Jpn J Ophthalmol 2007;51:258–264© Japanese Ophthalmological Society 2007

DOI 10.1007/s10384-007-0440-7

LABORATORY INVESTIGATION

Optical Quality of the Eye Degraded byTime-Varying Wavefront Aberrations with
Tear Film Dynamics
Yoko Hirohara1,3, Toshifumi Mihashi1,3, Shizuka Koh2, Sayuri Ninomiya1,
Naoyuki Maeda2, and Takashi Fujikado1
1

Department of Applied Visual Science, Osaka University Graduate School of Medicine, Osaka,
Japan; 2 Department of Ophthalmology, Osaka University Graduate School of Medicine, Osaka,
Japan; 3 Research Institute, Topcon Corporation, Tokyo, Japan

Abstract
Purpose: Wavefront aberrations (WFAs) of the eye vary with time because of the tear film dynamics.
We investigated, using a simulation method, the variation of optical quality with time-varying wavefront
measurements of 13 eyes with different refractions.
Methods: WFAs of 13 normal eyes of 13 subjects were measured every second for 10 s. First, we simulated
WFAs with conventional corneal laser refractive surgery by subtracting the second-order aberrations of
the least aberrated measurement from measured consecutive WFAs. Second, we simulated customized
refractive surgery by subtracting the second- to sixth-order aberrations of the least aberrated measurement from measured consecutive WFAs. We calculated Strehl ratios and retinal images from these corrected consecutive WFAs.
Results: In one eye, the root mean square (RMS) values of WFAs with a second-order correction weresometimes smaller than those of WFAs with a second- to sixth-order correction, when these were compared at the same time point after a blink. However, in the other 12 eyes, the RMS values with second- tosixth-order corrections were smaller than those with only a second-order correction. In eight eyes, theStrehl ratios with second- to sixth-order corrections were larger than those with second-order corrections. In the remaining five eyes, Strehl ratios with second- to sixth-order corrections were sometimes
smaller than those with second-order corrections.
Conclusions: In a simulation, the correction of time-invariant higher order aberrations usually reduced
RMS values, but it did not always result in higher Strehl ratios than those obtained with only secondorder corrections. Jpn J Ophthalmol 2007;51:258–264 © Japanese Ophthalmological Society 2007
Key Words: aberration, retinal image, Shack-Hartmann aberrometer, Strehl ratio, tear film

Introduction
Studies of wavefront aberrations (WFAs) of human eyes
have clearly revealed that the optical quality of the eyes isdegraded by monochromatic aberrations.1,2 This degradation is caused not only by static phenomena but also by
Received: July 18, 2006 / Accepted: March 1, 2007
Correspondence and reprint requests to: Takashi Fujikado, Department of Applied Visual Science, Osaka University Graduate School of
Medicine, Room G4, 2-2 Yamadaoka, Suita, Osaka 565-0871, Japane-mail: fujikado@ophthal.med.osaka-u.ac.jp

ocular dynamics.3,4 The typical ocular dynamics involved indegrading optical quality are tear film dynamics,5–7 ocular
accommodation,8,9 and cardiovascular activity.10
Tear film dynamics have been studied by consecutive
measurements of the cornea with a Placido’s disk videokeratoscope.11–14 Nemeth et al.11 measured changes in thesurface regularity index, the surface asymmetry index, andcorneal power. Goto et al.12 and Kojima et al.13 calculatedchanges in the break-up area and break-up time from
Placido’s disk images. Montes-Mico et al.14 evaluatedcorneal WFAs calculated from Placido’s disk images for dryeye patients. Recently, post-blink aberration changes were

Y. HIROHARA ET AL.
OPTICAL QUALITY WITH WAVEFRONT ABERRATIONS

259

investigated by measuring point-spread functions (PSF)5and ocular WFAs.6,7 Ocular WFAs increase several seconds
after a blink.5–7 Tutt et al.15 studied optical quality related
to tear film dynamics.
To investigate the changes in optical quality with tearfilm dynamics, we measured post-blink WFAs consecutively. We then employed simple computational models, as
described below, to simulate WFAs for eyes that had undergone conventional corneal laser refractive surgery or refractive correction with spectacles, as well as for eyes that had
undergone customized corneal laser refractive surgery or
refractive correction with customized contact lenses. We
simulated time variations of the PSF, Strehl ratio, and theretinal images of Landolt’s rings from these computed consecutive WFAs.

