Optics and Photonics Journal, 2011, 1, 204-215
doi:10.4236/opj.2011.14032 Published Online December 2011 (http://www.SciRP.org/journal/opj)
Information Encryption Based on Using Arbitrary
Two-Step Phase-Shift Interferometry
Chi-Ching Chang1, Wen-Ho Wu2, Min-Tzung Shiu2, Wang-Ta Hsieh2, Je-Chung Wang3, Hon-Fai Yau4
1Institute of Electro-Optical & Energy Engineering, Ming Dao University, Chinese Taipei
2School of Defense Science, National Defense University, Chinese Taipei
3Department of Applied Chemistry and Materials Science, National Defense University, Chinese Taipei
4Department of Optics and Photonics, National Central University, Chinese Taipei
Received August 4, 2011; revised November 8, 2011; accepted November 16, 2011
Abstract
A deterministic phase-encoded encryption system is proposed. A lenticular lens array (LLA) sheet with a
particular LPI (lenticular per inch) number is chosen as a modulator (key) instead of the random phase
modulator. The suggested encryption scheme is based on arbitrary two-step phase-shift interferometry (PSI),
using an unknown phase step. The encryption and decryption principle is based on an LLA in arbitrary un-
known two-step PSI. Right key holograms can be used to theoretically show that the object wavefront is the
only one left in the hologram plane and that all accompanying undesired terms are eliminated. The encrypted
image can therefore be numerically and successfully decrypted with the right key in the image plane. The
number of degrees of freedom of the encryption scheme increases with the distance from the object and the
LLA to the CCD, and also with the unknown phase-step and the LLA LPI number. Computer simulations are
performed to verify the encryption and decryption principles without a key, with the wrong key and with the
right key. Optical experiments are also performed to validate them.
Keywords: Deterministic Phase-Encoded Encryption, Arbitrary Two-Step Phase-Shift Interferometry,
Lenticular Lens Array
1. Introduction
In digital holography (DH), the hologram is digitized and
the object wavefront is numerically reconstructed using a
computer. This idea was proposed by Goodman and Law-
rence [1]. This approach has attracted increasing atten-
tion because a charged-couple device (CCD) camera was
first used to acquire holograms directly with off-axis and
in-line configurations [2,3]. This method, DH, enables
full digital hologram recording and processing without
chemical or physical development, increasing the flexi-
bility and speed of the experimental process.
Since the mid-1990s, DH has found several practical
and successful applications, including deformation ana-
lysis, particle tracking, microscopy, and encrypting in-
formation. One such DH application is the encryption of
information, which is done using various algorithms and
architectures [4-25], such that an unauthorized person
cannot easily access the secret information. The basic
principle of this encryption method is to encode the ob-
ject wavefront, the reference wavefront or a combination
of both wavefronts by positioning phase encoding com-
ponents in the input, Fresnel, or a Fraunhofer domain, or
a combination of both domains. Of the encryption ap-
proaches that have been developed in recent years, the
most often used and highly successful ones are based on
phase-shift interferometry (PSI), making good use of
available CCD resources to capture on-axis encrypted
digital holograms [6-8,12,14].
Since the standard PSI approach depends on a special
and constant phase shift, 2π/N, where the integer N ≥ 3,
this requirement is commonly difficult to meet precisely
in practice. Therefore, Meng et al. [17,20] presented an
algorithm for two-step PSI, with an arbitrary known
phase step δ, for image encryption with double ran-
dom-phase encoding in the Fresnel domain.
Unlike other works [17,20], this work proposes a de-
terministic phase-encoded encryption system that uses an
LLA as a phase modulator (key) [22]. The encryption
scheme is based on the arbitrary two-step PSI [26], but

