Focal shift in focused truncated pulsed-laser beam

Focal shift in focused truncated pulsed-laser beam

Ying ZhongState Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China

[email protected]

Received 11 April 2007; revised 26 June 2007; accepted 12 July 2007;posted 13 July 2007 (Doc. ID 82007); published 31 August 2007

The focal shift of a focused truncated pulsed-laser beam is investigated. In the case of the Fresnel approx-imation, the analytic expression of the time-averaged intensity distribution along the axis is derived basedon the series expansion. It shows that the focal shift of the pulsed beam can be completely determined bya series of normalized spectrum moments and the central Fresnel number defined according to the centralfrequency of the pulse. The absolute value of the focal shift of the pulsed beam decreases monotonously andslowly with the normalized spectrum width increasing and the central Fresnel number fixed, and itincreases monotonously with the central Fresnel number decreasing and the normalized spectrum widthfixed. Besides the central Fresnel number and the normalized spectrum width, the shape of spectralintensity of the pulse affects the focal shift too. © 2007 Optical Society of America

OCIS codes: 320.7090, 320.5550, 220.2560.

1. Introduction

For the focused monochromatic laser beam, the ac-tual focal point (defined as the point with the highestintensity along axis) predicted by diffraction theoriesis always closer to the lens than the focal point givenby geometrical optics [1]. Such a focal shift not onlyattracts theoretical interest but also has great impor-tance, because the precise location of the actual focalpoint of the focused laser beam is always required inits applications. The focal shift of the monochromaticlaser beam has been studied before, including theinfluencing factors of focal shift [2–4], focal shift ofvarious kinds of beams [5,6], as well as focal shiftpredicted by different theories [7,8].

The focused pulsed-laser beam is utilized for appli-cations in micromachining and ultrafast process de-tection, because of its ultrahigh instantaneous powerand ultrashort time of duration [9]. Such an ultra-short pulse in time domain leads to a broad spectrum,so its focusing property is quite different from that ofthe monochromatic beam [10–15]. In this paper, thefocal shift of the truncated pulsed-laser beam is stud-ied. With the monochromatic beam, the focal shiftis mainly determined by the Fresnel number andslightly affected by the numerical aperture [3], so the

discussion in this paper is restricted to small numer-ical aperture so that the Fresnel paraxial approxima-tion is valid. Further discussions considering theeffect of numerical aperture and other factors may bediscussed in future work.

In Section 2, the analytic expression of the time-averaged intensity distribution along the axis is ob-tained by series expansion. In Section 3, the focalshifts of truncated pulsed-laser beams with Gaussianspectral intensity and rectangular spectral intensityare discussed as two examples. In Section 4, conclu-sions are summarized.

2. Analytic Expression of the Time-Averaged IntensityDistribution along the Axis near the Focus

Suppose the Gaussian laser beam has been truncatedby the pupil and has become the uniform beam. Asshown in Fig. 1, according to the Fresnel diffractiontheory with paraxial approximation and rotationalsymmetry, for each monochromatic component of thefocused pulsed-laser beam, the complex field alongthe axis near the focus can be expressed as

U��, z� � �1

1 � z�f exp�i 2��

c �n� � z��exp�i

�R2�

c �1z �

1f �� 1, (1)0003-6935/07/256454-06$15.00/0

© 2007 Optical Society of America

6454 APPLIED OPTICS � Vol. 46, No. 25 � 1 September 2007

where each monochromatic component is assumedto be a homogenous planar wave at the lens aper-ture, � is the frequency of the monochromatic com-ponent, c is the velocity of light in vacuum, n, R, and� are the refractive index, aperture radius, andcentral thickness of the lens, respectively, z is theaxial coordinate with the back surface center O ofthe lens acting as the original point, and the geo-metrical focus F is located at the point z � f. Sincethe duration of the femtosecond laser pulse is muchshorter than the response time of the photo detector,the time-averaged intensity is detected [16,17], whichis given by

I�z� �1T

��

�

U��, z�2S��d�, (2)

where T is the pulse duration and S(�) is the spectralintensity of the laser pulse. For example, the Gauss-ian spectral intensity has

S�� exp�� 0�2

��2 �, (3)

where �0 is the central frequency and �� is the spec-trum width. For the rectangular spectral intensity,

