Method of optical frequency measurement

1. Field of the Invention

The present invention relates to a method of optical frequency measurement, and more particularly, to a method of measuring the frequency of an unknown laser by varying parameters of optical frequency combs (“frequency combs”).

2. Description of the Related Art

Since Professor Hänsch from Germany applied a femtosecond mode-locked laser to measure the frequency of the D 1 line of caesium atoms, mode-locked lasers have drawn more and more attention in the application of optical frequency measurement.

As shown in FIG. 1 , in the frequency domain, a mode-locked laser consists of frequency combs with uniform frequency intervals. The frequency of each comb line is equal to an integral multiple of the pulse repetition frequency f r plus an offset frequency f o . That is, the frequency f n of the n th comb line is represented by the following formula:

f n =n×f r +f o ;

where n is a positive integer, f r is the pulse repetition frequency (“repetition rate”), and f o is a carrier-envelope offset frequency (“offset frequency”). Usually, the offset frequency f o is measured with a self-referencing technique. FIG. 2 shows an f-2f self-referencing technique.

However, two possible values f o1 and f o2 smaller than f r will be obtained when the offset frequency f o of the mode-locked laser is detected with the self-referencing technique, and the two offset frequencies satisfy f o1 +f o2 =f r . According to different definitions, the offset frequency can also be defined as adding an integral multiple of the repetition rate to f o1 and f o2 respectively. Similarly, two beat frequencies f b1 and f b2 can be generated between the unknown laser and the adjacent comb lines (e.g., the n th and the (n+1) th comb lines) of the mode-locked laser, where f b1 +f b2 =f r . Therefore, the frequency f L of the unknown laser close to the n th comb line is represented by the following formula:

f L =n×f r +f o +f b ;

where f o =f o1 or f o2 , and f b =f b1 or f b2 .

The mode order number n of the frequency comb, f o , and f b in the above formula need to be determined so that the frequency f L of the unknown laser can be obtained. The following conventional methods can help to determine n, f o and f b , so as to determine the value of f L . The first method uses historical measured frequencies of the unknown laser, and the frequencies have to be accurate within ±f r /4 to determine the value of n, f o and f b . The second method is to determine the approximate frequency of the unknown laser with a wavelength meter. If the accuracy of the frequency measured by the wavelength meter is within ±f r /4, then the values of n, f o , and f b can be determined. The accuracy of a normal commercial wavelength meter is approximately 2×10 −7 , and this will cause an uncertainty of about 40 MHz for the frequency measurement of a laser with wavelength of 1550 nm. Therefore, for frequency combs with interval less than 160 MHz, the commercial wavelength meter is not accurate enough to determine the mode number.

The third method is disclosed by Long-sheng Ma et al., and does not need a wavelength meter to determine the mode number n of the frequency comb. According to their method, a frequency comb is applied to measure the beat frequency of an unknown laser under multiple groups of different repetition rates f r and f r ′, and the shifted mode number of the beating comb line is recorded when the repetition rate changes. The mode number n of the frequency comb is calculated according to the following formula:

n=[±f ′−(± f o )+ m f r ′±f b ′−(± f b )]/( f r −f r ′);

where m is the shifted mode number of the beating comb line when the repetition rate changes from f r to f r ′, and f b and f b ′ are the beat frequencies between the unknown laser and the beating comb line before and after the repetition rate being adjusted, respectively, and f o , f o ′ are the offset frequencies before and after the frequency comb is adjusted. In actual optical frequency measurement, each of f o , f o ′, f b , and f b ′ has two possible measured values. As the value of each of f o , f o ′, f b , and f b ′ cannot be determined uniquely, the ± sign in the above formula cannot be determined individually. Therefore, the measurement results of two or more different m values have to be compared and analyzed to identify the correct mode number n of the beating comb line.

The three methods of optical frequency measurement described above require adequate historical measurement data, accurate wavelength meters, and a complicated procedure of comparing multiple measurement results respectively to obtain the frequency of an unknown laser.

