Formulation of the rotational transformation of wave
fields and their application to digital holography
Kyoji Matsushima
Department of Electrical and Electronic Engineering, Kansai University,
Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan*Corresponding author: matsu@kansai-u.ac.jp
Received 17 September 2007; revised 11 February 2008; accepted 11 February 2008;
posted 15 February 2008 (Doc. ID 87623); published 2 April 2008

Rotational transformation based on coordinate rotation in Fourier space is a useful technique for simulating wave field propagation between nonparallel planes. This technique is characterized by fast computation because the transformation only requires executing a fast Fourier transform twice and a single
interpolation. It is proved that the formula of the rotational transformation mathematically satisfies
the Helmholtz equation. Moreover, to verify the formulation and its usefulness in wave optics, it is also
demonstrated that the transformation makes it possible to reconstruct an image on arbitrarily tilted
planes from a wave field captured experimentally by using digital holography. © 2008 Optical Societyof America
OCIS codes:
090.1995, 350.5500, 000.3860.

1. Introduction

Simulation in wave optics has become increasingly
more important because of the evolution of modern optical technologies such as diffractive optical elements
and digital holography. In these technologies, exact
and fast simulation techniques are always demanded,
especially for free space propagation. There arevarious formulations for spatial propagation or diffraction of coherent wave fields in a free space,including such well-known formulas as the Fresnel–
Kirchhoff integral or Fresnel and Fraunhofer diffraction. However, these share a common restriction: thereference screen must be parallel to the source planein which the wave field is initially given. Formulations
based on coordinate rotation in Fourier space [1,2] andthe Rayleigh–Sommerfeld integral [3] have been presented for removing this restriction. A similar formulation based on the Fresnel approximation is also
reported for numerical reconstruction in digitalholography [4].
Recently, the author and colleagues presented another formulation for propagation between nonparal0003-6935/08/19D110-07$15.00/0© 2008 Optical Society of America
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APPLIED OPTICS / Vol. 47, No. 19 / 1 July 2008lel planes [5]. This formula, referred to as rotationaltransformation, features exact and fast simulation.
Since this is a useful technique, applications using
the presented formula have been reported for creating surface objects [6] and hidden-surface removal [7]in computer-generated holograms and for the numerical reconstruction of digital holograms [8]. However,slight doubts regarding the mathematical reliabilityof the formula have not been dispelled, because thederivation in [5] is based on the physical interpretation of the angular spectrum of plane waves.
I aim to provide a brief proof that the formula rigorously satisfies the Helmholtz equation and that
there are no approximations included in the formulation. Another aim is to demonstrate the applicationof rotational transformation to digital holography.
Evolution of computer technology is realizing various
possibilities for digital signal processing of light. Digital signal processing makes it possible to process
light wave fields in a manner that is difficult or impossible to accomplish by using conventional opticalcomponents. Digital holography is a critical technology for acquisition of wave fields in digital signal processing of light. For example, De Nicola et al. useddigital holography and reported correct image recon-struction under conditions of severe anamorphism[9]. They also reported numerical reconstruction ofimages on tilted planes [8] using digital holography
and rotational transformation based on the formulation presented in an earlier paper of ours [5]. However, the results reported by De Nicola et al. are ofholographic microscopy for an object with dimensions of several micrometers. In this paper, reconstruction of a large object with dimensions of
4 cm × 1 cm is demonstrated for experimental verification of rotational transformation.

twhere R−1ξ ðθξ Þ and Rξ ðθξ Þ are the inverse and transposed matrix of a rotation matrix, respectively.
As a result, the inverse matrix of any transformation matrix defined by the product of individual rotation matrices in Eq. (3) is generally given byT−1 ¼ t T:

