1. Field of the Invention
The present invention generally relates to a method and an apparatus for optical frequency measurement and, more particularly, to a method and an apparatus using one or two frequency-stabilized mode-locked laser combs operating at different repetition rates to measure the frequency of a laser under measurement (LUM). The ordinal comb number difference can be obtained according to the beat frequencies, the offset frequencies and the repetition rates so as to measure the ordinal comb number and the frequency of the LUM.
2. Description of the Prior Art
Since 1999 T. W. Hänsch used femtosecond pulsed lasers in optical frequency measurement of the cesium D1 line, mode-locked lasers have attracted lots of attention in optical frequency measurement.
As shown in
fn=n×fr+fo
wherein n is a positive integer, fr is the pulse repetition frequency (or, in brief, the repetition rate) and fo is the carrier-envelope offset frequency (or, in brief, the offset frequency). The offset frequency fo is usually measured by using the self-referencing technique.
However, two possible values fo and fr−fo may be obtained between 0 and fr with the self-referencing technique to measure the offset frequency of a mode-locked laser. It is required to determine one of fo and fr−fo to be the correct offset frequency. By definition, either fo or fr−fo plus the integer multiples of the repetition rate fr can be defined as an offset frequency. Moreover, for a measured beat frequency fb, the beat frequency fb occurs when the frequency fL of the LUM is fb higher or lower than the comb line. Therefore, the frequency of the LUM can be fL=n×fr+fo±fb or fL=n×fr+(fr−fo)±fb=(n+1)×fr−fo±fb. Since n is left undetermined, the frequency of the LUM fL can be expressed as:
fL=n×fr±fo±fb
In the latter representation, the offset frequency can be −fo. Even though the ordinal comb numbers obtained using the two aforementioned methods may be different, the frequencies of the LUM thus measured are identical. In the specification of the present invention, the frequency of each comb line is integer multiples of the repetition rate fr plus a positive offset frequency fo, which is the same as the former representation.
The correct offset frequency and the sign of the beat frequency can be determined by changing the repetition rate or the offset frequency according to the corresponding variation of the beat frequency. The ordinal comb number can be determined based on the following methods. In the first method, historic measurement data of the frequencies of the LUM are used, wherein the n value is determined only when the precision is within ±fr/4. In the second method, a wavemeter is used to determine an approximate frequency of the LUM. If the precision of the frequency measured by the wavemeter is within ±fr/4, it can be decided which comb line is closest to the frequency of the LUM so as to determine the n value. The precision of a commercial wavemeter is about 2×10−7, which leads to 40 MHz inaccuracy for a 1550 nm laser. Therefore, the pitch of the OFC is at least 160 MHz so that it can work with the commercial wavemeter for optical frequency measurement.
In the third method, Long-Sheng Ma et al. used an OFC to determine the ordinal comb number without using a wavemeter by measuring the beat frequencies at multiple repetition rates fr and fr′ and recording the ordinal comb number variation m when the repetition rate changed. Accordingly, they derived an equation for calculating the ordinal comb number, which is expressed as:
n=[±fo′−(±fo)+m fr′±fb′−(±fb)]/(fr−fr′) (1)
wherein m is the ordinal comb number variation when the repetition rate changes from fr to fr′; fb, fb′ are the beat frequencies for the LUM and the OFC; fr, fr′ and fo, fo′ are the repetition rates and offset frequencies before and after the OFC is adjusted, respectively. At that time, the signs ± could not be determined, and therefore, the correct ordinal comb number could be obtained only after at least two m values were compared. Moreover, in this method, the ordinal comb number variation was recorded as the repetition rate changed. For low repetition rate laser, the number of recorded ordinal comb number is very large. Therefore, this method is not practical.
In the three foregoing methods, sufficient historic data, a high-precision wavemeter and gradually changing repetition rates are required to record all of the ordinal comb number variations so as to obtain the frequency of the LUM. Obviously, these methods are not explicit and not suitable for use in optical frequency measurement for all repetition rates.
Jin-Long Peng et al. disclosed a method for measuring the ordinal comb number by using mode-locked lasers, which is useful for a wide range of repetition rate. First, the sign of the beat frequency was determined according to the change of the repetition rate and the change of the beat frequency. Then the offset frequency was changed and the correct offset frequency of the mode-locked laser was determined according to the change of the beat frequency, as shown in
fL=n fr1+fo1±fb1 (2)
fL=(n+m)fr2+fo2±fb2 (3)
From Equations (2) and (3), n can be derived as
n=[m fr2+fo2−fo1±fb2−(±fb1)]/(fr1−fr2) (4)
In Equation (4), fr1, fr2, fo1, fo2, fb1, fb2 can be measured by a microwave frequency counter. The correct offset frequency fo1 and fo2 and the signs of fb1 and fb2 can be determined using the foregoing methods. Equation (4) is similar to the Equation (1) derived by Long-Sheng Ma et al. (L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S. Windeler, IEEE Journal of Selected Topics in Quantum. Electronics. 9, 1066 (2003)) even though the offset frequency is expressed by a different notation. Moreover, Jin-Long Peng et al. disclosed a method for measuring the ordinal comb
number difference m, which was derived from Equation (4) and expressed as:
The correct offset frequency and the signs of the beat frequencies could be determined using the foregoing method, wherein the repetition rate variation required for changing an ordinal comb number was used to measure fr1/n, as shown in
Therefore, there is need in providing a method and an apparatus for measuring the ordinal comb number variation without scanning the repetition rate so as to obtain the ordinal comb number and the frequency of an LUM.