205 C.-C. CHANG ET AL.
with an arbitrary unknown phase step Δφ. The LLA is
placed in the reference path to modulate and thus encode
the incident plane wave phase, which is reflected by a
mirror. The phase-encoded wavefront is then used as the
reference wave in encrypting the object wavefront.
The unknown phase-step Δφ of a reference wave is
controlled by inserting a glass plate between the mirror
and the LLA. The unknown Δφ can be estimated from
Ref. 26, with the assumption that the phase change due
to object illumination by the object beam is negligible.
This assumption obviously applies to image encryption.
Notably, the amplitude of the reference wave can be
greater or less than that of the object wave in the un-
known two-step PSI.
The encrypted hologram includes the dc term, virtual
image and twin image, because the object wavefront in-
terferes with the phase-encoded reference wave at the
hologram plane. The dc term and twin image (unwanted
terms) can usually be eliminated by applying the princi-
ple proposed in the cited work [26]. However, the virtual
image is then the product of the object wavefront and the
conjugate term of the phase-encoded reference wave. If
the conjugate term of the phase-encoded reference wave
is not accurately decoded from the virtual image. Then
the encrypted image cannot be decrypted and faithfully
recovered. The conjugate of the phase-encoded term can
only be completely cancelled and counteracted by the
phase-encoded term, which can be retrieved from two
right key holograms (with no object) using the algorithm
proposed in the work cited above [26]. Two right key
holograms are therefore required to implement the de-
cryption scheme and the encrypted image is numerically
decrypted and recovered satisfactorily.
The rest of this paper is organized as follows. Section
2 briefly reviews the physical properties of the phase-
encoded LLA component. Section 3 then elucidates the
encryption and decryption principle with an LLA in arbi-
trary unknown two-step PSI. In Section 4 computer
simulations are performed to confirm the encryption and
decryption scheme without a key, with the wrong key
and with the right key. In Section 5 Mach-Zehnder inter-
ferometry is used to implement the encryption system
and to validate the scheme. For comparison, numeric
decryptions of the encrypted information carried by the
object wavefront without a key, with the wrong key and
with the right key, are demonstrated. Conclusions are
finally drawn in Section 6, along with recommendations
for future research.
2. Brief Review of Phase-Encoded
Component—LLA
An LLA sheet is used to regulate the reference beam
phase to perform the encryption scheme phase-encoded
modulations. The LLA is comprised of several cylindri-
cal lenses of equal size and curvature, formed into a
one-dimensional array [27] (Figure 1).
Notably, the phase transformation of LLA can be ex-
pressed as [28]
2
()
(,; ,)exp(1)2R








LLA
ml
tml jkn

 (1)
where m = 0, ±1, ±2, ±3,…, which can be calculated by
taking the integral of the ratio x/l, m = int(x/l). The pa-
rameters k, n and R are the light source wave numbers (k
= 2π/λ), the refraction index and the lens radius of cur-
vature, respectively. The value l is given by l = 1/LPI,
where LPI is the LLA density.
The correlation s(x, y) between two phase functions
tLLA(ξ, η; m, l) and tLLA(ξ, η; m’, l’) [29],.
(,)(,; ,)(,;',')
LLA LLA
s
tmltml

 

 (2)
where the operator ○
× denotes correlation, and can be
evaluated. Various phase-encoded schemes can be easily
shown implemented using different LPI numbers, such
as LPI = 50, 62, 75 and others [30]. Additionally, the
phase code value and length can be varied and realized
using various LPI numbers. Figures 2(a) and 2(b) pre-
sent the autocorrelations of 62 LPI and 50 LPI LLA
phase functions. Figure 2(c) presents the cross-correla-
tion between the two functions. The phase code length
equals 1392. Under the ideality assumption [27] the pa-
rameters of the two LPI LLAs used to evaluate the cor-
relations s(ξ, η) are presented in Table 1 [30].
Table 1. Parameters of 50 and 62 LPI LLA.
50 LPI 62 LPI
n 1.58 1.58
width of lenticule (cm) 0.0508 0.0408
R (cm) 0.0254 0.0204
Figure 1. Side view of a lenticular lens array.