S�� rect�� 0

�� , (4)

where rect�x� � 1 if |x| 1�2 and rect�x� � 0 else,�0 is the central frequency, and �� is the spectrumwidth. For the monochromatic beam with frequency�0, S�� can be written as

S�� 0�, (5)

where � is the Dirac’s function.Equations (1) and (2) are inefficient for calculating

the focal shift numerically. In the following, an ex-pression of the time-averaged intensity distributionwill be derived based on series expansion, so that the

focal shift can be calculated through solving an alge-braic equation with a well provided starting value. Byexpanding the exponential function in Eq. (1) in se-ries expansion and inserting it into Eq. (2), the nor-malized axial intensity I�z� � I�z��I�f� can be given as

I�z� �2f2

z2 �n�2

� �1z �

1f �2n�4

��1�n�21

�2n � 2�!��R2

c �2n�4

�

��

�

S��2n�2d�

��

�

S��2d�

. (6)

Then, the central frequency � is defined as

� �

��

�

S��d�

��

�

S��d�

, (7)

and the normalized frequency is defined as � � ��.Thus, in Eq. (6),

��

�

S��2n�2d�

��

�

S��2d�

� �2n�4�n, (8a)

�n �

��

�

S��2n�2d�

��

�

S��2d�

, (8b)

where �n �n � 2, 3, . . .� is called the normalized spec-trum moment. In terms of � and �n, Eq. (6) can berewritten as

I�z� � 2�u � 1�2 �n�2

� ��1�n�2

�2n � 2�!��N� �2n�4�nu

2n�4

� �2 � �m�1

�

rmum, (9a)

where N � R2� f is called the central Fresnel num-ber, � c�� is the wavelength corresponding to thecentral frequency, u � f�z � 1 is the normalizedcoordinate, and

Fig. 1. Scheme of the focused laser pulse.

1 September 2007 � Vol. 46, No. 25 � APPLIED OPTICS 6455

The focal shift is defined as �f�f � �zf � f��f, wherez � zf is the position of the actual focus with the peakintensity. Assume that uf � f�zf � 1 is the u coordi-nate of the actual focus, so �f�f � �uf��uf � 1�. Thatis, �f�f is determined by uf. Therefore, Eq. (9) indi-cates that the focal shift of the pulsed-laser beam canbe completely determined by the central Fresnelnumber N and the normalized spectrum moments �n,because the normalized axial intensity distributionI�z� is completely determined by N and �n. However,as reported previously, the focal shift of the mono-chromatic laser beam in the case of small numericalaperture is completely determined by the Fresnelnumber [3].

For the convenience of numerical computation, theseries in Eq. (9a) is truncated, then,

I�z� � �2 � �m�1

M

rmum. (10)

According to Fig. 3 in Section 3, the curves plotted onthe basis of truncated series expressions agree wellwith the curves plotted on the basis of integral ex-pressions, so Eq. (9a) is convergent near the focus. IfM is large enough, the precision for calculating thefocal shift can be ensured.

Thus, the position uf of the actual focus with thepeak intensity can be obtained by solving the alge-braic equation of

dI�z�du � �

m�1

M

mrmum�1 � 0. (11)

Since Eq. (11) may have more than one solution, astarting value should be provided. According to com-putation experiences in Section 3, the focal shift issmall when N� � 0.3, and the solution of Eq. (11) withM � 3 is close to the actual focus. However Eq. (11)with M � 3 has two roots of u� � ��r2 ��r2

2 � 3r1r3��3r3�, and u� is found to be closer to theactual focus, so u� � ��r2 � �r2

2 � 3r1r3��3r3� isselected as the starting value for solving Eq. (11)with large M. When N � 0.3, the focal shift is quitelarge, and the starting value can be selected as thesolution of Eq. (11) with M � 5, which also needs astarting value and it is selected to be the solution u�

of Eq. (11) with M � 3, too.

3. Examples

For laser pulse with Gaussian, rectangular, or mono-chromatic spectral intensity as given by Eqs. (3)–(5),

the central frequency � defined by Eq. (7) can beeasily derived as

� � �0. (12)

In the appendix, the analytic expression of the nor-malized spectrum moments �n is derived for theGaussian spectral intensity as

�n �

1 � �m�1

n�1

�2n � 2�!��2n � 2 � 2m�!m!��2m

23m

1 � �1�4��2 ,

(13a)

for the rectangular spectral intensity as

�n � 1��2n � 1��

k�0

2n�2

�1 � ��2�2n�2�k�1 � ��2�k

1 � ��2�12,

(13b)

and for the monochromatic wave as

�n � 1, (13c)

where �� is the normalized spectrum width.Eqs. (13a) and (13b) show that �n is completely de-termined by �� for pulse with Gaussian or rectangu-lar spectral intensity. As shown in Section 2, the focalshift of any pulse can be completely determined bythe central Fresnel number N and the normalizedspectrum moments �n, so the focal shift of the pulsewith Gaussian or rectangular spectral intensity canbe completely determined by N and ��.