Therefore, the generalized transformation matrix below is used in formulations as follows:2

a1
T−1 ¼ 4 a4
a7

2. Formulation of Rotational Transformation
A. Coordinate Systems and Rotation Matrices

Two coordinate systems are defined as shown in Fig. 1.
One of the coordinate sets is referred to as the source
coordinate ðx; y; zÞ. A wave field is initially given in theplane ðx; y; 0Þ of the source coordinates. This plane isreferred to as the source plane. The reference coordinates ð^x; ^y; ^zÞ are the second set of coordinates, in whicha wave field is calculated by rotational transformationin the plane ð^x; ^y; 0Þ, referred to as the reference plane.
Both coordinate systems share the origin and are notparallel to each other.
Position vectors r ¼ ðx; y; zÞ and ^r ¼ ð^x; ^y; ^zÞ can be
mutually transformed by the transformation matrix
T as follows:^r ¼ Tr;−1^

r ¼ T r:ð1Þ

ð3Þ

where ξ and η denote axes x, y, or z and θξ and θη arethe angles of rotation around the axes ξ and η, respectively. Individual rotation matrices are shown in
Table 1. These commonly possess the following characteristics:tR−1ξ ðθξ Þ ¼ Rξ ð−θξ Þ ¼ Rξ ðθξ Þ;2

a1
T ¼ 4 a2
a3
B.

a2
a5
a8

3
a3
a6 5 ;
a9

ð6Þ

a4
a5
a6

3
a7
a8 5 :a9

ð7Þ

Rotational Transformation

When a wave field f ðx; y; 0Þ with wavelength λ is givenin the source plane, the source field at arbitrary positions is given by [10]f ðx; y; zÞ ¼

Z Z þ∞−∞

Fðu; vÞ exp½i2πfux þ vy

þ ðλ−2 − u2 − v2 Þ1=2 zgdudv¼ F −1 fFðu; vÞ exp½i2πwðu; vÞzg;ð8Þ

wðu; vÞ ¼ ðλ−2 − u2 − v2 Þ1=2 ;

ð9Þ

ð2Þ

The matrix T is, in general, given as a rotation matrix Rξ ðθξ Þ or the product of several rotation matrices
as follows:
T ¼ Rξ ðθξ Þ…Rη ðθη Þ;

ð5Þ

ð4Þ

where Fðu; vÞ and ½u; v; wðu; vÞ are the Fourier spectrum of f ðx; y; 0Þ and Fourier frequencies for ðx; y; zÞ,respectively. Note that the wave field f ðx; y; zÞ represented in Eq. (8) rigorously satisfies the Helmholtz
equation.
The wave field in the reference plane ^f ð^x; ^y; ^z ¼ 0Þis given by substituting Eq. (2) and T−1 of Eq. (6) into
Eq. (8) as follows:^f ð^x; ^y; 0Þ ¼ f ða1 ^x þ a2 ^y; a4 ^x þ a5 ^y; a7 ^x þ a8 ^yÞZ Z¼
Fðu; vÞ exp½fi2π½ða1 u þ a4 v þ a7 wÞ^xþ ða2 u þ a5 v þ a8 wÞ^ygdudv;

Fig. 1. Definition of the coordinate systems and their rotation.

ð10Þ

where the abbreviation w ¼ wðu; vÞ is used.
^, v^, and w^ ¼ wð^^ u; ^vÞ ¼
When Fourier frequencies u^2 − v^2 Þ1=2 are defined with respect to ^x, ^y, andðλ−2 − u^z, the vectors f ¼ ðu; v; wÞ and ^f ¼ ð^^; wÞ
^ in theu; vFourier space are transformed into each other by
the same transformation matrices T and T−1 as inreal space. Therefore, by using the transform^f ¼ Tf, the exponent in Eq. (10) can be reduced to^^yÞ, and the wave field in the referencei2πð^u^x þ v
1 July 2008 / Vol. 47, No. 19 / APPLIED OPTICS

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Table 1.