The present invention provides a simple and systematic method for absolute optical frequency measurement, which uses mode-locked laser combs to measure the frequency of an unknown laser. By observing the beat frequency variation while changing the repetition rate of the mode-locked laser, the relative position of between the unknown laser and the frequency comb, and hence the beat frequency between them, is uniquely determined. By observing the beat frequency variation while varying the measured offset frequency, the actual offset frequency of the mode-locked laser is correctly determined. By measuring the beat frequency of the unknown laser at different repetition rates, and measuring the difference between the mode numbers of the beating comb lines, the mode number of the frequency comb is uniquely determined. Thus, the frequency of the unknown laser is absolutely determined.

An object of the present invention is to provide a systematic method of optical frequency measurement using mode-locked lasers with any repetition rates, and more particularly, to definitely determine the relative position of the unknown laser to the beating comb line by observing the beat frequency variation while adjusting the repetition rate of the mode-locked laser, and to uniquely determine the value of the offset frequency by observing the beat frequency variation while varying the control parameters to change the offset frequency of the mode-locked laser. By measuring the beat frequency at two different repetition frequencies, the mode number of the frequency comb is uniquely determined, and hence the frequency of the unknown laser is absolutely measured. This method can be applied to mode-locked lasers with any repetition rates. Therefore, the present invention is a versatile and simple method of optical frequency measurement.

Another object of the present invention is to provide a method of measuring the mode number of the frequency comb and the optical frequency without being influenced by the frequency fluctuation of the unknown laser. The beat frequencies between the unknown laser and two mode-locked lasers with different repetition rates are measured simultaneously. The relative positions of various beat frequencies and various comb lines and the correct offset frequencies of various mode-locked lasers are determined according to the aforesaid methods. The frequency fluctuation of the unknown laser can be subtracted by simultaneously measuring the two beat frequencies or by measuring the beat frequency difference directly. The mode number determination is only related to the beat frequency difference. Therefore, the determination of the mode number of the frequency comb and the measurement of the optical frequency is not influenced by the frequency fluctuation of the unknown laser.

To achieve the aforementioned objects, the present invention discloses a method of optical frequency measurement, which is applicable to measure the frequency of an unknown laser. First, at least one mode-locked laser is beating with the unknown laser, and beat frequencies occur between the n th comb lines of the mode-locked lasers and the unknown laser. By varying the repetition rate, the relative frequency location between the unknown laser and the n th comb line is determined according to the direction of change in the beat frequency. By varying parameters of the mode-locked laser, such as the power of a pump laser, the offset frequency measured with a self-referencing technique is changed accordingly. According to directions of the variation of the offset frequency and the beat frequency, the accurate value of the offset frequency is obtained. Then, the repetition rate is adjusted, and the change in the mode number of the beating comb line that generates the beat frequency with the unknown laser is measured. The value of n is determined from the repetition rate, the difference between the mode numbers of the beating comb lines, the difference in offset frequency, the beat frequencies, and the difference in repetition rate.

In order to make the aforesaid and other objects, features and advantages of the present invention comprehensible, preferred embodiments accompanied with figures are described in detail below.

The invention will be described according to the appended drawings in which:

FIG. 1 shows the basic architecture for the frequency measurement of an unknown laser using a frequency comb in the frequency domain;

FIG. 2 is an illustrative diagram of measuring the offset frequency with an f-2f self-referencing technique;

FIG. 3 is a flow chart of the method of optical frequency measurement according to the present invention;

FIG. 4( a ) is a flow chart of determining the relationship between the beat frequency, the unknown laser and the beating comb line according to the present invention;

FIGS. 4( b )- 4 ( c ) are schematic diagrams of determining the relationship of the beat frequency, the unknown laser and the beating comb line according to the present invention;

FIG. 5( a ) is a flow chart of determining the accurate offset frequency according to the present invention;