To make the spectrum width of Gaussian spectralintensity comparable with that of the rectangularspectral intensity, the root-mean-square (RMS) spec-trum width is introduced as [9]

��RMS � ��

�

S�� 2d�

��

�

S��d� �1�2

(14)

and the normalized RMS spectrum width is definedas ��RMS � ��RMS��. It can be easily obtained that��RMS � ��2 for pulse with Gaussian spectral in-tensity and ��RMS � ��2�3 for pulse with rectangu-

rm ��2��1�n�2

�2n � 2�!��N� �2n�4�n �

2��1�n�1

�2n�! ��N� �2n�2�n�1, if m � 2n � 2, n � 2, 3, . . .

4��1�n�2

�2n � 2�!��N� �2n�4�n, if m � 2n � 3, n � 2, 3, . . . .

(9b)


lar spectral intensity. Then focal shifts of laser pulseswith Gaussian spectral intensity and rectangularspectral intensity are comparable, when their centralFresnel numbers or normalized spectrum widths areequal.

The spectrum width of the pulse is relative to itstemporal width, according to the time-frequency un-certainty relationship [9]. For a pulse with a constantspectral phase (Fourier transform limit), its temporalintensity i(t) can be determined by its spectral inten-sity S(�); that is,

i�t� ��

�

�S��exp��i2��t�d��2

. (15)

The full width at half maximum �tFWHM is used hereto describe the temporal width of a pulse, which isdefined as the temporal distance between the twopoints where the temporal intensity falls from thecentral maximum to its half. For a pulse withGaussian spectral intensity and constant spectralphase, i�t� � 2��2 exp��4�2��2t2� and �tFWHM �0.187��RMS. For a pulse with rectangular spectralintensity and constant spectral phase, i�t� � ��2

sinc2��t� and �tFWHM � 0.244��RMS. Figure 2 plotsthe relationship between the normalized RMS spec-trum width ��RMS and the temporal width �tFWHMwith the central frequency � corresponding to a wave-length of 800 nm. For a pulse with rectangular spec-tral intensity, the lowest frequency � � ��2 must bepositive, so ��RMS 0.58 should be satisfied. For apulse with Gaussian spectral intensity, the lowestfrequency �1 given by

S��1� � exp��1 � �0�2

��2 �� 0.01

should also be positive, so ��RMS 0.33 should besatisfied.

In Fig. 3, the normalized axial intensity distribu-tion I�z� calculated by integral expressions of Eqs. (1)and (2) is plotted, together with I�z� calculated by thetruncated series expressions of Eqs. (9b), (10), and(13). It shows that (1) when M � 15 in Eq. (10), theprecision for calculating the focal shift can be en-sured, so M � 15 is adopted in the following calcula-tions; (2) the solutions of Eq. (11) with M � 3 and 5can provide a good starting value not only for large N,but also for quite small N.

In Fig. 4, the focal shift �f�f is plotted as a functionof N with various ��RMS, and in Fig. 5, it is plotted asa function of ��RMS with various N. The two figuresshow that the focal shift �f�f is always negative,which means the actual focus is always closer to thelens than the geometrical focus. This is reasonable,

Fig. 2. Relationship between the normalized spectrum width��RMS and the temporal width �tFWHM, with the central frequency� corresponding to a wavelength of 800 nm. Curve G is for thepulse with Gaussian spectral intensity and constant spectralphase, and curve R is for the pulse with rectangular spectral in-tensity and constant spectral phase.