C.-C. CHANG ET AL.
206
(a) (b) (c)
Figure 2. Evaluations of correlation: (a) autocorrelation with 50 LPI LLA phase function; (b) autocorrelation with 62 LPI
LLA phase function; (c) cross-correlation between two phase functions.
Based on Figures 2(a)-(c) the encrypted information,
which is encoded using an LLA with a particular LPI
number, can only be decrypted using that (correct) LPI
number, which is the right key. It cannot be recovered
without a key or with the incorrect LPI number. For ex-
ample, if the encrypted image is encoded using a 62 LPI
LLA, decoding encrypted information without a key or
with a 50 LPI LLA is impossible. The number of degrees
of freedom of the encryption scheme increases with the
LLA LPI number.
3. Principle of Encryption and Decryption
using Arbitrary Unknown Two-Step PSI Figure 3. Schematic of the proposed encryption setup.
terms constitute the dc term, the third term corresponds
to the virtual image and the fourth corresponds to the real
image. Only the third term contains the desired informa-
tion to be decrypted. The other terms, which cause blur-
ring, must be removed and (or) suppressed before nu-
merical reconstruction. The unwanted terms are suppre-
ssed based on the principle in the work cited above [26].
In the proposed encryption scheme (Figure 3) a refer-
ence wave with known amplitude and phase, is modu-
lated using an LLA with a particular LPI. The modulated
reference wave is used as the deterministic phase- en-
coded reference wave. The object wavefront is generated
using a light that passes through or reflects from the object.
Let ψO and ψDPM(x, y; m, l) denote the object wave-
front to be recorded and the phase-encoded reference
wave at the hologram plane, respectively. The ψDPM(x, y;
m, l) can be mathematically expressed from the LLA
modulation and distance z propagation as [31]
22
(, ;,)
exp( )exp() ()d
2








DPM
LLA
xyml
jkz jk
txy
jz z

The dc term can be completely eliminated in typical
method [32], which separately captures the intensities of
the object wave and reference wave at the hologram
plane. The reduced intensity of the first hologram can be
written as
d
 

(3) 2
2
22
*
*
(, ;,)exp()
exp( )
exp( )
ENCO DPM
ODPM
ODPM
ODPM
Ixyml=|ψ+ψj|
=| ψ|+|ψ|
+ψψ j
+ψψ






(5)
where (x, y) denote coordinates in the hologram plane.
The intensity of the first encrypted hologram at the
hologram plane is
An unknown phase-step Δφ of the reference wave is
obtained by inserting a glass plate in the reference path.
The intensity of the second encrypted hologram can then
be expressed as
22 2
**
(, ;,)
ENC1O DPMODPM
ODPM ODPM
Ixyml=|ψψ |=|ψ|+|ψ|
+ψ+ψψ
 (4)
On the right-hand side of Equation (4) the first two

207 C.-C. CHANG ET AL.
2
2
22
*
*
(, ;,)exp()
exp( )
exp( )
ENCO DPM
ODPM
ODPM
ODPM
Ixyml=|ψ+ψj|
=| ψ|+|ψ|
+ψψ j
+ψψ






(6)
Similarly, when the dc term is completely suppressed,
the reduced intensity of the second hologram is given by
'*
2
*
(,;, )exp()
exp( )
ENCO DPM
ODPM
Ixyml=ψψj
+ψψ



 (7)
From the simple argument in the cited work [26], the
unknown phase-step can be easily estimated with limited
pixels near the optical axis (in discrete form) as
'(,)
cos[( , )]
'(,)
ENC2
ENC1
Ipq pq
Ipq
  (8)
where p = 1,…, M and q = 1,…, N and M × N is the
number of light-sensitive pixels on the CCD, respec-
tively. Alternatively, Δφ can be estimated using the Fou-
rier transform ratio of their reduced intensities with the
same approximations,


1
'cos( )
'
ENC2
ENC
I
I


 (9)
Let Equation (7) be multiplied by the term exp(–jΔφ)
and then subtracted from Equation (5). The following
equation is obtained.


*
*
'exp() '1exp(2)
(,)(,; ,)
ENC1 ENC2
ODPM
ODPM
IjI j
pq pqml



 

(10)
since Δφ can be regarded as fixed in the recording stage
in the arbitrary two phase-step approach, the twin-image
is satisfactorily and completely eliminated by the nu-
merical operations performed using the reduced intensi-
ties of the two holograms and the estimated value of Δφ.
However, the right-hand side of Equation (10) is now
seen to be composed of the object wavefront ψO and the
conjugate term of ψDPM(p, q; m, l), which is abbreviated
as ψDPM(m, l) hereafter. If the conjugate term of ψDPM(m,
l) cannot be determined by decoding the virtual image
and thus separated from ψO, the encrypted image will
then not to be decrypted or faithfully recovered from the
encrypted holograms. To accurately decode the conju-
gate term of ψDPM(m, l) from Equation (10), two addi-
tional right key holograms must be recorded using the
right key, which can be easily and mathematically con-
firmed by the arguments below. For comparison, the
encrypted information about the object wavefront will be
numerically decrypted without a key, with a wrong key
and with the right key.
3.1. Decryption without Key
For decryption with no key, the abbreviation
*
(,; ,)(,)(,)
ENCh ODPM
Upqmlpq ml   (11)
is applied in the hologram plane. Then, let UENC be di-
rectly reconstructed in the image plane using the convo-
lution approach [33]
 