Methods

RMS of WFAs (µm)

The WFAs were measured in 13 normal eyes of 13 subjects,
average age, 29.5 ± 4.2 years. The average ± SD of sphericalerror for the 13 eyes was −2.50 ± 3.28, and the average ±SD of break-up time was 7.31 ± 1.80. Written informed
consent was obtained from each subject. The procedures
were performed to conform to the tenets of the Declarationof Helsinki. WFAs were measured using a prototype ShackHartmann wavefront aberrometer (Topcon, Tokyo, Japan)
once every second for 10 s, immediately after blinking.
Lenslet size was 0.22 mm2 on the pupil plane, and the focal
length was 5 mm. WFAs were analyzed using normalized
Zernike polynomials defined over a 4-mm-diameter area inthe center of the pupil. The coefficients up to the sixth orderwere calculated.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

We simulated two types of refractive corrections: secondorder correction and second- to sixth-order correction.
Figure 1 shows the method of these corrections.
WFAs are expressed by the following equation:
6iW( x, y, t ) = ∑ ∑ ci− i + 2 j (t ) zi− i + 2 j ( x, y),(1)

i=2 j =0

where W(x, y, t) is a WFA, x and y are horizontal and vertical coordinates on the pupil, t is the time after blink, ci(t) is
the Zernike coefficient, and zi(x, y) is the Zernike polynomial. The tilt terms were omitted because they did not affectthe visual effect. The least aberrated wavefront measurement that had the smallest root mean square (RMS) valueof WFAs for the measurements in each trial was chosen forthe simulation. This minimal wavefront was chosen to represent the time-invariant, potentially correctable component of the total aberration. Depending upon the typeof correction, either only the second-order terms or all of
the terms of this wavefront were subtracted from all of
the measured wavefronts in the time series to simulate thecorrection.
The WFAs that had the smallest RMS value of measured
WFAs in each trial are described by
6iW( x, y, tmin ) = ∑ ∑ ci− i + 2 j (tmin ) zi− i + 2 j ( x, y),(2)i=2 j =0

where tmin is the time after blink when the RMS value of the
WFA becomes the smallest.
In the second-order correction, retinal images were
simulated by using WFAs for which only the secondorder aberrations were corrected. The second-order corrected WFA, W2nd_co(x, y, t), is expressed by the following
equation:

The least WFA

1.11

1

2.1 1

2

3.1 1

3

4.1 1

4

5.1 1

5

6.1 1

6

7.0 8

7

8.08

8

9.0 8

9

times after blinking [s]simulation of
second-ordercorrection
Subtract only second-ordercoefficients of the least WFAs
from the WFA coefficients ofeach time point.
Simulation

simulation of
second- to sixthorder correction
Subtract the second- to sixthorder coefficients of the least
WFAs from the WFA
coefficients of each time point.
Simulation

Figure 1. Methods of secondorder correction and second- tosixth-order correction. WFA,
wavefront aberrations.

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Jpn J Ophthalmol
Vol 51: 258–264, 2007i

6

W2 nd_ co ( x, y, t ) = ∑ ∑ ci− i + 2 j (t ) zi− i + 2 j ( x, y)i = 2 j =02

− ∑ c2−2 + 2 j (tmin ) z2−2 + 2 j ( x, y)2

j =0= ∑ {c2−2 + 2 j (t ) − c2−2 + 2 j (tmin )} z2−2 + 2 j( x, y)
j =0
6

i

+ ∑ ∑ ci− i + 2 j (t ) zi− i + 2 j ( x, y) .

