂φ∂y

⎡⎣⎢

⎤⎦⎥

2i sin2πmΔ + i sin2πnΔ( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Arnison, Larkin, Sheppard, Smith, Cogswell, J. Microsc. 214, 7-12 (2004)

• Integrate phase gradient to get phase (but still constant of integration)

�

φ = ∂φ∂x∫ dx + const.

Phase  reconstruc;on  from  AI-­‐DPC  

S Mehta, Thesis (2010)

Nomarski Differential interference contrast (DIC)

Encyclopedia of Modern Optics, RD Guenther, DG Steel, L Bayvel, eds, Elsevier, Oxford, 3, pp. 103-110

Nomarski  DIC  

• Uses  polariza;on  so  depends  on  birefringence  of  sample  

• Can  use  in  conven;onal  or  confocal  mode  

Birefringent

Phase-­‐stepping  DIC  

�

I = 2a2C(m;m) − 2a2C(m;m)cos(2πmΔ −φ0)

• Slowly-varying phase gradient 2πm

• Same form as normal interference pattern • Measure I for different values of bias retardation φ0

• Using phase-stepping algorithm, can recover phase gradient 2πm • Integrate phase gradient to get phase (but still constant of integration)

Cogswell, Smith, Larkin, Hariharan, Proc. SPIE 72, 2984 (1997)

�

φ = ∂φ∂x∫ dx + const.

Phase  gradient  from  phase  stepping  DIC  

Phase  measurement  using  phase-­‐stepping  DIC  

Measure yx ∂∂φ∂∂φ /,/

φ(x, y) = F−1

F ∂φ∂x

+ i ∂φ∂y

⎡⎣⎢

⎤⎦⎥

2i sin2πmΔ + i sin2πnΔ( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Arnison, Larkin, Sheppard, Smith, Cogswell, Linear phase imaging using differential interference microscopy, J. Microsc. 214, 7-12 (2004)

Phase  reconstruc;on  from  DIC  

S Mehta, Thesis (2010)

Phase  reconstruc;on  from  DIC  

Three  different  DIC  configura;ons  

Transfer  func;on  C(m;p)  

Mouse  intes;ne  

Become more similar

Op;cal  fibre,  S  =  0.4  

Op;cal  fibre,  S  =  1  

Phase  gradient  from  phase  stepping  DIC  and  TIE-­‐DIC  

Phase  reconstruc;on  from  TIE-­‐DIC,  π/4  and  3π/4  bias  

TIE  from  colour  (single  shot)  

HMVEC cells HeLa cells

Summary  • TIE  can  reconstruct  phase  from  2  sec;ons  • TIE  does  not  recover  3D  informa;on  

• DIC  can  reconstruct  phase  from  2  direc;ons  of  shear  

• DIC  has  problems  with  birefringent  objects  

• DPC  has  inferior  depth  imaging  performance    

               (may  be  an  advantage)  

• Possibility  to  use  combina;ons  of  DIC/TIE  etc