*
(,)(,)( )
-1
ODPMi ENC
pqml=Uhz d   (12)
where h (z = d) represents the impulse response
through a distance z = d. As expected, the left-hand side
of Equation (12) still contains the product of ψO and the
conjugate term of ψDPM(m, l), so the encrypted image is
not successfully decrypted in this case.
3.2. Decryption with Wrong Key - Incorrect LPI
LLA (m’≠m and l’≠l)
Let two wrong key (incorrect LPI LLA) holograms (with
no object), IKEY1(m’, l’) and IKEY2(m’, l’), be captured by the
arbitrary two-step PSI method. According to similar nu-
merical procedures derived from Equations (10)-(12), the
phase-encoded pattern term ψDPM(m’, l’ ) and reconstructed
object wavefront can be straightforward written down
( ',')'( ',')exp()'(',')
(',')
KEYh KEY1KEY2
DPM
UmlImlj Iml
ml



(13a)


 


 

*
,; ,,','
(,) ,','

Oi
-1
ODPM DPM
ψpqmlm l
=pqmlm
hz d
  l
(14a)
Based on the right-hand side of Equation (14a), if the
incorrect LPI LLA (wrong key) is used, the conjugate
term of ψDPM(m, l) will not be fully separated from the
object wavefront term. Since the conjugate term cannot
be completely cancelled by ψDPM(m’, l’), the encrypted
image cannot be recovered from the encrypted holo-
grams with the aid of a wrong key.
3.3. Decryption with Right Key - Correct LPI
LLA (m’=m and l’=l)
Let two wrong key (incorrect LPI LLA) holograms (with
no object), IKEY1(m’, l’) and IKEY2(m’, l’), be captured by
the arbitrary two-step PSI method. According to similar
numerical procedures derived from Equations (10)-(12),
the phase-encoded pattern term ψDPM(m’, l’) and recon-
structed object wavefront can be straightforward written
down

C.-C. CHANG ET AL.
208
(,)'(,)exp()'(,)
(,)
KEYh KEY1KEY2
DPM
UmlImlj Iml
ml



(13b)
  