(3)

i = 3 j =0

The second- to sixth-order correction was simulated bysubtracting the second- to sixth-order aberrations of theW(x, y, tmin) from each WFAs. The second- to sixth-ordercorrected WFA W2nd_6th_co(x, y, t) is expressed by the following equation:
6i

W2 nd_6 th_co ( x, y, t ) = ∑ ∑ ci− i + 2 j (t ) zi− i + 2 j ( x, y)i=2 j =06i− ∑ ∑ ci− i + 2 j (tmin ) zi− i + 2 j ( x, y)i=2 j =06i= ∑ ∑ {ci− i + 2 j (t ) − ci− i + 2 j (tmin )}zi− i + 2 j ( x, y).
i=2 j =0(4)
It is clear that W2nd_6th_co(x, y, t) becomes zero when t = tmin.
The series of residual RMS values for each WFA, PSF,
Strehl ratio, and retinal image of the Landolt’s rings after
second-order correction and after second- to sixth-ordercorrection were simulated.
The PSF was calculated from the pupil function by a fast
Fourier transform (FFT). When we constructed the pupil
function, we used the second-order corrected WFA andsecond- to sixth-order corrected WFA. Then the opticaltransfer function (OTF) was calculated from the PSF byagain using FFT. The Strehl ratio was calculated in theprocess of calculating the OTF.16 We compared the Strehl
ratio and the RMS of measured WFAs.

Results
The results for two eyes of two subjects, HT (24 years old)and YH (29 years old), are shown in Figs. 2 and 3. The RMS
of measured WFAs became a minimum at 2 s after blinking
for HT and 4 s after blinking for YH.
The RMS values of the W2nd_co ranged from 0.065 to
0.337 µm for HT and from 0.068 to 0.266 µm for YH. RMS
values of the W2nd_6th_co ranged from 0 to 0.204 µm for HT
and from 0 to 0.153 µm for YH.
The minimum Strehl ratios calculated from W2nd_co andW2nd_6th_co were 0.066 and 0.102, respectively, for HT. The
Strehl ratio calculated from W2nd_6th_co was always larger
than that calculated from W2nd_co. With the minimum Strehl
ratio, simulated Landolt’s rings of W2nd_6th_co were clearer
than those of W2nd_co for HT.
The minimum Strehl ratios calculated from W2nd_co andfrom W2nd_6th_co were 0.129 and 0.105 for YH. The Strehl

ratios calculated from W2nd_6th_co were sometimes smaller
than those calculated from W2nd_co. When the Strehl ratio
calculated from W2nd_6th_co was smaller than that calculated
from W2nd_co, the Landolt’s rings calculated from W2nd_6th_cowere more blurred than those calculated from W2nd_co. It wasmore difficult to recognize the gap in the simulated 20/20
Landolt’s rings of W2nd_6th_co than the gap in the rings ofW2nd_co.
Figure 4 shows graphs of the RMS values of W2nd_co andof W2nd_6th_co in the 13 normal eyes of 13 subjects. For 12 of
the 13 eyes, the RMS values of the W2nd_co were larger thanthose of the W2nd_6th_co in one series of measurements. Oneof the 13 eyes had RMS values of W2nd_co that were occasionally smaller that those of W2nd_6th_co (Fig. 4, arrow).
Figure 5 shows graphs of Strehl ratios calculated from
W2nd_co and W2nd_6th_co of the 13 normal eyes of the 13 subjects. For five of those 13 eyes, the Strehl ratios with theW2nd_co were larger than those with the W2nd_6th_co in one
series of measurements (Fig. 5, arrows). Three of these fiveeyes had this inverse relationship just after blinking. One of
the eyes had the inverse relationship between 3 and 7 s after
blinking. The other eye had the inverse relationship 8 s after
blinking, which was the last measurement of the timecourse.
The Strehl ratio and RMS of W2nd_6th_co were significantly
correlated for all 13 eyes (Pearson’s correlation coefficient,
P < 0.01), and the Strehl ratio and RMS of W2nd_co werecorrelated for 10 of the 13 eyes (Pearson’s correlation coefficient, P < 0.05).