Fig. 3. Normalized axial intensity distribution I�z� of the focusedlaser pulse with Gaussian spectral intensity, where ��RMS � 0.3,and N � 5, 1, 0.5, and 0.1, respectively. Curves I are calculated byintegral expressions (1) and (2), and the other curves are calculatedby truncated series expressions (9b), (10), and (13) with M � 3, 5,and 15.


because the focal shift of each monochromatic com-ponent of the focused pulsed-laser beam is negative.Furthermore, the absolute value of the focal shiftdecreases monotonously and slowly with the normal-ized spectrum width ��RMS increasing and the centralFresnel number N fixed, and it increases monoto-nously (and fast for N 4) with N decreasing and��RMS fixed. The absolute value of the focal shift of thepulse with Gaussian spectral intensity is always lessthan that of the pulse with rectangular spectral in-tensity even if their central Fresnel numbers or nor-malized spectrum widths are equal. It means that theshape of spectral intensity also affects the focal shift.

4. Conclusions

In this paper, the focal shift of the truncated pulsed-laser beam is studied. The series expression of thetime-averaged intensity distribution along the axis isobtained, which is more efficient for calculating thefocal shift than the integral expression. This seriesexpression indicates that the focal shift of the laserpulse is completely determined by the central Fresnelnumber N and a series of normalized spectrum mo-ments �n. Focal shifts of laser pulses with Gaussianspectral intensity and rectangular spectral intensityare discussed in detail as two examples. Numericalresults show that the actual focus is reasonably closerto the lens than the geometrical focus for the pulsed-laser beam, just like the case of the monochromaticlaser beam. The absolute value of the focal shiftdecreases monotonously and slowly when the nor-malized spectrum width increases and the centralFresnel number is fixed, and it increases monoto-nously (and fast when the central Fresnel numberis small) when the central Fresnel number de-creases and the normalized spectrum width is fixed.Besides the central Fresnel number and the nor-malized spectrum width, the shape of spectral in-tensity can also affect the focal shift remarkably.

Appendix

In this appendix, the analytic expression of �n forGaussian, rectangular, or monochromatic spectral in-tensity is derived. For the Gaussian spectral inten-sity, inserting Eq. (3) into Eq. (8b) yields

�n �

��

�

d� exp � 2�� 1�2��2��2n�2

��

�

d� exp � 2�� 1�2��2��2

, (A1)

where �� , and

��

�

d� exp��2�� 1�2

��2 ��2n�2 � �k�0

2n�2 �2n � 2�!��2��k�1

k!�2n � 2 � k�!

� ��k�1Ik, Ik ��

�

exp��2��kd�, (A2)

where � � �2�� 1��. In Eq. (A2), Ik � 0 if k is odd,and if k is even Ik can be calculated by a recursionformula of

I2m �2m � 1

2 I2m�2 �2m � 1

22m � 3

2 I2m�4

� · · · ��2m � 1�!!

2m I0, (A3)

where k � 2m, �2m � 1�!! � �2m � 1��2m � 3�� · · · � 3 � 1, and I0 � ��. From Eqs. (A1)–(A3), Eq.(13a) can be obtained. For the rectangular spectral

Fig. 4. Relationship between the focal shift �f�f and the centralFresnel number N with various normalized RMS spectrum widths��RMS. Curves G1 and G2 stand for pulses with Gaussian spectralintensities; curves R1 and R2 stand for pulses with rectangularspectral intensities. Curves G1 and R1 correspond to ��RMS

� 0.3; curves G2 and R2 correspond to ��RMS � 0.1; and curve M1corresponds to ��RMS � 0, i.e., the monochromatic beam.

Fig. 5. Relationship between the focal shift �f�f and the normal-ized RMS spectrum width ��RMS with various central Fresnel num-bers N� . Curves with ��RMS � 0, 0.33� represent pulses withGaussian spectral intensities, and curves with ��RMS � 0, 0.58�represent pulses with rectangular spectral intensities.


intensity, inserting Eq. (4) into Eq. (8b) provides

�n �

��

�

rect �� 1��2n�2d�

��

�

rect �� 1��2d�

�1��2n � 1� �

k�0

2n�2

�1 � ��2�2n�2�k�1 � ��2�k

1 � ��2�12.

(A4)

For the monochromatic wave, inserting Eq. (5) intoEq. (8b) yields

�n �

��

�

�1�� 1��2n�2d�

��

�

�1�� 1��2d�

� 1. (A5)

This research is supported by the Start-up Fund ofTianjin University for Ying Zhong, and by the OpenResearch Fund of State Key Laboratory of PrecisionMeasuring Technology and Instruments.

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Focal shift in focused truncated pulsed-laser beam