1

0

1
0
B
Rx ðθx Þ ¼ @ 0 cos θx
0 − sin θx

0C
sin θx A;
cos θx0
B
Ry ðθy Þ ¼ @Rotation Matricescos θy0sin θyplane is given as
^f ð^
x; ^
y; 0Þ ¼

Z Z
^ÞÞ
Fðαð^
u; ^vÞ; βð^u; v^^yÞdudv;
× exp½i2πð^u^x þ v

ð11Þ

where arguments u and v of spectrum Fðu; vÞ in
Eq. (10) are rewritten by using f ¼ T−1^f as follows:^ þ a2 v^ þ a3 wð^^ u; v^Þ;^Þ ≡ a 1 u
u ¼ αð^u; v^Þ ≡ a 4 u^ þ a5 v^ þ a6 wð^^ u; v^Þ:v ¼ βð^
u; v

ð12Þ

^, v
^ in the integration
Changing from variables u, v to uof Eq. (11) can be achieved by substituting dudv ¼
^Þjd^
jJð^
u; v
ud^
v and the Jacobian obtained by
^Þ ¼
Jð^
u; v

∂α ∂β ∂α ∂β
−
:
^∂v
^ ∂v
^∂u
^
∂u

ð13Þ

As a result, rotational transformation of the wave
fields is formulated as follows:
^f ð^
x; ^
y; 0Þ ¼

Z Z

^ u; v^Þ exp½i2πð^^^yÞd^Fð^u^x þ vud^v

^ u; v^Þg;¼ F −1 fFð^10 − sin θy
1
0 C
A;
0 cos θy0cos θzB
Rz ðθz Þ ¼ @ − sin θz
0sin θzcos θz
0

1
0
C
0A
1

origin in the source Fourier space ðu; vÞ is projected to^0 Þ ¼ ða7 λ−1 ; a8 λ−1 Þ in the reference Fourier spaceð^u0 ; vby the transform ^f ¼ Tf. Therefore, the second FFT
required in Eq. (14) does not work effectively because
the spectrum is sampled far from the origin. To use
FFTs for numerical calculation, shifted Fourier space~Þ ¼ ð^^0; v^ − ^v0 Þ should be introduced in order
ð~
u; v
u−u
to cancel these undesired offsets. When using theseshifted Fourier frequencies, the spectrum in the reference plane is rewritten as^ u; v^ uþu~ u; v^Þ ¼ Fð~^0; v~þv^0 Þ ¼ Fð~
~Þ:
Fð^

ð17Þ

An example of a sampling area in the shifted reference space is shown in Fig. 2(c).
Let us invert the procedure. Suppose that the spec~ u; v
~Þ is sampled in a square area centered in thetrum Fð~shifted Fourier space, as shown in Fig. 2(f), the origin of~Þ agrees with the center of the spectrum ð^
^Þ ¼
ð~
u; v
u; v
^0 Þ in the reference Fourier space, as shown inð^u0 ; v2(e), and the origin in the source Fourier space, in~Þ is projected to the source
2(d). The sampling point ð~u; vFourier space ðu; vÞ as follows:
^0; v
~þv
^0 Þ;
u ¼ αð~
uþu

ð18Þ

ð14Þ

^ u; v^ þ a2 v^ þ a3 wð^^ u; v^Þ; a4 u^ þ a5 v^^Þ ¼ Fða1 uFð^^ u; v^ÞÞjJð^^Þj;þ a6 wð^
u; v
^Þ ¼
Jð^
u; v

ð15Þ

ða2 a6 − a3 a5 Þ^u ða3 a4 − a1 a6 Þ^v
þ^ u; v^Þ^ u; v^Þwð^wð^þ ða1 a5 − a2 a4 Þ:

ð16ÞThe Jacobian approximates a constant and therefore
can be ignored in cases where the field is paraxial in
the source or reference coordinates [5]. It is worthemphasizing that ^f ð^x; ^y; 0Þ given by Eqs. (14)–(16)is a complete solution of the Helmholtz equation.
In addition, only a double Fourier transform isneeded to perform this transformation.
C. Resampling Fourier Spectrum

Since fast Fourier transforms (FFTs) are commonlyused for numerical calculation, the source spectrum
Fðu; vÞ is sampled in a square area, and the center ofthe sampling area is the origin of the Fourier space,as shown in Fig. 2(a). However, as shown in 2(b), theD112

APPLIED OPTICS / Vol. 47, No. 19 / 1 July 2008

Fig. 2. Example of sampling areas in three Fourier spaces: (a), (d)source, (b), (e) reference, and (c), (f) shifted reference Fourier
spaces. The left-hand column (a)–(c) is the case of a spectrum
sampled in an equidistant grid within the square area in thesource space, whereas the right-hand column (d)–(f) is for equidistant and square sampling in a shifted reference space. This example is for T ¼ Rx ð20°ÞRy ð30°Þ.