Discussion
The average RMS values of the time-varying W2nd_6th_co fora 4-mm pupil was 0.204 µm for HT and 0.153 µm for YH.
Considering that the average RMS for the normal population is 0.08 µm,17 these residual aberrations are large.
In most cases, the RMS values of the W2nd_6th_co were
smaller than those of the W2nd_co. However, one of the
13 eyes had RMS values of the W2nd_co that were occasionally smaller that those of the W2nd_6th_co. This could happen
when the higher order aberrations varied greatly fromthose for the minimum aberration that we used for thecorrection.
The Strehl ratio of W2nd_co was once larger than that ofW2nd_6th_co for YH. This did not happen for HT. Five of the
13 eyes we measured had the same results. The Strehl ratio
I is expressed as2

1 1 2 π i 2λπ Weρdρdθ
π 2 ∫0 ∫022

1 1 2π 2π1  2π = 2 ∫ ∫ 1 + i W + i W + ρdρdθ ,182 λ 
π 0 0 
λ


I=

(5)

where W is the wavefront aberration, ρ and θ are the radius
and azimuth in the polar coordinate system. If we assume
that the wavefront aberrations are so small that we may

Y. HIROHARA ET AL.
OPTICAL QUALITY WITH WAVEFRONT ABERRATIONS

times after
blinking (s)Measurement to subtract
1
2
3

261

4

5

6

7

8

residual
WFAs

1[µm]
0

WFAs RMS

0.081

0.065

0.103

0.109

0.132

0.223

0.157

0.337

0.546

0.626

0.448

0.388

0.320

0.088

0.167

0.066

-1

PSF
Strehlratio
Landolt’s
ring

20/40
20/20

(a)
1[µm]

residual
WFAs

0

WFAs RMS

0.037

0.000

0.062

0.063

0.083

0.138

0.122

0.204

0.861

1.000

0.741

0.647

0.499

0.119

0.290

0.102

-1

PSF
Strehlratio
20/40
Landolt’s
ring
20/20

(b)
Figure 2a, b. Simulation results for subject HT. a Second-order correction. b Second- to sixth-order correction. The first row shows time after
blinking (s), and the following row depicts color-coded maps of residual WFAs. RMS, root mean square; PSF, point-spread function. Strehl
ratio. Simulated Landolt’s ring images of targets for visual acuity of 20/40 and 20/20. The blue numbers in the yellow ground are maximum
Strehl ratios, and the red numbers in the light blue ground are minimum Strehl ratios in the series. At top of figure, the arrow indicates themeasurement to subtract.

neglect the third- and higher orders of wavefront aberrations in (5), then I is expressed as22π
I ≈ 1 −   W 2 − (W )  .
 λ 
2(6)

The second term of the right-hand side of the equation
represents the RMS. The larger the RMS is, the smaller the
Strehl ratio is. However, this is not true when the higher
order terms of equation (5) are included in the calculationof the Strehl ratio. Applegate et al.19 reported that the
impact on the retinal image of different aberration terms
differs; some combinations of aberration terms evenincrease visual performance, although those combinations
also increase the RMS error. Oshika et al.20 showed that theaberrated eye can show apparent accommodation. Chenget al.21 reported that spherical aberration, coma, and secondary astigmatism all reduce visual acuity and increase
depth of focus. Both reported that some higher order aber-rations combined with specific low-order aberrations canimprove, instead of reduce, image quality. We therefore
cannot determine all of the effects of aberrations on visualfunction from the RMS alone.
This 10-s measurement duration is longer than the usual
blink interval, which is around 4 to 5 s.22 This longer duration could increase aberrations and degrade images becauseof tear film break-up. Nine of the 13 eyes had smaller Strehl
ratios at the last measurement than at the first measurement(on average for 4 s just after blinking). The tear film conditions in the other eyes were not considered to be influenced
by the longer duration.
Three eyes had smaller Strehl ratios with the W2nd_6th_cothan with the W2nd_co just after blinking. The Strehl ratios
were the smallest or the second smallest toward the end of
the 10-s interval. The small Strehl ratio just after blinking
was most likely due to the inhomogeneous tear film causedby the blink. Neuronal visual function is suppressed just262