It is an object of the present invention to provide a method and an apparatus for optical frequency measurement, capable of measuring the ordinal comb number difference when frequency-stabilized mode-locked laser combs operating at different repetition rates without scanning the repetition rate so as to measure the ordinal comb number and the frequency of the LUM.
In order to achieve the foregoing object, the present invention provides a method for optical frequency measurement for measuring the frequency of an LUM, comprising steps of:
In order to achieve the foregoing object, the present invention further provides an apparatus for optical frequency measurement for measuring the frequency of a laser under measurement (LUM), comprising:
The objects, spirits and advantages of the preferred embodiments of the present invention will be readily understood by the accompanying drawings and detailed descriptions, wherein:
The method for optical frequency measurement according to the present invention is different from the aforementioned method using mode-locked lasers in measuring the ordinal comb number difference m. The method of the present invention uses a mode-locked laser comb operating at different repetition rates in turn to measure beat frequencies or two mode-locked laser combs operating at different repetition rates simultaneously to measure beat frequencies. The frequency of the LUM can be expressed as Equations (2) and (3) at different repetition rates and beat frequencies. The ordinal comb number and ordinal comb number difference m are expressed as Equations (4) and (5). The correct offset frequencies and the signs of the beat frequencies can be determined by observing the change of the beat frequencies while changing the repetition rates or the offset frequencies.
In Equation (5), fr1, fr2, fo1, fo2, fb1, fb2 can be precisely measured and m is a positive integer. If n can be approximately known and the relative uncertainty of n is much smaller than 1/m, then, this value of n is sufficient for precisely calculating the m value. Therefore, the wavelength of the LUM can approximately be known as long as a monochrometer or an optical spectrum analyzer is disposed in a two mode-locked laser combs system and the estimated n can be derived from the wavelength of the LUM and the repetition frequency of the laser comb.
In the present invention, the approximate frequency of the LUM can be obtained by dividing the light speed by the wavelength and the refractive index of air, which is 1 with precision within 1/1000. More accurate refractive index of air can be obtained using the Edlén Equation. Therefore, n can be approximately measured according to the approximate frequency and the repetition rates of the optical combs. m can be calculated from Equation (5). A correct m value is an integer closest to the value calculated from Equation (5). n can be obtained by using the correct m value in Equation (4). A correct n value is an integer closest to the value calculated from Equation (4). The frequency of the LUM can be obtained using Equations (2) or (3).
The ordinal comb number difference can also be obtained using the following method. The mode-locked laser combs operate at three different repetition rates. Two of the three repetition rates are almost the same so that the ordinal comb numbers of generating the beat frequencies are identical. For example, let the third repetition rate be fr3. If the difference between fr3 and fr1 is so small that the ordinal comb number where the beat frequency occurs is n. Assuming that the offset frequency is fo1, the frequency of the LUM can be expressed as:
fL=n×fr3+fo1±fb3 (6)
The ordinal comb number difference can be derived from Equations (2), (3) and (6):
The correct m value can be calculated from Equation (7) and the n value can be calculated from Equation (4). The ordinal comb number can also be derived from Equations (4) and (6):
n=[±fb3−(±fb1)]/(fr1−fr3) (8)
Since the LUM beats with the same comb line to generate the beat frequency, fr1−fr3 is very small. Generally, the uncertainty of ±fb3−(±fb1) is much larger than fr1−fr3. The uncertainty of the n value obtained from Equation (8) is much larger than 1. Therefore, the n value obtained from Equation (8) is an approximate ordinal comb number, which requires to be used in Equation (5) to calculate a correct m value. The correct m value is then used in Equation (4) so as to calculate a precise n value. Therefore, the present invention is characterized in that the correct ordinal comb number is calculated according to the measured ordinal comb number difference.