^0; v~þv^0 Þ:
v ¼ βð~uþu

ð19Þ

~ u; v~Þ and Fðu; vÞ are sampled in equidiWhen both Fð~
stant sampling grids, the sampling points in thesampled spectrum Fðu; vÞ generally do not agree with
~ u; v~Þ because of the distortion
the sampling points in Fð~of the sampling grid caused by the projection in
Eqs. (18) and (19). As a result, an interpolation is ne~ u; v
~Þ from Fðu; vÞ.
cessary for resampling Fð~~, v~
The inverse Fourier transform of Eq. (17) as to u
is given by using the shift theorem as~ u; v^ uþu~Þg ¼ F −1 fFð~
^0; v
~þv
^0 Þg
F −1 fFð~^ u; v~Þg exp½−i2πð^^0 ^yÞ:¼ F −1 fFð~u0 ^x þ v
ð20Þ^ u; v^Þ as to u^,
Since the inverse Fourier transform of Fð^^^ is identical to that of Fð~~Þ as to u
~ , ~v, we can rev
u; vwrite Eq. (14) as^f ð^~ u; ~vÞg exp½i2πð^^0 ^yÞ:u0 ^x þ vx; ^y; 0Þ ¼ F −1 fFð~is a virtual zone in the shape of a quadrangular pyramid in the object space, as shown in Fig. 5. An object
wave emitted from any object placed within the zone
can be recorded without any aliasing error. The
shape of the aliasing-free zone is obtained from anestimation of the maximum spatial frequency on
the sensor surface and is presented as the relationship between the object width w and the object distance d from the image sensor, as follows [12]:ð21Þ

In the case of a slight tilting of the source plane, theFourier spectrum in the reference plane can be ap~ u; v~Þ ≈ Fðu; vÞ;proximated to the source one, i.e., Fð~^0 ^yÞ is atherefore, ^f ð^x; ^y; 0Þ ≈ f ðx; y; 0Þ exp½i2πð^u0 ^x þ v
good approximation. This agrees with the theoreticalmethod providing analysis of the effect of the slighttilting motion of object surfaces [11].
In summary, the spectrum Fðu; vÞ of the source
wave field is first calculated by using a FFT in the
numerical implementation of the theory presented~ u; v~Þ is resampled fromhere. Next, the spectrum Fð~Fðu; vÞ by using an interpolation as follows:~ u; v~Þ ≅ Fðαð~
^0; v
~þv
^0 Þ; βð~
^0; v
~þv
^0 ÞÞ
Fð~
uþu
uþu
~þv
^0 Þj:
^0; v
jJð~
uþu

Fig. 3. Schematics for capturing the wave field from a tilted
plane.

ð22Þ

Finally, the wave field in the reference coordinates isobtained by inverse FFT in Eq. (21).
3. Application of Rotational Transformation to Digital
Holography

To verify the above formulation of the rotationaltransformation, an attempt to reconstruct clear
images of the pattern printed on a tilted plane was
performed by using digital holography. As shownin Fig. 3, the wave field emitted from the planar object, on which some letters are printed, was captured
on the image sensor by using phase-shifting digitalholography. The printed pattern (letters) covering
an area of 4 cm × 1 cm is shown in Fig. 4.