FIGS. 5( b )- 5 ( d ) are schematic diagrams of determining the offset frequency by observing the beat frequency variation while changing the offset frequency of the frequency comb according to the present invention;

FIG. 6( a ) is a flow chart of determining the accurate offset frequency according to the present invention;

FIGS. 6( b )- 6 ( d ) are schematic diagrams of determining the offset frequency by observing the beat frequency variation while changing the offset frequency of the frequency comb according to the present invention; and

FIGS. 7( a )- 7 ( b ) are schematic diagrams of determining the mode number n of the beating comb line by varying the repetition rate according to the present invention.

The present invention provides a systematic method of optical frequency measurement, which realizes an effective measurement of the frequency of an unknown laser simply and accurately. Several embodiments are given below to illustrate the present invention. However, the present invention is not limited to the embodiments disclosed here.

FIG. 3 is a flow chart for optical frequency measurement of an unknown laser according to the present invention. As shown in step S 31 , the repetition rate of a mode-locked laser is stabilized first; the offset frequency of the mode-locked laser is measured with a self-referencing technique; and the offset frequency is stabilized. A beat frequency is generated between the n th comb line of the mode-locked laser and the unknown laser. When a beat frequency is detected, both lasers with frequency lower or higher than that of the comb line can generate the same beat frequency. Thus, the accurate frequency location of the unknown laser needs to be further determined. Then, the repetition rate is varied and the relative frequency location between the unknown laser and the n th comb line is determined according to the direction of the beat frequency variation as stated in the instruction of step S 32 . As shown in step S 33 , the offset frequency of the mode-locked laser is varied, and the actual offset frequency is determined according to the direction of the measured offset frequency variation and the direction of the beat frequency variation. After the beat frequency and the offset frequency are determined, step S 34 is implemented to determine the value of the mode number n of the frequency comb. That is, the repetition rate is adjusted, and the difference in mode number between the comb lines generating the beat frequencies is measured. Afterward, the value of n is determined from the difference in mode number, the difference in offset frequency, the beat frequencies, the repetition rate, and the difference in repetition rate. By implementing the above steps, the beat frequency, the offset frequency, and the value of n are all uniquely determined. Thus, the accurate value of the frequency of the unknown laser can be absolutely calculated.

FIG. 4( a ) is a flow chart of determining the relation between the beat frequency, the unknown laser and the frequency comb according to the present invention. As shown in steps S 41 -S 43 , a beat frequency occurs between an unknown laser and a mode-locked laser with a repetition rate of f r , and the possible value of the offset frequency of the mode-locked laser measured with the self-referencing technique is f o1 and f o2 . Moreover, the frequency of the unknown laser is close to the frequency of the n th comb line, and the frequency of the n th comb line is f n =n×f r +f o , where f o =f o1 or f o2 with f o1 +f o2 =f r and let f o1 <f o2 . The beat frequencies f b1 and f b2 are generated from the unknown laser and the neighboring comb lines of the mode-locked laser, where f b1 +f b2 =f r and let f b1 <f b2 . As shown in FIG. 4 ( b ), the frequency f L of the unknown laser may be f L1 or f L2 . Therefore, the relative position between the unknown laser and the n th comb line needs to be predetermined.