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Vol 51: 258–264, 2007

1

times after
blinking (s)2

Measurement to subtract
3
4
5

6

7

8

9

1[µm]

residual
WFAs

0

WFAs RMS 0.257

0.163

0.084

0.068

0.093

0.094

0.145

0.154

0.266

0.288

0.579

0.610

0.524

0.486

0.311

0.293

0.149

-1

PSF
0.129

Strehlratio
20/40
Landolt’s
ring
20/20

(a)
1[µm]

residual
WFAs

0

WFAs RMS 0.153

0.090

0.040

0.000

0.050

0.049

0.081

0.101

0.148

0.379

0.834

1.000

0.764

0.759

0.462

0.322

0.148

-1

PSF
Strehl
ratio

0.105

20/40
Landolt’s
ring
20/20

(b)

Figure 3a, b. Simulation results for subject YH. a Second-order correction. b Second- to sixth-order correction. See legend of Fig. 2 for otherdetails.

Subject 1

2

3

4

5

6

7

8

9

10

11

( HT )( YH )

12

13

RMS value (µm)

0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

1 3 5 7 91 3 5 7 91 3 5 7 9 1 3 5 7

1 3 5 7 9 1 3 5 7 91 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 1 3 5 7 9Times after blinking (s)Figure 4. Graphs of RMS values after second-order correction and second- to sixth-order correction in 13 normal eyes of 13 subjects. —䊏— simulation of second-order correction. —䊊— simulation of second- to sixth-order correction.

Y. HIROHARA ET AL.
OPTICAL QUALITY WITH WAVEFRONT ABERRATIONS

Subject 1

2

3

4

5

6

263

7

8

9

10

11

( HT )( YH )

12

13

1
0.9

Strehl ratio

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1 3 5 7 91 3 5 7 91 3 5 7 9 1 3 5 7

1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7

1 3 5 7 9

Times after blinking (s)Figure 5. Graphs of Strehl ratios after second-order correction and second- to sixth-order correction of 13 normal eyes of 13 subjects. —䊏— simulation of second-order correction. —䊊— simulation of second- to sixth-order correction.

after blinking,23 and this suppression may help to preventthe transfer of bad retinal images to the brain.
In this research, we found that RMS error was always
more reduced after correction of the second- to sixth-orderaberrations than after correction of only second-order aberrations, but image quality was sometimes not improved
more by the former than by the latter. When customized
corneal refractive surgery is performed, it should be taken
into account that the aberrations of the eye fluctuate. It may
be preferable to measure aberrations several times andcorrect only those aberrations that are significantly larger
than the fluctuations in measurements.
In the simulation, we assumed that aberrations of the eye
can be corrected perfectly. However, it has been reported
that higher order aberrations after customized refractive
surgery are almost the same or increased.24–27 Therefore, thesimulations differed from the results of recent customized
refractive surgery. We are not certain that it will be possible
to develop a perfect corneal customized refractive surgery
to correct all aberrations in the future.
Thai et al.28 measured contrast sensitivity before precontact lens tear break-up by a stimulus presented 2 s after
the blink. Measurements of visual performance and tearbreak-up using a tear scope were made simultaneously inthis experiment. It was found that contrast sensitivity is
reduced after the pre-lens tear film break-up, and thiseffect is increased with higher spatial frequencies.28 Ishida
et al.29 measured monocular recognition acuity continuously during a 30-s blink-free period using continuous functional visual acuity measurement (FVAM), and they foundthat the mean FVAM results sometimes decreased during
testing in both dry eye patients and control subjects.29
Because we found that Strehl ratios and retinal images
fluctuated from tear film conditions in this study, we areplanning to compare these data with subjective data in the
future.

Acknowledgment. The authors thank Prof. Howard Howland (Cornell
University) for his helpful comments on the manuscript.

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