 
*
,,,

-1
OiODPM DPM
ψpq =pqmlml
hz d
  ,
(14b)
and with an arbitrary phase-step (Δφ = 1.30 radians),
respectively. After the dc term is eliminated from the
holograms, the Δφ values can then be estimated using
Equation (8) or (9) with the pixel array (515:524, 691:700)
close to the optical axis. The optimal value of Δφ is
1.304 radians (estimated using Equation (9) as presented
in Table 2.
From Equation (14b), the conjugate term of ψDPM(m, l)
is completely cancelled and counteracted by ψDPM(m, l),
and is therefore fully separated from the object wave-
front term in the hologram plane. Clearly, if the correct
LPI LLA is used, then theoretically only the recon-
structed object wavefront term will be left in the image
plane and the accompanying undesired term, which is the
conjugate term of ψDPM(m, l), is eliminated in the holo-
gram plane. Therefore, the encrypted image in this sug-
gested encryption system can be numerically and suc-
cessfully decrypted using arbitrary two-step PSI and
right key holograms.
Figure 4(d) presents the decrypted magnitude-contrast
image, which was directly reconstructed using Equation
(12) without a key. As expected, a comparison with the
original image does not easily reveal the hidden informa-
tion, the two Chinese characters, from the decrypted im-
age. Therefore, without further decoding the phase-en-
coded pattern of the reference wave using the right key,
the encrypted information cannot be decrypted.
Figures 4(e) and (f) present the two wrong key holo-
grams (with no object) simulated using the arbitrary
two-step (0 and 1.30 radians) PSI. The phase-encoded
pattern term ψDPM(m’, l’) is obtained numerically using
Equation (13a). Since the conjugate term of ψDPM(m, l)
cannot be completely cancelled by ψDPM(m’, l’), it is not
fully separated from the object wavefront term in the
hologram plane. Figure 4(g) presents the decrypted im-
age numerically reconstructed using Equation (14a) and
the wrong key. The hidden information can still not be
recovered from the encrypted holograms using the wrong
key, as shown in Figure 4(g).
4. Computer Simulations
Simulations are performed to verify the proposed ap-
proach. In the simulation procedure, the ratio of the in-
tensity of the object wave to that of the reference wave is
assumed to be 1:1. Figure 4(a) presents the original in-
put image of two Chinese characters “Chung Cheng”
(1392 × 1040 pixels).The 62 LPI and 50 LPI LLA are
used as the right key and the wrong key, respectively.
The two LLAs are placed at a distance d = 5 cm from the
hologram plane. The distance between the object plane
and the hologram plane is 20 cm and the illumination
wavelength λ = 0.6328 μm. The arbitrary phase-step Δφ
is set to 1.30 radians.
Figures 4(h) and 4(i) present the two right key holo-
grams simulated using the arbitrary two-step (0 and
1.30radians) PSI. The phase-encoded pattern term ψDPM-
(m, l) is obtained numerically using Equation (13b). The
conjugate term of ψDPM(m, l) is completely cancelled by
ψDPM(m, l), and now fully separated from the object
wavefront term in the hologram plane. Figure 4(j) pre-
sents the decrypted magnitude-contrast image that was
numerically reconstructed from Equation (14b) using the
Figures 4(b) and (c) present the two simulated en-
crypted holograms without a phase-step (Δφ = 0 radians)
(a) (b)

209 C.-C. CHANG ET AL.
(c) (d)
(e) (f)
(g)

C.-C. CHANG ET AL.
210
(h) (i)
(j)
Figure 4. Simulation results: (a) original input image; (b) first encrypted hologram without phase-step; (c) second encrypted
hologram with phase-step 1.30 radians; (d) decrypted image without key; (e) first wrong key hologram without phase-step; (f)
second wrong key hologram with phase-step 1.30 radians; (g) decrypted image with wrong key; (h) first right key hologram
without phase-step; (i) second right key hologram with phase-step 1.30 radians; (j) decrypted image with right key.
Table 2. Simulations of estimated phase-step value Δφ and
mean square error and SNR of the decrypted images.
right key. Unlike in Figures 4(d) and (g), the encrypted
USAF resolution chart information is clearly identifiable
in the center of Figure 4(j), although some artificial
blurring is evident. Notably, as determined by compari-
son with the original image in Figure 4(a), the least square
error and SNR (in dB) [34] for Figure 4(j) are around
0.0119 and 19.25 dB, respectively, as indicated in Table 2.
Arbitrary two-step PSI
Default   (radians) 1.30
Estimated   (radians) 1.304
Error (%) 0.3%
Mean square error (magnitude-contrast) 0.012
SNR (dB) (magnitude-contrast) 19.25
5. Experimental Setup and Results
Mach-Zehnder interferometry (Figure 5) was used to
realize the encryption system. The light source was a
He-Ne laser with a power of 20 mW and a wavelength of