We determine the relative frequency position of the unknown laser to the n th comb line by observing the beat frequency variation while changing the repetition rate. As shown in FIG. 4( c ), when the repetition rate is changed from f r to f r ′, the beat frequencies f b1 and f b2 shift to f b1 ′ and f b2 ′, respectively. As shown in step S 44 of FIG. 4( a ) and FIG. 4( c ), f r ′ is smaller than f r , i.e., Sign(f r ′−f r ) is a minus. In this case, if the measured beat frequency f b1 ′ is smaller than f b1 , or f b2 ′ is greater than f b2 , i.e., Sign(f b1 ′−f b1 )=−Sign (f b2 ′−f b2 ) is also a minus, then the unknown laser locates on the left side of the n th comb line. Here, the left side of the frequency comb indicates a lower frequency, and the right side of the frequency comb indicates a higher frequency. Thus, the result of step S 46 is obtained, i.e., f L =f L2 =f n −f b1 . On the contrary, if Sign(f b1 ′−f b1 )=−Sign(f b2 ′−f b2 ) is a plus, then the unknown laser locates on the right side of the n th comb line. Thus, the result of step S 45 is obtained, i.e., f L =f L1 =f n +f b1 . According to the equation f b1 +f b2 =f r , f L2 =f n−1 +f b2 and f L1 =f n+1 −f b2 can also be obtained. Certainly, f r can be adjusted to a greater value of f r ′, i.e., Sign(f r ′−f r ) is a plus; and the analysis procedure of steps S 44 -S 46 is also applicable.

Step S 44 determines that f L is equal to f L1 or f L2 . However, the offset frequency of the mode-locked laser still has to be decided to be f o1 or f o2 . FIG. 5( a ) is a flow chart of determining the offset frequency when f L is equal to f L1 . By varying the offset frequency, the correct offset frequency can be determined according to the change in the beat frequency. Referring to steps S 52 -S 54 and FIGS. 5( b )- 5 ( d ), when Sign(f o1 ′−f o1 ) is equal to Sign(f b1 ″−f b1 ), f o2 can be determined to be the accurate offset frequency; conversely, when Sign(f o1 ′−f o1 ) is not equal to Sign (f b1 ″−f b1 ), f o1 can be determined to be the accurate offset frequency. Here, Sign(f o1 ′−f o1 )=−Sign(f o2 ′−f o2 ), and Sign (f b1 ″−f b1 )=−Sign (f b2 ″−f b2 ). The verification principle is explained as follows. As shown in step S 51 , by varying the pump laser power of the mode-locked laser, or varying other factors influencing the offset frequencies, the measured offset frequencies change from f o1 to f o1 ′ and from f o2 to f o2 ′, and meanwhile, the measured beat frequencies change from f b1 to f b1 ″ and from f b2 to f b2 ′. Assume the change in the pump power makes f o1 increase, i.e. Sign(f o1 ″−f o1 ) is a plus. If f o1 is the correct offset frequency as shown in FIG. 5( c ), all frequency combs shift towards right as the current offset frequency increases, then f b1 ″ should be smaller than f b1 , i.e. Sign(f b1 ″−f b1 ) is a minus. Therefore, if Sign(f o1 ′−f o1 ) is not equal to Sign(f b1 ″−f b1 ), f o1 can be determined to be the accurate offset frequency as shown in step S 53 , i.e., f n =n×f r +f o1 . Conversely, if f o2 is the correct offset frequency, as shown in FIG. 5( d ), all frequency combs shift towards left as the current offset frequency reduces, thus f b1 ″ should be greater than f b1 , i.e. Sign(f b1 ″−f b1 ) is a plus. Therefore, if Sign(f o1 ′−f o1 ) is equal to Sign(f b1 ″−f b1 ), f o2 can be determined to be the accurate offset frequency as shown in step S 54 , i.e., the frequency of the n th comb line is f n =n×f r +f o2 .

FIG. 6( a ) is a flow chart of determining the offset frequency when f L is equal to f L2 . The method in FIG. 6( a ) is the same as the method in FIG. 5( a ), and can be derived from the above rules.