4λdw ≤ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;16δ2 − λ2ð23Þ

where δ is the sensor pitch. We can minimize the object distance and maximize the object size by using
the aliasing-free zone. As a result, the angle of thevisual field can be maximized to the limit imposedby the sensor pitch.
The experimental setup for capturing the wave
field by the phase-shifting lensless Fourier digital holography is shown in Fig 6. The planar object illuminated with coherent light (λ ¼ 532 nm) is slanted at
approximately 70° around the y axis, and the distance between the center of the object and the image
sensor is approximately 19:5 cm. The piezo phaseshifter is inserted into the reference path to change
the phase of the reference field. Furthermore, thespatial filter forms the reference field into a sphericalwave. A CMOS-type high-resolution image sensor
with sensor pitches of 6:0 μm × 6:0 μm is used to capture the fringe pattern. The number of pixels of theimage sensor is 2000 × 2000 pixels.
B.

Numerical Reconstruction by Rotational Transformation

A schematic procedure for reconstructing a pattern
on a tilted planar object is shown in Fig. 7. The complex image gðx0 ; y0 Þ was composed from four different

A. Experimental Setup

The wave field from the tilted planar object was captured by using the aliasing-free zone [12] in phaseshifting lensless Fourier digital holography. This isbecause the object is large compared with those usedin holographic microscopy [8]. The aliasing-free zone

Fig. 4. (Color online) Photograph of the pattern printed on theplanar object.
1 July 2008 / Vol. 47, No. 19 / APPLIED OPTICS

D113

Fig. 7. Procedure for numerical reconstruction of images on theplanar object from the complex image captured on the image
sensor.
Fig. 5. Schematic illustration of an aliasing-free zone.

phase-shifting fringe patterns captured by the image
sensor [13]. The wave field f 0 ðx; y; z0 ¼ −zR Þ was obtained in the plane parallel to the image sensor byusing a Fourier transform as follows:f 0 ðx; y; z0 ¼ −zR Þ ¼ F fgðx0 ; y0 Þgu¼−x=λzR ;v¼−y=λzR ; ð24Þ
where the origin of z0 is on the sensor surface and zR
is the distance between the sensor surface and thepinhole of the spatial filter inserted into the reference path, as shown in Fig. 6.
A close-up photograph of the tilted planar object
captured by an ordinary lens camera is shown in
Fig. 8(a). Similar images can be obtained as the amplitude image jf 0 ðx; y; −zR Þj, as shown in Fig. 8(b). In
this amplitude image, the number of samplings andsampling pitches have changed from that of the image sensor because of the Fourier transform in
Eq. (24). These are 2048 × 2048 pixels and 8:1 μm ×
8:1 μm in Fig. 8(b), respectively.
We cannot obtain clear images in both results in
Fig. 8 because of the depth of focus. Rotational transformation should be used to produce a clear image on
the surface of the planar object. However, if the center of the sampled wave field f 0 ðx; y; −zR Þ does not coincide with the surface of the planar object exactly,
we cannot obtain a clear image, because rotationaltransformation yields the wave field rotated on thecenter of the sampling area. Therefore, the fieldf 0 ðx; y; −zR Þ must be propagated to the position z0 ¼
−zR −Δz numerically so that the center of the field
is exactly placed on the surface of the planar object,

Fig. 6. Experimental setup for recording phase-shifting lensless
Fourier digital holograms: L, lens; M, mirror; BS, beam splitter;
SF, spatial filter.
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APPLIED OPTICS / Vol. 47, No. 19 / 1 July 2008as shown in Fig. 7. We can use any method for thispropagation. In this paper, the angular spectrum of
the plane wave in Eq. (8) is used for this numericalpropagation.
The wave field f 0 ðx; y; −zR −ΔzÞ is the starting
point of rotational transformation, i.e., f ðx; y; 0Þ ¼
f 0 ðx; y; −zR −ΔzÞ, and transformation matrix T ¼
Ry ðθy Þ was used in the calculation. First, the spectrum Fðu; vÞ is calculated by using an FFT. Next,
the spectrum in the shifted reference Fourier space~ u; v~Þ is obtained by resampling Fðu; vÞ, usingFð~
Eq. (22). Since the complex value of the spectrum^0; v~n þ v^0 Þ; βð~^ 0 ; ~vn þ ^v0 Þ must be obF½αð~um þ uum þ u~n Þ on the equitained for each sampling point ð~
um ; vdistant sampling grid, interpolation is needed
because of the nonlinearity attributed to Eq. (12).
Finally, we can calculate the wave field in the reference plane by use of Eq. (21). Since the purpose of thisexperiment is to calculate only the amplitude image,the phase factor in Eq. (21) can be ignored; i.e., the FFT~ u; v~Þ. The amplitude image
is performed only in Fð~^
j f ð^x; ^y; 0Þj after the rotational transformation at θy ¼
71:8° is shown in Fig. 9. The whole pattern on the surface of the tilted planar object appears clearly whencompared with the results in Fig. 8.
The case of two-axis rotation is shown in Fig. 10.
The planar object is slanted at 30° around the y axis
after rotation at −60° around the x axis. Therefore
the transformation matrix T ¼ Ry ð30°ÞRx ð−60°Þ isused to retrieve the original pattern.
4.