211 C.-C. CHANG ET AL.
Figure 5. Schematic diagram of the optical encryption setup
where SF: spatial filter, PBS: polarized beam splitter, BS:
beam splitter.
λ = 632.8 nm. Two λ/2 retarding wave plates and a po-
larized beam splitter (PBS) were used together to adjust
the object wave intensity ratio to that of the reference
wave. Both waves were collimated into a plane wave us-
ing spatial filters (SFs), apertures and lenses. Two LLAs,
one 62 LPI and the other 50 LPI, were used as the right
key and wrong key, respectively. Two LLAs were used to
modulate the incoming plane wave, which was reflected
by a mirror. A light-sensitive sensor (Pixera-150SS CCD
camera; 1040×1392 pixels, wsy = 0.484 cm and wsx=0.650
cm) was used to acquire the holograms.
The phase of the reference wave in the reference path
was first left unchanged during the first encrypted holo-
gram recording. An arbitrary phase-step of the reference
wave was then applied by inserting a glass plate into the
reference path before the second hologram was captured.
Two Chinese characters (Figure 6) were encrypted.
The distance between the object and CCD was d = 21.5
cm. The modulators (LLAs) were placed 7.6 cm from the
CCD. After the object was removed, the wrong key and
the right key holograms were obtained by recording the
fringe patterns of the modulated reference wave and
plane wave using the arbitrary two-step PSI technique.
Figures 7(a) and (b) present the two encrypted holo-
grams without a phase-step and with an arbitrary phase-
step, respectively, captured using a CCD in the hologram
plane. After the dc term was eliminated from the two
encrypted holograms, Δφ can be estimated using Equa-
tion (8) or (9) for the pixel array (515:524, 691:700)
close to the optical axis. The optimal Δφ value was esti-
mated from Equation (9) to be around 0.1380 radians.
For comparison the encrypted information about the
object wavefront was numerically decrypted without a
key, with a wrong key and with the right key. Figure 7(c)
presents the decrypted magnitude-contrast image that
directly reconstructed using Equation (12) without a key.
Figure 6. Original optical image to be encrypted.
di icthe
d a deterministic phase-encoding en-
ff ult to identify. Clearly without separating
phase-encoded reference wave pattern from the object
wavefront in the hologram plane, the encrypted informa-
tion in the image plane cannot be decrypted.
Figures 7(d) and (e) present the two wrong key (50
LPI LLA) holograms (with no object) without and with
the arbitrary phase-step, respectively. The phase-encoded
pattern term ψDPM(m’, l’) is then obtained numerically
using Equation (13a) but the effect of the conjugate term
of ψDPM(m, l) cannot be completely cancelled and
thereby fully separated from object wavefront term. Fig-
ure 7(f) presents the decrypted magnitude-contrast im-
age that was numerically reconstructed using Equation
(14a) and a wrong key. Therefore, the hidden informa-
tion cannot be recovered from the encrypted holograms
with the aid of the wrong key.
Figures 7(g) and (h) present the two right key (62 LPI
LLA) holograms without and with the arbitrary phase-
step, respectively. Similarly, the phase-encoded pattern
term ψDPM(m, l) is obtained using numerical operations
from Equatioin (13b). Figure 7(i) presents the decrypted
magnitude-contrast image that was numerically recon-
structed using Equation (14b) using the right key. Unlike
in Figure 7(c) and (f), Figure 7(i) clearly shows the en-
crypted “Chung Cheng” information. As expected, the
conjugate term of ψDPM(m, l) is cancelled by ψDPM(m, l)
and separated from the object wavefront term in the
hologram plane. Clearly, when the right key is used, the
undesired term, which is, the conjugate term of ψDPM(m,
l), is eliminated in the hologram plane and only the re-
constructed object wavefront term remains in the image
plane. The hidden information is numerically decrypted
using arbitrary two-step PSI with right key holograms
6. Conclusions
This work presente
cryption system that uses a lenticular lens array (LLA) as
a phase modulator. The encryption approach was imple
In Figure 7(c) the hidden information is invisible and

C.-C. CHANG ET AL.
212
(a) (b)
(c)
(d) (e)

213 C.-C. CHANG ET AL.
(f)
(g) (h)
(i)
Figure 7. Experimental results: (a) first encrypted hologram without phase-step; (b) second encrypted hologram with
phase-step; (c) decrypted image without key; (d) first wrong key hologram without phase-step; (e) second wrong key
hologram with phase-step; (f) decrypted image with wrong key; (g) first right key hologram without phase-step; (h) second
right key hologram with phase-step; (i) decrypted image with right key.

C.-C. CHANG ET AL.
214
mented using arbitrary two-step PSI with an unknown
phase-step. The encrypted information that was encoded
using some LPI number LLA was theoretically deter-
mined to be decryptable only using the correct LPI
number or the right key and cannot be recovered without
a key or with an incorrect LPI number. The number of
degrees of freedom in the encryption scheme increase
with the distances between the object and LLA from the
CCD, and also an unknown phase-step and the LPI num-
ber of the LLA. Therefore, the confidential information is
not easily accessible by an unauthorized person.
7. Acknowledgements
The authors would like to thank the National Science
Council of the Republic of China, Taiwan, for financially
ting thsupporis research under Contract No. NSC
100-2221-E-451-007.
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