After the correct beat frequency and offset frequency are obtained, the next step is determining the mode number n of the comb line. The value of n can be determined according to the beat frequencies measured with two mode-locked lasers set at different repetition frequencies. Assuming the measured beat frequency is f ba when the repetition rate is f ra and the offset frequency is f oa , the frequency f L of the unknown laser is represented by the following formula after the relative position of the unknown laser to the beating comb line is determined with the above method, and is as shown in FIG. 7( a ), then:

f L =nf ra +f oa +f ba   Formula (1)

When the repetition rate changes from f ra to f rb , the mode number of the beating comb line changes by m accordingly, and the beat frequency measured is f bb , as shown in FIG. 7( b ). To apply to more general condition, the current offset frequency is supposed to be f ob , and therefore the frequency of the unknown laser is represented by the following formula.

f L =( n+m ) f rb +f ob +f bb   Formula (2)

Then, the mode number can be derived from Formula (1) and Formula (2):

n =( m f rb +f ob −f oa +f bb −f ba )/( f ra −f rb )  Formula (3)

The value of m can be determined with two methods. The first method is that the repetition rate of the frequency comb gradually changes from f ra to f rb , and the change in the mode order number of the frequency comb is counted during the change in the repetition rate; the second method is that when the beat frequencies are on the same side of the beating comb line and f bb =f ba and f ob =f oa , if the difference in the repetition rate required to is change the mode number of a beating comb line by one is measured to be df rb , then m is an integer closest to |f ra −f rb |/|df rb |. The value of m can be 0, and in such case, the uncertainty of f ob −f oa +f bb −f ba and f ra −f rb needs to be small. Thus, following the above method, the mode number n of the beating comb line can be determined accurately according to Formula (3).

However, as the measured offset frequency and beat frequency have some uncertainty, the value calculated according to Formula (3) is not an exact integer. Therefore, the mode number n of the beating comb line is equal to the integer closest to (m f rb +f ob −f oa +f bb −f ba )/|f ra −f rb |. After the mode number n of the frequency comb is determined, the frequency of the unknown laser can be calculated according to Formula (1) or Formula (2).

Formula (1)-(3) are derived by assuming the beat frequencies is generated from the larger frequency side of the beating the comb line according to FIGS. 7( a ) and 7 ( b ). For other cases, the mode number of the beating comb line can be obtained according to the above method, and the signs of various relevant measured values in Formulae (1)-(3) also need to be properly changed according to corresponding inference processes.

The present invention also discloses a method of absolute optical frequency measurement using two frequency combs. The steps of the entire frequency measurement are similar to the above embodiment, and can also be illustrated with FIGS. 7( a ) and 7 ( b ). At this time, the repetition rate f ra of one of the frequency combs remains unchanged, while the repetition rate of the other frequency comb changes from f ra to f rb , and the frequency of the unknown laser measured simultaneously with the two frequency combs at different repetition frequencies can also be represented by Formula (1) and Formula (2) respectively. At this time, f ba and f bb can be measured simultaneously and the frequency disturbance of the unknown laser can be eliminated through f ba −f bb . Alternatively, the beat frequencies f ba and f bb can be mixed with a frequency mixer to obtain a signal of f ba −f bb , which can be directly measured by a frequency counter. Thus, even if the frequency of the unknown laser is not very stable, the uncertainty of f ba −f bb can be reduced as long as the frequencies of the frequency combs remain stable. In such case, the value of n can be measured accurately with small variation of the repetition rate. Usually, it is difficult to maintain mode-locked status when the repetition rate of the mode-locked laser changes greatly. In addition, the value of m can be measured with the same method described above.

These embodiments are explained on the basis of the beat frequency occurring between the unknown laser and the adjacent frequency comb lines. The first beat frequency and the first complementary beat frequency can also occur between the unknown laser and the next adjacent comb lines or the next to the next adjacent comb lines and so on. In such case, the sum of the beat frequency and the first complementary beat frequency is an odd multiple of the repetition rate. Furthermore, the current embodiment is explained on the basis that the sum of the first offset frequency and the first complementary offset frequency is equal to the repetition rate. As to another basis, the sum of the first offset frequency and the first complementary offset frequency can also be an odd multiple of the repetition rate.

The above-described embodiments of the present invention are intended to be illustrative only. Numerous alternative embodiments may be devised by persons skilled in the art without departing from the scope of the following claims.