In the foregoing method, fr3 can also be so close to fr2 that the ordinal comb number where the beat frequency is generated is n+m. Assuming that the offset frequency is fo2, the frequency of the LUM can be expressed as:
fL=(n+m)·fr3+fo2±fb3 (9)
The ordinal comb number difference can be obtained from Equations (2), (3) and (9):
Further extending the disclosure of the present invention, if fr3 is not so close to fr2 that the ordinal comb number where the beat frequency occurs is the same, the minimal ordinal comb number variation can be recorded as the repetition rate changes from fr2 to fr3. If the ordinal comb number is n+m+k, the frequency of the LUM can be expressed as:
fL=(n+m+k)·fr3+fo2±fb3 (11)
where the ordinal comb number variation k can be obtained as the repetition rate changes from fr2 to fr3. Meanwhile, the m value can be expressed as:
The foregoing method is still within the scope of the present invention. In the method for optical frequency measurement according to the present invention, the offset frequencies can be controlled to be identical, which leads to simplified calculation by eliminating the offset frequencies when the ordinal comb number or the ordinal comb number difference is being calculated.
The advantage in using two mode-locked laser combs for optical frequency measurement is that the beat frequencies at different repetition rates can be measured at the same time. For example, in the case of fr1 and fr3 are close to each other, fb1 and fb2 in Equation (7) are measured simultaneously by operating the repetition rates of the two laser combs at fr1 and fr2, then, fr1 is changed to fr3 and fr2 is keeping unchanged and another set of fb2 and fb3 are measured simultaneously. By selecting the proper beat frequencies and simultaneously measuring the beat frequencies at different repetition rates, the frequency perturbation of the LUM can be eliminated in the calculation of the ordinal comb number or the ordinal comb number difference. Therefore, the measurement of the ordinal comb number or the ordinal comb number difference is independent from the frequency perturbation of the LUM so as to achieve rapid measurement of optical frequency measurement. Even though the foregoing advantage does not exist when only one mode-locked laser comb is used, the measurement the frequency of the LUM can be performed when the integrating time is longer or the repetition rate variation is larger.
The apparatus for optical frequency measurement of the present invention comprises two mode-locked laser combs, as shown in
Please refer to
In
In Step 71, the computer controls the mode-locked laser controller to lock the repetition rate of the mode-locked laser comb 60a at fr1 and to lock the offset frequency at fo1, and to lock the repetition rate of the mode-locked laser comb 60b at fr2 and to lock the offset frequency at fo2. The measurement can be simplified when the correct offset frequency fo1 and fo2 are locked at the same frequency.
In Step 72, the mode-locked laser combs and the LUM are coupled. A dispersion device is used to introduce the light having the same path as the LUM into a photodetector to generate the beat frequencies. As shown in
In Step 73, the mode-locked laser combs operates at two different repetition rates so as to measure the first beat frequency and the second beat frequency. As shown in
In Step 74, the signs of the beat frequencies are determined. The repetition rates fr1 and fr2 are slightly changed so as to determine the signs of the beat frequencies fr1 and fr2 according to the change of the repetition rates fr1 and fr2.
In Step 75, the correct offset frequencies of the mode-locked laser combs are determined. The offset frequency fo1 is changed so as to determine the correct offset frequency fo1 according to the change of the beat frequency fb1. The offset frequency fo2 is changed so as to determine the correct offset frequency fo2 according to the change of the beat frequency fb2.
In Step 76A, the ordinal comb number difference is determined. The calibrated grating 63 determines an approximate frequency of the LUM 61. An approximate ordinal comb number can be calculated according to the approximate frequency and the repetition rate. The m value is calculated from Equation (5) in Step 77A. The signs of the beat frequencies need to be adjusted according to practical measurement. The correct m value is an integer closest to the calculated value from Equation (5).
The determination of the ordinal comb number difference is also performed in Step 76B by changing the repetition rate from fr1 or fr2 to the third repetition rate fr3 and measuring the third beat frequency. The repetition rate variation is so small that the ordinal comb number is not changed. The ordinal comb number difference can be calculated from Equation (7) or (10) in Step 77B.
In Step 78, the ordinal comb number is calculated from Equation (4). The signs of the beat frequencies are adjusted according to practical measurement. The correct n value is an integer closest to the calculated value from Equation (4).
In Step 79, the frequency of the LUM is calculated from Equation (2) or (3). The signs of the beat frequencies are adjusted according to practical measurement.
The method for optical frequency measurement using one mode-locked laser comb comprises similar steps. However, the beat frequencies at different repetition rates cannot be measured simultaneously. In this case, the beat frequencies are measured successively by a frequency counter.
Although this invention has been disclosed and illustrated with reference to particular embodiments, the principles involved are susceptible for use in numerous other embodiments that will be apparent to persons skilled in the art. This invention is, therefore, to be limited only as indicated by the scope of the appended claims.
Number | Date | Country | Kind |
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96138607 A | Oct 2007 | TW | national |
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6097487 | Kringlebotn et al. | Aug 2000 | A |
6853456 | Mells | Feb 2005 | B2 |
7564561 | Peng | Jul 2009 | B2 |
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08-101066 | Apr 1996 | JP |
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Number | Date | Country | |
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20090097035 A1 | Apr 2009 | US |