Discussion

In this paper rotational transformation is used to retrieve the original pattern printed on tilted planes. In
some cases, however, the inverse problem could be
more interesting. Figure 11 shows the case of tilting
the image sensor; i.e., the sensor surface is not perpendicular to the normal line of the object surface.
Even in this case, distortion of the captured field
is most likely compensated by the rotational trans-

Fig. 8. (Color online) (a) Close-up photograph of the object and (b)the amplitude image jf 0 ðx; y; −zR Þj numerically reconstructed in theplane parallel to the image sensor.

Fig. 9. Amplitude image of the wave field calculated in the tilted
plane by rotational transformation.

formation. The sequence of the numerical reconstruction is, however, different from the object rotation shown in Fig. 7. Assuming that the wave field
f ðx; y; 0Þ on the sensor surface is captured by some
type of digital holography, the wave field must be rotationally transformed prior to translational propagation. Once the field ^f ð^x; ^y; 0Þ is obtained in aplane parallel to the object surface, the wave field
^f ð^
x; ^
y; −ΔzÞ can be reconstructed by using ordinary
digital holography methods.
Reconstruction of the surface pattern of a slanted
opaque object is adopted for experimental verificationof the rotational transformation formulated in thispaper. However, numerical reconstruction of transparent objects by using rotational transformation
may be interesting for some biological applications.
A conceptual illustration in such a case is shown in
Fig. 12. When the object wave of a specimen withsparse contents or a slight change of refractive index
is captured by digital holography, we can reconstructtomographic images in planes parallel to the sensor
by ordinary techniques. Rotational transformation
makes it possible to reconstruct tomographic images
in any plane in the object space. The reconstruction
plane is no longer limited to the parallel ones. As well,
the computational complexity of the rotational transformation is only a double FFT and an interpolation.
Therefore, computation time is almost the same asthat for reconstruction in a parallel plane.
5. Conclusion

In conclusion, I proved that the rotational transformation formula presented in [5] satisfies the Helmholtz equation and therefore gives an exact spatial
solution of the wave equation. This rotational trans-

Fig. 10. Amplitude images (a) jf 0 ðx; y; −zR Þj in the parallel planeand (b) j ^f ð^x; ^y; 0Þj in the tilted plane reconstructed by using rotational transformation for two-axis rotation. The planar object isrotated at −60° around the x axis prior to rotation at 30° aroundthe y axis.

Fig. 11. Procedure for compensation for tilting a sensor surface
by rotational transformation.

Fig. 12. (Color online) Tomographic imaging of a transparent specimen by rotational transformation. Pn and Sn are planes paralleland nonparallel to the image sensor, respectively.

formation has the advantage of easy implementationand fast computation for numerical calculations because the transformation can be performed by just a
double FFT and an interpolation.
As rotational transformation is useful for wave optics, especially for digital holography, its formulation
and usefulness were verified by demonstrating thatimages in tilted planes can be clearly reconstructedfrom the wave field captured by phase-shifting lensless Fourier digital holography.
The author thanks T. Nakatsuji for his assistancewith the experiments.
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